OPTIMIZATION OF DIFFUSION-ENCODING GRADIENT SCHEME
FOR DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGING
OF NERVE FIBERS
By
Shantanu Majumdar

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Electrical Engineering
2011

ABSTRACT
OPTIMIZATION OF DIFFUSION-ENCODING GRADIENT SCHEME FOR
DIFFUSION-WEIGHTED MAGNETIC RESONANCE IMAGING OF NERVE
FIBERS
By
Shantanu Majumdar
Diffusion-Weighted Magnetic Resonance Imaging (DWMRI or DWI) is a specialized
imaging technique that can be used to quantify diffusivity of water molecules in biological
tissues. Nerve fibers in nervous tissues consist of axon bundles which are highly directional
microscopic tube-like structures with semi-permeable boundaries. Water molecules within
fibers exhibit diffusion anisotropy due to preferential movement of the molecules along the
direction of the fiber. The diffusion anisotropy can be measured by collecting data using
a series of diffusion-encoding gradients (diffusion-encoding gradient scheme) in the DWI
experiment and solving the inverse problem that characterizes the diffusion anisotropy
process. The direction of highest diffusivity gives the direction of the nerve fibers. Hence,
DWI provides a completely non-invasive technique to image nerve fibers and study nerve
connectivity in the brain, the spinal cord or the peripheral nervous system.
In model-based DWI methods (such as diffusion tensor imaging, DTI), the diffusion
process is characterized by a parametric relation between the measured DWI signal and
the diffusion model parameters (diffusivity, fiber orientation) as well as the experimental
parameters (gradient strengths and directions in the scheme). The model parameters are
estimated by solving the inverse problem corresponding to the diffusion process under a
given experimental setting. The estimated model parameters are further used to compute
secondary diffusion-related quantities (such as mean diffusivity and fractional anisotropy
which are potential biomarkers of the health of the nerve fibers) or to reconstruct fiber
tracts connecting different locations in the imaged structure (fiber tractography). It is
important that the diffusion model parameters are precisely estimated since these directly
affect any secondary processing step. The precision of the estimated model parameters

depends on the selection of the experimental parameters and thus can be improved by
optimal selection of these parameters.
In this work, a framework to optimize the diffusion-encoding gradient scheme is developed for model-based DWI methods. The framework reduces the estimation uncertainty of diffusion model parameters (thus improves precision) by optimally selecting the
diffusion-encoding gradients to minimize an analytical lower bound of the estimation uncertainty (known as the Cramer-Rao lower bound, CRLB). Focus has been on special
structures, such as the spinal cord, where the axon bundles are oriented in specific direction known a priori . This availability of a priori information of the fiber orientation
has been exploited and embedded into the optimization framework to reduce uncertainty
of parameter estimation. The framework uses subject-specific information on diffusion
parameters and also allows for a safety margin beyond the expected performance range
of diffusion parameters, thereby making it more relevant and less biased. The framework
has been validated via Monte Carlo simulations as well as by conducting DTI experiments on human subjects. Also results from fiber tractography show improvement in the
quality of tracked nerve fibers upon using the optimized gradient scheme. Thus, the use
of the optimization framework can improve the quality of DWI diagnostics by improving
precision of the imaging technique and encourage comparison of patient groups.

Copyright by
Shantanu Majumdar
2011

ACKNOWLEDGMENTS

I would like to convey my deepest gratitude to my advisors, Dr. Satish Udpa, Dr. L.
Guy Raguin and Dr. David C. Zhu for showing me the right direction in pursuing this
doctoral research. Their valued advice has always helped me in understanding the details
of this research. Above all, their professional and moral support gave me the strength
to reach this final stage of my doctoral research. I would also like to thank Dr. Lalita
Udpa and Dr. Jane Turner for their valuable inputs. My thanks to the faculty and staff
at the Department of Electrical and Computer Engineering, Michigan State University
for providing their help.
I am thankful to the Department of Radiology, Michigan State University, for letting
me use the 3T GE Signa HDx research MRI scanner. Special thanks to Dr. Jim Potchen,
Tom Cooper, Dr. Kevin Berger from the Department of Radiology. My thanks to all the
volunteers who participated in the human study.
I would also like to thank my colleagues from the Nondestructive Evaluation Laboratory, Michigan State University for providing me with helpful advice from time to time.
Finally, I thank my mother, my father and my sister for their unrelenting support in
helping me go through this journey.

v

TABLE OF CONTENTS

LIST OF TABLES

ix

LIST OF FIGURES

xi

LIST OF SELECTED ABBREVIATIONS

xviii

LIST OF SELECTED SYMBOLS
1 Introduction
1.1 Neuronal connectivity . . . . . .
1.2 Diffusion-weighted imaging . . . .
1.3 Optimizing DWI protocols . . . .
1.4 Motivation and scope of research
1.5 Outline . . . . . . . . . . . . . . .

xix

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2 A review of current techniques in neuroimaging
2.1 Neuronal connectivity: DWI and other competing technologies . . . . . .
2.2 Applications of DWI . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Prior work on DWI protocol optimization . . . . . . . . . . . . . . . . .

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3 Basic principles of nuclear magnetic resonance imaging
3.1 Nuclear magnetic resonance and signal generation . . . . .
3.2 MR Imaging . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Slice selection . . . . . . . . . . . . . . . . . . . . .
3.2.2 2D Fourier transform method for imaging . . . . .
3.2.3 Phase encoding . . . . . . . . . . . . . . . . . . . .
3.2.4 Frequency encoding . . . . . . . . . . . . . . . . . .
3.3 Using echoes . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 T1 and T2 weighting . . . . . . . . . . . . . . . . . . . . .

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4 Concepts in diffusion-weighted MRI
4.1 Self-diffusion of water in nervous tissue . . . . . . . . . . . . .
4.2 Pulsed gradient spin echo sequence for diffusion quantification
4.3 Propagator formalism . . . . . . . . . . . . . . . . . . . . . . .
4.4 Parametric DWI . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 DTI formulation . . . . . . . . . . . . . . . . . . . . .
4.4.2 ADTI formulation . . . . . . . . . . . . . . . . . . . .
4.4.3 QUAQ formulation . . . . . . . . . . . . . . . . . . . .
4.5 Non-parametric DWI . . . . . . . . . . . . . . . . . . . . . . .
4.6 Experimental DTI protocol used in this work . . . . . . . . . .
4.6.1 Pulse sequence . . . . . . . . . . . . . . . . . . . . . .
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4.7

4.6.2 Experimental settings . . . . . . . . . . . . . . . . . . . . . . . . .
Fiber assignment by continuous tracking . . . . . . . . . . . . . . . . . .

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5 Optimization of gradient scheme in diffusion-weighted imaging
5.1 General concepts in parameter estimation and optimization . . . . . . . .
5.1.1 Cramer-Rao Lower Bound . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Sensitivity matrix . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 Noise models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.4 Choice of estimators for different noise models . . . . . . . . . . .
5.1.5 Relation with FA, MD and α . . . . . . . . . . . . . . . . . . . .
5.2 CRLB for different noise models . . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Rician noise case . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.2 Gaussian noise case at high SNR . . . . . . . . . . . . . . . . . .
5.2.3 Interpretation of CRLB . . . . . . . . . . . . . . . . . . . . . . .
5.2.4 Sensitivity matrix computation . . . . . . . . . . . . . . . . . . .
5.3 Partitioning CRLB matrix for optimized estimation of selected parameters
5.3.1 Diffusivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Angular parameters . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Gradient scheme optimization framework . . . . . . . . . . . . . . . . . .
5.4.1 Reformulation of the gradient scheme . . . . . . . . . . . . . . . .
5.4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.3 Optimization Efficiency . . . . . . . . . . . . . . . . . . . . . . . .
5.4.4 b-factor optimization . . . . . . . . . . . . . . . . . . . . . . . . .
5.4.5 Additional constraints on FA and MD in gradient scheme optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Comparison of optimized gradient schemes and their predicted performances
5.5.1 Distribution of gradient directions . . . . . . . . . . . . . . . . . .
5.5.2 Performance curves . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Evaluation of estimation performance by simulations
6.1 Noise characterization . . . . . . . . . . . . . . . . . . . . .
6.1.1 Method . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Performance of Estimators . . . . . . . . . . . . . . . . . . .
6.2.1 Rician noise . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Gaussian noise . . . . . . . . . . . . . . . . . . . . .
6.3 Validation of performance curves . . . . . . . . . . . . . . .
6.4 Simulation for effect of b-factor, N and Λ in ADTI model . .
6.4.1 Performance indices . . . . . . . . . . . . . . . . . . .
6.4.2 Simulation parameters . . . . . . . . . . . . . . . . .
6.4.3 Effect of b-factor . . . . . . . . . . . . . . . . . . . .
6.4.4 Effect of number of diffusion gradient directions (N)
6.4.5 Effect of Cone angle (Λ) . . . . . . . . . . . . . . . .

vii

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6.5

6.4.6 Selection of b-factor, N and Λ . . . . . . . . . . . . . . . .
Comparison of performance indices for optimized gradient schemes
6.5.1 ADTI model . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.2 DTI model . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Spinal cord axisymmetric diffusion tensor imaging
125
7.1 A validation study for gradient scheme optimization . . . . . . . . . . . . 125
7.1.1 Experimental Protocol . . . . . . . . . . . . . . . . . . . . . . . . 126
7.1.2 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7.1.3 Results for optimization using cone of fibers obtained from a priori
information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.1.4 Results for optimization without a priori knowledge of cone of fibers133
7.2 Fiber tracking analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.2.2 Fiber tracking results . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3 A partitioned CRLB based optimization of b-factor and diffusion gradient
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4.1 Justification for the use of axisymmetric diffusion model for cervical
spinal cord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.4.2 Optimized distribution of gradient directions . . . . . . . . . . . . 148
7.4.3 Clinical relevance . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4.4 Fiber tracking in cervical spinal cord . . . . . . . . . . . . . . . . 152
7.4.5 Partitioned CRLB-based b-factor and gradient scheme optimization 153
7.4.6 Justification for the use of prior information . . . . . . . . . . . . 155
8 Conclusions
8.1 Summary . . . . . . . . . . . . . . .
8.1.1 Gradient scheme optimization
8.1.2 Simulation experiments . . . .
8.1.3 Spinal cord ADTI experiments
8.2 Contributions . . . . . . . . . . . . .
8.3 Future Work . . . . . . . . . . . . . .

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APPENDICES
168
A
Flowcharts and instructions for the overall optimization procedure . . . . 168
B
Gradient tables for the spinal cord imaging study . . . . . . . . . . . . . 180
BIBLIOGRAPHY

188

viii

LIST OF TABLES

6.1
6.2
6.3
6.4
6.5
6.6
7.1
7.2
7.3
7.4

7.5
7.6
7.7

7.8

Simulation results for the least-squares (LS) estimator performance assuming Rician noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

Simulation results for the least-squares with bias correction (LSC) estimator performance assuming Rician noise model . . . . . . . . . . . . . . .

110

Simulation results for the maximum likelihood (ML) estimator performance assuming Rician noise model . . . . . . . . . . . . . . . . . . . . .

110

Simulation results for the least-squares (LS) estimator performance assuming Gaussian noise model . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

Simulation results for the ADTI optimized gradient scheme for Rician noise
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

Simulation results for the DTI optimized gradient scheme for the Rician
noise model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

Average diffusivities and cone angles from preliminary DTI experiment
data of five subjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

Performance comparison of the optimized scheme (OPT30) and MF30 from
subject data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

eSNR comparison of the optimized scheme (OPT30) and MF30 from DTI
subject data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

Comparison of standard deviations (SD) of angular parameters at ROI
voxels from subject data using optimized gradient scheme (OPT30) and
MF30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

134

Comparison of standard deviations (SD) of diffusivities at ROI voxels from
subject data using optimized gradient scheme (OPT30) and MF30 . . . .

134

Comparison of standard deviations (SD) of FA at ROI voxels from subject
data using optimized gradient scheme (OPT30) and MF30 . . . . . . . .

134

Comparison of mean values of diffusivities and angular deviation at ROI
voxels from subject data using optimized gradient scheme (OPT30) and
MF30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

Comparison of standard deviations (SD) of estimated ADTI model parameters and FA at the ROI voxels for subject 1 with gradients optimized for
a completely uncertain fiber orientation (OPT30-90) and MF30. . . . . .

137

ix

7.9

Comparison of fiber tracking results for OPT30 and MF30 protocols using
simulated fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

140

7.10 Comparison of fiber tracking results for OPT30 and MF30 protocol for the
total number of reconstructed fibers (TF) through the ROI using 5 ADTI
subject data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

7.11 Comparison of fiber tracking results for OPT30 and MF30 protocol for the
average fibers per voxel (AF) using 5 ADTI subject data . . . . . . . . .

141

7.12 Comparison of fiber tracking results for OPT30 and MF30 protocol for the
average length (in mm) of fibers (AL) tracked using 5 ADTI subject data

141

7.13 Comparison of ADTI parameter standard deviations (RMS) in the ROI
voxels for data based on MF30 and OPT30 gradient schemes and one
healthy subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145

7.14 Comparison of ADTI parameter mean values in the ROI voxels for data
based on MF30 and OPT30 gradient schemes and one healthy subject . .

146

B.1 DTI Gradient Table for Subject 1 . . . . . . . . . . . . . . . . . . . . . .

181

B.2 DTI Gradient Table for Subject 2 . . . . . . . . . . . . . . . . . . . . . .

182

B.3 DTI Gradient Table for Subject 3 . . . . . . . . . . . . . . . . . . . . . .

183

B.4 DTI Gradient Table for Subject 4 . . . . . . . . . . . . . . . . . . . . . .

184

B.5 DTI Gradient Table for Subject 5 . . . . . . . . . . . . . . . . . . . . . .

185

B.6 DTI Gradient Table for Subject 1 with Λ = 90◦ . . . . . . . . . . . . . .

186

x

LIST OF FIGURES

1.1

1.2

1.3

3.1

3.2

3.3
3.4

3.5

(a) T2 -weighted image, (b) fractional anisotropy (FA) map and (c) mean
diffusivity (MD) map. The data for these images are acquired using a
DTI protocol (with MF30 gradient scheme and b = 1000 s mm−2 ) and
the images are processed using FSL software package (Analysis Group,
FMRIB, Oxford, UK). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Fiber tracking through Corpus Callosum (a) Axial view, (b) Coronal view.
The fibers shown are 3D reconstructed streamline fibers projected onto the
2D image views. The color coding indicates the fiber orientation on the
RGB scale, namely, red is right(R)-left(L), blue is superior(S)-inferior(I)
and green is anterior(A)-posterior(P) orientations respectively. Note that
the axial and coronal views are not on the same scale. The data for these
images are acquired using a DTI protocol (with MF30 gradient scheme
and b = 1000 s mm−2 ) and fiber tracking is performed using MedINRIA
software package (Asclepios Research Project, INRIA Sophia Antipolis,
France). For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this dissertation.

5

DTI results on a healthy adult human in the cervical spinal cord/brainstem
region. (a) Fiber orientations estimated by DTI superimposed over a T2 weighted image, (b) distribution of the angle between the fiber orientations
and the mean fiber orientation. . . . . . . . . . . . . . . . . . . . . . . .

8

(a) Alignment of spins with B0 (most are parallel, while some can be antiparallel to B0 . M is the net magnetization vector. (b) Separation of spin
energy into low and high levels when subjected to B0 . . . . . . . . . . .

21

(a) M when RF pulse, B1 , is applied as seen in the laboratory reference
frame (b) M nutated by flip angle θ as seen in the rotating reference
frame. (c) M during relaxation after RF pulse is turned off as seen in
the laboratory reference frame. Mz and Mxy are the longitudinal and
transverse magnetizations respectively. . . . . . . . . . . . . . . . . . . .

24

(a) T1 recovery of longitudinal magnetization and (b) T2 decay of transverse magnetization. M0 is the equilibrium magnetization. . . . . . . . .

25

Schematic showing slice (∆z) selection in z-direction using Gz field in (a)
and the sinc-modulated B1 RF signal in frequency (b) and time (c) domains
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Schematic showing k-space imaging using 2D Fourier transform method.

29

xi

3.6

3.7
3.8
4.1
4.2

4.3
4.4
4.5
4.6
5.1

5.2

5.3

(a) Schematic showing absence of spatial encoding when no gradient field is
used. (b) Schematic showing encoding of a fixed length (2L) in x-direction
using gradient field, Gx . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Gradient echo pulse sequence diagram showing the timings of the slice
selection, phase-encoding and frequency-encoding gradients. . . . . . . .

31

Spin-echo pulse sequence diagram showing the timings of the slice selection,
phase-encoding and frequency-encoding gradients. . . . . . . . . . . . . .

32

Brownian motion of water molecules in (a) a healthy axon fiber and (b)
an axon fiber with a rupture. . . . . . . . . . . . . . . . . . . . . . . . .

35

Schematic of the PGSE sequence showing only the timing of the diffusionencoding gradient and the RF pulses. Here, τ = TE /2, where TE is the
echo time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

DTI diffusivity ellipsoid indicating diffusivities (D , D⊥1 , D⊥2 ) and the
orientation angles (θF , φF , ψF ). . . . . . . . . . . . . . . . . . . . . . . .

46

ADTI diffusivity ellipsoid indicating diffusivities (D , D⊥ ) and the orientation angles (θF , φF ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

(a) A schematic of a typical echo-planar imaging sequence for gradient
echo. (b) The k-space trajectory for the EPI sequence. . . . . . . . . . .

52

(a) Single RF pulse with slice select gradient. (b) 12 RF pulses of the
spatio-spectral pulse sequence with the slice select gradients. . . . . . . .

53

Rician distribution at different SNRs (SNR = m/σ). Distribution are
generated by (a) varying m and fixing σ at 1.0 and (b) varying σ and
fixing m at 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Plot of (a) the normalized MR signal and (b) its square for the ADTI
model with respect to the gradient direction angles (θ, φ) with the mean
fiber direction is at (θF , φF ) = (0◦ , 0◦ ). The gray scale represents the
corresponding signal value. . . . . . . . . . . . . . . . . . . . . . . . . . .

74

Plot of the sensitivity values and their square for the ADTI model with
respect to gradient direction angles (θ, φ). Shown in the figure are sensitivity values and their squares w.r.t. D ((a) and (b)), D⊥ ((c) and (d)),
θF ((e) and (f)) and φF ((g) and (h)). The mean fiber direction is at
(θF , φF ) = (0◦ , 0◦ ). The gray scale represents the corresponding sensitivity values or its square. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

xii

5.4

Variation of the sensitivity values and the normalized MR signal w.r.t.
diffusion gradient direction angles (θ, φ) for nearly axisymmetric case for
the DTI model. Shown in the figure are sensitivity values w.r.t. (a) D ,
(b) D⊥1 , (c) D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the normalized MR
signal. The mean fiber direction is at (θF , φF ) = (90◦ , 0◦ ). . . . . . . . .

80

Variation of the squares of sensitivity values and the normalized MR signal
w.r.t. diffusion gradient direction angles (θ, φ) for nearly axisymmetric case
for the DTI model. Shown in the figure are squares of sensitivity values
w.r.t. (a) D , (b) D⊥1 , (c) D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the
normalized MR signal. The mean fiber direction is at (θF , φF ) = (90◦ , 0◦ ).

81

Variation of the sensitivity values and the normalized MR signal w.r.t.
diffusion gradient direction angles (θ, φ) for non-axisymmetric case for the
DTI model. Shown in the figure are sensitivity values w.r.t. (a) D , (b)
D⊥1 , (c) D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the normalized MR signal.
The mean fiber direction is at (θF , φF ) = (90◦ , 0◦ ). . . . . . . . . . . . .

82

Variation of squares of the sensitivity values and the normalized MR signal
w.r.t. diffusion gradient direction angles (θ, φ) for non-axisymmetric case
for the DTI model. Shown in the figure are squares of sensitivity values
w.r.t. (a) D , (b) D⊥1 , (c) D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the
normalized MR signal. The mean fiber direction is at (θF , φF ) = (90◦ , 0◦ ).

83

Gradient directions (white circles) on a 2D opened hemisphere showing a
reformulated scheme (for P = 3). Mean fiber orientation is at (θF , φF ) =
(0◦ , 0◦ ). The gray scale underlay shows the normalized MR signal variation
due to changing gradient directions (θ, φ). . . . . . . . . . . . . . . . . .

88

Robust optimization results for the ADTI model with Rician CRLB for
b = 1000 s mm−2 with P = 3 rings for fiber orientations within a cone
of axis along the z-axis and cone angle Λ = 35◦ . (a) Gradient directions
(white circles) an opened hemisphere with gray scale equal to the normalized MR signal. (b) Comparison of performance curves. . . . . . . . . . .

90

5.10 Variation of normalized hypervolume of uncertainty for 30-direction optimal gradient schemes for ADTI model. The cone angles vary as (Λ =
[10◦ − 90◦ ]) and Rician noise case is selected. Noise level, σ = 0.1. Normalization reference is MF30 gradient scheme. . . . . . . . . . . . . . . .

91

5.11 Plot of the ratio of hypervolumes for OPT30 w.r.t. MF30 scheme optimized
using Gaussian noise assumption and for the DTI model. The contour line
at unity value shows the cone of angles within which the optimization
performance is improved. . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.5

5.6

5.7

5.8

5.9

xiii

5.12 Optimized Gradient schemes for the ADTI diffusion model under the following optimality criteria: optimize for all parameter ((a) Rician and (b)
Gaussian), diffusivities only ((c) Rician and (d) Gaussian) and fiber orientation only ((e) Rician and (f) Gaussian). MF30 scheme is shown in (g).
The fiber orientation, (θF , φF ) = (0◦ , 0◦ ). The gray scale values represent
the normalized MR signal and the white circles are the locations of the
diffusion gradient unit vector on an opened unit hemisphere. . . . . . . .

98

5.13 Optimized Gradient schemes for the DTI diffusion model under the following optimality criteria: optimize for all parameter ((a) Rician and (b)
Gaussian), diffusivities only ((c) Rician and (d) Gaussian) and fiber orientation only ((e) Rician and (f) Gaussian). The fiber orientation, (θF , φF )
= (90◦ , 0◦ ). The gray scale values represent the normalized MR signal and
the white circles are the locations of the diffusion gradient unit vector on
an opened unit hemisphere. . . . . . . . . . . . . . . . . . . . . . . . . .

99

5.14 Performance curves for the ADTI diffusion model for optimization w.r.t.
all parameters ((a) Rician and (b) Gaussian), diffusivities ((c) Rician and
(d) Gaussian) and angular parameters ((e) Rician and (f) Gaussian). . .

100

5.15 Performance plots for the DTI diffusion model for optimization w.r.t. all
parameters ((a) Rician and (b) Gaussian), diffusivities ((c) Rician and (d)
Gaussian) and angular parameters ((e) Rician and (f) Gaussian). Gray
scale values indicate the ratio of CRLB-based hypervolume of OPT30 by
MF30 and the contour lines are at unity ratio. . . . . . . . . . . . . . . .

101

6.1

6.2

6.3

6.4

Comparison of the estimated and true values of the normalized MR signal
(E) for the cases of Rician and Gaussian fit to the distribution of S/S0
and S/m0 . Simulations are run at (a) NEX = 2 and (b) NEX = 4. . . .

104

ˆ
Variation of the estimated σE of the noisy normalized MR signal (E) for
the cases of Rician and Gaussian fits to the distributions of S/S0 and S/m0
respectively. Simulations are run at (a) NEX = 2 and (b) NEX = 4. . . .

105

Variation of ratio of hypervolume of uncertainty (DOP T /DM F , D =
det (ΣCR )) for ADTI model from Monte Carlo simulations w.r.t. angular deviation (α) at σ = 0.1, Λ = 35◦ . Gaussian case with LS estimator
(a) and Rician case with LS (b), LSC (c) and ML (d) estimators. . . . .

112

Effect of varying b-factor on the diffusion gradient optimization. Performance indices (a) normalized hypervolume for successful trials (µr1 ), (b)
percentage success rate (PS ), (c) effective SNR (eSNR) and (d) normal2
ized variance of FA (σF A ) are shown for different b-factors and for both
the optimized gradient schemes (OPT) and the MF30 scheme (MF). . . .

116

xiv

6.5

6.6

7.1

Effect of varying number of diffusion gradients (N) on the diffusion gradient optimization. Normalized hypervolume for successful trials (µr1 ) ((a)
and (b)), effective SNR (eSNR) ((c) and (d)) and normalized variance of
2
FA (σF A ) ((e) and (f)) are shown for different N and for both the optimized gradient schemes (OPT) and the MF-based schemes (MF). Figures
(a), (c) and (e) are for b = 1000 s mm−2 , whereas figures (b), (d) and (f)
are for b = 2500 s mm−2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

117

Effect of varying cone angle (Λ) on diffusion gradient optimization. Performance indices (a) normalized hypervolume for successful trials, (µr1 ), (b)
percentage success rate (PS ), (c) effective SNR (eSNR) and (d) normalized
2
variance of FA (σF A ) are shown for different Λ and for both the optimized
gradient schemes (OPT) and the MF30 scheme (MF). . . . . . . . . . . .

119

(a) T1 coronal view of the cervical spinal cord and brain stem near the C1
- C2 vertebral region. The boxed region is selected manually and further
magnified in (b) (e). (b) T2 coronal image showing the segmented spinal
cord within the initial ROI in (a) after removal of the CSF surrounding the
spinal cord by MD thresholding. (c) T2 coronal image showing the selected
white matter ROI voxels in (b) after removal of gray matter voxels by FA
thresholding. Similarly, the brain stem region is first manually selected in
(a), and then thresholded in two stages, (d) and (e), to extract the ROI
voxels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

128

7.2

Optimized gradient scheme (white circles) for 30 gradient directions on a
2D opened hemisphere for cone angles (a) Λ = 35◦ (b) Λ = 90◦ (completely
uncertain fiber orientation case). Underlying grayscale image shows the
normalized MR signal levels over the hemisphere changing with respect to
the diffusion gradient direction (θ, φ). Mean fiber orientation is at (0◦ , 0◦ ). 129

7.3

Region of interest (ROI) voxel distributions with respect to hypervolume
ratio (DOP T 30 /DM F 30) for different subjects. DOP T 30 and DM F 30 are
square roots of the determinants of covariance matrix of parameter estima√
tion ( detΣ for estimation (Est.) and detΣCR for prediction (Pred.))
for OPT30 and MF30 schemes respectively. Figs. (a) to (e) correspond to
subject number 1 to 5. Majority of the ROI voxels for each subject are
in the less than unity range indicating an overall uncertainty reduction in
the parameter estimation. . . . . . . . . . . . . . . . . . . . . . . . . . .

132

Distributions of relative differences for various estimated quantities under
the DTI and ADTI model. Differences of (a) λr = (λ2 +λ3 )/2 and D⊥ , (b)
λ1 and D , (c) FA (d) MD, (e) θF and (f) φF are shown. These results are
based on the voxels in the cervical spinal cord white matter tracts (C1-C2
region) from five subject data collected using the MF30 gradient scheme.

136

7.4

xv

7.5

Fiber tracking based on simulated fiber bundle surrounded by isotropic
medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

138

(a) Coronal view, (b) sagittal view and (c) axial view of T2-weighted image
of the cervical spinal cord with the seed ROI shown in red. . . . . . . . .

139

3D fiber tracking shown in the cervical spinal cord region using the DTIStudio software. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

Distribution of 30 gradient directions (white circles) on a opened unit hemisphere shown with an underlay image of the normalized MR signal at
b = 2062 s mm−2 and FA = 0.9. . . . . . . . . . . . . . . . . . . . . . . .

144

ROI voxel distribution with respect to hypervolume ratio
(DOP T 30/DM F 30 ) based on the ADTI datasets from MF30 and
OPT30 gradient scheme for one healthy subject. Estimated PS = 72.2%
and µr1 = 0.742 for 36 ROI voxels. Estimation is based on the covariance
of estimates and prediction is based on the Rician CRLB formulation. . .

145

7.10 Distributions of optimized diffusion gradient directions (indicated by white
circles) on an opened unit hemisphere. Underlying grayscale image shows
the normalized MR signal levels over the hemisphere changing with respect
to the diffusion gradient direction (θF , φF ). Optimized gradients are shown
for the ADTI model in (a) and the non-axisymmetric DTI model in (b) for
healthy white matter tracts with FA = 0.745, MD = 0.818 ×10−3 mm2 s−1 .
For the pathological case (cervical spinal cord in ALS patients), optimized
gradients are shown for the ADTI model in (c) and the non-axisymmetric
DTI model in (d) with FA = 0.45, MD = 0.96 ×10−3 mm2 s−1 . For all
cases, the fiber orientation angle is (θF , φF ) = (0◦ , 0◦ ). . . . . . . . . . .

148

7.11 An illustration of the FACT algorithm applied to a 2D distribution of fiber
orientations. Tracks are shown in orange. Each box represents a pixel with
the fiber orientation shown by the arrow. True fiber orientation is along the
vertical. Light red pixels are ones with more uncertain fiber orientation.
The green pixels are the seed (ROI) pixels. Only tracks penetrating the
seed pixels are retained. . . . . . . . . . . . . . . . . . . . . . . . . . . .

153

A.1 Flowchart for the processing of the preliminary DTI data to compute the
cone angle (Λ) and the mean of ADTI model parameters in the spinal cord
tract region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170

A.2 General overview of the overall gradient and b-factor optimization scheme.
Flowcharts for simulated annealing, robust optimization and b-factor optimization are shown in Figs. A.3–A.6. . . . . . . . . . . . . . . . . . . .

173

7.6
7.7
7.8

7.9

xvi

A.3 Flowchart describing the simulated annealing algorithm to find the suboptimal solution (not using the cone angle information). Ω = {gi ; i ∈
[1, N]}, N is the number of gradient directions (N = 30). g i is the ith
gradient direction vector: g i ≡ [gxi , gyi , gzi ] ≡ [θi , φi ]. Cost function,
S = detΣCR, f ull or detΣCR, partial . S is a function of Ω . S0 is the
initial value of cost function for gradient scheme Ω0 and it is later updated
in every iteration. ‘rand’ is a function that generates random numbers
between 0 and 1 with uniform probability. The flowchart for the decision
box for stopping criteria is shown in Fig. A.4. . . . . . . . . . . . . . . .

174

A.4 Flowchart for the stopping criteria in the simulated annealing algorithm.
Here, NtryT = number of trials at a fixed temperature T, max NtryT =
maximum limit of NtryT, Nsucc = success count, max Nsucc = maximum
limit of Nsucc, Nrej = consecutive rejection count, max Nrej = maximum
limit of Nrej, T = temperature(simulated), min T = minimum limit of T,
cool(T) = cooling method for T. . . . . . . . . . . . . . . . . . . . . . . .

175

A.5 Flowchart for the robust optimization procedure for the gradient directions
utilizing the cone angle information of fiber orientations. Ω = {g i ; i ∈
[1, N]}, N is the number of gradient directions (N = 30). g i is the ith gradient direction vector. Cost function, S = detΣCR, f ull or detΣCR, partial .
iter = number of iterations, iterMax = maximum limit of iter, del = absolute relative change in the cost function (abs(S − S0 /S0 )), delMin =
minimum limit of del, osc = number of oscillations in the cost function,
oscMax = maximum limit of osc. . . . . . . . . . . . . . . . . . . . . . .

176

A.6 Flowchart for the robust optimization of gradient directions and b-factor
using simulated annealing algorithm utilizing the cone angle information of fiber orientations. Here, Ω = {g i ; i ∈ [1, N]}, N is the
number of gradient directions (N = 30). g i is the ith gradient direction vector: g i ≡ [gxi , gyi , gzi] ≡ [θi , φi ]. Cost function, S =
detΣCR, f ull or detΣCR, partial . S is a function of Ω . S0 is the initial
value of cost function for gradient scheme Ω0 and it is later updated in
every iteration. The flowchart for the decision box for stopping criteria is
shown in Fig. A.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177

A.7 Flowchart for the processing of the multiple DTI 4D volume datasets obtained by using the optimized protocol and compared to the MF30 protocol.179
B.1 Diffusion gradient directions (white circles) for Subject 1 optimized using prior structural information. The underlay shows the normalized MR
signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . . . .

180

B.2 Diffusion gradient directions (white circles) for Subject 2 optimized using prior structural information. The underlay shows the normalized MR
signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . . . .

182

xvii

B.3 Diffusion gradient directions (white circles) for Subject 3 optimized using prior structural information. The underlay shows the normalized MR
signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . . . .

183

B.4 Diffusion gradient directions (white circles) for Subject 4 optimized using prior structural information. The underlay shows the normalized MR
signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . . . .

184

B.5 Diffusion gradient directions (white circles) for Subject 5 optimized using prior structural information. The underlay shows the normalized MR
signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . . . .

185

B.6 Diffusion gradient directions (white circles) for Subject 1 optimized using
completely uncertain fiber orientation. The underlay shows the normalized
MR signal w.r.t. gradient direction angles (θ, φ). . . . . . . . . . . . . . .

186

xviii

LIST OF SELECTED ABBREVIATIONS

ADTI

axisymmetric diffusion tensor imaging

CRLB

Cramer-Rao lower bound

CSF

cerebrospinal fluid

DTI

diffusion tensor imaging

DWI

diffusion-weighted imaging

eSNR

effective signal-to-noise ratio

FA

fractional anisotropy

FACT

fiber assignment by continuous tracking

FID

free induction decay

LS

least-squares

MD

mean diffusivity

MF

minimum force

MLE

maximum likelihood estimator

MRI

magnetic resonance imaging

NEX

number of excitations

PGSE

pulsed gradient spin-echo sequence

RF

radio frequency

ROI

region of interest

SE-EPI

spin-echo echo-planar imaging

xix

LIST OF SELECTED SYMBOLS

TR

time of repetition, ms

TE

time of echo, ms

T1

longitudinal magnetization recovery time constant, ms

T2

transverse magnetization decay time constant, ms

γ

gyromagnetic ratio, rad s−1 T−1

δ

diffusion encoding gradient pulse width, ms

∆

time interval between diffusion encoding gradient pulses, ms

b

b-factor or diffusion sensitivity factor, s mm−2

D

diffusion tensor matrix, mm2 s−1

D

longitudinal diffusivity in ADTI model, mm2 s−1

D⊥

transverse diffusivity in ADTI model, mm2 s−1

θF

zenith angle for fiber orientation in ADTI model, rad

φF

azimuthal angle for fiber orientation angle in ADTI model, rad

E

normalized MR signal or echo attenuation, dimensionless

ΣCR

Cramer-Rao lower bound of the covariance matrix of estimated parameters, unit depends on the diffusion model

X

sensitivity matrix, unit depends on the diffusion model

σ

noise standard deviation in normalized MR signal, dimensionless

N

number of gradient directions, dimensionless

Λ

cone angle of fiber directions, deg

PS

percentage success rate, %

µr1

mean ratio of hypervolumes, dimensionless

xx

CHAPTER 1
Introduction
1.1

Neuronal connectivity

The human body comprises of a complex network of neural connections between different
organs and the nervous system. Through these neural connections, the functioning of the
different organs is controlled by the central nervous system (CNS). The CNS comprising
of the brain and the spinal cord itself contain neural networks by which different regions
of the CNS communicate with each other. The neural connections are formed by a large
number of nerve cells consisting of a cell body and a tail-like structure called axon. Axons
are specialized structures responsible for conducting action potentials (neural electrical
impulses) from a nerve cell to another. Axons connect with other nerve cells via synapses.
Bundles of axons or axonal fiber tracts (or neural fiber tracts) within the brain, the spinal
cord and the peripheral nervous system provide a communication network for transmitting
a plethora of control signals to a variety of organs. The integrity of the neural network
formed by the neural fiber tracts is extremely important for the normal functioning of the
human body. Thus, there is a need for non-invasive methods for imaging the neural fiber
tracts for proper evaluation of the health of the tracts and the diagnosis of pathology.
The integrity of the neural network can be analyzed by neuronal connectivity studies
which can either be structural (based on neuro-anatomy) or functional (based on neurophysiology) studies. Some of the techniques based on measuring neuro-physiological phe-

1

nomena (as critically discussed in [1, 2]) are Electroencephalography (EEG, which measures the electrophysiology in the brain), Magnetoencephalography (MEG, which measures the induced magnetic field caused by the electrophysiology in the brain), functional
Magnetic Resonance Imaging (fMRI, which measures the hemodynamic response (blood
oxygen level dependent (BOLD) signal [3]), generally in the brain) and Fludeoxyglucose
(FDG) Positron Emission Tomography (FDG-PET, which measures the metabolic absorption rate in the whole body), O-15 PET (measures cerebral blood flow). It should
be noted that physiological phenomena are largely due to changes in the nerve cell body
rather than the fiber tracts themselves and the connectivity is established in different
parts of the nervous system by either analyzing the temporal correlation of these physiological processes or testing a hypothesis based on a cause-effect relation (causality).
Hence, it’s an indirect way of establishing neuronal connectivity as compared to a more
direct way in neuro-anatomical imaging.
Early techniques in neuro-anatomical imaging were ex vivo and only applicable postmortem, such as tract tracing using fluorescent dyes (for example, [4]). However, recent
advances in imaging technology especially in magnetic resonance imaging, have provided
completely non-invasive in vivo methods for studying neuronal connectivity anatomically.
To understand the anatomy of fiber tracts, there is a need to precisely and accurately
visualize the fibers causing little or no damage to the structure. Stejskal and Tanner [5, 6]
showed that nuclear magnetic resonance (NMR) can be used to obtain information about
diffusion of water molecules when confined within restrictive boundaries. They demonstrated that by applying a pair of diffusion-encoding gradients in a spin-echo experiment,
the NMR signal can be weighted by an additional attenuation factor that depends on the
diffusivity of water molecules. After the advent of magnetic resonance imaging (MRI),
the use of the Stejskal-Tanner pulse sequence [5] (or sequences based on it) during MR
imaging would be known as Diffusion-Weighted MRI (DW-MRI or DWI). An important
observation from the DWI is that the MR signal attenuation due to diffusion depends on
the direction of the diffusion-encoding gradient. Thus, applying diffusion-encoding gradients along the direction of high water diffusivity produces higher MR signal attenuation

2

compared to along any other direction. Hence, DWI can be used to measure diffusion
anisotropy.
In the axon bundles in nerve fibers, the water molecules follow preferential movement
along the fiber direction. This preferential movement is due to barriers to molecular
motion such as myelin (fat) sheath in neural fibers that causes restricted or hindered
diffusion towards the boundary of the fibers and unrestricted diffusion along the longitudinal direction of the fiber. Such anisotropy in the local diffusion process can be measured
by DWI to infer tissue microstructure in the neural fibers. By optimally selecting the
strength and directions of the diffusion-encoding gradient, neural fiber tracts can be more
precisely demarcated. This leads to a more comprehensive study of the neural fiber tracts
and helps to improve the DWI protocols in clinical studies.

1.2

Diffusion-weighted imaging

In diffusion-weighted imaging, the diffusion-weighted MR images are acquired by using
specialized pulse sequences such as the Stejskal-Tanner Spin Echo sequence [5] which
uses additional diffusion-encoding gradients. This pulse sequence can capture the effect
of local water diffusion and attenuate the MR signal according to the diffusion process.
It should be noted that the DWI images themselves are less informative. However, from
post-processing of the DWI image data, quantities characterizing the diffusion process
are estimated which provide more direct information about the imaged structure and
are more commonly used in the analysis or diagnosis of diseases. These quantities can
either be scalar so as to generate a secondary parametric map or these can be angular
information which can be directly used for tracking neural fibers.
The diffusion process can be characterized either by a parametric model or by a nonparametric process, such as defined by a transform of the DWI data. This distinction
differentiates the post-processing techniques. For model-based post-processing (e.g., DTI
[7], QUAQ (quantitative analysis of q-space) [8], CHARMED (composite hindered and
restricted model of diffusion) [9]), the MR signal is modeled by a parametric relation

3

between the signal and diffusion model parameters (such as the local diffusion coefficients
or diffusivities and the fiber orientation angles). This is in contrast to non-parametric
methods (e.g., diffusion spectrum imaging [10], q-ball imaging [11]), where the signal is
processed to obtain probability distributions of quantities of interest. For instance, the
orientation distribution function in q-ball imaging is obtained by performing the FunkRadon transform of the DWI data collected under certain experimental settings [11]. The
focus of this work is on model-based methods, such as DTI.
In model-based post-processing approaches, the DWI data is fitted to a diffusion model
to extract specific quantitative information. Diffusion tensor imaging (DTI, [7, 12]), the
most often used model, is based on an anisotropic hindered diffusion model with six
parameters, and yields a local “apparent diffusion tensor”, whose principal eigenvector
provides the fiber orientation. DTI model uses a single tensor to characterize diffusion
within the fiber tracts and as such it reconstructs a single (unidirectional) fiber bundle
within each volume element (voxel) of the image. The DTI model can be modified by
assuming axisymmetric diffusion inside fibers (ADTI), reducing the number of model
parameters to four. A physics-based approach, called QUAQ (quantitative analysis of
q-space [8]), models fibers as impermeable axisymmetric cylinders, and uses the solution
of the diffusion equation inside a cylinder which also contains four parameters.

Figure 1.1. (a) T2 -weighted image, (b) fractional anisotropy (FA) map and (c) mean
diffusivity (MD) map. The data for these images are acquired using a DTI protocol
(with MF30 gradient scheme and b = 1000 s mm−2 ) and the images are processed using
FSL software package (Analysis Group, FMRIB, Oxford, UK).

4

For neuronal connectivity, diffusivities and fiber orientation maps are investigated since
these can indicate presence and connection of fibers in the imaged region. For example, a
secondary quantity computed based on the DTI processing is fractional anisotropy (FA)
which is a dimensionless number represented by a ratio of diffusion coefficients as defined
in [13]. It is indicative of the local diffusion anisotropy (see Fig. 1.1(b)). FA images
are widely used in connectivity studies (for example, [14]). Another scalar quantity
derived from the DWI data is the mean diffusivity (MD) which provides an average of
diffusivities calculated from the apparent diffusion tensor. The MD is more indicative
of local diffusion isotropy (see Fig. 1.1(c)). For fiber tractography (or tracking), fiber
orientation information as well as other diffusion coefficient based quantities (such as FA)
are collected as the output of DWI post-processing and used in tractographic algorithms
([15, 16]) (see Fig. 1.2). Since fiber tractography is a secondary processing stage, its
precision and accuracy (and, in general, quality) depends on the precision and accuracy
of the post-processing results of DWI.

Figure 1.2. Fiber tracking through Corpus Callosum (a) Axial view, (b) Coronal view.
The fibers shown are 3D reconstructed streamline fibers projected onto the 2D image
views. The color coding indicates the fiber orientation on the RGB scale, namely, red
is right(R)-left(L), blue is superior(S)-inferior(I) and green is anterior(A)-posterior(P)
orientations respectively. Note that the axial and coronal views are not on the same scale.
The data for these images are acquired using a DTI protocol (with MF30 gradient scheme
and b = 1000 s mm−2 ) and fiber tracking is performed using MedINRIA software package
(Asclepios Research Project, INRIA Sophia Antipolis, France). For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic
version of this dissertation.

5

In DWI, there are a number of MRI experimental parameters that can be optimally
chosen to improve the precision of the estimated fiber orientation and the diffusion model
parameter. These include the diffusion-encoding gradient strength, gradient directions
and timing parameters, such as the diffusion gradient pulse duration and time interval between the diffusion gradients. Several optimization criteria and gradient-encoding
schemes have been discussed in [17]. DWI protocol optimization can also be based on a
specific structure to be imaged. If adequate knowledge about the structure is available,
optimal schemes based on the a priori knowledge can be designed.

1.3

Optimizing DWI protocols

In the formulation of an optimization problem, it is required to define the optimized
parameters, the optimization cost function which is minimized during the optimization
procedure, other additional parameters which are part of the formulation but not varied,
constraints on the optimized parameters and finally the prior knowledge of the parameters of the problem. After the optimization problem is formulated, the next steps include
the development and implementation of the algorithm that solves the optimization problem and computes the optimized parameters. Finally, the performance of the optimized
parameters is validated using either simulations or experiments.
A DWI experimental protocol consists of a prescribed list of settings for the different
experimental parameters to be used during imaging. For a Stejskal-Tanner spin-echo
pulse sequence, these include parameters related to the diffusion-encoding gradient (its
strength, direction and timings). In the context of the optimization problem, these are
the optimized parameters. However, a variety of cost functions have been defined by
researchers which address different aspects of the DWI experiment. The optimization
problem depends on the post-processing of the DWI data. Since the focus is on modelbased post-processing (especially DTI), I will discuss some of the optimization done for
the DTI protocol.
A number of DTI diffusion gradient optimization techniques have been proposed previ-

6

ously. Numerically optimized schemes (minimum force, minimum energy, minimum condition number), heuristic schemes (orthogonal encoding) and geometric schemes (icosahedral polyhedra) are approaches that only optimize diffusion gradient directions [17].
There are schemes based on optimizing the specific cost function, such as in [18–21].
Hasan et al. [17] summarized several DTI-based or model-independent optimization cost
function to be minimized (variance for the diffusion tensor components, condition number [19], force [22], Coulomb energy) and compared the resulting schemes via Monte
Carlo simulations. Besides diffusion gradient directions, optimization can also be on the
selection of the diffusion gradient strength, pulse duration and pulse interval between
diffusion gradients and number of diffusion gradient directions, as shown by Alexander et
al. [23] and Brihuega-Moreno [24]. In DTI, the diffusion gradient strength and the timing
parameters are combined into b-factor [12].
In the discussion so far, the various DTI protocol optimization mentioned focus on the
selection of the optimization cost function while not exploring the use of prior knowledge
of the diffusion model parameters which can be effectively used to improve the optimization performance. Use of the prior knowledge constrains the optimization problem.
Hence, the prior should be chosen cautiously so as to not overly constrain the problem.
Typically, the prior knowledge of the imaged structure can be utilized for optimization.
For example, in spinal cord tracts in normal healthy subjects, the nerve fibers are mostly
in the superior-inferior orientation and enclosed within a narrow range of fiber angles.
To illustrate this point, DTI data was collected using standard protocol and analyzed in
the cervical spinal cord region of a healthy adult.
Fig. 1.3(a) shows the coronal view of the spinal cord/brain stem region. The underlying
image is a T2 -weighted MRI image used for demarcation of the spinal cord region (dark
region) within the cerebrospinal fluid (CSF) surroundings (bright regions). The overlying
arrows show the fiber orientation (projected on the 2D image) estimated using DTI at
each pixel of the image. The distribution of the deviation angle α w.r.t. the average
orientation of the nerve fibers is plotted in Fig. 1.3(b). It is observed that the majority
(∼85%) of the fiber orientations is contained within a narrow range of fiber angles (cone

7

z (mm)

30
20
10
0

20
40
x (mm)

(a)

Distribution ( % )

8
6
4
2
0
0
(b)

20

40
60
α(°)

80

Figure 1.3. DTI results on a healthy adult human in the cervical spinal cord/brainstem
region. (a) Fiber orientations estimated by DTI superimposed over a T2 -weighted image,
(b) distribution of the angle between the fiber orientations and the mean fiber orientation.
of half-angle 35◦ ).

1.4

Motivation and scope of research

Tissue structural information is well documented from anatomical studies for both healthy
([25]) and pathological tissues ([26]). The available knowledge of expected structure
can be represented by mathematical models. For a simple case, the structure can be
modeled as a cone of structural projections with a specified mean orientation and a range
of orientation angles about the mean orientation, as shown for the spinal cord nerve
fibers case in Fig. 1.3. The mean and range values for the cone model can be obtained
from previous DTI studies (as was performed for Fig. 1.3 and other studies such as
8

[27, 28]). Similar cone-based structural model can also be applied to other tissues that
show directional nature with long range projections (without sharp bending) and where
the structural information necessary for the model is available from previous DTI studies,
such as median nerves in the wrist [29], optic nerves in the brain [30], peroneal nerves
in the leg [31] and muscle fibers [32]. Apart from the structural information, diffusivity
properties of the tissue can also be obtained from previous DTI studies. Hence, there is
a prevalence of tissues in the human body with available a priori structural information
and this information can be effectively exploited to optimize the DTI protocol.
The research work in this thesis focuses on utilizing the prior information of the imaged
tissue available from previous studies to optimally select the DTI experimental parameters (gradient directions and the b-factor) so as to improve the precision (or reduce
uncertainty) of the estimated diffusion model parameters. The optimization framework
developed in this work is based on D-optimality [33, 34] which minimizes the determinant of the Cramer-Rao lower bound (CRLB) of the covariance matrix of the estimated
model parameters. The use of prior information constrains the optimization problem to
improve the precision of the parameter estimation. Such optimized protocol will lead to
more precise estimation of FA, MD and improved fiber tracking.
The optimized DTI protocol can be applied to a number of scenarios. It can be used
for aging related studies of normal subjects where the structural orientation of the tissue
remain consistent and diffusivities for different age groups can be obtained from previous
DTI studies, for example, changes in cervical spinal cord white matter organization in
normal subjects and the correlation with function such as dexterity as demonstrated in
[35]. A second scenario for the application of the optimized protocol could be assessment
of nerve regeneration [36]. In this case, the expected orientation angles of the growing
nerve fibers can be based on previous DTI studies on normal subjects and thus a prior
information based optimized protocol can be used to monitor the regeneration process.
A third scenario could be in the application to the detection of neurodegenerative diseases, such as multiple sclerosis (MS). MS results in demyelination of nerve fibers causing
reduced capability to conduct nerve impulses. Also, MS lesions that are caused due to

9

scarring result in transection of fiber tracts. Thus, in MS, the FA values show a significant
decrease as compared to normal subjects [37, 38]. The DTI protocol can be optimized at
a reduced FA value corresponding to MS (rather than normal subjects) and can be used
to improve the detectability of MS.
Although a number of studies have been reported on the optimization of DTI experimental parameters (as summarized by Hasan et al. [17]), few have utilized the prior
structural information. Two recent works on using prior information for DTI protocol
optimization are Peng et al. [21] and Gao et al. [39]). Both of them used the cone model
for fiber orientations. However, each of their techniques have certain shortcomings that I
intend to address in this research work. Their techniques have very specific cost functions,
thus lacking flexibility in selecting the diffusion model parameters for which the protocol
is optimized. The optimization framework developed here provides a number of choices to
select the optimization cost function, such as optimizing for improved precision of all diffusion model parameters, diffusivities only, fiber orientation angles only, variance of FA,
variance of MD, etc. All of these cost functions are derived from the CRLB formulation.
Moreover, diffusion-weighted images suffer from low SNR and since these are magnitude
images, the DTI measurement noise is non-additive Rician rather than additive Gaussian which has been used in these previous works. This research work develops both
the Rician and Gaussian CRLB-based optimization frameworks. Finally, the previous
works have largely used the tensor-based diffusion model for the optimization. However,
the framework developed in this work can optimize for any diffusion model (DTI, ADTI,
CHARMED, QUAQ etc.) although focus has been on DTI only. Thus, this work aims
to provide a more generalized optimization framework for the experimental protocol.

1.5

Outline

This thesis is mainly divided into three sections. The first section comprising of Chapters
2 – 4 consists of a review of the various DWI (and other neuroimaging) techniques, then
a brief description of MRI and DWI principles.

10

The second section is mainly in Chapter 5 where the theoretical framework of the
DWI protocol optimization is described in details. This chapter introduces the concept
of CRLB and its use for optimal design of experiments. Also, various noise models, such
as Rician and Gaussian noises, and their corresponding CRLB definitions are derived
in this chapter. The optimization algorithm is also described here. An aspect of DWI
optimization for improvement in the precision of selective diffusion model parameters
is also introduced. Finally, optimized gradient schemes for DTI and ADTI models are
generated and there performance analyzed.
Finally, the third section which are in Chapters 6 and 7 discuss the simulation results
and the spinal cord ADTI study on human subjects. Chapter 6 discusses noise characterization, the simulation results for the effect of various experimental parameters on the
estimation of diffusion model parameters and performance of various estimators. Chapter
7 describes the human study for spinal cord ADTI where the gradient scheme optimization procedure and the validation process are described in details. Next, a study on the
effect of optimization on fiber tracking is presented. Finally, an experimental technique
for optimizing both the b-factor and the gradient directions is given which optimizes the
experimental parameters for improved precision of selected diffusion model parameters.
Chapter 8 concludes this thesis by summarizing the results and contributions and
discussing future research works that can benefit from the work done in this research.

11

CHAPTER 2
A review of current techniques in
neuroimaging
2.1

Neuronal connectivity: DWI and other competing technologies

Neuronal connectivity can be classified as structural, functional or effective. While structural connectivity is a more direct visualization of neural connections, functional and
effective connectivity are related to neurophysiological processes mainly originating from
the nerve cell body instead of the axons. Functional connectivity is defined as the temporal correlations between spatially remote neurophysiological events while effective connectivity is defined as the influence one neural system exerts over another, either at a
synaptic or cortical level [40].
At a conceptual level, both functional and effective connectivity can be observed using
either neuroimaging (such as fMRI, O-15 PET) or electrophysiology (such as EEG, MEG)
[40]. However, at the practical level, neuroimaging and electrophysiological methods are
fundamentally different due to difference in time scales and the measured neurophysiological phenomenon. It is important to note that functional connectivity only provides a
statistical correlation-based relationship without any causal implications, while effective
connectivity measures causal neural connections that can decipher the directionality of

12

the neural processes. However the causal relations are generally based on an assumption
of a causality model or framework, such as structural equation modeling [41], dynamic
causal modeling [42], Granger causality mapping [43], multivariate autoregressive modeling [44].
Neuroimaging methods in fMRI measure hemodynamic response (BOLD contrast [45])
and in O-15 PET measure cerebral blood flow [46], while electrophysiological methods
in EEG measure volume conduction [47] or associated effects, such as induced magnetic
field due to volume conduction in MEG [48]. Volume conduction is caused by spiketrains of neural electrical impulses in active cortical regions. Neuroimaging methods have
better spatial resolution (for example, in fMRI, sub-millimeter voxels can be imaged)
than temporal resolution (for example, in fMRI, temporal resolutions are of the order of
seconds). However, electrophysiological methods have better temporal resolution (order
of milliseconds), but poor spatial resolution (tens of millimeters of separability of cortical
sources).
For studying structural connectivity, techniques can be either based on imaging or
histological studies. One of the histological method for neuroanatomical studies is tract
tracing which involves injecting the tissue sample with lipophilic flourecent dyes [4] and
tracing the labelled neural tracts. This technique is applicable post-mortem and the tissue
staining process generally takes times of the order of more than 24 hours. In the context
of the research work in this thesis, neuroimaging techniques can be broadly classified into
MRI-based and non-MRI based methods. MRI-based neuroimaging methods include T1 weighted MRI, T2 -weighted MRI, MR microscopy (MRM) and finally DWI. While T1
and T2 weighted MRI are currently part of routine clinical MRI, MR microscopy and
DWI are generally research protocols. MRM is essentially ultra high resolution (order of
10–100 microns) MRI which involves the use of ultra-high magnetic field (e.g., 9.4 or 11.7
Tesla) and very high gradients [49]. In-vivo MR microscopy is limited to small animals.
MRM is also reported on excised human tissue samples (post-mortem) from the cortical
region [50]. Amongst non-MRI based methods, X-ray based computed tomography (CT)
methods with contrast enhancement have been reported for neural tracking [51]. Another

13

research reported use of scanning electron microscopy (SEM) along with reconstruction
techniques to explore neural connections [52].
With the improvements in MRI technology, diffusion-weighted imaging is more frequently used for neuroanatomical imaging. Neural fiber tracts can be traced in-vivo
from one region of the brain to another completely non-invasively and within a reasonable scan time (order of tens of minutes). Thus, DWI provides the anatomical aspect
to neuroimaging which is not present in neurophysiology-based methods. Functional or
effective connectivity studies can be either validated or augmented by structural connectivity studies based on DWI. Integration of DWI-based information with functional or
effective connectivity is fast becoming a means of studying different neurological processes
as well as diseases such as Alzheimer’s disease [53] or accessing tumors [54].

2.2

Applications of DWI

After introducing the various techniques in neuroimaging which are contemporary to
DWI-based methods, I will focus on DWI and its various applications. DWI in an MR
imaging method where the MR signal is sensitize to the local diffusion process. Structures
in nerve (axonal) fiber tracts are highly organized and oriented in a particular direction.
Owing to myelin sheath covering the axons, the self-diffusion of water molecules in these
structures is anisotropic in nature with high diffusivity along the direction of fibers and
low otherwise. DWI can measure the diffusion anisotropy by collecting a series of DW
images at different gradient directions. The diffusion anisotropy can be modeled by a
diffusion tensor as in DTI ([7]) or using non-parametric methods as in q-ball imaging
([11]).
In DTI, after the acquisition of DWI data, the diffusion tensor is estimated on a voxelby-voxel basis. Next, two types of secondary post-processing can be conducted to obtain
knowledge of neuronal connectivity. The first processing is computation of scalar quantities that are representative of the local diffusion process, such as the eigenvalues of the
diffusion tensor, mean diffusivity, fractional and relative anisotropy [13]. These scalar

14

metrics can be used for further quantitative analysis in studying connectivity. These
have been used for detection of stroke [55] and multiple sclerosis [56]. The second processing is fiber tracking which is generally the visual representation of reconstructed nerve
fiber tracts. Fiber tracking uses the local fiber orientation (in the form of direction angles) calculated from the DWI data processing and applies either deterministic tracking
([16, 57, 58]) or probabilistic tracking algorithms ([59, 60]). Deterministic tracking algorithms estimate a streamline fit using continuous tracking to reconstruct the fibers using
local fiber orientation information and additional constraints (on FA, for example). This
algorithm require a seed region to initiate the tracks. Probabilistic methods provide
probability maps of connectivity of a seed region to the rest of the brain (or specified
targets in the brain). Two common techniques are “FACT” (fiber assignment by continuous tracking, [16] used by DTIStudio software package) which gives deterministic
reconstructed fiber tracks and “Probtrack” [59] (part of FSL package) which provides
a quantitative voxel-by-voxel probability of connection of seed to targets. Tracking can
also be performed by using thresholded FA maps which are projected on to a standard
space and computing statistical quantities such as mean, variance or Z-score as shown in
Tract-Based Spatial Statistics (TBSS [14]) method.
Based on the scalar metrics derived from the post-processing of DWI data (such as FA,
MD, diffusivities) or fiber connectivity information, DWI can be applied for the diagnosis
of pathology. For example, DTI-based metrics have been used as biomarkers for detecting
spinal cord diseases, such as multiple sclerosis [56], amyotrophic lateral sclerosis [61], aging
and spondylosis-related changes [62], as well as spinal cord compression [27]. Many of
these studies involve characterizing the difference in the DTI-based biomarkers between
normal and pathological population which helps to identify the range of values of these
metrics. In the context of DWI protocol optimization in this thesis, such prior knowledge
of range of values can be used to optimize the DWI protocol to diagnose specific diseases.
Applications of DWI can also be based on the imaged target organs in the body. DWI
can be for neuroimaging in the brain, spinal cord and the peripheral nervous system. DWI
can also be extended to any tissue which exhibit an organized and oriented structure,

15

such as skeletal muscle fibers. DWI applications for skeletal muscle fibers are especially
interesting in this context since the DWI protocol optimization discussed in the thesis
can be extended for imaging muscle tissues. Some of the recent work conducted on DWI
of muscle include use of DTI-based fiber tracking for in vivo three-dimensional (3D)
architecture of skeletal muscles in mice hind leg [63], application of DTI fiber tracking
in human skeletal muscle and test the cause of the heterogeneity in pennation angle
(fiber orientation) [64] and creating biomechanical models of the quadriceps mechanism in
humans using DTI fiber tracking information [32]. The DTI-based fiber tracking provides
information on the local orientation of the muscle fibers (pennation), fiber length and
cross-section of the fiber bundles. These information correlate to muscles physiological
cross sectional area which can be used to predict the muscular force produced by the
muscle [32].
The work in this thesis will focus on the spinal cord. DTI has been used for studies
on the spinal cord [65–67] and have shown promising results. Ries et al. [65] applied
DTI to subjects suffering from narrowing of the cervical canal and observed substantial
differences in diffusion characteristics to detect lesions in the spine. Mottershead et al. [66]
demonstrated the existence of a strong correlation between myelin content and axonal
density with diffusion anisotropy. Ducreux et al. [67] reconstructed 3D fiber tracts to
visualize the deformation of the posterior spinal cord lemniscal and corticospinal tracts.
DTI has been proven to be a valuable diagnosis tool for spinal cord diseases, such as
multiple sclerosis [56], amyotrophic lateral sclerosis [61], aging and spondylosis-related
changes [62], as well as spinal cord compression [27].
ADTI diffusion model has been used in the cervical spinal cord experiment in this
work. Previously, the axisymmetric diffusion assumption has been used by Anderson [68]
to estimate fiber diffusion properties through high angular resolution diffusion imaging
(HARDI), and by Assaf et al. [9] with the composite hindered and restricted model of
diffusion (CHARMED) to model diffusion in intra-axonal compartments. The mean of
the secondary and tertiary eigenvalues of the diffusion tensor has also been used previously
to compute the transverse diffusivity to investigate regional differences in white matter

16

tracts in the cervical spinal cord funiculli in humans [69], as well as in rat spinal cord
studies [70, 71].

2.3

Prior work on DWI protocol optimization

DTI protocol can be broadly divided into selection of the b-factor (which combines diffusion gradient strength and the timing parameters) and selection of the diffusion gradient
directions. For the diffusion gradient directions, a number of gradient scheme have been
discussed by Hasan et al. [17]. These gradient scheme are either selected based on geometry (such as icosahedral polyhedra) or heuristics (orthogonal encoding) or numerical
optimization (by minimizing a cost function, such as minimum energy, minimum force).
Some other examples of numerically optimized gradient scheme is discussed next. Papadakis et al. [18] isolated the effect of gradient directions from the weighting b-factor
for the diffusion tensor estimation. They defined an “index of DTI”, which relates the
variance of the measured data to the total variance of the measured tensor components
and used it as a measure of optimality of the diffusion gradient directions. Skare et al.
[19] proposed the condition number as a means of studying the noise propagation, and
showed improvements in the estimation of the FA by minimizing condition number of the
transformation matrix for a given gradient scheme. Batchelor et al. [20] showed that noise
propagation in DTI is anisotropic and used the standard deviation of FA as a measure of
optimality. Variance of FA was also used by Peng and Arfanakis [21] to compare several
gradient schemes. Hasan et al. [17] used several DTI-based or model-independent optimization metrics to be minimized (variance for the ADT components, condition number
[19], force [22], Coulomb energy) and compared the resulting schemes via Monte-Carlo
simulations. A common gradient directions optimization scheme is the minimum force
(MF) based scheme by Jones et al. [22]. They proposed that in absence of any prior
knowledge of the tensor to be estimated, gradients can be uniformly distributed in 3-D
gradient space by minimizing Coulomb’s force between unit charges on a sphere where
charges represent the gradient directions. The MF-based gradient schemes do not assume

17

any DTI signal model. Besides diffusion gradient directions, optimization can also be on
the selection of the diffusion gradient strength, pulse duration and pulse interval between
diffusion gradients and number of diffusion gradient directions, as proposed by Alexander
et al. [23].
So far I discussed optimization by the selection of cost function. Next, I will discuss
optimization using prior knowledge of the structure. Thus, these will fall under constrained optimization procedures. Work on the optimization of estimation of DTI model
parameters using prior structural information has been previously discussed by Peng et
al. [21], Gao et al. [39] and Yanasak et al. [72]. Each of these works attempted to improve certain aspects of the DTI model parameter estimation. Peng et al. [21] reduced
the total variance of FA by using a 30◦ cone of fibers as a prior structural information
for the fiber bundles in the corticospinal tract. Gao et al. [39] showed an optimization
procedure based on simulated annealing to simultaneously optimize various DTI experimental parameters. They used 20◦ cone configurations in their procedure and proposed
to apply it to neonatal DTI. Simulated annealing is a stochastic minimization technique
[73] which is based on the annealing process (slow cooling process) in metallurgy and is
known to be robust with respect to local minima problems common in gradient-based
minimization methods. Yanasak et al. [72] proposed a gradient scheme to improve the
precision of angular measurements in DTI. They showed that restricting the zenith angles
of diffusion gradient directions within certain bands can improve the precision of angular
measurements significantly.
In this thesis, I have extensively used CRLB to derive the cost function to predict
the uncertainty of the diffusion model parameters so that an optimized DWI protocol
parameters can be obtained. CRLB is defined by both the noise and the signal model.
Use of CRLB has been previously demonstrated by Brihuega-Moreno et al. [24] for the
optimal selection of b-values (or b-factors) by minimizing the CRLB of diffusion coefficient
with respect to the b-values. They assumed Gaussian noise model for magnitude MR
images in the definition of CRLB. However, the Rician noise model is a more accurate
noise model for magnitude MR image data. Alexander [23] demonstrated the use of

18

Rician CRLB in the optimization of acquisition parameters based on the CHARMED
model [9]. Both the Rician CRLB [23, 74]) and the Gaussian CRLB has been derived
and analyzed in this work. Through this work a number of scenarios of signal and noise
models have been analyzed. For each case, the CRLB and the cost function has been
obtained and optimization was validated, either by simulations or by DTI experiments.

19

CHAPTER 3
Basic principles of nuclear magnetic
resonance imaging
3.1

Nuclear magnetic resonance and signal generation

Atoms with an odd number of protons and/or neutrons (odd mass number) have nuclear
spin angular momentum. The nucleus of these atoms can be considered as spinning
charged sphere which can possess small magnetic moment. In biological specimen, the
most common nucleus that is used for NMR is Hydrogen (1 H) which has a single proton.
It is abundantly found in water (H2 O) and fat or other organic molecules. Examples of
other common nuclei used in MR studies are Phosphorus (31 P), Sodium (23 Na), Carbon
(13 C).
From a quantum mechanical perspective [75, 76], the spin angular momentum of a
nucleus is quantized and can exist in only certain states. For hydrogen, the nuclear
spin can take only two quantum states of angular momentum which are given by Iz ,
where Iz = ±1/2 and

is the reduced Planck’s constant ( = 1.054 × 10−34 J s). Thus,

two populations of spins of hydrogen nuclei can exist in a specimen. In the absence
of any external magnetic field, the spin populations are equal in number and cancel
each other’s magnetic moments. Thus, the net magnetic dipole moment of the overall

20

population is zero. When the specimen is exposed to a constant external magnetic field,
B0 (with magnitude B0 and direction along +Z-axis in the laboratory reference frame
by convention), the spins tend to align with the field in either parallel or anti-parallel
fashion and attain potential energy due to the interaction between magnetic moment and
the external applied field, B0 (Fig. 3.1 (a)). Since, the spins are at two distinct states of
angular momentum (and thus magnetic moment), the application of the external field,
B0 , results in spins at two distinct potential energy levels separated by an energy gap,
∆E, given by ∆E = γ B0 , where γ is gyromagnetic ratio (Fig. 3.1 (b)).

(a)

(b)

Figure 3.1. (a) Alignment of spins with B0 (most are parallel, while some can be antiparallel to B0 . M is the net magnetization vector. (b) Separation of spin energy into
low and high levels when subjected to B0 .
Although, the spins tend to occupy the lower energy state (parallel), the energy gap,
∆E, is easily overcome due to the thermal energy of the nuclei (from body temperature).
The ratio of the spin population in the two energy states in presence of an external
21

magnetic field can be given by n+ /n− = e∆E/kT , where n+ , n− are the spin populations
at lower and higher energy states respectively and k is the Boltzmann constant (k =
1.38 × 10−23J K−1 ). Typically, this corresponds to about 3 parts per million per Tesla
at 310 K temperature of excess lower energy spins. This excess spin population causes a
net magnetic moment along the direction of the applied field. Sum total of the magnetic
moments in a voxel (or volume element) is denoted by net magnetization, M , which is a
vector and it is also aligned with the applied magnetic field, B0 , at equilibrium. Thus,
at equilibrium, only longitudinal magnetization exist.
Any longitudinal magnetization (along B0 ) due to magnetic resonance is difficult to
measure since it’s contribution can not be distinguished from that of B0 while taking
field measurements. In order to generate and measure a unique MR signal, the net magnetization, M , needs to be reoriented or flipped away from the longitudinal direction. To
achieve this flip, spins in the lower energy state must be shifted to the higher energy state
by providing energy equal to ∆E. Based on quantum mechanical model, the frequency
equivalent of ∆E is called Larmor frequency, ω0 . Magnetic resonance is exhibited when
external magnetic field at frequency equal to Larmor frequency, ω0 , is applied to a spin
system thereby causing energy transfer from the source field to the spin system. Larmor
frequency is given by,
ω0 = γB0
or
f=

γ
B
2π 0

(3.1)

,where ω0 is known as the Larmor frequency and γ is gyromagnetic ratio. For 1 H, γ/2π
= 42.576 Mhz T−1 .
The net magnetization, M , and the applied external magnetic field, B0 , follow a
relation given by the Bloch’s equation (without the relaxation and diffusion terms),
dM
= M × γB0
dt

(3.2)

Since, the net magnetization (M ) is related to the angular momentum of the spins, the
solution to Bloch’s equation gives the angular frequency with which spins would precess
22

about B0 if not aligned with it. And it turns out to be the same as Larmor frequency,
ω0 . Thus, the rate at which the spins precess about the direction of the external static
magnetic field, B0 , is also given by the Larmor frequency, ω0 .
Going back to the process of flipping the magnetization vector, an additional external
high frequency (radio frequency, RF) magnetic field pulse, B1 (apart from the static
magnetic field, B0 ) tuned to the Larmor frequency, ω0 , when applied transverse to B0 ,
would cause the spins and the net magnetization, M , to flip towards the transverse
direction of B0 (Fig. 3.2 (a)). Since the RF field is at frequency ω0 , it carries energy
equivalent to ∆E which is necessary to transfer spins to the higher energy state. While
flipping the spins, the spins and net magnetization, M , would follow a precessing motion
about B0 at frequency ω0 . This can be explained using Eq. 3.2 in the context of B1
where B1 creates a torque that rotates the M vector towards the transverse direction
(Fig. 3.2(b)). Generally, B1 is of the order of a few Gauss in strength and milliseconds
in duration.
B1 = B1 cos(ω0 t)i − B1 sin(ω0 t)j

(3.3)

where B1 is the magnitude of B1 and i, j are the unit vectors along +X and +Y directions
in laboratory reference frame and t is the time. If the RF pulse is applied to spins at
equilibrium, then right after flipping, the spins would have phase coherence (i.e., the spins
are rotating in phase). Also, the net magnetization vector, M , is nutated towards the
X-Y plane. By careful design, an RF pulse can be applied to flip M completely to 90◦ .
Thus, M will have no longitudinal component. Such an RF pulse is called a 90◦ pulse.
The MR signal is a measure of the transverse magnetization and is collected after the
RF pulse is turned off. In absence of the RF pulse, magnetization relaxation initiates and
the net magnetization, M , tends to return to the equilibrium state and towards the B0
direction. The relaxation process consists of the growth of the longitudinal magnetization
and it is accompanied by the decay of the transverse magnetization (Fig. 3.2 (c)). The
former follows T1 -recovery (Fig. 3.3 (a)), while latter follows T2 -decay (Fig. 3.3 (b)).
T1 -recovery corresponds to spin-lattice relaxation where the spins return to lower energy
state while releasing energy, ∆E, to the surrounding lattice. T2 -decay, on the other hand,

23

(a)

(b)

(c)

Figure 3.2. (a) M when RF pulse, B1 , is applied as seen in the laboratory reference
frame (b) M nutated by flip angle θ as seen in the rotating reference frame. (c) M
during relaxation after RF pulse is turned off as seen in the laboratory reference frame.
Mz and Mxy are the longitudinal and transverse magnetizations respectively.

24

Mz / M0

1

0.5

0
0
(a)

T1 recovery
5
Time (s)

Mxy / M0

1

10
T2 decay

0.5
0
−0.5
−1
0

(b)

5
Time (s)

10

Figure 3.3. (a) T1 recovery of longitudinal magnetization and (b) T2 decay of transverse
magnetization. M0 is the equilibrium magnetization.
corresponds to the dephasing of the precessing spins while returning to lower energy
states. The dephasing can be attributed to spin-spin interactions owing to local field
interactions due to individual magnetic fields of each spin. When the RF pulse is applied
to spins in equilibrium, then the moment after the RF pulse is turned off, the spins are
in phase coherence (or in phase). However, due to the spin-spin interactions, the phase
coherence is lost resulting in the T2 -decay of transverse magnetization. The time constant
characterizing the return of the magnetization vector along the longitudinal direction is
called T1 , while the time constant characterizing the decay of the magnetization vector
component in the transverse plane is called T2 . In human tissue, T1 values range from
180–2000 ms (at 1 T) whereas for T2 values, the range is 40–300 ms (at 1 T) [77]. The
25

MR signal is proportional to the magnitude of the transverse magnetization which is in
turn dependent on the density of the nucleus in the volume of the specimen as well as
the relaxation time constants. Thus, the contrast mechanism in the MR signal between
tissues is mainly dictated by the spin density and the relaxation time constants (T1 and
T2 ). Additional signal weighting can be achieved by specialized pulse sequences, such
as diffusion-weighting, which is described in the next chapter. The signal right after the
excitation pulse (90◦ pulse) is known as free induction decay (FID). The signal can be
represented mathematically by,
S(x, y) = Kρ(x, y)[1 − e−TR /T1 (x,y) ]e−TE /T2(x,y)

(3.4)

where S(x, y) is the signal at (x, y) location, ρ is density of the nuclei, T1 and T2 are the
relaxation and decay time constants, respectively. K is any other gain constants lumped
together, such as digitizer gain, coil sensor gain. TR is the pulse repetition time which
corresponds to the time after which the sequence of RF excitation signal is repeated
to take measurement at different locations in the specimen. TE is echo time and it
corresponds to the time at which the data is acquired. Echoes will be discussed later.
In practice, the resonant frequency is not uniform all through the specimen due to
main field inhomogeneity, susceptibility effects, chemical shifts and the application of the
linear gradient field itself. Due to additional dephasing effects on the spins, these effects
cause additional loss of the transverse magnetization resulting in a faster decay rate and
∗
a shorter time constant (called T2 ) as compared to the T2 decay. The expression for the

MR signal is updated as,
∗

S(x, y) = Kρ(x, y)[1 − e−TR /T1(x,y) ]e−TE /T2 (x,y)

3.2

(3.5)

MR Imaging

MR imaging method can be classified into two main types, namely, 2D multi-slice and
3D volumetric. In 2D multi-slice imaging, the images are obtained from exciting slices
of the target object (using slice-selective RF excitation and linear gradient field) while in

26

3D volumetric imaging, the whole object is excited (using nonselective RF excitation).
Following the RF excitation, the spins are spatially encoded with slightly different resonant frequencies and phases by using linear gradient magnetic fields. The MR signal
measured by the receiver coil is finally due to the summation of the signals from all the
spatially encoded spins. After the measurement of the MR signal, image reconstruction
is performed which can also be of two types, namely, projection-reconstruction and 2D
Fourier transform methods. I will focus on the 2D multi-slice and 2D Fourier transform
based imaging since this is commonly used in diffusion-weighted imaging.

3.2.1

Slice selection

For 2D multi-slice MR imaging, the signal is collected from specific slices excited by
applying special RF pulses and linear gradient fields transverse to slicing plane. In order
to excite a slice perpendicular to the longitudinal axis (Z-axis), of thickness ∆z, the
gradient Gz (z)k is turned on which provides spatial encoding along Z-direction (k is
the unit vector along +Z direction). Gz = ∂Bz /∂z, ω = γBz and thus, slice frequency
bandwidth, ∆ωz = γGz ∆z. Alongside, B1 must be applied and be tuned to the Larmor
frequencies of the spins only in the slice of interest. Since the slice requires a frequency
band (∆ωz ) which is equal to γGz ∆z, B1 (magnitude of B1 ) has to be designed to carry
a frequency band that matches the frequencies in the slice (Fig. 3.4). This is commonly
achieved by providing a sinc-modulated RF pulse. The B1 excitation takes the form of,
B1 = B1 (t)cos(ω0 t)i − B1 (t)sin(ω0 t)j

3.2.2

(3.6)

2D Fourier transform method for imaging

After the slice selection process, it can be assumed that all of the signal comes from
spins in the slice section only. The spins in the slice plane (X-Y plane) are further
spatially encoded for frequency and phase such that the signal collected can be represented
in k-space. k-space represents the spatial frequency domain corresponding to Fourier
transform of an image in spatial variables, (x, y, z). Since this is a 2D process, only x, y
27

(a)

(b)

(c)

Figure 3.4. Schematic showing slice (∆z) selection in z-direction using Gz field in (a) and
the sinc-modulated B1 RF signal in frequency (b) and time (c) domains respectively.
variables are used in the expression assuming z to be fixed. The measured demodulated

28

Figure 3.5. Schematic showing k-space imaging using 2D Fourier transform method.
MR signal can be expressed as,
s(t, ty ) =

x y

m(x, y)e−iγGy yty e−iγGx xt dxdy

= F2D {m(x, y)}, kx = (γ/2π)Gx t, ky = (γ/2π)Gy ty

(3.7)

where F2D refers to the 2D Fourier transform in k-space and kx and ky represent the
spatial frequencies (Fig. 3.5).

3.2.3

Phase encoding

From Eq. 3.7, it is seen that phase encoding of the spins can be achieved by applying a
gradient along the +Y-direction, Gy (Gy = ∂Bz /∂y), for a fixed time interval, ty . This
step warps the spins and provides the necessary phase offset (and thus sets the spatial
frequency ky ) required for the Fourier transform representation (in Eq. 3.7) before signal
measurement (data acquisition) is performed. Note that in Eq. 3.7, Gy is assumed to be
constant. For a more general representation of phase encoding,
ky = (γ/2π)

ty
0

Gy (t′ )dt′

(3.8)

where Gy (t′ ) is the gradient along +Y-direction varying with time, t′ .

3.2.4

Frequency encoding

After the phase encoding step, the spins are set to a particular spatial frequency, ky ,
along the Y-direction. However, in order to differentiate between the spins along the
29

X-direction (Fig. 3.6), spins are frequency encoded by applying a gradient along the
+X-direction, Gx (Gx = ∂Bz /∂x), such that ω(x) = γ(B0 + Gx x) = ω0 + γGx x. The
measured MR signal, s(t, ty ), is a function of time and the time dependence is reflected
in kx = (γ/2π)Gx t. During data acquisition, sampling of the time-dependent MR signal
represents sampling of k-space along the kx spatial frequency.

(a)

(b)

Figure 3.6. (a) Schematic showing absence of spatial encoding when no gradient field is
used. (b) Schematic showing encoding of a fixed length (2L) in x-direction using gradient
field, Gx .

3.3

Using echoes

During the frequency encoding of spins, the k-space is often sampled from the most
negative to the most positive spatial frequency value. In terms of the gradient, this can
be achieved by placing a dephasing gradient lobe of negative field strength preceding the
main gradient (rephasing) lobe of positive field strength. The time at which the phase

30

accumulation due to the dephasing lobe is canceled by that of the rephasing lobe is the
time of echo (TE ) which is also the time at which the center of k-space is reached. This
kind of echo is called gradient echo since such an echo can cancel the effect of dephasing
caused by itself. Generally, the sampling of k-space is symmetric about the center of
k-space such that the dephasing lobe is half the time duration as the rephasing lobe (as
∗
shown in Fig. 3.7). Gradient echo based sequences are fast, but still produce T2 -weighted

images since the gradient echo does not compensate for the MR signal loss due to field
inhomogeneity sources.

Figure 3.7. Gradient echo pulse sequence diagram showing the timings of the slice selection, phase-encoding and frequency-encoding gradients.
Spin-echo sequences are used to cancel the effect of field inhomogeneity at a voxel.
Spin-echo pulse sequence is a specialized pulse sequence where the additional dephasing
due to field inhomogeneity sources is reversed by application of a second RF pulse with
∗
180 flip angle. As shown in Fig. 3.8, the initial signal dies at a rate given by the T2 decay.

Then, a 180◦ pulse is applied which causes phase reversal and after a time called echo
31

time (TE ), the spins undergo complete rephasing (constructive interference) to produce
an MR signal called spin-echo. The spin-echo signal is stronger and the signal level is
∗
only limited by the intrinsic T2 decay (as shown in Fig. 3.8) as compared to T2 decay in

absence of any spin-echo sequence.

Figure 3.8. Spin-echo pulse sequence diagram showing the timings of the slice selection,
phase-encoding and frequency-encoding gradients.

3.4

T1 and T2 weighting

As shown in Eq. 3.4, the signal at any location can be represented in terms of the density
of the nuclei or spins (ρ), T1 and T2 . When a spin-echo sequence is used, the signal
expression remains the same as Eq. 3.4 even though the field inhomogeneity sources are
taken into consideration.
In order to obtain T1 -weighted images, TR ≈ T1 of the target tissue and TE is short
32

such that,
S(x, y) ≈ Kρ(x, y)[1 − e−TR /T1(x,y) ]

(3.9)

This is a fast scan and typical TE = 20 ms and TR = 600 ms for B0 = 1.5 T.
For a T2 -weighted image, TR is much longer than the T1 in the target tissue and
TE ≈ T2 of target tissue. Signal is represented as,
S(x, y) ≈ Kρ(x, y)e−TE /T2(x,y)

(3.10)

These are long scans due to longer TR and typical TR = 2500 ms and TE = 80 ms at B0
= 1.5 T.

33

CHAPTER 4
Concepts in diffusion-weighted MRI
4.1

Self-diffusion of water in nervous tissue

Water (H2 O) molecules (hence, protons H+ in water) exhibit self diffusion. Molecular
self-diffusion is different from regular diffusion since the diffusion of particles occurs within
itself (i.e., the diffusing particle, protons in this case, and the medium are the same) and
it is due to the kinetic energy of the particle from thermal agitation (Brownian motion).
Fig. 4.1 (a) depict the case of Brownian motion of water molecules for a healthy axon
fiber where it can be observed that the preferential movement of water along the fiber
orientation and restricted motion in the transverse direction. Note, in this case, an
impermeable axon boundary has been defined. Alternately, a semi-permeable boundary
can also be defined. Fig. 4.1 (b) shows a case of a ruptured axon fiber and its effect
on the movement of water molecules. It can be seen that at the location of the rupture,
the water molecules are less restricted in motion towards the transverse direction to fiber
orientation. These examples show that the molecular self-diffusion delineates the tissue
microstructure due to its preferential movement.
Bundles of nerve fibers form tracts in the central nervous system (e.g., corticospinal or
pyramidal tract travel between the cerebral cortex and the spinal cord) or nerves in the
peripheral nervous system (e.g., common peroneal nerve in the legs). Myelinated nerve
fibers are covered by layers of specialized membrane known as myelin, which contain

34

lipids (fats) and provide natural barrier to the movement of water molecules. A bundle
of nerve fibers can be treated as bundles of tubes carrying water. At microscopic length
scale, which is of the size of the diameter of nerve fibers, the water diffusion is isotropic
but restricted or hindered by barriers. However, at macroscopic length scale (such as few
millimeters as in the spatial resolution of MRI), the diffusion process can be modeled
as homogeneous, unrestricted and anisotropic [7], such as in DTI/ADTI. In QUAQ [8],
on the other hand, diffusion is modeled as a spatially restricted but isotropic process
representing impermeable cylinders of water.

(b)

(a)

Figure 4.1. Brownian motion of water molecules in (a) a healthy axon fiber and (b) an
axon fiber with a rupture.
Principles of molecular diffusion are governed by Fick’s laws. Fick’s first law relates
the diffusive flux (amount of particles passing through a fixed area in a short time) with
the concentration gradient and the constant of proportionality is defined as the diffusion
coefficient.
J = −D∇c

(4.1)

where J is the diffusive flux vector (unit: mol mm−2 s−1 ), c is the concentration (unit:
mol mm−3 ) of the particles at a position (x, y, z) , D is the diffusion coefficient or diffusivity (unit: mm2 s−1 ) and ∇ is the derivative operator (∇ = (i∂/∂x, j∂/∂y, k∂/∂z)).
Fick’s first law shows that whenever a spatial concentration gradient is created, the par-

35

ticles will diffusion in the opposite direction to the concentration gradient to equalize the
concentrations. Combining the law of conservation of mass (continuum principle) with
Fick’s first law, Fick’s second law of diffusion is obtained. The second law shows the
effect of diffusion on changes in concentration, c, with time, t.
∂c
= ∇ · (D∇c)
∂t

(4.2)

For constant D with respect to space, the above equation simplifies to ∂c/∂t = D∇2 c,
where ∇2 = ∇ · ∇.
Historically, the first mathematical description of Brownian motion and its relation
with molecular diffusion was done by Albert Einstein (in 1905 [78]). Einstein provided
a statistical description of the Fick’s second law of diffusion and defined a probability of
finding a particle at a position at a particular time when under diffusion. His work showed
that Brownian motion of particles in a fluid medium was a thermodynamic phenomenon
which can be characterized statistically by a probability function. Einstein reformulated
Fick’s second law as [78, 79],
∂Ψ(r, t)
= D∇2 Ψ(r, t)
∂t

(4.3)

where Ψ(r, t) is the probability of finding a particle at position r at time t. Let the
conditional probability (or propagator), Ps (r 0 |r, t), be the probability that a particle
starting at r 0 would diffuse to r in time t and that Ψ(r, t) =

Ψ(r 0 , 0)Ps (r 0 |r, t), such

that,
∂Ps
= D∇2 Ps
∂t

(4.4)

Solving the reformulated Fick’s law for a freely diffusing particle under the initial condition Ps (r 0 |r, t) = δ(r − r 0 ), (δ being the Dirac delta function in this case),
Ps (r 0 |r, t) =

(4πDt)−3/2exp

(r − r 0 )2
−
4Dt

(4.5)

Thus, the root mean squared distance traveled after a time t during free diffusion in any
√
direction is given by 2Dt. Einstein also defined the diffusion coefficient, D = kT /b
where k is the Boltzmann’s constant, T is the temperature and b is the drag coefficient.
Torrey [80] later included the diffusion term in the famous Bloch equation using the
results from Einstein’s work (see Eq. 4.6).
36

4.2

Pulsed gradient spin echo sequence for diffusion
quantification

Stejskal and Tanner [5, 6] proposed the famous pulsed gradient spin-echo (PGSE) experiment to quantify diffusivity (or diffusion coefficient) of fluids using NMR. In this
experiment, a pair of diffusion-encoding gradients is placed on both sides of the 180◦
RF pulse. A typical example with rectangular gradient pulse is shown in Fig. 4.2. In
absence of diffusion, the phase accumulation in the spins due to the first gradient pulse
is canceled by the second gradient pulse resulting in only a T2 -weighted signal. However,
due to diffusive movement of spins, the MR signal suffers additional attenuation which
can be expressed as a function of diffusivity. PGSE experiment is a generic experiment
for the quantification of diffusivity of fluids since it follows the physics of diffusion and
links it with the signal acquisition process.

Figure 4.2. Schematic of the PGSE sequence showing only the timing of the diffusionencoding gradient and the RF pulses. Here, τ = TE /2, where TE is the echo time.
Torrey introduced the diffusion term in Bloch’s equation [80] which is given by
Mxy
∂M (r, t)
M − Mz
= γM (r, t) × B + 0
k−
n + ∇ · D∇M (r, t)
∂t
T1
T2

(4.6)

where M (r, t) is the net magnetization vector at a position r at time t, B is the applied
magnetic field (along the +Z direction, k), D is the diffusivity. M0 , Mz are the initial
magnetization and the longitudinal magnetization (along the +Z direction) component
37

respectively. Mxy is the transverse magnetization component perpendicular (along the
n direction) to the applied field B. T1 , T2 are the relaxation time constants. ∇ is the
gradient operator. Note that the velocity term in the Bloch Torrey equation (Eq. 4.6) is
ignored.
Eq. 4.6 assumes diffusion to be isotropic and uniform, hence, representing the microscopic nature of self-diffusing water molecules. The transverse component of the magnetization vector, Mxy , after application of a gradient (G(t) = {gx (t), gy (t), gz (t)}), is
given by,
1
Mxy (r, t) = m(r, t) exp −(jω0 + )t
T2

(4.7)

∂m(r, t)
= −jγ(r · G(t)) m(r, t) + D∇2 m(r, t)
∂t

(4.8)

where ∇2 = ∇ · ∇, is the Laplacian operator.
The solution for m(r, t) in Eq. 4.8 when D = 0 is of the form [75],
m(r, t) = A exp (−jγr · F (t))

(4.9)

where
t

F (t) =
0

G(t′ )dt′

(4.10)

The G(t) is assumed to be only a function of time, t, and is constant in space and A is
a constant. When D = 0, A → A(t) in Eq. 4.9 assuming D is only a function of time,
t, and is constant within the imaging voxel space. Substituting Eq. 4.9 in Eq. 4.8 and
simplifying,
∂A(t)
= −γ 2 D(F (t) · F (t))A(t)
∂t

(4.11)

Thus,
ln
=

−γ 2 D

t
A(t)
= −γ 2 D
F (t′ ) · F (t′ ) dt′
A(0)
0
t′

t
0

0

G(t′′ )dt′′

·

t′
0

G(t′′ )dt′′

(4.12)

dt′

where A(t) and A(0) are the echo signal intensities at times t and t = 0 respectively.
For a spin echo sequence, let the 90◦ pulse be applied at time t = 0, and the 180◦ pulse
be applied at time t = τ . The gradient, G(t), can be turned on as follows (shown in Fig.
38

4.2):

G(t) =



 0, 0 < t ≤ t


1





 g, t < t ≤ t + δ


1
1





 0, t + δ < t < τ

1

(4.13)


 0, τ ≤ t ≤ t + ∆


1





 g, t + ∆ < t ≤ t + ∆ + δ


1
1





 0, t + ∆ + δ < t ≤ 2τ

1

where g is a constant vector not dependent on time. Thus, a pair of diffusion-encoding
gradients is applied on both sides of the 180◦ pulse. Note that in practice, a perfect
rectangular gradient pulse is not possible and generally a trapezoid pulse is practical
which has a rising and falling edge (whose slope is decided by the slew rate of the MRI
scanner).
The integral, F (t), is calculated for the different time intervals as follows:


 0,

0 < t ≤ t1






 g(t − t ),

t1 < t ≤ t1 + δ

1





 gδ,

t
t1 + δ < t < τ
′ )dt′ =
F (t) =
G(t

0
 −gδ,

τ ≤ t ≤ t1 + ∆






 −gδ + g(t − t − ∆), t + ∆ < t ≤ t + ∆ + δ


1
1
1





 0,

t1 + ∆ + δ < t ≤ 2τ

(4.14)

Note that at time, t = τ , the 180◦ pulse causes phase reversal.

Using Eq. 4.14 in Eq. 4.12 and simplifying the results leads to the famous StejskalTanner diffusion equation for echo signal attenuation, as given by,
A(t = 2τ ) = A(0)exp −γ 2 g 2δ 2 ∆ −

δ
3

D = A(0)exp(−bD)

(4.15)

where g = |g| (magnitude of the g vector). Note that when the diffusion gradient is turned
off (g = 0), the spin-echo signal intensity is simply the T2 -weighted signal. It should be
noted that imaging gradients also contribute to the diffusion sensitization of the MR signal
39

although their contribution can be considered negligible when relatively large diffusionencoding gradients are applied [81]. Thus, even the T2 -weighted reference image is slightly
diffusion-weighted. However, when imaging gradients are also considerably large, such as
in MR microscopy, the diffusion coefficient has to be accurately estimated by considering
the effect of imaging gradients.

4.3

Propagator formalism

Since self-diffusion of water molecules can be hindered by barriers (such as myelin sheath
in nerve fibers) or restricted spatially, a formalism is necessary where appropriate boundary conditions can be applied on diffusion. In this case, the probabilistic approach to
describing diffusion in space and time is favorable as was introduced by Einstein while
explaining Brownian motion. Such approach can be used to represent both restricted and
unrestricted (or free) diffusion.
The propagator is defined by the probability P (r 0 |r, t) that a particle initially at
position r 0 will have moved to a position r after a time interval t. The echo signal
attenuation, E, [6, 82] in terms of the propagator is given by,
E(g, t) =
V0

P0 (r 0 )

V

P (r0 |r, t)exp[−jγδ(r − r0 ) · g]dV dV0

(4.16)

where g is the diffusion-encoding gradient and g ≡ {gx , gy , gz }. Note that the above
expression of the echo attenuation assumes narrow gradient pulse, i.e., ∆ ≫ δ such
that the average displacements during the gradient pulse is negligible compared with
the average displacements between the gradient pulses. P0 (r 0 ) is the probability of the
diffusing particle at r 0 at the time of the first gradient pulse. γ is the gyromagnetic ratio
and δ is the duration the diffusion gradient pulse.
This relation can be approximated assuming r 0 does not affect the echo attenuation.
This holds true for narrow pulsed field approximation where the diffusion pulse duration
is small enough so that diffusion during the encoding period is ignored. Also, omitting
the integral for P0 (r0 ) gives a Fourier relationship between the echo attenuation and the

40

propagator as given [6],
E(g, t) =
V

Pa (r, t)exp[−jγδr · g]dV

(4.17)

where Pa (r, t) = P (0|r, t). The solution of echo attenuation relies on obtaining an
expression for the propagator Pa (r, t) and then applying the Fourier relation to obtain
the expected echo attenuation, E.
Callahan [83] suggested the use of q-space formulation to represent the echo attenuation
and making use of the Fourier relation with respect to the propagator. Here, q represents
the reciprocal space vector corresponding to the displacement vector (r − r 0 ) in a Fourier
transform pair. The echo attenuation has been rewritten as [83],
E(q, ∆) =

ρ(r 0 , 0)P (r0 |r, ∆)exp[j2πq · (r − r 0 )]dr0 dr

(4.18)

where ρ(r 0 , 0) is the starting spin density. The reciprocal space vector q is defined
as a function of the diffusion gradient as (2π)−1 γδg. For the case of the restricted
diffusion of diffusivity, D, confined within a pore of size a, the echo attenuation can
be approximated by a Fourier relation [83], E(q, ∆ → ∞) = |S(q)|2 , where S(q) is
the Fourier transform of ρ(r). This is under the assumption that ∆ ≫ a2 /D such
that P (r0 |r, ∆ → ∞) → ρ(r) and also ρ(r 0 , 0) → ρ(r 0 ). Physically, the assumption
∆ ≫ a2 /D means that sufficient time is given between the diffusion encoding gradient
pulses for the diffusing water molecules to reach the impermeable boundary of the pore,
otherwise for shorter ∆, the diffusion will become unrestricted. The Fourier relation
explains the famous diffusive diffraction pattern seen in PGSE experiments with small
impermeable pores of uniform size [83].
So far I have shown that the echo attenuation is the Fourier transform of the propagator
over space. Thus, in order to obtain an expression for the echo attenuation, the expression
of the propagator has to be available. The propagator is generally solved from Fick’s law
by applying appropriate boundary conditions required by the problem ([83]). The Fick’s
law can be applied directly to the propagator (P (r0 |r, t)),
D∇2 P =
41

∂P
∂t

(4.19)

This is equation can be solved using standard eigenmode expansion, the initial condition
that P (r 0 |r, t = 0) = δ(r 0 − r) and proper boundary conditions.

4.4

Parametric DWI

In complex structures, such as nerve fibers, it is difficult to characterize the exact nature
of diffusion on a macroscopic length scale. Hence, a number of parametric models have
been suggested to interpret the echo attenuation signal observed in the PGSE experiment.
Three of such types are described in the following subsections, namely, DTI, ADTI and
QUAQ.

4.4.1

DTI formulation

At macroscopic length scales, diffusion can be modeled as an anisotropic and homogeneous
process [7]. Diffusion anisotropy can be represented by a rank 2 tensor (also called a
dyad). A rank 2 tensor is a 3 × 3 matrix. The generalization to 3D anisotropic diffusion
process is best understood from Fick’s first law. From Fick’s first law, it is known that
J = −D∇c, where J is the diffusive flux, c is the concentration at a position and ∇c
gives the concentration gradient vector. For an isotropic diffusion case where D is a scalar
quantity, the diffusive flux, J , is along the direction of the concentration gradient vector,
∇c. However, when diffusion is anisotropic, a diffusion tensor, D is used in place of the
scalar D and Fick’s first law becomes, J = −D · ∇c. Now, the diffusive flux, J, need
not be along the direction of the concentration gradient vector, ∇c, and this direction
is governed by the tensor operation (D) on the concentration gradient vector (∇c). An
analogy of such an anisotropy is stress tensor and force relation. It is known that stress
can be tensile (normal) and shear (tangential). Thus, force (F) exerted due to stress (T)
on a surface (s) is given by, F = T · s where the force (F) need not be in the direction of
the surface normal s.
The use of diffusion tensor was presented by Basser et al. [7, 12] who also introduced the
technique Diffusion Tensor Imaging (DTI). In the case of axonal fiber tracts, the tensorial

42

notation interprets the hindrance of diffusion of water molecules in the transverse direction
to the fiber orientation (due to semi-permeable myelin sheath membrane) as anisotropic
diffusion and the eigenvectors of the tensor matrix indicate the principal directions of
diffusion. However, this model does not consider restricted diffusion as will be discussed
later.
The DTI signal formulation can be derived by generalizing Eq. 4.12 and using a
diffusion tensor D for the diffusion process. Thus,
t
A(t)
= −γ 2
F (t′ ) · D · F (t′ ) dt′
A(0)
0

ln
= −γ

t′

t

2
0

′′

′′

·D·

G(t )dt

0

t′

′′

′′

G(t )dt

0

(4.20)

′

dt

The above step can be written using matrix form as [12],
ln

t
A(t)
= −γ 2
F (t′ )T DF (t′ ) dt′
A(0)
0

= −γ 2

t
0

F (t′ )F (t′ )T dt′ : Deff

(4.21)

= −b : Deff
where b is the b-matrix and Deff is the “effective” diffusion tensor. “:” is the generalized
dot product (applicable to tensors). T represents the transpose operation. The effective
diffusion tensor gives an equivalent tensorial representation of the diffusion process for
the time period [0, TE ] assuming a medium with a fixed diffusion tensor during this
time. Thus, Deff has no time dependence although the actual diffusion process is timedependent. Also, the tensor is assumed to have no spatial variation within the voxel.
The effective diffusion tensor can be written as a product of the b-matrix and D eff as
A(t)
ln
=−
A(0)

3

3

eff
bij Dij

(4.22)

i=1 j=1

eff
where bij and Dij are the ijth elements of the b and the D eff matrix respectively. For
the PGSE sequence given in Fig. 4.2, the DTI equation can be rewritten in matrix form

43

as,
δ
A(TE )
= −γ 2 δ 2 ∆ −
g T Deff g
A(0)
3
δ
ˆ
ˆ
= −γ 2 δ 2 ∆ −
g 2 g T Deff g
3

ln

(4.23)

ˆ
ˆ
= −b g T Deff g
ˆ
ˆ
where g is the unit column vector along the gradient direction and g ≡ [ˆx , gy , gz ]T . b
g ˆ ˆ
is the b-factor and g is the gradient column vector as defined before. This is the classic
echo signal formulation for the DTI model. For the sake of simplicity of notation, D eff
will be written as D only although wherever applicable, the proper tensor will be used.
From a probabilistic standpoint, as was introduced earlier in this chapter, Eq. 4.23
shows that water molecules follow simple anisotropic unrestricted diffusion governed by
a 3D Gaussian probability distribution of displacement, P (r0 |r, t).
P (r 0 |r, t) =

1
(4πt)3 |D|

exp[−

(r − r 0 )T D−1 (r − r 0 )
]
4t

(4.24)

This shows that the diffusion tensor is proportional to the 3D variance of the probability
of displacements. This result has been shown by a number of researchers [6, 7].
The effective diffusion tensor matrix (D) is a symmetric 3 × 3 matrix with six unique
parameters which quantifies the anisotropic

Dxx

D =  Dxy
Dxz

diffusion within each voxel in the 3D space.

Dxy Dxz

(4.25)
Dyy Dyz 
Dyz Dzz

ˆ
and g = [gx gy gz ]T = the diffusion gradient direction unit vector (column vector) written
ˆ ˆ ˆ
in matrix format. The expression for the echo signal attenuation was defined as [7],
E(g) =
2

2

A(TE )
= exp −bˆ T Dˆ
g
g
A(0)

(4.26)

2

= exp −b gx Dxx + gy Dyy + gz Dzz + 2gx gy Dxy + 2gx gz Dxz + 2gy gz Dyz
ˆ
ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ ˆ
In a fiber, there is preferential diffusion along the longitudinal direction which gives
the fiber orientation. Thus, after estimating the D matrix from the DWI data using the
model in (4.23), the fiber orientation is given by the direction of principal eigenvector of
the D matrix. Thus, from eigen decomposition,
DE = EΛ
44

(4.27)

where E is the matrix of eigenvectors (column vectors) (e1 , e2 , e3 ) and Λ is the eigenvalue
matrix,
E = [e1 |e2 |e3 ]


λ1 0 0


Λ =  0 λ2 0 
0 0 λ3

(4.28)
(4.29)

If λ3 > λ2 > λ1 , then e3 is the principal eigenvector and the fiber orientation is given
by the direction of e3 . The diffusion coefficient along this direction will be given by the
eigenvalue, λ3 . For the sake of clarity, it is assumed that principal fiber direction is along
the +Z-axis of the diffusion space and define, D = λ3 , D⊥1 = λ2 , D⊥2 = λ1 .
From the perspective of model parameter estimation, for the DTI formulation, the
model is composed of the 6 diffusion coefficients of the effective diffusion tensor matrix. Also, an additional A(0) signal is required. However, these model parameters
are dependent of the coordinate axis. To obtain a set of parameters which are independent of the coordinate axis, the set of eigenvalues and the Euler angles as model
parameters are collected. Thus, if the set of model parameters is defined as β, then,
β ≡ {D , D⊥1 , D⊥2 , θF , φF , ψF }, where D , D⊥1 , D⊥2 are the longitudinal and transverse diffusivities and θF , φF , ψF are the Euler angles. This is also a 6-parameter model.
The diffusion process could be best described by an diffusion ellipsoid diagram shown in
Fig. 4.3.
Decomposing D into its eigenvectors and eigenvalues,


D⊥2
0
0


D⊥1 0 
D = RT D 0 R, where D0 =  0
0
0
D

(4.30)

and R = R(θF , φF , ψF ) represents the combined rotation matrix which can be decom-

posed into component rotation matrices as shown.
R = Rz (ψF ) ∗ Ry (θF ) ∗ Rz (φF )



cos(θF ) 0 −sin(θF )


Ry (θF ) = 
0
1
0

sin(θF ) 0 cos(θF )
45

(4.31)

(4.32)

Figure 4.3. DTI diffusivity ellipsoid indicating diffusivities (D , D⊥1 , D⊥2 ) and the orientation angles (θF , φF , ψF ).


(4.33)



(4.34)


cos(φF ) sin(φF ) 0


Rz (φF ) =  −sin(φF ) cos(φF ) 0 
0
0
1


cos(ψF ) sin(ψF ) 0


Rz (ψF ) =  −sin(ψF ) cos(ψF ) 0 
0
0
1

4.4.2

ADTI formulation

For axisymmetric DTI model [8, 84, 85], transverse diffusivity is assumed isotropic (Fig.
4.4). Thus, D⊥1 = D⊥2 = D⊥ . Also, D ≥ D⊥ and ψF is identically zero. This leads
to a 4-parameter model. Let β be the parameter set, β ≡ {D , D⊥ , θF , φF }. D is
decomposed into its eigenvalues and eigenvectors as:


D⊥ 0
0


D = RT D 0 R, where D 0 =  0 D⊥ 0 
0
0 D
46

(4.35)

and R is the transpose of the matrix of eigenvectors. If R = I, a 3 × 3 identity matrix,
then D = D 0 and the fiber orientation, given by the principal eigenvector direction, is
along the +Z-direction. So, for the fiber orientation to be at the spherical coordinates
(θF , φF ), the tensor (D0 ) will also be oriented along the fiber direction. Thus, in case
of a fiber oriented at the spherical coordinates (θF , φF ), R = R(θF , φF ) will be the
rotation matrix which reorients the coordinate axis along the fiber direction. It can be
decomposed into component rotation matrices as shown.
R = Ry (θF )Rz (φF )

(4.36)



(4.37)


cos(θF ) 0 −sin(θF )


Ry (θF ) = 
0
1
0

sin(θF ) 0 cos(θF )

cos(φF ) sin(φF ) 0


Rz (φF ) =  −sin(φF ) cos(φF ) 0 
0
0
1


(4.38)

The expression for the echo attenuation or the normalized MR signal for the ADTI
model in matrix format can be simplified as,
E = exp −bˆ T Dˆ
g
g
= exp −bˆ T RT D 0 Rˆ
g
g
= exp

−b(Rˆ )T D
g

(4.39)

g
0 Rˆ

ˆ
= exp −bˆ ′T D0 g ′
g
ˆ
ˆ
ˆ
where column vector g ′ = Rˆ and g ′ ≡ [gx ′ , gy ′ , gz ′ ]T and g ≡ [ˆx , gy , gz ]T as before. T
g
ˆ ˆ ˆ
g ˆ ˆ
is the transpose of a matrix. b is the b-factor. Expanding the exponent,
ˆ
ˆ
g ′T D 0 g ′ = gz D + (ˆx + gy )D⊥
g 2 ˆ2
ˆ2

(4.40)

Since R is the transpose of the matrix of eigenvectors which reorients the coordinate
ˆ
ˆ
axis along the fiber, g ′ = Rˆ is the projection of the gradient direction g on to the eigen
g
ˆ
directions. Let f be the unit column vector along the fiber direction given by spherical
coordinates (θF , φF ). Then, projection of the gradient direction on the fiber direction,
47

Figure 4.4. ADTI diffusivity ellipsoid indicating diffusivities (D , D⊥ ) and the orientation
angles (θF , φF ).
ˆ ˆ
ˆ
g = g T f , is the same as gz . And the transverse component is given by, g⊥ =

gx + gy .
ˆ2 ˆ2

Thus, the ADTI signal equation can be rewritten as,
2
E = exp −b(g 2 D + g⊥ D⊥ )

(4.41)

2
Also, g⊥ = 1 − g 2 . D and D⊥ are the diffusion coefficients in the parallel and

perpendicular directions to the fiber orientation, respectively.

4.4.3

QUAQ formulation

In quantitative analysis of q-space [8] (QUAQ), the diffusion process is modeled by unrestricted diffusion on the longitudinal direction to the fiber and restricted diffusion along
transverse direction. Each fiber population is assumed to consist of an impermeable
48

cylindrical shape, with self-diffusion coefficient D inside the cylinder (and no diffusion
outside), an average or “apparent” radius a and orientation angles in spherical coordinates of the individual fibers (φF , θF ). The MR signal attenuation is formulated in the
q-space, where q is the reciprocal space vector [83] corresponding to a position r. The
echo attenuation at time, ∆, in terms of q is given by,
E(q, ∆) = E (q , ∆) E⊥ (q⊥ , ∆)

(4.42)

where E (q , ∆) and E⊥ (q⊥ , ∆) are the contributions due to the longitudinal and transverse components of diffusion respectively. Here, q = γδg /2π, g = g · f and f is the
unit vector along the fiber direction defined in spherical coordinates by (θF ,φF ). And,
q⊥ =

1 − q2.

For the longitudinal unrestricted diffusion, the simple Gaussian propagator can be used
and the echo attenuation is given by,
E (q ) = exp −4π 2

∆−

δ
3

D q2

(4.43)

For the transverse restricted diffusion, the solution for the echo attenuation for a cylindrical pore using the propagator formalism is shown in [8, 83]. The narrow pulse-width
assumption has been applied in the derivation (∆ ≫ δ). Since the transverse component
of echo attenuation is being calculated, the coordinate axis is aligned with the fiber orientation such that the cylinder is along Z-axis with (r, θ) as the cylindrical coordinates
with respect to the central axis. The echo attenuation for the transverse component is
given by[8],
′
J0 (2πa q⊥ )
E⊥ (q⊥ ) = 4
2πa q⊥

+8

∞

∞

n=1 k=1

2

+4

∞
k=1

′
2πa q⊥ J0 (2πa q⊥ )
2
(2πa q⊥ )2 − β0,k

′
2πa q⊥ Jn (2πa q⊥ )
2
(2πa q⊥ )2 − βn,k

2

2
2
exp −β0,k

D∆
a2

2
βn,k

2 D∆
exp −βn,k 2
2
a
βn,k − n2

2
where q⊥ = γδg⊥ /2π and g⊥ = 1 − g 2 . βn,k is the kth root of the first derivative of the
′
Bessel function of order n, Jn , such that (Jn (βn,k ) = 0) [8].

While, in DTI, the diffusion tensor that is estimated is an “apparent” or “effective”
diffusion tensor that approximates the actual diffusion process in each voxel as a unre49

stricted (barrier-free) anisotropic diffusion process, in QUAQ, the actual isotropic diffusion process within a network of impermeable cylinders is considered. Hence, in DTI, the
estimated diffusivities strongly depend on the acquisition parameters, while, in QUAQ,
the estimated parameters have a physical interpretation in direct correlation with the
physical problem under investigation.

4.5

Non-parametric DWI

An alternate method of analyzing the diffusion-weighted data is by solving for the conditional probability function (P (r0 |r, t)) or the propagator for the diffusion process as
is without using a parametric model for the propagator. Such methods are known as
q-space imaging (QSI) or diffusion spectrum imaging (DSI) [86]. In QSI, the sampling
is performed in the Fourier space of displacements, i.e., q-space over a regular grid. The
reconstruction is performed via Fast Fourier transform and a discrete representation of
the probability function is obtained [87]. The need for QSI arose due to the need to
resolve fiber crossings where a simple single fiber model can not provide accurate results.
QSI requires dense sampling on the 3D Cartesian grid (thus is very time consuming) and
uses large pulsed gradients. Total scan time depends on the number of samples of the
q-space and thus depends on the number of diffusion-weighted images acquired which is
significantly higher for non-parametric DWI than model-based DWI (such as DTI). For
example, a typical full brain DTI with 15 diffusion directions can be around 2 min (for
TR = 6.4 s, NEX = 1), while a full brain QSI (such as Q-ball imaging) with 252 directions
can be around 27 min [11] (for TR = 6.4 s, NEX = 1). Due to long scan times, QSI is
not clinically viable. Since the focus is on resolving the fiber orientations in a crossing
fiber situation, a cumulative probability with respect to the propagator called orientation
distribution function (ODF, φ(ˆ)) is estimated.
x
∞

φ(ˆ) =
x

P (αˆ)dα
x

(4.44)

0

where x is the unit vector in the direction of displacement vector x. Hence, the ODF
ˆ
is the radial projection of the propagator on to a unit sphere. It has higher values
50

at a particular orientation when the presence of a fiber is more probable. The ODF
can be obtained by acquiring data on spherical shells in q-space and reconstructing the
ODF using fitting methods as in high angular resolution diffusion imaging (HARDI [88])
or transform methods, such as Funk-Radon transform in Q-ball imaging (QBI [11]).
Although a non-parametric approach is applied in the estimation of the propagator and
the ODF, the interpretation of the ODF often involves fitting the ODF to model-based
functions (e.g., spherical harmonics [89, 90]) in order to identify fiber tracts.

4.6

Experimental DTI protocol used in this work

4.6.1

Pulse sequence

For the DTI experiment, the sequence used is based on the PGSE experiment suggested by
Stejskal and Tanner but with a number of modifications. The sequence is a spin-echo echoplanar imaging (SP-EPI) sequence. This sequence belongs to the fast imaging methods.
Since the DTI models unrestricted diffusion, the narrow pulse assumption (∆ ≫ δ) is
not required in the pulse sequence [82]. The following are the salient features of the DTI
pulse sequence used in the GE Signa HDx 3T scanner (GE Healthcare, Waukesha, WI):
1. Use of echo-planar imaging: Echo planar imaging is one of the fast imaging methods
in MRI and it was first introduced by Mansfield [91]. The use of EPI drastically
reduced the MRI scan times and made imaging large parts of the body possible
within a tolerable amount of scan time. In EPI, the k-space plane is sufficiently
sampled to reconstruct an image after a single RF excitation. The planar k-space
sampling is done by alternating the readout gradient which produces a train of
gradient echoes and phase encoding the echoes using intermediate short gradient
pulses (blips). Fig. 4.5 shows the schematic for a typical EPI sequence along with
the k-space trajectory for MR signal acquisition.
2. Use of spatio-spectral RF pulse: In a conventional RF pulse (such as the 90◦ pulse),
the spin flip is achieved by a single pulse. Additionally, for slice selection, the slice

51

(a)

(b)

Figure 4.5. (a) A schematic of a typical echo-planar imaging sequence for gradient echo.
(b) The k-space trajectory for the EPI sequence.
selected gradient (Gz ) is turned on in case of 2D multi-slice MR imaging. However,
the chemical shift effect where protons in the fat (or molecules other than water)
precess at a slightly different Larmor frequency as compared to protons in water

52

leads to image artifacts, such as ghosting [92]. In order to minimize this effect, a
spatially as well as spectrally (frequency) selective RF pulse sequence is used [92].
The spectral selection is done using a sequence of RF pulses (instead of a single
RF pulse) where each RF pulse is tuned at the Larmor frequency of protons in
water and the pulses are spaced at time intervals equal to half of the time period
corresponding to the frequency difference between Larmor frequencies of protons in
water and fat [92]. This way the RF excitation of protons in fat is suppressed. Note
that the effective flip angle of the combined RF pulse sequence is the cumulative
sum of the individual flip angles. Fig. 4.6 shows the traditional single RF pulse and
12 pulse spatio-spectral pulse sequence used in the GE scanner for DTI experiments.
Flip angles of the RF pulses in the spatio-spectral pulse in Fig. 4.6(b) add up to
90◦ .

(a)

(b)

Figure 4.6. (a) Single RF pulse with slice select gradient. (b) 12 RF pulses of the
spatio-spectral pulse sequence with the slice select gradients.

53

3. Use of dual spin-echo: Diffusion-weighted imaging is prone to field gradient eddy
current effects due to the fast switching of strong gradient fields. Such switching
induces eddy current on the conductive surfaces of the MRI scanner and the current
can linger even after the switching event is past. The presence of eddy current
causes spatial magnetic field distortion which results in phase error during k-space
sampling resulting in numerous image artifacts, such as contraction, dilation, shift,
shear and ghosting. The problem could be severe in diffusion-weighted imaging
since diffusion-weighting is performed with varying gradient field strengths and
could result in image misregistration between the diffusion-weighted images [93]. In
a dual-spin echo sequence, a second 180◦ pulse is applied so as to refocus the spins
twice. Such a twice-refocused spin-echo sequence is less prone to field gradient eddy
current effects. Eddy current effect is generally modeled as an exponential decay
with a decay constant. If the eddy current decays slowly, then during readout the
eddy current induced residual field will attenuate the MR signal additionally. This
effect is also more pronounced in EPI sequences. The pulse durations (δs) can be
optimally selected to completely cancel the effect of eddy currents at a particular
decay constant [93], thereby reducing the eddy current induced residual field effect
on the MR signal. After twice refocusing, the eddy current build-up at the time of
acquisition is reduced to zero. In the DTI framework, such modifications in the pulse
sequence results in a modified expression of the b-factor and the b-factor has to be
recalculated in terms of the new timing parameters. Although analytical expressions
of the b-factor exists under different assumptions of the timing parameters (for
example, [94]), on the GE scanner, the b-factor is numerically estimated [95] by
using the definition of the b-matrix which is based on gradient waveform integration
T
equation shown in Eq. 4.21: b = γ 2 0 E F (t′ )F (t′ )T dt′ . The integration based

technique, however, assumes piecewise linear segments of the gradient waveform.
During the DTI experiment, the user prescribes a b-factor for the scan. Using
the integration technique [95], the calculated b-factor is matched with the userprescribed b-factor to estimate the gradient strengths along the different gradient

54

coordinate axes. The calculated b-factor is adjusted iteratively until the value is
within a tolerance range of the prescribed b-factor.

4.6.2

Experimental settings

Apart from the standard settings for Spin-Echo pulse sequence such as TE , TR , diffusionrelated experimental parameters are the b-factor, the diffusion-encoding gradient directions (vector g) and the number of gradient directions (N). Since b-factor is a function
of the diffusion-encoding gradient strength (G), diffusion pulse duration (δ) and pulse
separation or diffusion time (∆), these can also be separately set during experiments.
Note that the time of echo, TE , is dependent on the selection of the ∆ and δ. And for the
dual-spin echo sequence, an effective b-factor based on the individual timing parameters
is calculated and used in the DTI model. From an optimization perspective, in this work,
focus is on the diffusion-encoding gradient vector, g, and its distribution in gradient space
(or q-space). Also, the selection of the b-factor on the whole and its influence on the estimation performance is investigated without going into the dependence of the b-factor
on the individual timing parameters.

4.7

Fiber assignment by continuous tracking

Estimated diffusion model parameters in DTI (or ADTI) can be used to track white
matter connectivity from a seed region to the rest of the imaged tissue by estimating
white matter tract trajectories. In general, fiber tracking algorithms can be based on a
deterministic framework, such as ones proposed by Mori et al. [16], Conturo et al. [58]
and Basser et al. [96], or on a probabilistic framework, such as ones proposed by Poupon
et al. [97], Parker et al. [98] and Behrens et al. [59]. While deterministic algorithms are
fast and more often used clinically [99], probabilistic techniques could potentially be more
accurate (for example, as shown by Behrens et al. [59]) but tend to be computationally
intensive (hence slow) and generally require more data. A commonly used deterministic
fiber tracking algorithm is fiber assignment by continuous tracking or FACT. Alternately,

55

a common example of probabilistic fiber tracking is implemented in the software package
FSL which is called “probtrack” [59, 100] which uses the estimates of posterior probabilities of local fiber orientations obtained from “Bayesian Estimation of Diffusion Parameters
Obtained using Sampling Techniques” (BEDPOST) [59, 101] to calculate a probabilistic
connectivity map from a seed to different targets in the imaged structure.
The general notion for the deterministic fiber tracking based on DTI is to define a seed
region (for example, a seed voxel) in image space and trace a streamline by following the
direction of the principal eigenvector of the diffusion tensor (which represents the direction
of the fastest diffusion within a voxel) from the seed voxel to other voxels within the 3D
image under certain constraints based on the local diffusion anisotropy and curvature
of the estimated streamlines. Mathematically, this is equivalent to solving an ordinary
differential equation (ODE) numerically for a curve in 3D space given an initial condition
(seed) [96, 99] and where the unit tangent vector at a point on the trajectory is equated
to the principal eigenvector of the estimated diffusion tensor. The numerical integration
for solving the ODE can be a first order method (as in the Euler’s method [16]) or higher
order method (as in the Runge-Kutta method [96]).
FACT method introduced by Mori et al. [16, 102] is one of the most common methods
for deterministic fiber tracking. The tracking is initiated from seed points (generally
voxel center) and the direction of the track is changed from the current voxel’s principal
eigenvector direction to that of the neighboring voxel at the point on the boundary of
the voxels where the track leaves the voxel and enters the next. This ensures that tracks
are reconstructed more realistically than by merely connecting the centers of the voxels
[16]. The tracking algorithm uses a brute-force approach where tracks originate from seed
points from the uniformly sampled 3D image space (for example, voxel centers). From
this ensemble of reconstructed fibers, specific tracks are visualized by selecting specific
ROI which the tracks originate from or pass through [102, 103]. The FACT method
is implemented in the DTIStudio software [103] which is used for the fiber tracking
analysis in this work. FACT-based tracking has been previously used for visualizing
white matter connectivity for in vivo DTI experiments, such as in the human brain for

56

cortical association tracts [104] and corpus callosum [105] and in the spinal cord for
detecting spinal cord compression [27] and astrocytomas [67].
In the FACT algorithm, generation of arbitrary tracks is restricted by constraining the
trajectory paths with thresholds based on the local diffusion anisotropy (FA threshold)
and curvature of the evolving fiber track (curvature threshold). High FA threshold makes
sure that tracking is performed within the highly anisotropic white matter regions and
not the low anisotropy gray matter or CSF. The curvature threshold ensures that sharp
changes in the fiber track are avoided [102]. Selection of the threshold values is generally
based on previous studies in the region of interest and is user defined, although studies on
the sensitivity of these thresholds to the fiber tracking result can help select the thresholds
more robustly [106].
While FACT-based fiber tracking provides a visual interpretation of the white matter
tracts in the form of estimated streamlines, certain tract-based metrics can also be defined
to quantify the quality of the tracked fibers and compare across subjects. Correia et al.
[107] defined a number of such metrics. DTIStudio computes some of these metrics as
part of the output of the tracking analysis, such as, the total number of fibers tracked
(TF), the average number of fibers per voxel in the tracked fibers (AF) and the average
length of fibers tracked (AL). These metrics can be ROI specific (calculated for tracks that
are either originating from or passing through the ROI voxels) or for the whole image.
The length-based metrics (AL) and number of fibers based metrics (TF, AF) provide
independent information about the white matter integrity [107]. For example, higher AL
indicates longer fiber tracks and thus better long range white matter connectivity and
integrity. On the other hand, higher TF indicates more fiber tracks from the ROI and
thus indicates better chance of connectivity. AF indicates the track density with respect
to an ROI. These metrics have been previously used for fiber tracking analyses, such as
in studying age-related degradation in the central nervous system where both TF and
AF showed good correlation (negative correlation) with age for different white matter
tracts [108]. FACT-based algorithms are commonly used for clinical studies and hence
this work will explore the effect of gradient optimization on FACT-based fiber tracking.

57

CHAPTER 5
Optimization of gradient scheme in
diffusion-weighted imaging
The gradient scheme in diffusion-weighted imaging consists of the diffusion gradient directions and the b-factor which is calculated based on the parameters related to the gradient
strength and time durations. In this work, I have focused on the optimal selection of the
gradient directions. However, I have also studied the b-factor optimization. The optimization is based on D-optimality [33, 34] which minimizes the Cramer-Rao lower bound
(CRLB) on the estimation variance and uses the Fisher information matrix [109]. The
optimization procedure will only improve precision (or reduce estimation uncertainty).
The procedure assumes that the parameter estimator is unbiased, i.e., there is no difference (or no bias) between the expected and true values of the parameters. In case
the estimation is actually biased, minimizing the CRLB could still minimize the uncertainty in the estimation, but the analytical CRLB will not be equal to the actual lower
bound of the uncertainty. The effect of the optimization on the bias depends on the problem formulation, especially the signal and the noise model. The optimization is applied
to DTI (and ADTI) signal model which assumes that the diffusion phenomenon in the
nerve fibers strictly as an unrestricted diffusion and is described by a three-dimensional
Gaussian distribution. Another signal model, called the QUAQ model, assumes that the
diffusion occurs in impermeable tubes with unrestricted diffusion along the direction of

58

the tube and restricted diffusion in the transverse direction. The noise model can be
considered as Gaussian (at SNR > 5) or in general Rician. The Gaussian noise model,
which is more popular, has a simpler formulation than the Rician model. Both noise
models have been considered in this work.

5.1

General concepts in parameter estimation and
optimization

5.1.1

Cramer-Rao Lower Bound

Given the model-based normalized MR signal E and the variance σ 2 of the noise in the
MRI data, the experimental measurement of the normalized MR signal (noisy signal),
ˆ
ˆ
E, follows a probability distribution function (pdf), p(E|E, σ 2 ). For a set of N diffusionweighted images with different gradients, the joint pdf for the measurement column vector
ˆ ˆ ˆ
ˆ
ˆ
(E = [E1 , E2 , E3 , ..., EN ]T ) will be
N

ˆ
p(E) =
i=1

ˆ
p(Ei |Ei , σ 2 )

(5.1)

ˆ
assuming each measurement of the normalized MR signal, Ei , to be independent and
having the same variance (σ 2 ) but different model signals, Ei . The model signal, Ei ,
depends on the diffusion model parameters, denoted by a column vector of parameters
β := [β1 , β2 , β3 , ..., βM ]T (for M diffusion model parameters), as well as the MRI experˆ
ˆ
imental parameters, denoted by α vector. Thus, p(E) can be rewritten as p(E|β, α).
The experimental parameters (α) can be separated into the b-factor and the set of gradient directions or gradient scheme. Each ith gradient direction, g i , can be represented
as g i ≡ {gxi , gyi , gzi} ≡ {θi , φi }. The (θi , φi ) representation is preferred since there
are fewer parameters in this form. In matrix form, the gradient scheme consisting of N
gradient directions, Ω , is defined as, Ω := [θ1 , θ2 , θ3 , ..., θN , φ1 , φ2 , φ3 , ..., φN ]T . Thus,
α := [b, Ω ]T .
The Cramer-Rao Lower Bound (CRLB) [109] on the variance of the estimated model

59

ˆ
ˆ ˆ ˆ
ˆ
parameters (β = [β1 , β2 , β3 , ..., βM ]T ) can be given by the inequality,
ˆ
Σ(β) − I−1 (β) ≥ 0

(5.2)

ˆ
ˆ
where Σ(β) is the covariance matrix of the diffusion model parameter estimates (β)
defined as,
ˆ
ˆ
ˆ ˆ
ˆ
Σ(β) = (β − β )(β − β )T
where

(5.3)

is the expectation operation. I(β) is the Fisher information matrix for the true

parameters (β) and ‘≥ 0’ refers to a matrix being positive semidefinite. The Cramer-Rao
covariance lower bound is given by,
ΣCR (β) = I−1 (β)

(5.4)

and the jkth element of the Fisher information matrix is defined as,
[I(β)]jk = −

ˆ
∂ 2 lnp(E)
∂βj ∂βk

(5.5)

where βj and βk are the jth and kth diffusion model parameters from β, respectively.
The expectation operation (

ˆ
) is taken with respect to p(E) (Eq. 5.1). Both j ∈ [1, M]

and k ∈ [1, M], where M is the number of diffusion model parameters in β. The Fisher
ˆ
information matrix is a function of p(E) and thus depends both on β and α. So, I(β)
can be rewritten as I(β, α). Thus, ΣCR (β) can also be written as ΣCR (β, α) and
depends on both the diffusion model parameters and the MRI experimental parameters.
By optimizing the experimental parameters (α), the covariance bound on the diffusion
model parameter estimates can be minimized. This covariance bound is achieved by any
unbiased and minimum variance estimator. Thus, an optimal experimental design can
be obtained. As shown later, using determinant of the CRLB matrix as a cost function
for the optimization problem, an overall minimal uncertainty can be achieved in the
estimation.

5.1.2

Sensitivity matrix

As will be discussed later, the definition of CRLB (and the Fisher information matrix)
involves the sensitivity matrix (X) which is essentially Jacobian matrix of the signal at
60

different gradients with respect to the different diffusion model parameters. The jkth
element of the sensitivity matrix, X, is given by,
[X]ij :=

∂E(β; g i ; b)
∂βj

(5.6)

where i ∈ [1, N] and j ∈ [1, M], N and M are the number of gradient directions used
in the DTI experiment and the number of diffusion model parameters respectively. In
matrix form, the sensitivity matrix can be defined as,

∂E(β ;g 1 ;b)
∂E(β ;g 1 ;b)
∂E(β ;g 1 ;b)
...
∂β1
∂β2
∂βM

 ∂E(β ;g ;b) ∂E(β ;g ;b)
∂E(β ;g 2 ;b)
2
2

...
∂β1
∂β2
∂βM
X=


...
...
...
...
 ∂E(β ;g ;b) ∂E(β ;g ;b)
∂E(β ;g N ;b)
N
N
...
∂β
∂β
∂β
1

2

M










(5.7)

Sensitivity matrices are critical to the definition of CRLB since it links the covariance
bound to the sensitivities of the signal with respect to different model parameters. Although sensitivity matrix depend exclusively on the signal model, but, as will be shown
later, modified sensitivity matrices for Rician noise case would also depend on the noise
variance. In later sections, the detailed formulation for the sensitivity matrix for ADTI
and DTI model is given. For signal models with simple analytical expressions, such
as ADTI and DTI, the sensitivity matrices are computed analytically. Otherwise, the
sensitivities can be calculated numerically.

5.1.3

Noise models

MRI images are reconstructed from the complex k-space data by taking the magnitude of
the Fourier inverse of the k-space data. Assuming that both the real and imaginary part
of the Fourier inverse data contain Gaussian noise, the noise in the magnitude images is
Rician in nature. Thus, if x is the noisy signal following Rician pdf, then x =

xc

(magnitude of the complex signal, xc ), where xc = m + nr + jni and m is the magnitude
of the complex signal without noise and nr and ni are Gaussian noise signals with zero
mean and σ 2 variance. Through a rotation of the quadrature detector [110], all the signal
intensity is shifted to the real component of the complex signal. Thus, the imaginary

61

component of the signal is only noise, ni , and has a zero mean and σ 2 variance. The
Rician pdf is given by [111, 112],
x
x2 + m2
xm
p(x|m, σ) = 2 exp(−
) I0 ( 2 )
σ
2σ 2
σ

(5.8)

where x is the random variable following the Rician pdf which is defined by the two
parameters, the magnitude (m) and variance (σ 2 ). I0 is the zero order modified Bessel
function of the first kind. Note that x, m and σ are strictly non-negative. An important
observation regarding the Rician pdf is that the mean and the variance of the Rician pdf
depend on both the m and σ parameters, as given by,
π
−m2
L1/2 ( 2 )
2
2σ

Mean(x) = σ

Var(x) = 2σ 2 + m2 −

πσ 2 2 −m2
L1/2 ( 2 )
2
2σ

(5.9)
(5.10)

where L is the Laguerre polynomial. L1/2 can be written in terms of the modified Bessel
functions as,
x
−x
−x
L1/2 (x) = exp( )[(1 − x)I0 (
) − xI1 (
)]
2
2
2

(5.11)

where I0 and I1 are the modified Bessel functions of the zero and first order respectively.
There are two important approximations to the Rician pdf which depend on the SNR.
At very low SNR (when m ∼ 0), the Rician pdf reduces to Rayleigh distribution which
is given by,
x
−x2
p(x|σ) = 2 exp( 2 )
σ
2σ

(5.12)

On the other hand, at high SNR ((m/σ) > 5) [113, 114], the noise tends to be more
Gaussian (Fig. 5.1) following the pdf given by,
p(x|m, σ) = √

1
2πσ 2

exp(−

(x − m)2
)
2σ 2

(5.13)

Fig. 5.1 shows different Rician pdf when m and σ are varied. It is interesting to observe
the non-central nature of the Rician pdf. The pdf becomes more Gaussian as the signal
level (m) increases or the noise level (σ) decreases. From these plots, it can be observed
that when a Gaussian pdf is assumed at low SNR instead of the Rician pdf, the mean
value of the pdf is affected and this shows as a bias in the estimated parameters of the
pdf.
62

0.5

SNR=1
SNR=2
SNR=3
SNR=4

p(x)

0.4
0.3
0.2
0.1
0
0

2

4
x

(a)

0.8

8

SNR=1
SNR=2
SNR=3
SNR=4

0.6
p(x)

6

0.4
0.2
0
0

2

4
x

(b)

6

8

Figure 5.1. Rician distribution at different SNRs (SNR = m/σ). Distribution are generated by (a) varying m and fixing σ at 1.0 and (b) varying σ and fixing m at 2.5.

5.1.4

Choice of estimators for different noise models

The selection of estimator for the parameter estimation largely depends on the amount of
information available about the estimation problem. In case when the noise model is not
known, a least-squares estimator is preferred. For a non-linear signal model, a non-linear
least-squares estimator can be used for estimation such as least-squares estimator based
on the Levenberg-Marquardt (LM) algorithm [115, 116].
In this work, both the Rician and the Gaussian noise cases have been considered. For
the Gaussian noise case (which is generally assumed in absence of any noise information),
a least-squares estimator (LS) based on the LM algorithm is used. Since for the Rician

63

case the parameter estimates are biased due to the non-linear nature in which Rician noise
is injected into the signal, two methodologies were followed. From previous works such as
Henkelman [110], Gudbjartsson, Patz [113], Koay, Basser [114], a correction scheme has
been suggested to remove the bias in the estimation. For the first method, a correction
scheme suggested by Gudbjartsson, Patz [113] is used since the assumption is that the
data SNR are high enough (SNR>3) to assume a Gaussian noise. So, the correction
fixes the bias only and does not affect the variance of the estimates. This estimator
will be called least-squares with correction (LSC). The second methodology is based on
probabilistic methods and the maximum likelihood estimator (MLE) for the Rician noise
model is used. Such estimators have been discuss previously by Sijbers et. al. [117] and
these require the knowledge of the noise model explicitly. The advantage of using MLE
is that there is no need to apply any bias correction to the data since the correction is
inherent in the estimator.

5.1.5

Relation with FA, MD and α

The minimization of the determinant of CRLB provides an optimized gradient scheme
with respect to the uncertainty in the estimation of diffusion model parameters. Since
the diffusion model parameters are sometimes not the quantities of interest, it would be
important to find the relation of the performance of the schemes (in terms of CRLB) with
clinical biomarkers used for the identification of diseases, such as FA, MD and the fiber
angular deviation (α). FA and MD are functions of only the longitudinal and transverse
diffusivities in the diffusion model. FA is a dimensionless, normalized quantity that
indicates the local anisotropy in the imaging target [13]. MD, on the other hand, is the
mean of the diffusivities (trace of the diffusion tensor divided by 3). Finally, the angular
deviation is the deviation from the mean fiber orientation. This is a single quantity that
represents the angular position of the fiber in the voxel with respect to the mean fiber
orientation, instead of the two fiber direction parameters (θF , φF ). FA, MD and the
angular deviation are quantities that can be interpreted more easily in a pathological
condition than the diffusion model parameters.

64

In this section, the relationship of the performance of the diffusion gradient scheme
with the variances of FA, MD and angular deviation is studied. For the DTI model, MD
is defined as
MD =

D + D⊥1 + D⊥2
3

= Dav

(5.14)

and FA is defined as [13],
FA =

3 (D − Dav )2 + (D⊥1 − Dav )2 + (D⊥2 − Dav )2
2
2
2
D 2 + D⊥1 + D⊥2

(5.15)

where D , D⊥1 and D⊥2 are the eigenvalues of the diffusion tensor, D. By convention,
D is the diffusivity along the fiber orientation and D⊥1 and D⊥2 are the transverse
diffusivities. For the ADTI model, the expressions for FA and MD are simplified by
setting D⊥1 = D⊥2 = D⊥ , as given by
FA =

D − D⊥

2
D 2 + 2D⊥

(5.16)

and
MD =

D + 2D⊥
3

= Dav

(5.17)

The angular deviation (α) is defined as
α = cos−1 (f · f 0 )

(5.18)

where f represents the voxel fiber direction unit vector and f 0 is the mean fiber direction
unit vector defined as the average fiber orientation in voxels in the region of interest. Fiber
direction is the direction of the principal eigenvector of the diffusion tensor matrix, D.
The dot operation (·) represents the scalar product.
The propagation of uncertainty principle is used to find the variance of FA, MD and α
in terms of the variance bounds from the CRLB matrix. Assuming each metric, f (FA
or MD) is expanded by Taylor series expansion upto first order, for the DTI model, the
expansion is given by,
f ≈ f0 + D

∂f
∂f
∂f
+ D⊥1
+ D⊥2
∂D
∂D⊥1
∂D⊥2

65

(5.19)

Thus,
2
σf

≥

2
σD

∂f
∂D

2
2
+ σD
⊥1

2
∂f
2
+ σD
⊥2
∂D⊥1

+ 2 cov(D⊥1 , D⊥2 )

2
∂f ∂f
∂f
+ 2 cov(D , D⊥1 )
∂D⊥2
∂D ∂D⊥1

∂f
∂f ∂f
∂f
+ 2 cov(D , D⊥2 )
∂D⊥1 ∂D⊥2
∂D ∂D⊥2
(5.20)

where cov() is the covariance function.
For FA, the variance can be computed from Eqs. 5.15 and 5.20. The following can be
computed easily from the FA definition (Eq. 5.15),
∂F A
= (2 A − B − C − 2 D F A2 )/(2 SS F A)
∂D
∂F A
= (2 B − A − C − 2 D⊥1 F A2 )/(2 SS F A)
∂D⊥1
∂F A
= (2 C − B − Q − 2 D⊥2 F A2 )/(2 SS F A)
∂D⊥2

(5.21)
(5.22)
(5.23)

where
A = D − Dav
B = D⊥1 − Dav

(5.24)

C = D⊥2 − Dav
2
2
SS = D 2 + D⊥1 + D⊥2

For MD, the expressions are simpler than that of FA as shown below (Eq. 5.14).
∂MD
∂MD
1
∂MD
=
=
=
∂D
∂D⊥1
∂D⊥2
3

(5.25)

For variance of angular deviation, the first-order Taylor series approximation is given
by
2
2
σα ≥ σθ
F

∂α
∂θF

2

2
+ σφ

F

∂α 2
∂φF

(5.26)

where
∂α
=−
∂θF
and
∂α
=
∂φF

cos(φF )cos(θF )
1 − (cos(φF )sin(θF ))2
sin(φF )sin(θF )
1 − (cos(φF )sin(θF ))2
66

(5.27)

(5.28)

This is assuming that the covariance between θF and φF is zero. Note that the expression
for the variance is the same for DTI and ADTI model, since α only depends on the θF , φF
parameters (in f ) and f 0 does not depend on the model.
The variances and covariances of the diffusion parameters are replaced by the bounds
obtained from the CRLB of the variance of the parameters. For DTI, the model parameters are β = {D , D⊥1, D⊥2 , θF , φF , ψF }. The CRLB matrix can be decomposed into
diagonal and off-diagonal matrices as follows,
ΣCR = Σ1 + Σ2 + Σ3
where,







Σ1 = 






and






Σ2 = 




(5.29)

2
σD

0

0

0

0

0

0

2
σD

0

0

0

0

0

0

2
σD

0

0

0

0

0

0

2
σθ

0

0

0

0

0

0

2
σφ

0

0

0

0

0

0

2
σψ

⊥1

⊥2

F

F

F














(5.30)


0 cov(D , D⊥1 ) cov(D , D⊥2 ) cov(D , θF ) cov(D , φF ) cov(D , ψF )
0
0
cov(D⊥1, D⊥2 ) cov(D⊥1 , θF ) cov(D⊥1 , φF ) cov(D⊥1, ψF ) 


0
0
0
cov(D⊥2 , θF ) cov(D⊥2 , φF ) cov(D⊥2, ψF ) 

0
0
0
0
cov(θF , φF )
cov(θF , ψF ) 

0
0
0
0
0
cov(φF , ψF ) 
0
0
0
0
0
0
(5.31)

and Σ3 = ΣT , where T is the transpose operation.
2
Thus, the variance bounds of the estimates are computed from CRLB, ΣCR , as follows:
2
2
σD = ΣCR [1, 1]; σD

⊥1

2
= ΣCR [2, 2]; σD

⊥2

= ΣCR [3, 3]

cov(D , D⊥1) = ΣCR [1, 2]

(5.32)

cov(D⊥1 , D⊥2) = ΣCR [2, 3]
cov(D , D⊥2) = ΣCR [1, 3]
Also, the variance bounds of θF and φF are obtained as
2
2
σθ = ΣCR [4, 4]; σφ
F

F

67

= ΣCR [5, 5]

(5.33)

Similarly, for the ADTI model, the model parameters β = {D , D⊥ , θF , φF }. The
ΣCR can also be written in expanded matrix form as
ΣCR = Σ1 + Σ2 + Σ3
where

and






Σ1 = 






Σ2 = 


(5.34)

2
σD

0

0

0

0

2
σD

0

0

0

0

2
σθ

0

0

0

0

2
σφ

⊥

F

F









0 cov(D , D⊥ ) cov(D , θF ) cov(D , φF )
0
0
cov(D⊥ , θF ) cov(D⊥ , φF )
0
0
0
cov(θF , φF )
0
0
0
0

and Σ3 = ΣT , where T is the transpose operation.
2

(5.35)







(5.36)

Thus, the variance bounds of the estimates are computed from CRLB, ΣCR , as follows:
2
2
σD = ΣCR [1, 1]; σD = ΣCR [2, 2]; cov(D , D⊥ ) = ΣCR [1, 2]
⊥

(5.37)

And, the variance bounds of θF and φF are obtained as,
2
2
σθ = ΣCR [3, 3]; σφ = ΣCR [4, 4]
F
F

5.2
5.2.1

(5.38)

CRLB for different noise models
Rician noise case

In case of DTI data, the Rician noise model is incorporated as follows: the magnitude
data is computed from the complex data
Ec = (E(β; α) + nr ) + jni

(5.39)

where E(β; α) is the modeled MR signal (real quantity), nr , ni are zero mean and σ 2
variance Gaussian distributions, β; α are the diffusion model parameters and experimental parameters respectively, as defined before. It should be noted that the complex signal
68

is rotated by a rotation of the quadrature detector towards the real axis [110, 114]. Hence,
the imaginary component only contains the noise while the real component contains the
ˆ
signal and the noise. The measured signal is E = Ec

which has a Rician pdf given by,

ˆ
ˆ
ˆ
E2 + E2
EE
E
ˆ
)
p(E|E(β; α), σ 2 ) = 2 I0 ( 2 ) exp(−
σ
σ
2σ 2

(5.40)

When a set of N measurements is taken at different diffusion-encoding gradient direcˆ
ˆ
tions (g vectors), the measurement vector, E = {Ei , i ∈ [1, N]}, follows a multivariate
ˆ
Rician distribution p(E|β; α) ∼ R(µ(β; α), C(β; α)) where µ(β; α) is the mean vector of
ˆ
expected value of the measured normalized MR signal, E, and C(β; α) is the covariance
matrix of measurements.
The jkth element of the Fisher information matrix for Rician pdf [118] can be shown
to be

N

[I(β, α)]jk =
i=1

1 ∂Ei ∂Ei 2
(E − Zi )
σ 4 ∂βj ∂βk i

(5.41)

where
Zi =

∞
0

ˆ
ˆ
−2 E E
2 2 EE
ˆ ˆ
Ei I1 ( i2 ) I0 ( i2 ) p(E)dE
σ
σ

(5.42)

ˆ
ˆ
and p(E) = p(E|E, σ 2 ) (Rician pdf). I0 and I1 are the zero-order and first-order modified
Bessel functions of the first kind respectively. Eq. 5.41 can be rewritten using Zi = Wi2 ,
N

[I(β, α)]jk =
i=1

1
∂E
∂E
{(Ei − Wi ) i } {(Ei + Wi ) i }
∂βj
∂βk
σ4

(5.43)

Since Zi are all positive as seen in Eq. 5.42, Wi are all real-valued. Let X 1 and X 2 be
the modified model sensitivity matrices and their ijth elements are defined as follows:
[X 1 ]ij := (Ei − Wi )∂Ei /∂βj

(5.44)

[X 2 ]ij := (Ei + Wi )∂Ei /∂βj

(5.45)

In matrix-expanded form, the modified sensitivity matrices can be written as,


(E1 − W1 )∂E1 /∂β1
(E1 − W1 )∂E1 /∂β2 ... (E1 − W1 )∂E1 /∂βM


(E2 − W2 )∂E2 /∂β2 ... (E2 − W2 )∂E2 /∂βM 
 (E2 − W2 )∂E2 /∂β1
X1 = 

...
...
...
...


(EN − WN )∂EN /∂β1 (EN − WN )∂EN /∂β2 ... (EN − WN )∂EN /∂βM
(5.46)
69

and




X2 = 


(E1 + W1 )∂E1 /∂β1
(E1 + W1 )∂E1 /∂β2
(E2 + W2 )∂E2 /∂β1
(E2 + W2 )∂E2 /∂β2
...
...
(EN + WN )∂EN /∂β1 (EN + WN )∂EN /∂β2


... (E1 + W1 )∂E1 /∂βM

... (E2 + W2 )∂E2 /∂βM 

...
...

... (EN + WN )∂EN /∂βM
(5.47)

X 1 and X 2 are both N × M matrices and depend on β, Ω and the b-factor. i ∈ [1, N]
and j ∈ [1, M]. Also Ei = E(g i ; b; β) corresponds to the i th diffusion-weighted signal
obtained using the diffusion gradient g i , b-factor and parameter set β. In terms of the
original sensitivity matrix (X), the modified sensitivity matrices can be written as,
X 1 = E1 X

(5.48)

X 2 = E2 X
where

and

and



(E1 − W1 )
0
0
(E2 − W2 )
...
...
0
0

...
...
...
... (EN

0
0
...
− WN )



(5.49)



(E1 + W1 )
0
0
(E2 + W2 )
...
...
0
0

...
...
...
... (EN

0
0
...
+ WN )



(5.50)



E1 = 



E2 = 


 ∂E
1
∂β
 ∂E1

2
 ∂β1
X=
 ...


∂EN
∂β1

∂E1
∂β2
∂E2
∂β2

...
∂EN
∂β2

Eq. 5.43 can be written as,


∂E
... ∂β 1
M 
∂E
... ∂β 2 

M 
... ... 

∂EN
... ∂β










(5.51)

M

1
I(β, α) = 4 (X T X 2 )
1
σ

(5.52)

Thus, the corresponding Rician CRLB in matrix form is given as,
ΣCR (β, α) = I−1 (β, α) = σ 4 (X T X 2 )−1
1

(5.53)

Note that (X T X 2 ) is an M × M matrix. Using the determinant identity, det(rQ) =
1
r M det(Q) and also det(Q−1 ) = 1/det(Q) for an M × M matrix Q and a scalar r, the
70

determinant of Eq. 5.53 gives,
det ΣCR = det σ 4 (X T X 2 )−1 =
1

5.2.2

σ 4M
det (X T X 2 )
1

(5.54)

Gaussian noise case at high SNR

At high SNR, the Gaussian approximation holds good. In such a case, an additive white
Gaussian noise case with zero mean and σ 2 variance is considered. The observed signal
can be expressed as,
ˆ
E = E(β; α) + n

(5.55)

where n is zero mean and σ 2 variance Gaussian distribution and
ˆ
p(E|E(β; α), σ 2 ) = √

1
2πσ 2

exp(−

ˆ
(E − E)2
)
2σ 2

(5.56)

When a set of N measurements is taken at different diffusion-encoding gradient direcˆ
ˆ
tions (g vectors), the measurement vector, E = {Ei , i ∈ [1, N]}, follows a multivariate
ˆ
Gaussian distribution p(E|β; α) ∼ N(µ(β; α), C(β; α)) where µ(β; α) is the mean vecˆ
tor of expected value of the measured normalized MR signal, E, and C(β; α) is the
covariance matrix of measurements.
ˆ
For normally distributed measurements, p(E; β; α) has a mean vector µ(β; α) and
covariance matrix σ 2 I . Since the measurements are uncorrelated and has the same variance, σ 2 , the covariance matrix is a diagonal matrix with σ 2 as the diagonal element.
The jkth element of the Fisher information matrix can be computed by [109],
[I(β; α)]jk = [

∂µ(β; α) T 1 ∂µ(β; α)
] 2I[
]
∂βj
∂βk
σ

(5.57)

where I is identity matrix of size N x N, N being the number of DTI images acquired. T
represents transpose operation. j, k ∈ [1, M] where M is the number of diffusion model
parameters (for example, M = 4 for ADTI).
Since the covariance bound is given by ΣCR (β; α) = I−1 (β; α) and simplifying Eq.
5.57 [109],
ΣCR = σ 2 (X T X)−1

71

(5.58)

where X is called the model sensitivity matrix. Sensitivity matrix, X, are based on the
partial derivatives of the model E(β; α) w.r.t. the parameters: ηj (β; α) := ∂E(β; α)/∂βj
with j ∈ [1, M].
For N diffusion-encoding gradient directions used during DTI experiment, a sequence
of N diffusion-encoded images is collected. The sensitivity matrix X ∈ RN ×M is defined
as X(β; α) := Xi,j N ×M where Xi,j := ηj (g i ; b; β) [119].
From Eq. 5.58, applying determinant operation on both sides of the equations,
det ΣCR =

5.2.3

σ 2M
det (X T X)

(5.59)

Interpretation of CRLB

The determinant of CRLB (ΣCR ) corresponds to the product of the variance bounds of
the estimated parameters, assuming the parameters to be orthogonal to each other. It
is proportional to the bound on the hypervolume of uncertainty in the estimation of β
parameters, the hypervolume being defined as the product of the standard deviations of
the estimated parameters. The lower the hypervolume, the smaller will be the overall
uncertainty of estimation. From Eq. 5.54, it can be observed that the cost function is
a function of both experimental noise (σ) and the model sensitivity matrices (X 1 and
X 2 ). For the Gaussian case (Eq. 5.59), the relation between the sensitivity matrix and
noise variance is decoupled [119]. However, this is not the case for the Rician noise model
(Eq. 5.54) since the modified sensitivity matrices (X 1 and X 2 ) depend on σ 2 also.
The model sensitivity matrices are functions of the gradient scheme (Ω ), the b-factor
and the estimation parameter set (β). Thus, in order to reduce the determinant of ΣCR
by optimizing the gradient scheme(Ω ) only, the gradient scheme should be chosen to
maximize the determinant of X T X 2 (or minimize 1/ det (X T X 2 )), assuming a fixed
1
1
σ, thereby creating an experimental design for which the bound of the hypervolume of
uncertainty (

det ΣCR ) is minimized. If a bound on uncertainty can be optimized by

design, then any minimum variance unbiased (MVU) and efficient estimator [109] can be
shown to attain the bound asymptotically while performing the parameter estimation.
Such a determinant-optimal or simply D-optimal [33, 34] experimental design forms the
72

basis for designing optimal ADTI experiments in this work.

5.2.4

Sensitivity matrix computation

The definition of the CRLB involves the sensitivity matrix which is the Jacobian matrix
of the signal with respect to the model parameters at different gradient directions. The
following show the derivation for the sensitivity matrix for the ADTI and DTI model.
ADTI model
The expression for the normalized MR signal, E, for the ADTI model can be written as,
2
E = exp(−b(g 2 D + g⊥ D⊥ ))

(5.60)

where g = g · f and f is the unit vector along the fiber direction defined in spherical
2
coordinates (θF , φF ). Also, g⊥ = 1 − g 2 . The ADTI diffusion model parameter set,

β = {D , D⊥ , θF , φF }.
A row of the sensitivity matrix, X, is defined as,
[X]i. =

∂Ei
∂D

∂Ei
∂D⊥

∂Ei
∂θF

∂Ei
sin(θF )∂φF

(5.61)

Using Eq. 5.60 and 5.61,
∂E
= −bg 2 E
∂D

(5.62)

∂E
2
= −bg⊥ E
(5.63)
∂D⊥
∂E
= −bE(2g )(D − D⊥ )(sin(θ)cos(θF )cos(φ − φF ) − cos(θ)sin(θF )) (5.64)
∂θF
∂E
= −bE(2g )(D − D⊥ )(sin(θ)sin(φ − φF ))
(5.65)
sin(θF )∂φF
where θ, φ are the spherical coordinate angles for the gradient direction (g ≡ {θ, φ}).
To account for the curvature of the spherical coordinate system, the partial derivative
w.r.t. φF is divided by sin(θF ). Since the b-factor is fixed, the gradients are sampled
on a sphere of constant radius. These analytical expressions provide considerable insight
into the sensitivity of the MR signal w.r.t. to the diffusion model parameters. The first

73

observation is that all the sensitivities (derivatives) are weighted by the signal and the
b-factor. Thus, gradients sampling the regions of high signal or where high b-factor is
used will always provide better estimates of the parameters. However, high b-factor
causes the signal to reduce due to more signal attenuation. Hence, there has to be a
compromise solution between the choice of signal level and the b-factor. Next, for the
diffusivities, the sensitivities are proportional to the projections of the gradient direction
vector onto the fiber direction, i.e., g and g⊥ . Thus, the sampling locations selected
along the longitudinal direction will lead to a more sensitive estimation of D and the
ones along the transverse direction lead to a more sensitive estimation of D⊥ . For the
angular parameters, the interpretation is not direct. Apart from signal and b-factor,
the sensitivities w.r.t. angular parameters are also proportional to the difference of the
diffusivities (D − D⊥ ), which is indicative of the local diffusion anisotropy.

0

0

0.8

50

0.6

50
θ

θ

0.6
100

0.4

100

0.4
150
−100
(a)

0.2

150
0
100
φ sin θ

0.2

−100
(b)

0
100
φ sin θ

Figure 5.2. Plot of (a) the normalized MR signal and (b) its square for the ADTI model
with respect to the gradient direction angles (θ, φ) with the mean fiber direction is at
(θF , φF ) = (0◦ , 0◦ ). The gray scale represents the corresponding signal value.
Figs. 5.2 and 5.3 show the variation of the normalized MR signal and the sensitivities
(and squares) with changing gradient directions (given by θ, φ). The following model
parameters are used to generate the sensitivity and normalized MR plots: D = 1.62 ×
10−3 mm2 s−1 , D⊥ = 0.148 × 10−3 mm2 s−1 , θF = 0◦ , φF = 0◦ , σ = 0.1 (FA =
0.9, MD = 0.638 × 10−3 mm2 s−1 ), b = 1000 s mm−2 . The axisymmetric nature of
the signal model shows up nicely on the plot for MR signal. The axisymmetry is also
observed in the sensitivities w.r.t. D and D⊥ , but not in the sensitivity for angular
74

0

0

50

−100

100

−200
−100

(a)

0
100
φ sin θ

0

1
0
100
φ sin θ
x 10

0

−200
−400

100
150
−100

(c)

0
100
φ sin θ

4

100

2

150

−800

−100
(d)

0

5

6

50
θ

θ

2

−100

−600

0
100
φ sin θ

0

0
0.5

50

50

0
−0.5
−100

(e)

θ

100
150

θ

100

(b)

0

50

0.4

100

0.2

150

0
100
φ sin θ

−100
(f)

0

0
100
φ sin θ

0

0
0.5

50

50

0
−0.5
−100

θ

100
150

θ

3

150

−150
150

4

4
θ

θ

0

−50

50

(g)

x 10

0.4

100

0.2

150

0
100
φ sin θ

−100
(h)

0
100
φ sin θ

0

Figure 5.3. Plot of the sensitivity values and their square for the ADTI model with
respect to gradient direction angles (θ, φ). Shown in the figure are sensitivity values and
their squares w.r.t. D ((a) and (b)), D⊥ ((c) and (d)), θF ((e) and (f)) and φF ((g)
and (h)). The mean fiber direction is at (θF , φF ) = (0◦ , 0◦ ). The gray scale represents
the corresponding sensitivity values or its square.
75

parameters. The squares of the sensitivities are also shown to demonstrate the model
axisymmetry and also since the definition of CRLB uses the square of the sensitivity
matrices (i.e., X T X 2 for Rician case and X T X for Gaussian case), the square of the
1
sensitivity determines the optimized gradient distribution based on the CRLB.
DTI model
In this section, the sensitivity matrix computation is shown for the general 6-parameter
non-axisymmetric DTI or simply DTI case. Here, β = {D , D⊥1 , D⊥2 , θF , φF , ψF } and
number of estimation parameters (M) is 6. The normalized MR signal, E, is represented
in terms of the diffusion model and experimental parameters as
T T
E = e−bg R D 0 Rg

(5.66)

The DTI model sensitivity matrix is composed of partial derivatives of the signal with
respect to parameters. A row of the sensitivity matrix, X, is given by,
[X]i. =

∂Ei
∂D

∂Ei
∂D⊥1

∂Ei
∂D⊥2

∂Ei
∂θF

∂Ei
sin(θF )∂φF

∂Ei
∂ψF

(5.67)

The partial derivatives are analytically computed using signal equation Eq. 5.66 as
follows:
∂D 0
∂E
= −bEg T RT
Rg
∂D
∂D
where


0 0 0
∂D 0 

= 0 0 0 
∂D
0 0 1


(5.68)

(5.69)

This follows from the definition of the signal model (Eq. 5.66) and that D 0 is only
dependent on the diffusivities (D , D⊥1, D⊥2 ). Similarly,
∂E
∂D 0
= −bEg T RT
Rg
∂D⊥1
∂D⊥1
where


0 0 0
∂D 0


= 0 1 0 
∂D⊥1
0 0 0


76

(5.70)

(5.71)

where

∂E
∂D 0
= −bEg T RT
Rg
∂D⊥2
∂D⊥2

(5.72)


1 0 0
∂D 0


= 0 0 0 
∂D⊥2
0 0 0


(5.73)

For the partial derivatives with respect to Euler angle parameters (θF , φF , ψF ), the
chain rule for the derivatives is applied as follows:
∂E
= −bEg T
∂θF

∂R T
∂R
D 0 R + RT D 0
∂θF
∂θF

(5.74)

g

where
∂Ry (θF )
∂R
= Rz (ψF )
Rz (φF )
∂θF
∂θF

(5.75)

and



−sin(θF ) 0 −cos(θF )
∂Ry (θF ) 

=
0
0
0

∂θF
cos(θF ) 0 −sin(θF )

(5.76)

For computing the partial derivative with respect to φF , the reduction in the latitudinal
distance of the sphere due to the curvature has to be considered. Thus, an additional
sin(θF ) must be divided as shown:
∂E
−bE T
=
g
sin(θF )∂φF
sin(θF )

∂R T
∂R
D 0 R + RT D 0
∂φF
∂φF

g

(5.77)

where
∂R
∂Rz (φF )
= Rz (ψF )Ry (θF )
∂φF
∂φF

(5.78)

and



−sin(φF ) cos(φF ) 0
∂Rz (φF ) 

=  −cos(φF ) −sin(φF ) 0 
∂φF
0
0
0

(5.79)

Finally, the partial derivative computed with respect to ψF is as follows:
∂E
= −bEg T
∂ψF

∂R
∂R T
DR + RT D
∂ψF
∂ψF

g

(5.80)

where
∂R
∂Rz (ψF )
=
Ry (θF )Rz (φF )
∂ψF
∂φF
77

(5.81)

and



−sin(ψF ) cos(ψF ) 0
∂Rz (ψF ) 

=  −cos(ψF ) −sin(ψF ) 0 
∂ψF
0
0
0

(5.82)

Since, by definition, ψF (the third Euler angle) is the angle that the x-axis of the
rotated transverse diffusion plane makes with the line of nodes, this rotation is not on a
spherical latitude and the division by sin(θF ) is not required.
Note that the earlier analysis of the ADTI case was performed at θF = 0◦ . But, this
value cannot be used for the DTI case since at θF = 0◦ , φF and ψF are indistinguishable.
Using the definition of rotation matrix (R) above,
Ry (θF ) = I
R = Rz (ψF ) ∗ Ry (θF ) ∗ Rz (φF ) = Rz (ψF ) ∗ Rz (φF ) = Rz (ψF + φF )

(5.83)
(5.84)

where I represents the identity matrix of size 3 × 3. The problem with the above condition
is that the derivatives of E w.r.t. φF and ψF become equal and X T X has a rank less
than M (number of parameters, ie, 6). From the definition of CRLB, the CRLB becomes
indefinite. This problem can be solved by computing the derivatives at an angle other
than θF =0◦ , say θF =90◦ .
The variation of the signal and the partial derivatives with respect to different DTI
model parameters under an axisymmetric and non-axisymmetric condition are shown
in Figs. 5.4 and 5.6 and their squares in Figs. 5.5 and 5.7 . The parameters values
used to generate the sensitivities and the normalized MR signal for the axisymmetric
case in Fig. 5.4 are: D = 2 × 10−3 mm2 s−1 , D⊥1 = 0.2 × 10−3 mm2 s−1 , D⊥2 =

0.1999 × 10−3 mm2 s−1 , θF = 90◦ , φF = 0◦ , ψF = 0◦ (FA = 0.891, MD = 0.8 ×
10−3 mm2 s−1 ), b = 1000 s mm−2 , N = 30. For the non-axisymmetric case in Fig. 5.6,
the parameter values used are: D = 2 × 10−3 mm2 s−1 , D⊥1 = 0.2 × 10−3 mm2 s−1 ,
D⊥2 = 0.05 × 10−3 mm2 s−1 , θF = 90◦ , φF = 0◦ , ψF = 0◦ (FA = 0.935, MD =

0.75 × 10−3 mm2 s−1 ), b = 1000 s mm−2 , N = 30. The axes of the plot represents

the diffusion gradient direction in spherical coordinates, i.e., θ and φ on a unit sphere.
For the nearly axisymmetric case, the plots look similar to the ADTI sensitivity plots.
78

This validates that the DTI formulation can incorporate the axisymmetric case also.
Note that these cases are slightly non-axisymmetric (or nearly axisymmetric) so that the
CRLB matrix is not singular. For a perfectly axisymmetric case, ηψ will vanish making
F
the CRLB matrix singular since its determinant will be zero. For the non-axisymmetric
case, the normalized MR signal and the sensitivities w.r.t. to diffusivities are no longer
axisymmetric as expected. Thus, the DTI formulation provides for a more general model
although being more complicated than the ADTI formulation. Similar to the ADTI case,
variation of the square of the sensitivities and the normalized MR signal can indicate how
the final distribution of the optimized gradient distribution will be. CRLB is inversely
proportional to the determinant of the square of the sensitivity matrix and thus optimal
gradient directions will sample regions in the gradient space where the product of the
sensitivities will be higher.

79

0

0

0

50

−200

−100

100

−400

−150

150

100

θ

−50

150
−100 0 100
φ sin θ
0

(b)

θ

50

−200

100

−400

(c)

0
−0.5

−800

−100 0 100
φ sin θ

(d)

−5

x 10

0

0.5

50

5

50

0
−0.5
−100 0 100
φ sin θ

(f)

100

0

150

θ

100
150

θ

100
150

0

(e)

0.5

50

−600
−100 0 100
φ sin θ

−800

0

0

150

−600
−100 0 100
φ sin θ

θ

θ

50

(a)

0

−5
−100 0 100
φ sin θ

0.8

50
θ

0

0.6

100

0.4

150
(g)

0.2
−100 0 100
φ sin θ

Figure 5.4. Variation of the sensitivity values and the normalized MR signal w.r.t. diffusion gradient direction angles (θ, φ) for nearly axisymmetric case for the DTI model.
Shown in the figure are sensitivity values w.r.t. (a) D , (b) D⊥1 , (c) D⊥2 , (d) θF ,
(e) φF , (f) ψF and (g) the normalized MR signal. The mean fiber direction is at
(θF , φF ) = (90◦ , 0◦ ).

80

5

4

x 10
2.5
2
1.5
1
0.5

θ

50
100
150

50
4
100
2
150

−100 0 100
φ sin θ

(a)

−100 0 100
φ sin θ

(b)
5

θ

θ

4
100
0

−100 0 100
φ sin θ

(d)

−9

0.6
0.4

50

100

θ

θ

x 10
6

0

50
0.2

4
100
2

150
(e)

100
150

150

0

0.4
0.2

2

(c)

0.6

50

50

−100 0 100
φ sin θ

0

0

x 10
6

0

x 10
6

0

θ

0

150

−100 0 100
φ sin θ

0
(f)

0

−100 0 100
φ sin θ

0

0.6

θ

50
0.4
100
0.2

150
(g)

−100 0 100
φ sin θ

Figure 5.5. Variation of the squares of sensitivity values and the normalized MR signal
w.r.t. diffusion gradient direction angles (θ, φ) for nearly axisymmetric case for the DTI
model. Shown in the figure are squares of sensitivity values w.r.t. (a) D , (b) D⊥1 , (c)
D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the normalized MR signal. The mean fiber direction
is at (θF , φF ) = (90◦ , 0◦ ).

81

0

0

0

50

−100
−150
(b)

1

50
θ

0

100

−600

150

−800
−100 0 100
φ sin θ

(c)

−800

0

−400
100

−600
−100 0 100
φ sin θ

−200

150

−1

−100 0 100
φ sin θ

(d)

0

0
0.1

0.5

50

50

0
−0.5
−100 0 100
φ sin θ

(f)

100

0

150

θ

100
150

θ

−400

0

50

(e)

100
150

−100 0 100
φ sin θ
0

θ

θ

100

−200

−50

150

θ

50

(a)

0

−0.1
−100 0 100
φ sin θ

0
0.8

θ

50

0.6
100

0.4

150
(g)

0.2
−100 0 100
φ sin θ

Figure 5.6. Variation of the sensitivity values and the normalized MR signal w.r.t. diffusion gradient direction angles (θ, φ) for non-axisymmetric case for the DTI model. Shown
in the figure are sensitivity values w.r.t. (a) D , (b) D⊥1 , (c) D⊥2 , (d) θF , (e) φF , (f) ψF
and (g) the normalized MR signal. The mean fiber direction is at (θF , φF ) = (90◦ , 0◦ ).

82

5

4

x 10
3

0

50

2
θ

θ

50

x 10
6

0

100

4
100

1

2

150

150
−100 0 100
φ sin θ

(a)

−100 0 100
φ sin θ

(b)
5

0

x 10
8

0

100

4

100

2

150

θ

6

150

θ

1
0.8
0.6
0.4
0.2

50

50

−100 0 100
φ sin θ

(c)

0

0

−100 0 100
φ sin θ

(d)

0.6

0

0.4

50

0.015

100

θ

θ

50
0.2
150
(e)

0

0.01

100

0.005

150

−100 0 100
φ sin θ

0
(f)

0

−100 0 100
φ sin θ

0.8
0.6

100

0.4

150

θ

50

(g)

0

0.2
−100 0 100
φ sin θ

Figure 5.7. Variation of squares of the sensitivity values and the normalized MR signal
w.r.t. diffusion gradient direction angles (θ, φ) for non-axisymmetric case for the DTI
model. Shown in the figure are squares of sensitivity values w.r.t. (a) D , (b) D⊥1 , (c)
D⊥2 , (d) θF , (e) φF , (f) ψF and (g) the normalized MR signal. The mean fiber direction
is at (θF , φF ) = (90◦ , 0◦ ).

83

5.3

Partitioning CRLB matrix for optimized estimation of selected parameters

Generally, in an estimation problem, such as the estimation of DTI/ADTI model parameters, there are only certain parameters that are of interest and the rest can be assumed
to be “nuisance” parameters which are although estimated, but are not used or is not of
interest for further analysis. In the case of DTI, if the analysis is based on determining
the FA or MD in the voxel, only the diffusivities are of interest. Similarly, if the purpose
of the analysis is fiber tracking, only angular parameters, such as θF , φF are of interest.
Thus, a gradient optimization scheme that can optimize the model parameter estimation
to reduce the uncertainty of selective parameters is highly desirable.
In the context of DTI/ADTI model parameter estimation, there are two kinds of parameters, namely, diffusivity and angular parameters. I will focus on each of the kind of
parameters individually and come up with the gradient scheme optimization for selective
parameter estimation of each kind. The basic idea is factorization of the CRLB matrix
and optimize for the determinant of the sub-matrix which carries the variances of the
parameters of interest. This has been discussed in [33, 120]. Based on the discussion on
the derivation of the CRLB matrix, the Rician CRLB is given by
ΣCR = σ 4 (X T X 2 )−1
1

(5.85)

and the Gaussian CRLB is given by
ΣCR = σ 2 (X T X)−1

(5.86)

where X, X 1 and X 2 are the sensitivity matrices. Columns of these matrices contain partial derivatives with respect to diffusion model parameters computed at different
experimental settings, such as diffusion gradients.
Let a row of the sensitivity matrix be
[X]i. =

∂Ei
∂β1

∂E

...... ∂β i
M1

∂Ei
∂βM +1
1

∂E

...... ∂β i
M

(5.87)

where the vertical line divides the matrix into partial derivatives of M1 diffusivities and
M − M1 angular parameters. Thus, the X T X can be partitioned into four sub-matrices,
84

as given by
X TX =

A1 A2
A3 A4

(5.88)

where A1 , A2 , A3 and A4 are the sub-matrices after matrix squaring. Here, A1 is the
autocorrelation matrix of the partial derivatives with respect to only diffusivities and
A4 is the autocorrelation matrix of the partial derivatives with respect to only angular
parameters. A2 and A3 are cross-correlation matrices between the diffusivity and angular
matrices. Similarly, the CRLB matrix can be partitioned into four sub-matrices as given
by,
ΣCR =

B1 B2
B3 B4

=

σ2

A1 A2
A3 A4

−1

(5.89)

where B1 and B4 are the CRLB sub-matrices with respect to only diffusivity and angular
parameters, respectively. Thus, in order to minimize the CRLB of estimation variance of
diffusivities only, determinant of B1 can be minimized instead of the entire CRLB matrix.
Using Woodbury’s matrix inversion formula [120, 121], B1 is computed as,
B1 = σ 2 (A1 − A2 A−1 A3 )−1
4

(5.90)

The inversion is under the condition that A4 is not singular (i.e., det(A4 ) = 0). Scharf
et al. [120] shows another way of partitioning the CRLB matrix which is applicable only
to Gaussian CRLB case. However, Eq. 5.90 is a more general form of partitioning the
CRLB matrix and is useful for Rician CRLB case where the non-symmetric matrix is
partitioned, as given by
A1 A2
A3 A4

X TX 2 =
1

(5.91)

Here X 1 and X 2 are the modified sensitivity matrices which can also be split into the
partial derivatives of the diffusivity and angular parameters as shown for the sensitivity
matrix (X) case.

5.3.1

Diffusivities

For DTI case, model parameters are β = {D , D⊥1, D⊥2 , θF , φF , ψF }. In order to optimize the gradient scheme only for the diffusivities, CRLB matrix is partitioned to contain

85

the variance of only diffusivities. Thus, the sensitivity matrix is partitioned to separate
the diffusivities as given
∂Ei
∂D

[X]i. =

∂Ei
∂D⊥1

∂Ei
∂D⊥2

∂Ei
∂θF

∂Ei
sin(θF )∂φF

∂Ei
∂ψF

(5.92)

Next, using Eq. 5.90, the B1 matrix will only contain the variances of the diffusivities.
Finally, the cost function for the optimization of gradient directions becomes det(B1 ).
For ADTI model, the parameter set is β = {D , D⊥ , θF , φF }. Thus, to optimize the
diffusivities, the sensitivity matrix is partitioned as
[X]i. =

∂Ei
∂D

∂Ei
∂D⊥

∂Ei
∂θF

∂Ei
sin(θF )∂φF

(5.93)

As before, using Eq. 5.90, the B1 matrix is computed and the cost function is the
determinant of the B1 matrix.
For Rician CRLB matrix, the partitioning is similar except that instead of X, modified
sensitivity matrices (X 1 and X 2 ) are partitioned individually using Eq. 5.91 and Eq.
5.90 is used to compute the B1 matrix.

5.3.2

Angular parameters

For DTI model, selective optimization for angular parameters is done by partitioning the
sensitivity matrix as
∂Ei
∂θF

[X]i. =

∂Ei
sin(θF )∂φF

∂Ei
∂D

∂Ei
∂D⊥1

∂Ei
∂D⊥2

∂Ei
∂ψF

(5.94)

Note that only the θF , φF parameters are considered in the angular parameters set since
these directly affect the fiber orientation and α. ψF is treated as a nuisance parameter
along with the diffusivities. Finally, B1 is computed according to Eq. 5.90 and the
optimization cost function is det(B1 ).
For ADTI model, the sensitivity matrix is also partitioned to separate the angular
parameters as given by
[X]i. =

∂Ei
∂θF

∂Ei
sin(θF )∂φF

86

∂Ei
∂D

∂Ei
∂D⊥

(5.95)

As before, using Eq. 5.90, the B1 matrix is computed and the cost function is the
determinant of the B1 matrix.
Rician CRLB case is handled as mentioned in the previous section, but with the new
partitions of the modified sensitivity matrices.

5.4

Gradient scheme optimization framework

The proposed optimization framework is based on D-optimality and its application for
the design of optimal experiments [33, 34]. As described earlier, optimization is defined by optimal selection of experimental parameters by minimizing the determinant
of CRLB matrix. The analysis has focussed on optimizing the gradient scheme (Ω )
for different signal models (such as DTI, ADTI) and noise models (Rician and Gaussian). For the DTI model ([7]), the rotationally invariant diffusion model parameters
are β = {D , D⊥1 , D⊥2 , θF , φF , ψF } (M = 6). For the ADTI model, from axisymmetry
assumption, the model parameters are β = {D , D⊥ , θF , φF } (M = 4).
The optimal diffusion gradient scheme is obtained by solving the following equation,
Ωopt = arg

min
Ω

max

{θF ,φF }∈Λ

f

(5.96)

where f := 1/ det (X T X 2 ) (for the Rician noise case) or f := 1/ det (X T X) (for the
1
Gaussian noise case). Since f is a scaled form of det ΣCR , minimizing f is equivalent
to minimizing det ΣCR . In Eq. 5.96, the maximum value of f (worst case of f ) within
a cone (Λ) that includes the majority of fiber directions is searched and then this value
is minimized w.r.t. the gradient scheme (Ω ) to reach the optimal gradient design (Ωopt ).
The formulation is robust since the gradient scheme is optimal for an uncertainty range
of fiber orientations. The a priori structural information is incorporated in Eq. 5.96 by
measuring Λ from preliminary DTI scan on a subject. Since the optimization is local in
parameter space, the mean values of diffusion model parameters (β) (also obtained from
preliminary DTI scan) are used to calculate f .

87

5.4.1

Reformulation of the gradient scheme

To reduce the computation time for gradient scheme optimization, the definition of the
gradient scheme is reformulated as Ω = {(θr , ∆φr , Nr ); r ∈ [1, P ]}, where P is the number
of rings, each located at inclination angle (θr ) with azimuthal offset angle (∆φr ) and Nr
uniformly distributed points [84, 85]. Under this formulation, the number of optimization
parameters is greatly reduced. For example, for a 30-gradient direction problem (Fig.
5.8), instead of defining Ω in 60 variables, Ω = {(θi , φi ); i ∈ [1, 30]}, it can be redefined
in 9 variables, Ω = {(θr , ∆φr , Nr ); r ∈ [1, 3]} for P = 3 (3-ring configuration). P is
selected by trying a number of configurations (such as P = 3, 4, 5, 6 and 7) and choosing
the best in terms of cost function, i.e., the one that gives the minimum cost function.
This framework helps improve the optimization speed significantly (when P is less) and
also provides flexibility in designing complex schemes (for large P ).

0

0.8

θ(°)

20
40

0.6

60

0.4

80
−100

0
φ sin θ ( ° )

100

0.2

Figure 5.8. Gradient directions (white circles) on a 2D opened hemisphere showing a
reformulated scheme (for P = 3). Mean fiber orientation is at (θF , φF ) = (0◦ , 0◦ ). The
gray scale underlay shows the normalized MR signal variation due to changing gradient
directions (θ, φ).
The representation of the gradient scheme in a ring-like formation originates from
the observation that upon application of the axisymmetric condition on the diffusion
model such that the diffusion in the plane transverse to the fiber direction is isotropic,
the optimal sampling locations in the gradient space collapse onto axisymmetric rings
about the mean fiber direction [39, 72, 119]. This phenomenon can be seen in analytical
derivations as well as by numerical simulations. Thus, sets of gradient directions can be

88

grouped into ring-like distribution and defined as given in the reformulated definition of
Ω . Although this representation puts constraints on the gradient distribution, by adding
additional parameters such as ∆φr in the definition of Ω , even non-axisymmetric cases
can be represented. So, a range of distribution patterns of gradient directions can be
achieved by the reformulated gradient scheme.

5.4.2

Algorithm

The robust optimization problem defined in Eq. (5.96) is solved for the DTI and ADTI
models. A preliminary optimization using θF = 0◦ and Λ = 0 is performed to provide a
starting point, Ω0 := arg [minΩ f ], for the robust optimization algorithm. This strategy is
preferred to choosing an arbitrary gradient scheme and results in shorter computational
times. For the computation of Ω0 , a simulated annealing method, which performs a
stochastic exploration of the sampling space controlled by a temperature parameter (T ),
is implemented using the full set of variables [122]. Simulated annealing is a stochastic
minimization technique [73] which is based on the annealing process (slow cooling process)
in metallurgy and is known to be robust with respect to local minima problems common
in gradient-based methods. Details of this algorithm are given in Appendix A (Fig. A.3
and Fig. A.4). Solutions satifying the T -dependent Metropolis criterion [73] are accepted,
while others are rejected. The exploration stage is necessary to avoid local optima (the
cost function f is nonconvex), and the step-size for exploration is progressively reduced
after a maximum number of rejection steps are reached [119].
In the second stage, the robust optimization problem (Eq. 5.96) is solved using the
reduced-order parameterization with P rings. The ring locations, {θi ∈ [0, π/2]; i ∈
[1, P ]}, and number of points, {Ni ∈ [1, N − P ]; i ∈ [1, P − 1]}, are initialized based on
the clustering of the q-space sampling scheme Ω0 . These parameters are optimized by
iterating on possible values for Ni and P and using a deterministic gradient-based “minimax” algorithm [123] for θi . Eq. (5.96) is solved iteratively by updating the parameters
of Ω until the changes in these parameters do not change the cost function significantly.
The optimization is implemented in Matlab (Mathworks Inc., Natick, MA) using routines

89

from the Optimization Toolbox.

0

0.8

20
0.6

60

θ

40

0.4

80
−150 −100
(a)

1.1

x 10

−50

0
φ sin θ

50

100

0.2

150

−11

Hypervolume

MF30
OPT30
1.05

1

0.95

0.9
0
(b)

20
40
Angular deviation, α ( ° )

60

Figure 5.9. Robust optimization results for the ADTI model with Rician CRLB for
b = 1000 s mm−2 with P = 3 rings for fiber orientations within a cone of axis along the zaxis and cone angle Λ = 35◦ . (a) Gradient directions (white circles) an opened hemisphere
with gray scale equal to the normalized MR signal. (b) Comparison of performance curves.

5.4.3

Optimization Efficiency

ADTI model
Eq. 5.54 (Rician noise case) or Eq. 5.59 (Gaussian noise case) can be used to predict the
performance of the diffusion-encoding gradient scheme (Ω ) before performing the ADTI
experiment. A gradient scheme has a better performance when det ΣCR is lower with
respect to another scheme. Also, a scheme is more robust (in terms of fiber angular
deviation (α) from the mean fiber orientation) than another scheme when det ΣCR is
90

1.2

10°
20°
°

DOPT/DMF30

35
1.1

°

40

°

90
1

0.9

0.8
0

20
40
Angular deviation, α ( ° )

60

Figure 5.10. Variation of normalized hypervolume of uncertainty for 30-direction optimal
gradient schemes for ADTI model. The cone angles vary as (Λ = [10◦ − 90◦ ]) and Rician
noise case is selected. Noise level, σ = 0.1. Normalization reference is MF30 gradient
scheme.
low over a broader range of α. This performance range (of α) is determined by the value
of the cone angle (Λ) used during the optimization procedure.
Fig.

5.10 shows the variation of normalized hypervolume of uncertainty

(DOP T /DM F 30, DOP T = det ΣCR,OP T , DM F 30=

det ΣCR,M F 30 ) for the optimal

30-direction gradient schemes at different cone angles (Λ). Normalization is w.r.t. hypervolume of uncertainty for the MF30 scheme. The MF30 is a 30-direction MF techniquebased gradient scheme [22] which does not use any a priori angle information (Λ). For
smaller cone angles (Λ = 10◦ ), the performance of the optimized gradient scheme is better
than MF30 only when the fiber orientation is close to the mean fiber orientation (α = 0◦ ).
As the cone angle is increased, the performance becomes worse and eventually at Λ = 90◦
(completely uncertain case) DOP T /DM F 30 remains always less than unity, albeit close
to unity, indicating a mildly improved performance compared to that of MF30.

91

DTI model
The performance prediction of an optimized gradient scheme for DTI case is done by
computing the determinant of CRLB (at some fixed noise, σ 2 ) for different values of θF
and φF . Only these two parameters are chosen since the scheme is made robust with
respect to a range of fiber orientations. Thus to verify if the robustness is achieved, the
CRLB is computed in a range of fiber orientations given in terms of θF and φF . The
Fig. 5.11 shows the 2D performance plots for a non-axisymmetric diffusion case under
Gaussian noise.
0

1
0.95

θF ( ° )

50

0.9
0.85

100

0.8
150

0.75
−50

0
φF ( ° )

50

0.7

Figure 5.11. Plot of the ratio of hypervolumes for OPT30 w.r.t. MF30 scheme optimized
using Gaussian noise assumption and for the DTI model. The contour line at unity value
shows the cone of angles within which the optimization performance is improved.
The interpretation of the Fig. 5.11 is straightforward. Since a reduced uncertainty for
a range of fiber orientations is desired when compared to the performance of the standard
scheme such as MF30, it is verified if the robust optimal scheme is theoretically robust
(based on the CRLB) in performance within an angular deviation defined by Λ from
the mean fiber orientation. Fig. 5.11 shows the plot of the ratio of the cost function
(detΣCR ) for the optimized scheme w.r.t. MF30 scheme. The optimization is performed
at Λ = 35◦ . Thus, the ratio should be less than unity for angles within the cone angle
range as observed in Fig. 5.11. Note, the true fiber orientation is at (θF , φF )=(90◦ ,0◦ ).
Since Λ is specified during optimization and its value collected from the preliminary DTI

92

scan on the subject, the robust optimization is subject specific, and thus more accurate.

5.4.4

b-factor optimization

In the discussion so far, I focussed on the optimization of the gradient directions to
minimize the cost function (i.e., determinant of ΣCR ) within a range of fiber orientation.
However, in all of these cases, the b-factor has been kept constant. In this section, I discuss
the optimization of the b-factor as well as the gradient scheme using a modified form of
Eq. 5.96. Here the b-factor is included as an optimized parameter in the optimization
problem which can be written as,
αopt = arg

min
b,Ω

max

{θF ,φF }∈Λ

f

(5.97)

where, α ≡ [b, Ω ], f := 1/ det (X T X 2 ) (for Rician noise case) or f := 1/ det (X T X) (for
1
Gaussian noise case). Note that the sensitivity matrix, X (and the modified sensitivity
matrices, X 1 and X 2 ) are functions of both the gradient scheme (Ω ) and the b-factor.
Under this optimization scheme, the b-factor is considered on the whole rather than
considering its individual components, such as the diffusion gradient strength, gradient
pulse durations and the diffusion pulse interval. While optimizing for the b-factor, it
will be assumed that the values that are obtained are achievable and within limits of
the MRI scanner. This results in additional constraint on the optimization problem. A
change in the b-factor could be accomplished by changing any of its components. But,
it is preferred that the change in the b-factor is caused by the change in the diffusion
gradient strength only, while keeping the timing parameters fixed. However, in general,
the diffusion gradients are set at the maximum by the hardware already. Thus, the bfactor needs to increased by increasing the pulse duration or the inter-pulse intervals.
This leads to increase of TE . In addition to longer diffusion gradients, the SNR will
drop in the diffusion-weighted images. This limits the upper value of b-factor on clinical
scanners.
Inclusion of b-factor in the optimization framework result in a modified algorithm for
the framework. The modifications include:

93

1. Using simulated annealing method for the combined optimization of the b-factor
and the gradient scheme: the algorithm discussed previously uses a preliminary
exploratory stage based on simulated annealing method [73] to avoid the local
minima problem (see Appendix A (Fig. A.3 and Fig. A.4) for flowchart of the
optimization procedure) and then uses the reformulated gradient scheme (discussed
in the section ‘Reformulation of the gradient scheme’ and Appendix A (Fig. A.5))
to search for the optimized gradient scheme. When the b-factor is included in the
optimization, the local minima problem is severe since the b-factor scales the model
parameter sensitivities directly and thus has more influence over the CRLB matrix
than the gradient directions. Hence, when optimizing for the b-factor, a simulated
annealing method with slower cooling rate and finer step size adjustment is used
(see Appendix A (Fig. A.6 for flowchart and detailed steps)).
2. No use of the reformulated gradient scheme: Owing to local minima problem and
significant variations in the echo signals while changing the b-factor, the reformulated gradient scheme could pose additional constraints and as such is not used.
3. No use of the exploitation stage: In the previous algorithm, the second stage based
on the reformulated gradient scheme used a gradient descent-based method to reach
an optimal location quickly. However, due to local minima issues, the gradient
descent technique could not be applied.
From the derivation of the sensitivity matrix for both the ADTI and DTI model, it
is observed that sensitivities are proportional to the b-factor and hence if the b-factor
is varied, the optimization algorithm will attain higher b-factors to reduce the CRLB
variance. On the other hand, regions where determinant of CRLB matrix is low generally
correspond to regions where signal levels are high for a fixed b-factor. But, in order for
the signal level to be high, the b-factor has to be low. Thus, there is a definite region
where an optimal b-factor which satisfies both the high signal criterion as well as the high
sensitivity criterion is attained by the optimization algorithm.
In optimizing the b-factor, an important observation is that at high b-factor values, the

94

signal will be low and as such the SNR will also be low. Under such conditions, the Rician
noise model and use of the maximum likelihood estimator is most appropriate. However,
the SNR can be improved by increasing the number of excitations and thus using more
averaging to reduce the noise. So, at high b-factor, even though the signal level is low,
but a good SNR can be obtained by averaging of more images. The disadvantage of this
method is that the total scan time greatly increases. Still, the most common practice
while using high b-factor is increasing the number of averages in the MR data acquisition.

5.4.5

Additional constraints on FA and MD in gradient scheme
optimization

After discussing the gradient scheme optimization and the b-factor optimization, in this
section, I will discuss the inclusion of additional constraints on FA and MD into the
optimization problem. The optimization cost function (determinant of ΣCR ), so far, was
computed for a fixed value of diffusion model parameters and optimized within a range
of fiber directions. This notion can be extended to FA and MD values range as well. In
other words, the cost function can be optimized within a range of both fiber directions
as well as a range of FA and MD values. The range of FA and MD essentially sets a
constraint on the diffusivity values. The optimization problem can now be modified as
follows,
αopt = arg min
b,Ω

max f
C

(5.98)

where constraints are denoted by C and given by,
C ≡ {θF , φF } ∈ Λ & F A ∈ [F A0 , F A1 ] & MD ∈ [MD0 , MD1 ]

(5.99)

where, as before, f := 1/ det (X T X 2 ) (for Rician noise case) or f := 1/ det (X T X) (for
1
Gaussian noise case). The range of FA values is given by [F A0 , F A1 ] and the range of
MD values is given by [MD0 , MD1 ]. Also, the fiber angular range is given, as before,
by {θF , φF } ∈ Λ, where Λ is the cone of fiber orientations. This is the optimization
problem formulation in a very generalized form and could incorporate prior information
of both the fiber orientation as well as the diffusivities (in the form of FA and MD). The
95

optimization algorithm is the same as the one used for b-factor optimization. However,
due to a large number of constraints, the computation time for the optimization is much
higher as compared to just optimizing the gradient directions.

5.5

Comparison of optimized gradient schemes and
their predicted performances

This section presents a comparison of spatial distributions and the predicted performances
for various optimized schemes. Both ADTI and DTI diffusion model and Rician and
Gaussian noise model have been considered for the gradient optimization.
For the ADTI case, the parameter values and noise level used for the gradient optimization process are: D = 1.62×10−3 mm2 s−1 , D⊥ = 0.148×10−3 mm2 s−1 , θF = 0◦ ,

φF = 0◦ , σ = 0.1. This corresponds to FA = 0.9, MD = 0.638 × 10−3 mm2 s−1 Exper-

imental settings are : b = 1000 s mm−2 , N = 30. Prior structural information used:
Λ = 35◦ .
For the DTI case, the parameter values and noise level used for the gradient optimization process are: D

= 2.2 × 10−3 mm2 s−1 , D⊥1 = 0.4 × 10−3 mm2 s−1 ,

D⊥2 = 0.3657 × 10−3 mm2 s−1 , θF = 90◦ , φF = 0◦ , ψF = 0◦ , σ = 0.1. This cor-

responds to FA = 0.802, MD = 0.989 × 10−3 mm2 s−1 Experimental settings are :
b = 1000 s mm−2 , N = 30. Prior structural information used: Λ = 35◦ .

5.5.1

Distribution of gradient directions

Fig 5.12 shows the optimized gradients distribution on an opened unit hemisphere for the
ADTI diffusion model and Rician and Gaussian noise models under different optimality
criteria. The MF30 scheme is also shown for comparison. The CRLB-based D-optimality
criterion is inversely related to the determinant of the square of the sensitivity matrices. Hence, by minimizing the determinant of the CRLB for the gradient directions, the
optimization framework distributes these gradients in regions where the square of the
sensitivity is higher. Determinant of the square of the sensitivity matrices is propor96

tional to the product of the individual parameter sensitivities, such as sensitivities w.r.t.
D , D⊥ , θF and φF for ADTI, and the optimization framework looks for regions where the
majority of the model sensitivities are high. Referring to the sensitivity plots in Fig. 5.3
which shows the spatial variation of the sensitivities caused due to the prior assumption
of the fiber structure, for the all parameters case, the region transverse to the fiber orientation corresponds to higher overall square sensitivity values and hence are sampled more
in this region by optimization (shown in Fig. 5.12 (a) and (b)). For the diffusivity only
case, both the regions transverse and towards the fiber orientation are sampled (shown
in Fig. 5.12 (c) and (d)) and for the angular parameters only case, only transverse region
is sampled after gradient optimization (shown in Fig. 5.12 (e) and (f)). Note that the
sensitivity being a function of the normalized MR signal which is also higher transverse
to the fiber orientation also influences the distribution of the gradient directions, pushing
these towards the transverse direction. Results of Rician and Gaussian cases are similar
but Rician CRLB optimized cases are more dispersed than the Gaussian CRLB optimized
cases which could be due to the rescaling of the sensitivity matrix in the Rician case as
shown in Eq. 5.48. MF30 (Fig. 5.12 (g)) scheme uniformly samples the gradient space
since it does not use any optimization based on prior structural knowledge.
Fig. 5.13 shows the distribution of the optimized gradient directions on an opened
sphere for the DTI model. Here the true fiber orientation is along +X-zxis to avoid the
singularity in the CRLB definition when the fiber is at +Z-axis. While the reasoning
for the distribution is same as in the ADTI case, i.e., regions of higher overall square
sensitivities are selected by the optimization framework, the distribution is consistently
towards the transverse orientation, even for the diffusivities only case. This can be
explained from the sensitivity plots for the DTI model (see Figs. 5.4 – 5.7) where the
square sensitivities of most of the DTI model parameters (except for D ) are higher
transverse to the fiber orientation. This effect is less pronounced in the ADTI since it
has fewer diffusion model parameters.

97

(b)

θ

0

(c)

0
100
φ sin θ

θ
(e)

−100

0
100
φ sin θ

50
−100

(f)

θ

0
50
−100
(g)

0
100
φ sin θ

0

0.8
0.6
0.4
0.2

50
−100

50

(d)

0

0
100
φ sin θ

0

0.8
0.6
0.4
0.2

50
−100

50
−100

θ

(a)

0
100
φ sin θ

θ

50
−100

0

0.8
0.6
0.4
0.2

θ

θ

0

0
100
φ sin θ

0
100
φ sin θ

0.8
0.6
0.4
0.2

0.8
0.6
0.4
0.2

0.8
0.6
0.4
0.2

0.8
0.6
0.4
0.2

Figure 5.12. Optimized Gradient schemes for the ADTI diffusion model under the following optimality criteria: optimize for all parameter ((a) Rician and (b) Gaussian),
diffusivities only ((c) Rician and (d) Gaussian) and fiber orientation only ((e) Rician and
(f) Gaussian). MF30 scheme is shown in (g). The fiber orientation, (θF , φF ) = (0◦ , 0◦ ).
The gray scale values represent the normalized MR signal and the white circles are the
locations of the diffusion gradient unit vector on an opened unit hemisphere.

5.5.2

Performance curves

Fig. 5.14 shows the performance curves of various optimized gradient schemes for the
ADTI model. The CRLB-based hypervolume has been reduced for the OPT30 scheme
compared to the MF30 scheme in the desired angular deviation from the fiber orientation
defined by the cone angle during optimization. The curve is smooth for the all parameter
case, but fluctuates for the diffusivities only or angular parameters only cases within the
cone angle even though there is reduction in the hypervolume compared to MF30.
98

0

0
0.6

0.6
50

0.5
0.4

100

θ

θ

50

0.5
0.4

100

0.3
150

0.3
150

0.2
−100

(a)

0
φ sin θ

100

0.2
−100

(b)

0

0
φ sin θ

100

0
0.6
50

0.5
0.4

100

θ

θ

50

0.6
0.5
0.4

100

0.3
150

0.3
150

0.2
−100

(c)

0
φ sin θ

100

0.2
−100

(d)

0

0
φ sin θ

100

0
0.6

0.6
50

0.5
0.4

100

θ

θ

50

0.5
0.4

100

0.3
150

150

0.2
−100

(e)

0.3

0
φ sin θ

100

0.2
−100

(f)

0
φ sin θ

100

Figure 5.13. Optimized Gradient schemes for the DTI diffusion model under the following
optimality criteria: optimize for all parameter ((a) Rician and (b) Gaussian), diffusivities only ((c) Rician and (d) Gaussian) and fiber orientation only ((e) Rician and (f)
Gaussian). The fiber orientation, (θF , φF ) = (90◦ , 0◦ ). The gray scale values represent
the normalized MR signal and the white circles are the locations of the diffusion gradient
unit vector on an opened unit hemisphere.
Fig. 5.15 shows the performance plots of various optimized gradient schemes for the
DTI model. The contour line indicates the unity ratio level which coincides with the cone
angle set during optimization. Thus, a optimized scheme perform as designed within the
cone angle. The reduction is more significant for the all parameter case (Fig. 5.15 (a) and
(b)) than other cases (Fig. 5.15 (c) – (f)). The region of overall higher square sensitivities
99

is more when more diffusion parameters are selected for improved precision since there
is a possibility of more overlapping regions of higher individual sensitivities which is not
the case for fewer model parameters, such as diffusivities only or angular parameters only
cases.
−11

1.05

−11

1.1 x 10

MF30
OPT30

Hypervolume

Hypervolume

1.1 x 10

1
0.95
0.9
0

20
40
Angular deviation, α ( ° )

(a)

1.05
1
0.95
0.9
0

60

−9

−9

Hypervolume

Hypervolume

4.6
4.4

(c)

20
40
Angular deviation, α ( ° )

5
4.8
4.6
4.4
4.2
0

60
(d)

−3

MF30
OPT30

Hypervolume

Hypervolume

60

x 10

2.2
2
0

(e)

20
40
Angular deviation, α ( ° )
−3

x 10
2.4

MF30
OPT30

5.2

4.8

4.2
0

60

x 10
MF30
OPT30

5

20
40
Angular deviation, α ( ° )

(b)

x 10
5.2

MF30
OPT30

20
40
Angular deviation, α ( ° )

60

2.4
2.2
2
0

(f)

MF30
OPT30

20
40
Angular deviation, α ( ° )

60

Figure 5.14. Performance curves for the ADTI diffusion model for optimization w.r.t. all
parameters ((a) Rician and (b) Gaussian), diffusivities ((c) Rician and (d) Gaussian) and
angular parameters ((e) Rician and (f) Gaussian).

100

0
50
100

0.8
150

(a)

0
50
φF ( ° )

0

0.8

−50
(b)

0
50
φF ( ° )

0

100
0.8
150

0.7

1

50

0.9

θF ( ° )

θF ( ° )

100

0.7

1

50

0.9

100
0.8
150

−50
(c)

0
50
φF ( ° )

0

0.7

−50
(d)

0
50
φF ( ° )

0

1

50
100

0.8
150

0.7

1

50

0.9

θF ( ° )

θF ( ° )

0.9

150
−50

0.9

100
0.8
150

−50
(e)

1

50

0.9

θF ( ° )

θF ( ° )

0

1

0
50
φF ( ° )

0.7

−50
(f)

0
50
φF ( ° )

0.7

Figure 5.15. Performance plots for the DTI diffusion model for optimization w.r.t. all
parameters ((a) Rician and (b) Gaussian), diffusivities ((c) Rician and (d) Gaussian) and
angular parameters ((e) Rician and (f) Gaussian). Gray scale values indicate the ratio of
CRLB-based hypervolume of OPT30 by MF30 and the contour lines are at unity ratio.

101

CHAPTER 6
Evaluation of estimation
performance by simulations
In this chapter, various Monte Carlo simulations are performed to (a) evaluate the noise
models, (b) evaluate the performance of different parameter estimators and (c) evaluate
the effect of the non-optimized experimental parameters in terms of performance indices.
Finally, simulations are performed to study the performance of the optimized gradient
schemes for the ADTI and the DTI models with Rician and Gaussian noise models. Cases
for gradient optimization based on selected model parameters are also considered, such
as diffusivities only or angular parameters for fiber directions only, for different diffusion
and noise models.

6.1

Noise characterization

The normalized MR signal (or echo attenuation) due to diffusion is defined as the ratio
of two magnitude MR signals obtained with and without the application of diffusion
gradients respectively. Assuming that measured complex MR signals contain additive
white Gaussian noise, the corresponding magnitude MR signals (MR image intensities)
will contain Rician noise [110, 114] and the normalized MR signal being defined as the
ratio of two magnitude MR signals will follow a ratio pdf of two Rician distributed
random variables. The ratio pdf can be approximated by a Rician pdf or a Gaussian pdf.
102

In this section, an evaluation of the Rician approximation of the ratio pdf is performed
and sources of error are analyzed. For comparison, the Gaussian approximation is also
included in the evaluation.

6.1.1

Method

Let the measured (noisy) normalized MR signal, E, be defined as E = S/S0 , where S
and S0 are the measured magnitude MR signals with and without diffusion-weighting.
Let Sc = m + nr + jni and S = |Sc |, where Sc is the measured complex MR signal with
diffusion-weighting, nr , ni are N(0, σ) (Gaussian pdf). Also, S0c = m0 + n0r + jn0i and
S0 = |S0c | where S0c is the measured complex MR signal without diffusion-weighting,
n0r , n0i are N(0, σ0 ) (Gaussian pdf). Thus, S ∼ R(m, σ) and S0 ∼ R(m0 , σ0 ) are the
Rician pdfs of the measured magnitude signals.
For the simulations, a set of values for m, m0 , σ, σ0 are selected. Also, each realization
of the random variables can be averaged simulating the number of excitations (NEX) in
MRI experiments. 106 realizations of S and S0 are generated to compute the realizations
of E. In the first simulation, the distribution of E is fitted to a Rician pdf by the method
of moments (matching the expected value and the variance) to estimate mE and σE for
the pdf of E. In the second simulation, S0 is considered a fixed non-random variable
equal to m0 under the assumption m0 >> σ0 and the E is regenerated and fitted to a
Rician pdf by the method of moments to estimate mE and σE for the pdf of E. The
distribution of E from both simulations are also fitted to Gaussian pdfs by the method
of moments to estimate mE and σE for the pdf of E for the Gaussian fit.
For both simulations, m0 = 697 and σ0 = 100. These values are estimated from the
T2-weighted MR signals for the cervical spinal cord region in the white matter tract
voxels for one subject (imaging protocol is given in the next chapter) based on techniques
described in [124]. In ADTI or DTI model, the range of the normalized MR signal
values is limited by the maximum and minimum diffusivities for a fixed b-factor. The
intermediate values of the signal are obtained due to variation in the diffusion gradient
direction. The limits of m can be calculated using, m = m0 exp(−bD), where b is the

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b-factor and D is the diffusivity. From previous studies, two values for the diffusivity D,
1.62 × 10−3 mm2 s−1 and 0.1485 × 10−3 mm2 s−1 were selected. This gives the range of
m as m ∈ {0.1979m0 , 0.8620m0}. Both simulations are performed at 10 values in this
range of m values. Each of the simulations are repeated for NEX = 2 and 4.

6.1.2

Results
1

Rician fit (S/S0)

Fitted E

Gaussian fit (S/S0)
Rician fit (S/m0)

0.5

0
0
(a)

Gaussian fit (S/m0)

0.5
True E

1

1

Rician fit (S/S0)

Fitted E

Gaussian fit (S/S0)
0.5

0
0
(b)

Rician fit (S/m0)
Gaussian fit (S/m0)

0.5
True E

1

Figure 6.1. Comparison of the estimated and true values of the normalized MR signal (E)
for the cases of Rician and Gaussian fit to the distribution of S/S0 and S/m0 . Simulations
are run at (a) NEX = 2 and (b) NEX = 4.
Fig 6.1 shows the comparison of the estimated and true values of the normalized MR
signal (E) for the cases of Rician and Gaussian fits to the distribution of S/S0 and
S/m0 . For each case of NEX, the mean and the standard deviation (SD) of the relative
difference of the estimated normalized MR signal (E) with respect to the true value of E,

104

i.e., E −(m/m0 ), are calculated. For NEX = 2, the mean and the SD of relative difference
are (−1.02±0.75)×10−3 (for the Rician fit of S/S0 ), (1.52±0.62)×10−2 (for the Gaussian
fit of S/S0 ), (−2.89±6.89)×10−5 (for the Rician fit of S/m0 ) and (1.23±0.71)×10−2 (for
the Gaussian fit of S/m0 ). For NEX = 4, the mean and the SD of relative difference are
(−2.27 ± 1.26) × 10−4 (for the Rician fit of S/S0 ), (7.42 ± 2.87) × 10−3 (for the Gaussian
fit of S/S0 ), (−4.49 ± 7.05) × 10−5 (for the Rician fit of S/m0 ) and (6.02 ± 3.38) × 10−3
(for the Gaussian fit of S/m0 ). It is clear from these results that the Rician fit is better
than the Gaussian fit of S/S0 as the Gaussian fit is more biased as seen from the mean
values of the relative difference. A similar trend is seen for the case of Rician fit of S/m0
compared to the Gaussian fit. The bias reduces with increasing NEX from 2 to 4.

σE of fitted pdf

0.14

Rician fit (S/S0)
Gaussian fit (S/S0)
Rician fit (S/m0)

0.12

Gaussian fit (S/m0)
0.1
0.2 0.4 0.6 0.8
True normalized MR signal (E)

(a)

σE of fitted pdf

Rician fit (S/S0)
0.09

Gaussian fit (S/S0)
Rician fit (S/m0)
Gaussian fit (S/m0)

0.08

0.07
(b)

0.2
0.4
0.6
0.8
True normalized MR signal (E)

ˆ
Figure 6.2. Variation of the estimated σE of the noisy normalized MR signal (E) for
the cases of Rician and Gaussian fits to the distributions of S/S0 and S/m0 respectively.
Simulations are run at (a) NEX = 2 and (b) NEX = 4.

105

Fig. 6.2 shows the variation of the estimated σE for the noisy normalized MR signal for
the cases of Rician and Gaussian fits to the distributions of S/S0 and S/m0 respectively
at two values of NEX. These results indicate that while using the distribution of S/S0
for the noisy normalized MR signal, the estimated σE increases with respect to the value
of the true normalized MR signal E. This is true for both the Rician fit and Gaussian fit
to the S/S0 distributions. The Gaussian fit underestimates σE compared to the Rician
fit of S/S0 in general and more so at lower signal levels. For the NEX = 4 case, σE is
further decreased indicating that more averaging could reduce the dependence of σE on
E. However, when assuming m0 >> σ0 such that S/S0 can be approximated to S/m0 ,
the estimated σE for the Rician fit does not change with the true normalized MR signal.
However, for the Gaussian fit of the distribution of S/m0 , the σE parameter is again
underestimated, the effect becoming worse at lower signal levels. In this work, I used
the S/m0 distribution and select the value of σE = 0.1 (for NEX = 2 case) based on
the result of these simulations. However, in order to use the S/S0 distribution and fit a
Rician distribution to it, a signal-dependent σE has to be used. The covariance matrix
definition in the CRLB formulation has to be corrected to incorporate this effect.

6.1.3

Discussion

S and S0 are both signals with a Rician pdf since these are obtained from magnitude
MR images. The exact nature of the distribution of S/S0 has not been explored in these
simulations. However, the effect of fitting the distribution of S/S0 to a Rician and a
Gaussian pdf has been demonstrated. While fitting these distributions to that of S/S0
results in the dependence of the σE on the true normalized MR signal, the estimated
value of E is largely unaffected since the bias shown in the estimates is negligible. The
dependence of σE on E is further reduced by increasing NEX during signal acquisition.
In this work, it is assumed that m0 >> σ0 such that S/S0 is approximately equal to
S/m0 and the Rician fit of this distribution is more appropriate. The value of m0 is
measured by a single acquisition of S0 at NEX = 2. As shown in the simulations, using
the assumption m0 >> σ0 , σE can be assumed constant with respect to E. Thus, in

106

the CRLB formulation, the covariance matrix (Σ) of the measurements of the normalized
MR signal (E) can be assumed to be σE I (i.e., Σ = σE I), where I is an identity matrix of
size N × N, with N being the number of measurements (equal to the number of diffusion
gradient directions in the DTI experiment). However, using this assumption, the CRLB
will underestimate the bound of the covariance of parameter estimation. To correct the
CRLB formulation, the covariance matrix of the measurements can be redefined as a
diagonal matrix to include the dependence of σE on E, i.e., ijth term of the covariance
matrix becomes, [Σ]ij = {σEi for i = j, 0 otherwise}.

6.2

Performance of Estimators

Different estimators selected to perform the parameter estimation are evaluated for precision (uncertainty) and accuracy (bias) in the estimation of the model parameters. The
selection of the estimator is essential since the performance of the model parameter estimation is determined by the estimator although the performance can be predicted analytically using the CRLB formulation. The CRLB gives a bound on the estimation
uncertainty which is attained when the estimator is minimum variance, unbiased and
efficient [109]. However, this is only true asymptotically (for infinite signal samples and
infinitesimally small tolerance settings of the estimator). In reality, a finite tolerance
setting is set for the convergence to the final solution and also a finite number of signal samples are available for the parameter estimation. Thus, estimator performance in
terms of its precision and accuracy needs to be evaluated by means, such as Monte Carlo
simulations and proper settings for tolerances has to be made before using the estimator
for the analysis of experimental data.
DTI data being MRI magnitude data always contain Rician noise. However, if a high
SNR is assumed (SNR > 5), the Gaussian approximation is reasonable. This approximation simplifies the analysis significantly and hence is more popular although inaccurate.
In the simulations, different scenarios of signal and noise models are considered and the
corresponding analysis is presented for them. In this section, the performance of different

107

estimators, namely least-squares (LS), least-squares with bias correction (LSC) and maximum likelihood estimator (MLE), are evaluated in terms of its uncertainty (precision)
and the bias (accuracy) in the estimation of diffusion model parameters.
For the estimator performance evaluation, ADTI signals (using a particular gradient
scheme) are injected with noise (Rician or Gaussian) and a number of trials are simulated
where in each trial the parameter estimation is performed by different estimators. After
the estimation, the estimation covariance matrix and its determinant are computed. Also
the bias is calculated from the true parameter values. A total of 20000 realizations of
the ADTI signal are generated for each set of parameter values. The following parameter
values are used for the simulation: ADTI diffusion model parameters: D = 1.62 ×
10−3 mm2 s−1 , D⊥ = 0.148 × 10−3 mm2 s−1 , φF = 0◦ , θF = 90◦ , σ = 0.1. FA =

0.9, MD = 0.638 × 10−3 mm2 s−1 Experimental settings are : b = 1000 s mm−2 , N = 30.
Two gradient schemes are used: MF30 and OPT30. OPT30 is the optimized gradient
scheme obtained at the same parameter values and a cone angle of Λ = 35◦ . Same b-

factor and N are used for the optimization. Note that the θF is set to 90◦ since at 0◦ , the
distribution of θF is one-sided and non-symmetric. Thus, to display the full distribution
of θF , the fiber is directed towards the +X axis instead of the +Z axis. The estimators
were: LS, LSC and MLE (for the Rician noise case) and LS only (for the Gaussian
noise case). For each estimator, the settings are: maximum number of iterations = 105 ,
tolerance in variable values = 10−6 , use Levenberg-Marquardt algorithm. The purpose
of this simulation exercise is to study the estimation performance of each estimator in
terms of the uncertainty (standard deviation in the model parameter estimates), bias
(mean signed difference (MSD)) and overall uncertainty (hypervolume of uncertainty =
determinant of covariance matrix).

6.2.1

Rician noise

For the Rician noise case, noise is injected in the complex signal and ADTI signal is the
magnitude of the noisy complex signal. Although the MLE seems to be the most suitable
estimator for this noise model, use of LS and LSC is also considered due to their popularity

108

in usage. Tables 6.1, 6.2 and 6.3 show the estimation results (parameter estimates, the
ˆ
ˆ
mean signed difference (MSD = mean(β − β), where β and β are the estimated and true
values of the model parameters) and the hypervolume of uncertainty (

det(Σ)) for the

three estimators and for both MF30 and OPT30 gradient schemes. The performance of
MLE is the best in terms of bias correction in the Rician noise injected data. Although,
MLE does not have the lowest hypervolume of uncertainty, which is obtained for the LS
case. This is because the estimator cost function in LS is the square of fitting error and
it will give the better result for covariance matrix. However, there is more bias in the LS
case as compared to MLE case. LSC estimator has better bias correction as compared
to LS, but is worse in the uncertainty. Its performance is somewhat in between LS and
MLE. Based on these simulations, MLE is most suitable for Rician noise based data since
it can achieve comparable precision with higher accuracy.
Table 6.1. Simulation results for the least-squares (LS) estimator performance assuming
Rician noise model
True

β
D

(×10−3

mm2 s−1 )

D⊥ (×10−3 mm2 s−1 )
θF (◦ )
φ F (◦ )
Hypervolume (× 10−11)

1.620
0.149
90.000
0.000

MF30
Est.
1.564 ± 0.125
0.143 ± 0.037
89.982 ± 2.775
0.025 ± 2.741
0.970

MSD
-0.057
-0.006
-0.018
0.025

OPT30
Est.
MSD
1.571 ± 0.133 -0.049
0.142 ± 0.038
89.985 ± 2.636
-0.008 ± 2.647
0.931

-0.006
-0.015
-0.008

MSD is the mean signed difference between estimates and the true values, MSD
ˆ
ˆ
= mean(β − β), where β and β are the estimated and true values of the model
√
parameters. Hypervolume = detΣ, where Σ is covariance matrix of estimates.

6.2.2

Gaussian noise

For the Gaussian noise case, the noise is additive and zero mean, σ 2 variance. So, it should
not create any bias in the estimates. Also, under this case, the least-squares estimator
is the same as the maximum likelihood estimator [109]. Hence, for this case, only the
least-squares estimator performance in shown. The Gaussian approximation holds good
for SNR > 5 [114]. This is not always observed in DTI data. However, for other MRI

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Table 6.2. Simulation results for the least-squares with bias correction (LSC) estimator
performance assuming Rician noise model
True

β
D

(×10−3

mm2 s−1 )

D⊥ (×10−3 mm2 s−1 )
θF (◦ )
φ F (◦ )
Hypervolume (× 10−11)

MF30
Est.

OPT30
Est.

1.620

1.636 ± 0.136

0.016

1.637 ± 0.144

0.017

0.149
90.000
0.000

0.146 ± 0.038
89.986 ± 2.780
0.028 ± 2.742
1.078

-0.003
-0.014
0.028

0.146 ± 0.038
89.984 ± 2.647
-0.007 ± 2.650
1.034

-0.003
-0.016
-0.007

MSD

MSD

MSD is the mean signed difference between estimates and the true values, MSD
ˆ
ˆ
= mean(β − β), where β and β are the estimated and true values of the model
√
parameters. Hypervolume = detΣ, where Σ is covariance matrix of estimates.
Table 6.3. Simulation results for the maximum likelihood (ML) estimator performance
assuming Rician noise model
β

True

MF30
Est.

OPT30
Est.

D (×10−3 mm2 s−1 )

1.620

0.005

1.627 ± 0.141

0.007

D⊥ (×10−3 mm2 s−1 )
θF (◦ )
φ F (◦ )
Hypervolume (× 10−11)

1.626 ± 0.140

0.149
90.000
0.000

0.145 ± 0.038
89.986 ± 2.781
-0.032 ± 2.768
1.129

-0.003
-0.014
-0.032

0.145 ± 0.038
89.998 ± 2.670
0.007 ± 2.634
1.027

-0.003
-0.002
0.007

MSD

MSD

MSD is the mean signed difference between estimates and the true values, MSD
ˆ
ˆ
= mean(β − β), where β and β are the estimated and true values of the model
√
parameters. Hypervolume = detΣ, where Σ is covariance matrix of estimates.
based analysis where the image has better SNR, Gaussian approximation is commonly
used.

6.3

Validation of performance curves

Performance curves are validated by Monte Carlo simulations for the performance of
the optimized 30-direction gradient scheme as well as the standard MF30 scheme by estimating the diffusion model parameters using different estimators and computing the
covariance and its determinant. This process is repeated for a range of fiber orientation angles and the performance curves based on estimated uncertainties are generated. These are finally compared with the curves generated by CRLB formulation.

110

Table 6.4. Simulation results for the least-squares (LS) estimator performance assuming
Gaussian noise model
β

True

MF30
Est.

D (×10−3 mm2 s−1 )

1.620

0.014

1.636 ± 0.140

0.016

D⊥ (×10−3 mm2 s−1 )
θF (◦ )
φ F (◦ )
Hypervolume (× 10−11)

1.634 ± 0.134

0.149
90.000
0.000

0.146 ± 0.038
90.020 ± 2.747
0.001 ± 2.737
1.042

-0.003
0.020
0.001

0.146 ± 0.038
89.987 ± 2.621
-0.012 ± 2.625
0.993

-0.002
-0.013
-0.012

MSD

OPT30
Est.

MSD

MSD is the mean signed difference between estimates and the true values, MSD
ˆ
ˆ
= mean(β − β), where β and β are the estimated and true values of the model
√
parameters. Hypervolume = detΣ, where Σ is covariance matrix of estimates.
For each simulation for each fiber angle, 20000 realizations of the ADTI signal under
noise level of σ = 0.1 (Gaussian and Rician) are generated using both standard and
optimized gradient schemes. The model parameter values used in the simulation are:
D 0 = 1.62 × 10−3 mm2 s−1 , D⊥0 = 0.148 × 10−3 mm2 s−1 , θF = [90 − 25]◦ in steps of
5◦ and φF = 0◦ . The optimized gradient scheme was computed at the same parameter

values as mentioned except θF = 90◦ . Thus, in terms of the angular deviation from the
+X axis (θF = 90◦ ), the range of α = [0 − 65]◦ .
To verify the robustness of the scheme, the simulations were run at different fiber orientations and hence angular deviations (α) from the original fiber orientation of (90◦ , 0◦ )
(Fig. 6.3). Three estimators are compared, namely, LS, LSC and MLE . The simulations
were run to verify that the predicted improvement in performance shown by the CRLB
formulation is also observed in practice with the selected estimators. Also by running
the simulation at different angular deviations the robustness of the scheme w.r.t. angular
deviation (α) is verified. Finally, the results would also show that predicted performance
relate well with the estimated performance and thus to analyze a scheme, a prediction is
good enough instead of performing lengthy Monte Carlo simulations.
The prediction represents a bound on the cost function. In practice, for the simulations,
√
the estimated hypervolume ( detΣ) would be different from the predicted bound. This
difference will depend on the estimator tolerance settings and the number of realizations
of the signal. The smaller the tolerance and the larger the number of realizations, the

111

closer will be the estimation to the CRLB prediction. It will also depend on the choice
of the estimator since estimation will approach the predicted bound asymptotically when
the estimator is a minimum variance, unbiased and efficient estimator. But, a similar
trend is visible in the estimated hypervolume as compared to the hypervolume predicted

1.1
1
0.9
0

20

Hypervolume ratio

(a)

α (°)

40

1
0.95

Pred.
Est.
α (°)

1.05
1
0.95

Pred.
Est.
20

(b)

1.05

20

1.1

0.9
0

60

1.1

0.9
0
(c)

Hypervolume ratio

Pred.
Est.

Hypervolume ratio

Hypervolume ratio

from the CRLB formulation. These are observed in the Fig. 6.3.

40

(d)

40

60

1.1
1.05
1
0.95
0.9
0

60

α (°)

Pred.
Est.
20

α (°)

40

60

Figure 6.3. Variation of ratio of hypervolume of uncertainty (DOP T /DM F , D =
det (ΣCR )) for ADTI model from Monte Carlo simulations w.r.t. angular deviation
(α) at σ = 0.1, Λ = 35◦ . Gaussian case with LS estimator (a) and Rician case with LS
(b), LSC (c) and ML (d) estimators.
In Fig. 6.3, the cases of Gaussian and Rician noise models and the effect of optimization
on the hypervolume of uncertainty has been demonstrated. For the Gaussian case (Fig.
6.3(a)), the least-squares (LS) estimator has been used and it is seen that the prediction
and estimation correlate well. For the Rician noise case (Fig. 6.3(b–d)), the performance
is compared for the least-squares estimator(LS) with that of least-squares with noise
correction (LSC) and maximum likelihood (ML) estimators. Under similar tolerance
settings for each of the estimators, it is observed that MLE has performed well for the
optimized scheme. Additionally, it estimates the expected values of the parameters better
than LSC estimator since it models noise directly into the estimator as has been shown

112

in the previous section.

6.4

Simulation for effect of b-factor, N and Λ in ADTI
model

Via simulations, the effect of various parameters, including the b-factor, the number
of gradient directions (N) and the cone angle (Λ) of fiber directions, on the overall
uncertainty of ADTI parameters estimation was investigated. Although the focus is
on the optimization of the diffusion gradient directions while keeping b-factor and the
number of gradient directions fixed and assuming a certain prior cone angle (Λ) for the
fiber directions, these simulations indicates the effect of other non-optimized experimental
parameters.
For each simulation, a particular setting of b-factor, N, Λ and the gradient schemes
are used and multiple trials are simulated. In each trial, the diffusion model parameters
(D , D⊥ , θF , φF ) and the noise level (σ) are chosen from a distribution and the cost
function based on the Rician CRLB and the variance of FA are computed based on the
chosen model parameters. The distribution from which model parameters are chosen
incorporate spatial (voxel-to-voxel) variation observed in biological tissues. Finally, after
the simulation is completed, performance indices are computed over all trials for each
simulation.

6.4.1

Performance indices

Certain indices are defined to assess whether the optimized gradient scheme worked better
than a standard (commonly used) gradient scheme, such as MF30 scheme [22]. These are
defined in the following:
1) µr1 : the mean ratio of hypervolume of uncertainties in successful trials. µr1 =
mean(r1), r1 = DOP T /DM F , where D =

det ΣCR at each successful trial (r1 < 1)

and OPT indicates the optimized scheme and MF indicates the MF-based scheme used as
a reference for normalization. Since µr1 is computed in successful trials only, it is always
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in the range [0, 1]. A lower µr1 indicates better performance.
2) PS : percentage success rate denotes the number of trials (in percentage) out of
total trials where the normalized hypervolume is less than unity (r1 < 1).
3) eSNR : effective SNR is defined as the ratio of the mean of all diffusion-weighted
signals calculated in all trials over the root mean squared noise values provided in all
trials. Since there are different diffusion gradient directions, there are different diffusionweighting (and signal levels) in the DTI data. Thus, an effective signal level (averaging
all diffusion-weighted signals) is used to compute the eSNR. The eSNR can indicate how
effectively the diffusion gradients sample the gradient space.
2
4) σF A : variance of FA is computed using the propagation of error theory [125]. It is

computed as a function of the variances of the diffusivities (D and D⊥ ) which are given
by the Rician CRLB. It is normalized with respect to the corresponding value for the
reference scheme.

6.4.2

Simulation parameters

The following diffusion model parameters and noise levels were used in the simulations
(percentage variation are shown in parentheses): D = 1.6204 ± 0.081 × 10−3 mm2
s−1 (5%) ; D⊥ = 0.14852 ± 0.0074 × 10−3 mm2 s−1 (5%) ; (θF , φF ) = (0◦ , 0◦ ); σ =
0.1 ± 0.005 (5%); Rician CRLB; Number of trials = 10000. Other constraints include:
0.6 ≤ FA ≤ 1 and MD ≤ 1.2 × 10−3 mm2 s−1 . All varying parameters are uniformly
distributed about their mean values.
D = (1.6204 ± 0.081) × 10−3 mm2 s−1 and D⊥ = (0.14852 ± 0.0074) × 10−3 mm2 s−1
was used for the simulation results. These values correspond to high FA voxels (average
FA = 0.9) in the white matter tracts between the C1 and C2 levels of the cervical spinal
cord. The values of the diffusivity parameters are based on a preliminary DTI data obtained using a standard protocol (b = 1000 s mm−2 and MF30 gradient scheme). Similar
values (λ1 = 1.7 × 10−3 mm2 s−1 , λ2 = λ3 = 0.2 × 10−3 mm2 s−1 and a corresponding
FA = 0.87) have been used in previous simulation experiments in the works of Alexander
[23], Peng et al. [21] and Gao et al. [39]. The simulations in this work were performed

114

using diffusivity values with high FA since these values are representative of the white
matter tract regions and not are contaminated by the gray matter or CSF regions.

6.4.3

Effect of b-factor

Simulations were run using diffusion model parameters defined previously and varying
the b-factor between b = 1000 s mm−2 and 4500 s mm−2 (b = [1000, 1200, 1500, 2000,
2500, 3000, 3500, 4000, 4500] s mm−2 ). The cone angle, Λ = 35◦ and number of diffusion
gradients, N = 30 were used for all the cases. For each case of b-factor, an optimized
gradient scheme was generated and its performance indices were computed using MF30
at b = 1000 s mm−2 as the normalization reference scheme. Similarly, simulations were
run with MF30 at different b-factors and the indices were computed with respect to the
same reference scheme (i.e., MF30 at b = 1000 s mm−2 ).
For both the optimized and MF30 schemes, the performance (based on indices µr1 and
PS ) only improves within a certain range of b-factor (Fig. 6.4 (a) and (b)). However,
for the optimized case, the range of high performance is broader than the MF30 case.
A similar trend is seen for variance of FA. This effect was also demonstrated by Gao
et al. [39] where they have showed that when a cone angle of fiber distribution is used
for optimizing the diffusion gradient, the b-factor range increased as compared to an
un-optimized (MF-based) scheme.
Generally, changes in the b-factor is restricted by the MRI scanner gradient strength
limitation and the SNR. It can be clearly seen from the simulation (Fig. 6.4 (c)) that the
effective SNR decreases with the increase of the b-factor. However, the optimized scheme
shows better SNR compared to MF30 at higher b-factors. A commonly used b-factor in
DTI experiment is 1000 s mm−2 (for example, in [62]) and this was used in the ADTI
experiments. The optimized gradient scheme depends on the selection of the b-factor and
the performance of the optimized scheme varies with the changes to the b-factor as shown
in these simulations.

115

100

MF
OPT
PS ( % )

1.5

µr1

1
0.5
1000 2000 3000 4000
b ( s mm−2 )

(a)

7

(b)

2

MF
OPT

1.5

5
4
3

(c)

MF
OPT
0
1000 2000 3000 4000
b ( s mm−2 )

σ2
FA

eSNR

6

50

MF
OPT

1
0.5

2
1000 2000 3000 4000
b ( s mm−2 )

(d)

0
1000 2000 3000 4000
b ( s mm−2 )

Figure 6.4. Effect of varying b-factor on the diffusion gradient optimization. Performance
indices (a) normalized hypervolume for successful trials (µr1 ), (b) percentage success rate
2
(PS ), (c) effective SNR (eSNR) and (d) normalized variance of FA (σF A ) are shown for
different b-factors and for both the optimized gradient schemes (OPT) and the MF30
scheme (MF).

6.4.4

Effect of number of diffusion gradient directions (N )

Studies based on DTI showing the effect of number of gradient directions on noise propagation indicate that a higher number of gradient directions is preferred over signal averaging when characterizing noise sensitivity for a DTI gradient scheme [125]. In this
section, the effect of number of diffusion gradient directions on the overall performance
of the parameter estimation is investigated under the Rician CRLB formulation.
Simulations were run using diffusion model parameters defined previously and a varying
number of diffusion gradients (N) from 20 to 80 (N = [20, 30, 40, 50, 60, 80]). For each case
of N, an optimized schemes was generated and its performance indices were computed
with MF15 (N=15) as a reference. Similarly, the MF-based schemes were simulated at

116

various N values and MF15 was used as the reference. Two b-factors, b = 1000 s mm−2
and 2500 s mm−2 and a cone angle of Λ = 35◦ in all cases were used.

0.6

0.6

MF
OPT
µr1

0.4

µr1

0.4
0.2
0
20

0.2

40

(a)

N

60

0
20

80

N

60

80

6
eSNR

8

6
eSNR

40

(b)

8

4
2
0
20

40

(c)

60

N

2

MF
OPT
80

0
20
1

MF
OPT

40

40

(d)

0.5

0
20

4

σ2
FA

σ2
FA

1

(e)

MF
OPT

N

60

(f)

MF
OPT

0.5

0
20

80

60

N

MF
OPT
80

40

N

60

80

Figure 6.5. Effect of varying number of diffusion gradients (N) on the diffusion gradient
optimization. Normalized hypervolume for successful trials (µr1 ) ((a) and (b)), effective
2
SNR (eSNR) ((c) and (d)) and normalized variance of FA (σF A ) ((e) and (f)) are shown
for different N and for both the optimized gradient schemes (OPT) and the MF-based
schemes (MF). Figures (a), (c) and (e) are for b = 1000 s mm−2 , whereas figures (b), (d)
and (f) are for b = 2500 s mm−2 .
Based on Fig. 6.5, µr1 in general decreases with the increase in N for both the optimized

117

and MF30 schemes. This is expected since more samples lead to a lower CRLB. But,
at a higher b-factor (b = 2500 s mm−2 ), better performances are achieved with a lower
number of gradients as compared to the corresponding MF case. This result suggests
that the number of gradient directions and equivalently the scan time can be reduced
to obtain the same performance as the MF scheme. The effective SNR is steady with
respect to changes in N. Normalized variance of FA correlates well with µr1 as the
number of diffusion gradients increases. N = 30 was selected for the ADTI experiments.
This provides good performance with µr1 and also eSNR. This is also a commonly used
number of diffusion gradients reported in other studies.

6.4.5

Effect of Cone angle (Λ)

In the diffusion gradient scheme optimization, an a priori range of fiber orientation angles is assumed as defined by the cone angle (Λ). It signifies the uncertainty of fiber
orientations in the ROI voxels. This uncertainty can vary from a small angle (Λ = 10◦ )
to a completely uncertain case (Λ = 90◦ ). The performance will vary depending on the
choice of the Λ parameter during optimization.
For the simulations, diffusion model parameters defined previously were used along with
Λ varying from 10◦ to 90◦ (Λ = [10 20 35 40 60 90]◦ ). A b-factor of 1000 s mm−2 and N
= 30 are used for all simulations. For each value of Λ, an optimized scheme is generated
and performance indices are computed with MF30 as a reference. The normalization
reference was MF30 with b = 1000 s mm−2 . As shown in Fig. 6.6, µr1 increases with the
cone angle. Both PS and the effective SNR decrease with the increase in the cone angle.
But these do not change for MF30. The changes in the performance indices (µr1 and PS )
reach steady state after approximately a cone angle of 40◦ and remains approximately
steady till 90◦ . At completely uncertain case of Λ = 90◦ , the optimized gradient scheme
performs similarly as MF30. The variance of FA shows an opposite trend as compared to
µr1 . It decreases with the increase in the cone angle (Λ) till about 40◦ where it reaches
a plateau and then begins to increases. No changes are seen for MF30. This could be
explained from the fact that when smaller cone angles are used for gradient optimization,

118

the optimization can get too specific and reduces the uncertainty of the fiber direction
parameters (θF and φF ) and not diffusivities (D and D⊥ ). However, after Λ = 40◦ , this
effect is reduced. The variance of FA is a function of the variances of the diffusivities (D
and D⊥ ). The optimization need not provide a reduction of uncertainty in estimation of
both the angular parameters (θF and φF ) and the diffusivities (D and D⊥ ) concurrently.
However, it does provide a scheme with reduced overall uncertainty. This can be explained
since D-optimality uses determinant of the CRLB which is essentially the product of the
variances of the diffusion model parameters. Thus, although the determinant of CRLB
is reduced by optimization, all individual variances are not concurrently reduced.

µr1

PS ( % )

0.9

0.85

0.8

20

(a)

60
40

80

20

(b)

1.3

MF
OPT

5.8

40
60
Λ (°)

80

MF
OPT

1.2

6

1.1
1

5.6
20
(c)

80

σ2
FA

eSNR

6.2

MF
OPT
40
60
Λ (°)

MF
OPT

40
60
Λ (°)

80

20
(d)

40
60
Λ (°)

80

Figure 6.6. Effect of varying cone angle (Λ) on diffusion gradient optimization. Performance indices (a) normalized hypervolume for successful trials, (µr1 ), (b) percentage
2
success rate (PS ), (c) effective SNR (eSNR) and (d) normalized variance of FA (σF A ) are
shown for different Λ and for both the optimized gradient schemes (OPT) and the MF30
scheme (MF).

119

6.4.6

Selection of b-factor, N and Λ

The selection of the experimental settings for b-factor and the number of gradient directions (N) as well as the prior structural information in cone angle Λ is vital for the
performance of the gradient optimization procedure. As shown in the previous section, bfactor can have an optimal value for best estimation performance in uncertainty reduction
under the diffusion model parameter values used for the simulation. This observation was
exploited in the b-factor optimization in Chapter 5, section “b-factor optimization”. Due
to hardware constraints on the MRI scanner, the optimal b-factor need not be achievable.
Also, it is observed that higher b-factor, lower is the eSNR. Since the cost function (determinant of the CRLB matrix) does not include any SNR term directly, an optimization for
b-factor based on CRLB only need not have the best eSNR performance. Based on these
arguments, it is suggested that while performing b-factor optimization, an additional constraint on eSNR can be applied to the optimization problem (for example, eSNR > 4).
Also, MRI hardware limitations can also be incorporated as a limit on the values b-factor
can take during it’s optimization.
For the selection of the number of diffusion gradients, it is observed that the higher
the number of gradients, the better the estimation performance is in terms of estimation
uncertainty. However, more gradients requires a longer acquisition time. N = 30 is
used commonly and will be used for this work. Also, as shown before, similar estimation
performance can be achieved by fewer gradients when gradient optimization is performed.
This can be used to reduce the acquisition time.
The prior structural information as given by the cone angle (Λ) is best obtained from
a preliminary scan on each subject. A Λ = 35◦ is generally found in most subjects.
Although, smaller cone angle would mean lower uncertainty in the fiber orientation estimation, but performance of the diffusivities estimation is affected at this angle. On the
other hand, at higher cone angles, the uncertainty in fiber orientation is affected. Using
a preliminary DTI scan makes sure that the subject-to-subject variability is considered
during gradient scheme optimization and also that extremes in Λ are not selected, such
as Λ = 0◦ (completely certain fiber orientation) or Λ = 90◦ (completely uncertain fiber

120

orientation).

6.5

Comparison of performance indices for optimized gradient schemes

In this section, gradient schemes designed for the ADTI and the DTI diffusion model with
Rician noise and optimized for all parameters, diffusivities only and angular parameters
only cases (in Chapter 5, section 5.5) are evaluated in terms of the performance indices,
such as PS , µr1 and eSNR. Also, the CRLB of variance of individual diffusion model
parameters and FA, MD and α are compared in the evaluation of the gradient scheme.
A series of simulations are performed for the evaluation with each simulation consisting
of a number of trials. At each trial, the normalized MR signal and the sensitivities are
calculated based on the Rician CRLB which uses diffusion model parameter values and the
experimental settings (b-factor, gradient scheme). Diffusion model parameter values are
obtained from a uniform distribution with specified means and variances. This simulates
the spatial variation of the diffusion parameter values in a tissue. Experimental settings
are kept fixed within a simulation. Simulations are performed for both the OPT30 and
MF30 schemes and finally, for each simulation, the performance indices and variance
bounds are computed.
For the distribution of diffusion model parameter values, the mean values are specified
in Chapter 5, section 5.5 and the standard deviation for diffusivities is 5% of the mean.
The fiber angles are uniformly distributed within the cone angle. Also the noise standard deviation is 5% of central value which simulates the spatial variation of the noise
parameter, σ, within a tissue.
Three cases of gradient optimization are considered, namely, gradients optimized for all
parameters (“All”), for diffusivities only (“Diffusivities only”) and for angular parameters
(“Angles only”) for fiber orientation only.

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Table 6.5. Simulation results for the ADTI optimized gradient scheme for Rician noise
model
All
PS
74.3
1.1

µr1
0.959
0.995

Diffusivities Angles only
only
PS
µr1
PS
µr1
65.8 0.962 84.1 0.950
71.3 0.980 0.3 0.998

32.3

0.942

70.0

0.884

59.5

0.969

2
σF A

26.6

0.949

72.8

0.876

42.6

0.976

2
σM D
2
σθ
F
2
σφ
F
2
σα

4.3

0.999

21.4

0.956

0.0

0.0

99.8

0.899

7.2

0.975

71.4

0.949

56.0

0.932

7.1

0.957

76.9

0.931

65.0

0.925

6.1

0.958

56.8

0.946

HyperVol
2
σD
2
σD

6.5.1

⊥

ADTI model

In Table 6.5, the simulation results show the effect of using optimized gradients on the
variance of the individual diffusion model parameters, FA, MD and α using performance
indices. For the “All” parameters case, the overall reduction in the hypervolume of
uncertainty (as shown by a 74.3% success rate at 0.959 hypervolume ratio) also results
in the reduction of variance of the angular parameters and α. However, the effect on the
variance of diffusivities is not significant. The eSNRs are 5.737 for MF30 and 5.859 for
OPT30 schemes for this case. For the “Diffusivities only” case, the percentage success
is smaller than the “All” parameter case. However, the effect on the variance of D and
D⊥ is significant. Since the optimization cost function is to minimize the uncertainties
in the diffusivities only, the variances of angular parameters are not reduced sufficiently
by gradient optimization. The eSNRs are 5.736 for MF30 and 5.702 for OPT30 for
this case. The variance of FA is also reduced sufficiently. But, the variance of MD is
not reduced sufficiently. For the “Angles only” case, the reduction in the cost function
results in the reduction in the expected variances of θF and φF . This is an expected and
a desired result. Effect on the variances of the diffusivities is less significant which is also
expected. The eSNRs are 5.740 for MF30 and 5.872 for OPT30 for this case. Except for
the “Diffusivities only” case, the eSNR is always improved by gradient optimization.
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6.5.2

DTI model

Table 6.6. Simulation results for the DTI optimized gradient scheme for the Rician noise
model
All

HyperVol
2
σD
2
σD
2
σD

⊥1

⊥2
2
σF A
2
σM D
2
σθ
F
2
σφ
F
2
σα

PS
100.0
0.0

µr1
0.759
0.0

Diffusivities
only
PS
µr1
100.0 0.797
0.0
0.0

Angles only

100.0

0.806

100.0

0.768

81.3

0.957

100.0

0.816

100.0

0.688

100.0

0.814

28.1

0.976

25.6

0.952

40.9

0.955

0.0

0.0

0.0

0.0

40.6

0.965

85.9

0.910

78.2

0.809

98.1

0.872

89.5

0.911

53.9

0.937

68.1

0.958

79.8

0.915

55.7

0.874

92.8

0.921

PS
99.6
32.8

µr1
0.923
0.962

From the simulation results in Table 6.6, it is observed that while the hypervolume of
uncertainty is reduced with PS > 99% for all the cases, the reduction is not consistent in
all the diffusion model parameters, especially D and MD where the uncertainty is not
reduced for the “All” and the “Diffusivities only” cases compared to MF30. This can be
explained referring to the sensitivity plots in Figs. 5.4 – 5.7 where it is observed that the
majority of the high square sensitivity regions lie transverse to the fiber orientation (for
all parameters except D ) and as such the optimization framework selects the transverse
region predominantly. The high square sensitivity region for D lies towards the fiber
orientation which the framework does not sample. This effect is more pronounced in
the “All” parameters and “Diffusivities only” cases since here the majority of the higher
square sensitivity regions are transverse to the fiber orientation. Interestingly, in a counterintuitive sense, this effect is less observed for the “Angles only” case due to an overlap
in the high square sensitivity regions for the fiber angles (θF , φF ) and D (see Figs. 5.4
– 5.7). This effect was not significantly observed in the ADTI case since the high sensitivity regions are more overlapping than DTI and hence a common optimal region can
be obtained by optimization. Thus, using D-optimality criterion need not optimize the
123

gradients to improve the precision of all diffusion model parameters simultaneously, but
it will reach an overall optimal state w.r.t. the majority of the chosen model parameters.
The eSNR values were 4.735 for “All” case, 4.627 for “Diffusivities only” case and 4.485
for the “Angles only” case compared to 4.216 for MF30, indicating an improvement in
SNR.

124

CHAPTER 7
Spinal cord axisymmetric diffusion
tensor imaging
7.1

A validation study for gradient scheme optimization

In this section, a diffusion gradient optimization procedure is developed that is based on
D-optimality [33, 34] for the ADTI model which reduces the overall uncertainty in the
estimation of diffusion model parameters in the cervical spinal cord and brain stem region.
The optimized gradient scheme is designed to perform within a cone of fiber directions
with the mean fiber orientation being the axis of the cone. The cone angle is determined
by the a priori knowledge of the spinal cord structure obtained from preliminary DTI
experiments. The performance of the optimal scheme was compared with the MF-based
gradient scheme in terms of various performance indices defined in Chapter 6 previously
and also directly comparing the estimated standard deviations of different diffusion model
parameters. Also, the Rician noise model [23, 74] than Gaussian noise model has been
used to improve the estimation accuracy.

125

7.1.1

Experimental Protocol

To experimentally validate the optimization framework, MRI data was collected with the
diffusion gradient directions optimized for a cone of fiber directions with the b-factor and
the number of gradient directions fixed (as used in routine MRI experiments). This study
was approved by the Institutional Review Board (IRB) for conducting research on human
subjects. Five healthy subjects (4 males, 1 female, average age 29 years) participated
in this study and provided their signed IRB-approved informed consent. The following
steps were performed:
1) Preliminary scan to collect a priori information: A preliminary DTI scan was performed on the upper spinal cord and the brain stem region with a 15-direction MF-based
gradient scheme (MF15). The T2 and diffusion-weighted images were acquired using a
dual spin-echo EPI sequence on a 3T GE Signa HDx scanner (GE Healthcare, Waukesha,
WI), equipped with an 8-channel head coil with the following parameters: 22 contiguous
3-mm axial interleaved slices, TR = 7000 ms, TE = 77.4 ms, matrix size =128 x 128, FOV
= 16 cm x 16 cm, number of excitations = 1, parallel imaging acceleration factor = 2, b
= 1000 s mm−2 , 15 diffusion gradient directions (MF15) and scan time = 1 min 52 sec.
The subject exited the scanner and was scanned again after the optimized gradients were
calculated. For the post-processing of the preliminary data, MD, FA, the ADTI model
parameters (D , D⊥ , θF , φF ) and angular deviation (α) from the mean fiber orientation
were computed for each voxel. After visually locating the spinal cord from the T2 image,
an automatic ROI voxel selection was performed by applying the following thresholds to
extract the spinal cord tracts: FA ≥ 0.6 (for high anisotropy), MD ≤ 2.5 × 10−3mm2 s−1
(for removing CSF) and α corresponding to 80% of the distribution of fiber orientations
(to select majority of the tract voxels). Fig. 7.1 shows the process of selection of the
ROI voxels in the white matter tracts of the cervical spinal cord and brain stem regions.
For the ROI voxel extraction (shown in Fig. 7.1), a cuboidal region enclosing the spinal
cord below the C1 vertebral level and up to the C2 vertebral level was initially selected
manually. However, this region contains white matter tracts as well as the gray matter
in the spinal cord and CSF in the surroundings. To eliminate the gray matter and CSF

126

region voxels, an FA (FA > 0.6) and MD (MD < 2.5 ×10−3 mm2 s−1 ) based thresholding
was applied to the manually selected region. FA values of 0.6 or above have been previously reported [28, 69] in the cervical spinal cord region of healthy subjects. The high
FA threshold ensures minimal contamination of the ROI voxels with gray matter which
cover a significant portion of the spinal cord. The presence of gray matter or CSF in the
ROI voxels will result in lower overall FA estimation since gray matter or CSF regions are
inherently less anisotropic in terms of diffusion than the white matter tracts. Since the
main focus of the study is on normal healthy subjects, this thresholding method worked
well in extracting the white matter tracts and reducing contamination by gray matter
and CSF regions. Also, for the brain stem region, this method was applied to extract only
the healthy white matter tracts (shown in Fig. 7.1). Table 7.1 shows the mean values of
diffusivity and the cone (Λ) values used for the design of the optimized gradient table for
different subjects.
Table 7.1. Average diffusivities and cone angles from preliminary DTI experiment data
of five subjects
Subjects

D × 10−3

D⊥ × 10−3

1.819
1.494
1.388
1.791
1.909

0.233
0.159
0.176
0.178
0.321

(mm2 s−1 )

1
2
3
4
5

(mm2 s−1 )

Λ
(◦ )
35
35
35
35
40

2) Gradient scheme optimization: The optimized gradient schemes were designed offline based on the information obtained from the preliminary DTI experiment and solving
the cost function for robust optimization (minimizing det(ΣCR )). A b-factor = 1000 s
mm−2 and σ = 0.1 was selected for the procedure. The selection of noise level (σ) is
specific to the MRI scanner used since it depends on the state of the scanner. From
previous DTI studies not reported in this paper, σ was found to be approximately 0.1
and has been used in the gradient optimization. Fig. 7.2(a) shows the designed gradient
scheme for one of the subjects. The scheme for this subject was designed with Λ = 35◦
and the number of rings used in this design was 7 (P = 7, Nr ={2, 2, 2, 4, 4, 6, 10}).
127

(a)

60
I − S (mm)

I − S (mm)

60
40
20
0
0
(b)

20
40
R − L (mm)

(c)

20
40
R − L (mm)

60

20
40
R − L (mm)

60

60
I − S (mm)

I − S (mm)
(d)

20
0
0

60

60
40
20
0
0

40

20
40
R − L (mm)

40
20
0
0

60
(e)

Figure 7.1. (a) T1 coronal view of the cervical spinal cord and brain stem near the C1 C2 vertebral region. The boxed region is selected manually and further magnified in (b)
(e). (b) T2 coronal image showing the segmented spinal cord within the initial ROI in (a)
after removal of the CSF surrounding the spinal cord by MD thresholding. (c) T2 coronal
image showing the selected white matter ROI voxels in (b) after removal of gray matter
voxels by FA thresholding. Similarly, the brain stem region is first manually selected in
(a), and then thresholded in two stages, (d) and (e), to extract the ROI voxels.

128

θ(°)

0
30

0.6

60

0.4

90
−180

0.2
−90

(a)

0
90
φ sin θ ( ° )

180

θ(°)

0
30

0.6

60

0.4

90
−180
(b)

0.2
−90

0
90
φ sin θ ( ° )

180

Figure 7.2. Optimized gradient scheme (white circles) for 30 gradient directions on a 2D
opened hemisphere for cone angles (a) Λ = 35◦ (b) Λ = 90◦ (completely uncertain fiber
orientation case). Underlying grayscale image shows the normalized MR signal levels over
the hemisphere changing with respect to the diffusion gradient direction (θ, φ). Mean
fiber orientation is at (0◦ , 0◦ ).
The procedure took about 20 minutes for each subject.
3) Data collection and analysis with the optimized ADTI protocol: The main ADTI
experiment was conducted with the same MRI protocol as the preliminary scan except
that the 30-direction optimized diffusion gradient scheme (OPT30) was used and number
of excitations = 2 (for additional SNR improvement). The scan time per dataset was 7
min 21 sec. For comparison purpose, an equivalent dataset was acquired with the MF30
gradient scheme. Six data sets were acquired with each gradient scheme on each subject
to have sufficient data for covariance computation. For subject-wise post-processing,
the data sets were first linearly registered (using FLIRT in FSL [126]) to align the spinal
cords. Next, the automatic ROI selection was performed by applying thresholds described
in step 1 to identify the common voxels amongst the 12 data sets. The normalized MR
signal (E) was calculated by normalizing the diffusion-weighted MR signals with a nondiffusion-weighted (T2 -weighted) value for each ROI voxel.

129

7.1.2

Statistical Analysis

The sample size was increased from originally 6 data sets (for each gradient scheme) to
6000 data sets using a repetition bootstrap resampling technique [127]. Due to limitations
on collecting large amounts of data for each subject, bootstrap resampling techniques are
performed to compute statistics of a representative sample set of the subject data. Theoretically, if the subject were to be scanned indefinitely, only then the true values of
the parameters can be estimated. However, in reality it is not possible. Via computer
simulations, a large data set can be generated. But, in the case of DTI experiment,
large dataset can not be collected due to time constraints. Thus, only a representative
data set (for example, 6 data sets for each gradient scheme and each subject) is collected
and by resampling, a large dataset is generated which can be used to compute statistics,
such as the covariance matrix. The resampling technique in this work incorporated a
sampling with replacement within the original data sets. For each subject, 6 datasets
of 30-direction diffusion-weighted data were collected. Thus, for each diffusion-weighted
volume corresponding to a diffusion gradient direction, there are 6 volume data. During
repetition bootstrap, secondary datasets of 30-direction diffusion-weighted data are generated by selecting with replacement diffusion-weighted volumes from the 6 original data
set. This process can be repeated for 6000 times to generate the bootstrapped datasets.
Also, while performing the bootstrapping, the mean of the measured DTI signal from
the original datasets was preserved such that the sample mean of the final dataset (with
6000 data) was the same as the original dataset. This ensured that the technique did not
inject an artificial bias into the measured DTI signal.
The bootstrapped data from the common voxels (ROI voxels) were used for estimating
the diffusion parameters and their mean and covariances. Maximum likelihood estimator
(MLE) [117] using the Rician noise model was used for the diffusion parameter estimation.
The covariance matrix (for estimated model parameters), its determinant and square root
(hypervolume of uncertainty, for example , DOP T = det ΣOP T ) were computed at the
ROI voxels. The performance indices, PS and µr1 , and the eSNR were computed over
the ROI voxels for each subject data.

130

Unpaired t-tests (left-tailed and assuming unequal variance) were performed for comparing the estimated hypervolume of uncertainty and the standard deviations (root-meansquared (RMS)) of the model parameters (D , D⊥ , θF and φF ) and FA over the ROI
voxels. The mean values of diffusivities (D and D⊥ ) and the angular deviations (α) and
the standard deviations of model parameters and FA over ROI voxels are also reported.

7.1.3

Results for optimization using cone of fibers obtained
from a priori information

The reduction in estimation uncertainty is demonstrated in Fig. 7.3 through the distributions of the ratio of the hypervolumes of uncertainty for five subjects. DOP T 30 and
DM F 30 are the hypervolumes of uncertainty for the optimized and MF30 schemes respectively. For the estimation, the hypervolumes are computed from the estimated covariance
matrix using MLE. For the prediction, these correspond to the hypervolume bound defined by the Rician CRLB. Table 7.2 shows the estimation results on a subject-by-subject
basis and supports the following observations:
1) Reduction in uncertainty: More than 62% voxels demonstrate estimation uncertainty
reduction for all subjects as indicated in PS in Table 7.2. The mean ratio in successful
voxels, µr1 , is much less than unity in all cases showing improvement in overall uncertainty reduction. Table 7.2 also shows that the improvement in uncertainty is accurately
predicted based on the CRLB formulation. DOP T 30 is significantly less than DM F 30 on
three subjects and approach significance in two subjects.
2) Standard deviations of model parameters: For the angular model parameters, θF
and φF , the standard deviations (RMS) over the ROI voxels always show improvement
for the optimized ADTI protocol compared to MF30 scheme (Table 7.4). The t-test
shows that the results approach significance. The standard deviations for the diffusivities
and FA indicate improvement, but not consistently for all the cases (Table 7.5). The
optimization reduces the overall uncertainty but not necessarily the uncertainty for all
model parameters concurrently.
3) Mean of the estimates: The mean of the estimates of diffusivity values (D and
131

15

30

Voxel count

Voxel count

40

Est.
Pred.

20
10
0
0

1

(a)

2
3
DOPT30/DMF30

Est.
Pred.
5

0
0

4

1

(b)

25

2
3
DOPT30/DMF30

4

15
Voxel count

Voxel count

10

20
Est.
Pred.

15
10

10

Est.
Pred.

5

5
0
0
(c)

1

2
3
DOPT30/DMF30

0
0

4

1

(d)

2
3
DOPT30/DMF30

4

Voxel count

20
15

5
0
0

(e)

Est.
Pred.

10

1

2
3
DOPT30/DMF30

4

Figure 7.3. Region of interest (ROI) voxel distributions with respect to hypervolume
ratio (DOP T 30/DM F 30 ) for different subjects. DOP T 30 and DM F√ are square roots of
30
the determinants of covariance matrix of parameter estimation ( detΣ for estimation
(Est.) and detΣCR for prediction (Pred.)) for OPT30 and MF30 schemes respectively.
Figs. (a) to (e) correspond to subject number 1 to 5. Majority of the ROI voxels for each
subject are in the less than unity range indicating an overall uncertainty reduction in the
parameter estimation.
D⊥ ) and angular deviation (α) averaged over ROI voxels for the optimized and MFbased protocol are similar (Table 7.7). The mean % relative differences between these

132

two protocols (absolute difference divided by the mean) for all five cases in Table 7.7 are
3.52 % for D , 8.29 % for D⊥ and 7.47 % for the mean α. The effect of optimization
on the accuracies of the estimated diffusivities and angular deviation is less than 10%
with respect to the MF30 protocol and can be considered as not affecting the estimation
accuracy (or bias) sufficiently.
4) Signal-to-noise ratio: The optimized protocol has consistently higher eSNR as compared to the MF-based protocol as shown in Table 7.2, suggesting that the optimized
gradient scheme samples the gradient space better than the MF30 case with respect to
the eSNR.
Table 7.2. Performance comparison of the optimized scheme (OPT30) and MF30 from
subject data
√
Subjects Total voxels
PS (%)
µr1
D = detΣ × 10−11
Est. Pred. Est. Pred. MF30 OPT30 p-value
1
47
74.5 78.7 0.521 0.541 5.550
3.267
0.002
2
16
62.5 75.0 0.457 0.536 7.273
4.785
0.021
3
37
64.9 67.6 0.539 0.563 4.688
3.482
0.088
4
19
68.4 63.2 0.544 0.608 6.982
3.627
0.108
5
19
78.9 73.7 0.453 0.551 13.969 6.779
0.029
D = hypervolume of uncertainty; PS = 100 × Voxels (DOP T 30 /DMF 30 < 1) / Total voxels; µr1 =
Mean (DOP T 30 /DMF 30 ) in successful voxels; eSNR = effective SNR

Table 7.3. eSNR comparison of the optimized scheme (OPT30) and MF30 from DTI
subject data
Subjects
1
2
3
4
5

7.1.4

eSNR
MF30 OPT30
3.780
4.286
3.709
3.934
4.106
4.363
3.945
4.340
3.401
3.828

Results for optimization without a priori knowledge of
cone of fibers

Healthy spinal cord tract fibers tend to be oriented within a narrow angular range. However, pathological spinal cord might have more uncertain fiber orientations. The uncertain
133

Table 7.4. Comparison of standard deviations (SD) of angular parameters at ROI voxels
from subject data using optimized gradient scheme (OPT30) and MF30
Subjects
MF30

1
2
3
4
5

3.234
3.800
3.333
3.320
4.341

SD of θF (◦ )
OPT30 p-value
MF30
(SDOP T 30 <
SDM F 30 )
2.774 0.0006
3.262
3.360 0.0274
3.864
2.976 0.0027
3.334
2.961 0.0998
3.360
3.746 0.0387
4.281

SD of φF (◦ )
OPT30 p-value
(SDOP T 30 <
SDM F 30 )
2.758 0.0001
3.289 0.0089
3.115 0.0430
2.965 0.0628
3.851 0.1153

Table 7.5. Comparison of standard deviations (SD) of diffusivities at ROI voxels from
subject data using optimized gradient scheme (OPT30) and MF30
SD of D × 10−3

Subjects
MF30

1
2
3
4
5

0.315
0.318
0.269
0.330
0.387

(mm2 s−1 )
OPT30 p-value
MF30
(SDOP T 30 <
SDM F 30 )
0.291 0.0836
0.064
0.278 0.1035
0.066
0.239 0.0809
0.060
0.278 0.1796
0.064
0.295 0.0449
0.071

SD of D⊥ × 10−3

(mm2 s−1 )
OPT30 p-value
(SDOP T 30 <
SDM F 30 )
0.058 0.0047
0.066 0.6012
0.059 0.3832
0.055 0.0757
0.058 0.0067

Table 7.6. Comparison of standard deviations (SD) of FA at ROI voxels from subject
data using optimized gradient scheme (OPT30) and MF30
Subjects
MF30

1
2
3
4
5

0.046
0.054
0.045
0.046
0.057

SD of FA
OPT30 p-value
(SDOP T 30 <
SDM F 30 )
0.039 0.0003
0.051 0.2514
0.046 0.6365
0.045 0.4522
0.047 0.0154

fiber orientation scenario was experimentally simulated by first generating the optimized
gradient scheme with Λ = 90◦ (OPT30-90) for a healthy subject (subject 1). Next, the
ADTI experiment was conducted using the OPT30-90 and MF30 schemes and the ROI
voxels (62 voxels) were selected in the brain stem region (beyond the spinal cord region,

134

Table 7.7. Comparison of mean values of diffusivities and angular deviation at ROI voxels
from subject data using optimized gradient scheme (OPT30) and MF30
Subjects
1
2
3
4
5

D × 10−3 mm2 s−1

MF30
2.334
1.989
2.017
2.282
2.059

OPT30
2.354
2.103
2.009
2.127
1.983

D⊥ × 10−3 mm2 s−1

MF30
0.269
0.224
0.217
0.283
0.223

OPT30
0.259
0.256
0.250
0.304
0.230

α (◦ )
MF30
4.90 ± 1.00
6.70 ± 1.30
5.00 ± 1.00
6.90 ± 1.60
8.10 ± 1.10

OPT30
4.50 ± 1.30
7.20 ± 1.30
5.80 ± 1.70
7.30 ± 3.40
8.20 ± 1.90

see Fig. 7.1) where the mean angular deviation of fibers are higher than the upper spinal
cord tracts (α ∼ 19 ± 9◦ (using MF30)). From the data analysis, it is found that the PS
(61.3% (estimated) and 62.9% (predicted)) is less than for the previous human studies
(for Λ < 90◦ cases), but the eSNR of OPT30-90 (4.180) is still better than MF30 (3.808).
The standard deviations of the estimated ADTI model parameters and FA are shown in
Table 7.8. From the analysis results, it is observed that the use of OPT30-90 reduced
the uncertainty of the diffusivity parameters and FA significantly while not changing the
uncertainties in the fiber orientation angles sufficiently. This is expected for the fiber
orientation estimation since the fiber orientation is completely uncertain from a priori
knowledge. The mean diffusivities of OPT30-90 and MF30 are approximately equal to
each other. D = 1.575 × 10−3 mm2 s−1 (for MF30) and 1.570 × 10−3 mm2 s−1 (for
OPT30-90), D⊥ = 0.540 × 10−3 mm2 s−1 (for MF30) and 0.514 × 10−3 mm2 s−1 (for
OPT30-90). Thus, an overall reduction of the estimation uncertainty via the optimization is achieved. This procedure suggests that the optimization process can be applied
to pathological cases with a higher uncertainty in fiber orientation.

7.2

Fiber tracking analysis

In this section, a quantitative analysis of the effect of the diffusion gradient optimization
on the tracking metrics is studied. Firstly, a simulation experiment is performed where
fiber tracking is conducted based on data from OPT30 and MF30 gradient schemes on a
simulated fiber bundle and is used to compare fiber tracking metrics (TF, AF and AL)

135

Distribution ( % )

Distribution ( % )

40
30
20
10
0
−40 −20 0 20 40 60 80
Relative difference in λ vs D (%)
r

(a)

⊥

30
20
10
0
−40
−20
0
20
40
Relative difference in λ1 vs D|| (%)

(b)

30

Distribution ( % )

Distribution ( % )

60

20
10
0

(c)

(d)

−20 −10
0
10
20
Relative difference in MD (%)

20
Distribution ( % )

Distribution ( % )

20

(e)

20

0

−20 −10
0
10
20
Relative difference in FA (%)

15
10
5
0
−10

40

−5
0
5
Difference in θ (°)
F

15
10
5
0
−10

10
(f)

−5
0
5
Difference in φ (°)

10

F

Figure 7.4. Distributions of relative differences for various estimated quantities under the
DTI and ADTI model. Differences of (a) λr = (λ2 + λ3 )/2 and D⊥ , (b) λ1 and D , (c)
FA (d) MD, (e) θF and (f) φF are shown. These results are based on the voxels in the
cervical spinal cord white matter tracts (C1-C2 region) from five subject data collected
using the MF30 gradient scheme.

136

Table 7.8. Comparison of standard deviations (SD) of estimated ADTI model parameters
and FA at the ROI voxels for subject 1 with gradients optimized for a completely uncertain
fiber orientation (OPT30-90) and MF30.
Model parameters

MF30

OPT30-90

p-value
(SDOP T 30−90 < SDM F 30)

D × 10−3 ( mm2 s−1 )

0.236

0.187

0.047

0.068
6.551
7.852
0.086

0.062
6.745
9.239
0.080

0.028
0.306
0.246
0.024

D⊥ × 10−3( mm2 s−1 )
θF (◦ )
φ F (◦ )
FA

under the two schemes. Secondly, ADTI experimental data on healthy subjects collected
in the previous section are used for fiber tracking and comparing tracking metrics. The
results from both the simulations and experimental data indicate that by optimization
of the diffusion encoding gradients, fiber tracking in general can be improved.

7.2.1

Method

A simulation experiment is performed where a synthetic fiber bundle of radius 6 mm and
length 46 mm oriented along the +Z direction and surrounded by a 12 mm radius cylinder
of isotropic medium is simulated. 10 sets of ADTI data are generated using both MF30
and OPT30 schemes. For the fibers, the following diffusion model parameters were used:
D = 1.6204 × 10−3 mm2 s−1 , D⊥ = 0.148 × 10−3 mm2 s−1 , θF = 0◦ , φF = 0◦ (FA =

0.9, MD = 0.639 ×10−3 mm2 s−1 ). For the surrounding medium, isotropic diffusion was

assumed with MD = 2.6 × 10−3, FA = 0. Rician noise is added to the data at σ = 0.1.
b = 1000 s mm−2 and N = 30 gradient directions. ADTI data is processed using diffusion
tensor calculations and fiber tracking modules in DTIStudio (version 3.0.2) (Copyright
Mori, Jiang, Radiology department, Johns Hopkins University, Baltimore, MD, USA)
[103]. Fiber tracking settings were: starting FA = 0.6, stopping FA = 0.6 and curvature
limit = 40◦ . TF, AF and AL metrics were calculated for each dataset. Fig. 7.5 shows
the fiber tracking done on one of the MF30-based simulated data.
For the fiber tracking analysis of experimental data, the five subject ADTI data was
used which was collected using the MF30 and OPT30 protocols (described in the section
137

Figure 7.5. Fiber tracking based on simulated fiber bundle surrounded by isotropic
medium.
7.1). The 12 volume data collected for each subject are co-registered using FLIRT in
FSL package [126]. The seed ROI is drawn manually for each subject case in the inferior
end of the cervical spinal cord below the C2 vertebral level (shown for one subject case
in Fig. 7.6) and used for fiber tracking in all the 12 co-registered volume data. The
following FACT settings were used: starting FA = 0.6, stopping FA = 0.6 and angle of
curvature limit = 40◦ . FA value of 0.6 has been reported for the cervical spinal cord region
previously [28, 69]. The FACT curvature limit equal to or similar to 40◦ has been used
previously [67, 103]. Fibers tracked using the seed ROI for one subject are shown in Fig.
7.7. After the tracks were reconstructed, the metrics TF, AF and AL were calculated.

7.2.2

Fiber tracking results

Table 7.9 shows the fiber tracking results for the simulation experiment. It is observed
that both the TF and AF metrics are significantly higher for OPT30 than MF30. The
AL is almost equal for both the cases.

138

(a)

(b)

(c)

Figure 7.6. (a) Coronal view, (b) sagittal view and (c) axial view of T2-weighted image
of the cervical spinal cord with the seed ROI shown in red.

Figure 7.7. 3D fiber tracking shown in the cervical spinal cord region using the DTIStudio
software.

139

Table 7.9. Comparison of fiber tracking results for OPT30 and MF30 protocols using
simulated fibers.
Metric
TF (number of fibers)
AF (fibers per voxel)
AL (mm)

MF30
1025 ± 30
15.1 ± 0.4
45.4 ± 0.05

OPT30
1066 ± 28
15.5 ± 0.4
45.4 ± 0.04

p-value
0.003
0.014
0.335

Table 7.10 shows the estimated TF metric from the same seed ROI for each subject in
the 12 datasets under the two gradient schemes (OPT30 and MF30). It is observed that
the total number of fibers is higher on an average for the OPT30 protocol as compared to
the MF30 protocol. Table 7.11 shows the AF metric for all the subjects. This quantity
is also higher on an average for the 5 subjects for OPT30 protocol compared to MF30
protocol. Finally, Table 7.12 shows the estimated AL metric for all the subject data. The
average length of fibers tracked is moderately higher for the OPT30 protocol than that
of the MF30 protocol.
The OPT30 scheme is designed to improve the overall precision of the estimated diffusion model parameters which is expected to improve any secondary processing done based
on these parameters. In the previous section, the fiber orientation estimation showed better precision (less standard deviation) when using the OPT30 scheme. Also, FA = 0.6
was used in the design of the OPT30 scheme (for the extraction of white matter voxels).
In the FACT algorithm, FA = 0.6 was also used and all other FACT parameters, seed
ROI as well as the imaging parameters (except for the gradient scheme) were same for
both the OPT30 and MF30 cases. Thus, an unbiased comparison of the fiber tracking was
performed for the two cases. Since the FACT algorithm traces streamlines in the image
space based on the estimates of the local fiber orientation and FA, number of streamlines
traced reduces if these estimates are less precise. Thus, improvement in the precision of
diffusion parameter estimation shows up as an increase in the number of fibers tracked
(TF). The increase in TF also increases the AF metric. The AL metric was only moderately high for the OPT30 case indicating less effect of the gradient optimization on the
length of the fibers tracked.

140

Table 7.10. Comparison of fiber tracking results for OPT30 and MF30 protocol for the
total number of reconstructed fibers (TF) through the ROI using 5 ADTI subject data
Subjects

Grad. Sch.
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30

1
2
3
4
5

227
158
140
70
202
110
203
164
140
167

Datasets 1 –
166 224 219
133 175 124
186 115 134
88 134 135
176 167 161
123 142 120
176 178 190
146 132 125
157 173 172
105 127 139

6
168
128
132
111
164
149
184
160
186
106

Mean ± SD
199 ± 28
143 ± 20
140 ± 24
105 ± 26
179 ± 19
130 ± 15
185 ± 10
146 ± 15
164 ± 16
126 ± 24

189
137
134
93
203
133
179
146
156
110

Table 7.11. Comparison of fiber tracking results for OPT30 and MF30 protocol for the
average fibers per voxel (AF) using 5 ADTI subject data
Subjects
1
2
3
4
5

Grad. Sch.
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30

7.6
6.6
5.7
4.1
6.1
5.4
7.1
4.7
4.9
5.4

Datasets 1 –
6.4 7.4 7.1
5.5 7.2 5.8
6.5 5.1 5.4
3.8 5.0 5.4
5.5 5.8 5.4
5.2 5.7 4.7
6.0 5.7 6.1
5.3 4.6 4.4
5.1 6.3 5.6
4.6 4.6 5.0

6
6.4
6.1
5.8
4.2
5.4
6.2
5.5
5.4
5.9
4.4

6.6
5.9
5.0
4.6
5.9
6.1
5.2
4.8
5.7
4.8

Mean ± SD
6.9 ± 0.5
6.2 ± 0.6
5.6 ± 0.5
4.5 ± 0.6
5.7 ± 0.3
5.5 ± 0.6
5.9 ± 0.6
4.9 ± 0.4
5.6 ± 0.5
4.8 ± 0.4

Table 7.12. Comparison of fiber tracking results for OPT30 and MF30 protocol for the
average length (in mm) of fibers (AL) tracked using 5 ADTI subject data
Subjects
1
2
3
4
5

Grad. Sch.
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30
OPT30
MF30

30.7
27.9
24.1
17.8
27.1
25.2
28.6
21.1
20.6
22.4

Datasets 1 –
25.7 27.6 26.6
25.1 32.1 27.8
26.6 23.8 21.6
18.1 21.4 20.7
24.9 24.9 23.8
22.8 24.6 22.2
27.8 24.4 26.4
20.8 19.5 19.2
24.4 26.9 23.5
21.6 22.4 24.3

141

6
26.0
28.4
23.0
19.7
23.1
26.0
24.4
20.2
26.4
20.8

29.2
25.5
21.4
20.7
24.7
26.0
24.5
21.3
25.9
24.0

Mean ± SD
27.6 ± 1.9
27.8 ± 2.5
23.4 ± 1.9
19.7 ± 1.5
24.8 ± 1.4
24.4 ± 1.6
26.0 ± 1.9
20.3 ± 0.9
24.6 ± 2.3
22.6 ± 1.3

7.3

A partitioned CRLB based optimization of bfactor and diffusion gradient scheme

Optimization of DTI experimental parameters depends on the selection of the cost function that is minimized during the optimization process. Hasan et al. [17] discuss a number
of cost functions which have been previously reported, such as the total variance of estimates of the diffusion tensor matrix elements [18], the Coulomb’s force or energy assuming
the gradient directions are point charges on a unit sphere [22], the product of the inverse
of the squares of singular values of the encoding matrix, condition number of the encoding matrix, variance of secondary metrics such as FA [21] or ADC (apparent diffusion
coefficient) [24].
The CRLB provides a theoretical lower bound of the variance of the estimated diffusion
model parameters in terms of the signal noise and the DTI experimental parameters
provided the noise pdf is known and follows regularity condition [109]. Previous use
of CRLB can be seen in the work by Brihuega-Moreno et al. [24] for optimizing the
b-factor for improved precision of ADC estimation and Alexander [23] for optimizing
the diffusion gradient strength, the pulse interval between the diffusion gradient pulses
and the diffusion pulse duration for reducing the overall uncertainty of the simplified
CHARMED model parameters [9]. For a multi-parameter diffusion model, such as DTI
or ADTI, the CRLB matrix can be partitioned to select the uncertainty bounds of only
a subset of the diffusion model parameters (as shown in Chapter 5, section 3). The
determinant of the sub-matrix corresponding to the subset of the model parameters can
be minimized with respect to experimental parameters to obtain D-optimal experimental
parameters (gradient scheme or b-factor or both).
For the spinal cord tracts, the axon bundles are oriented in the superior-inferior orientation within a narrow range of fiber directions. Thus, the range of fiber angles is fairly
known a priori which can be used to improve the precision of the diffusion parameter estimation. As described in Chapter 5, the subsets of the set of diffusion model parameters
can be in general grouped into diffusivities (such as D and D⊥ for the ADTI model)

142

and angular parameters for the fiber orientation (θF and φF ). In this section, a combined b-factor and gradient scheme optimization technique is developed for the cervical
spinal cord tracts where the precision in the estimation of only diffusivities is improved
by using a partitioned CRLB as the cost function and the a priori fiber orientation range
information.

7.3.1

Method

Definition of the partitioned CRLB
The cost function for the partitioned CRLB matrix for the ADTI model for diffusivities
only case has been defined previously in Chapter 5, section 5.3.1. The determinant of
the sub-matrix B1 can be minimized with respect to both the b-factor and the gradient
directions such that the overall uncertainty of the diffusivities is reduced.
Optimization procedure
For the optimization of the b-factor and gradient scheme, the prior information used were
as follows: D = 1.6 × 10−3 mm2 s−1 , D⊥ = 0.148 × 10−3 mm2 s−1 , (θF , φF ) = (0◦ , 0◦ )
(+Z-axis). This corresponds to an FA = 0.9 and MD = 0.639 × 10−3 mm2 s−1 . The

range of fiber orientation (cone angle), Λ = 25◦ . The cone angle value is smaller than the
values used previously since this corresponds to voxels with a high mean FA = 0.9. The
number of gradient directions is fixed to 30 directions. The optimization was performed
by minimizing the cost function (det(B1 )) within the cone angle of fiber orientation using
simulated annealing technique [73] (flowchart for this step is described in Appendix I).
The optimized b-factor was found to be 2062 s mm−2 and the gradient scheme is shown
in Fig. 7.8.
Experiment
This study was approved by the Institutional Review Board (IRB) for conducting research
on human subjects. One healthy subject (male, age 29 years) participated in this study
and provided his signed IRB-approved informed consent. The experiment was conducted
143

θ

0
20
40
60
80

0.8
0.6
0.4
−100

0
φ sin θ

100

0.2

Figure 7.8. Distribution of 30 gradient directions (white circles) on a opened unit hemisphere shown with an underlay image of the normalized MR signal at b = 2062 s mm−2
and FA = 0.9.
using OPT30 and MF30 schemes at the C1-C2 vertebral location of the cervical spinal
cord with the following settings: T2 and diffusion-weighted images were obtained using a
dual spin-echo EPI sequence on a 3T GE Signa HDx scanner (GE Healthcare, Waukesha,
WI), equipped with an 8-channel head coil with the following parameters, 12 contiguous
3-mm axial interleaved slices, TR = 4500 ms, TE = 90.3 ms, matrix size =128 x 128, FOV
= 16 cm x 16 cm, number of excitations = 4, parallel imaging acceleration factor = 2, b
= 2062 s mm−2 , 30 diffusion gradient directions and scan time per dataset = 9 min 23
sec. Maximum likelihood estimator was used for the ADTI model parameter estimation
assuming Rician noise model. The post-processing technique is same as described in
section 7.1.1.

7.3.2

Results

Fig. 7.9 shows the ROI voxel distribution with respect to the hypervolume ratio and
it is observed that majority of the voxels (PS = 72.2% of 36 ROI voxels) have reduced
overall uncertainty in the diffusivities estimation (µr1 = 0.742). The prediction based on
the partitioned Rician CRLB matrix shows similar results with PS = 69.4% and µr1 =
0.786. The eSNR is 2.736 for the MF30 and 2.881 for the OPT30 case, thus indicating
improvement in the overall SNR. These eSNR values are less than the previously reported
values in section 7.1 (which were around 5) due to the use of a higher b-factor (b =
2062 s mm−2 ) in this experiment than the previous case (b = 1000 s mm−2 ). Simulations
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Voxel count

30

20
Est.
Pred.
10

0
0

1

2
3
DOPT30/DMF30

4

Figure 7.9. ROI voxel distribution with respect to hypervolume ratio (DOP T 30 /DM F 30)
based on the ADTI datasets from MF30 and OPT30 gradient scheme for one healthy
subject. Estimated PS = 72.2% and µr1 = 0.742 for 36 ROI voxels. Estimation is based
on the covariance of estimates and prediction is based on the Rician CRLB formulation.
in Chapter 6 section 6.4.3 also demonstrate the decreasing trend of eSNR with increasing
b-factor.
Table 7.13. Comparison of ADTI parameter standard deviations (RMS) in the ROI voxels
for data based on MF30 and OPT30 gradient schemes and one healthy subject
Model parameters

MF30

OPT30 (b opt.)

p-value
SDOPT30 < SDMF30

× 10−3

mm2 s−1

0.562

0.527

0.122

D⊥ × 10−3 mm2 s−1

0.046

0.043

0.045

10−3

mm2 s−1

0.169

0.159

0.141

FA

D

MD ×

0.0543

0.0536

0.495

θF

(◦ )

2.992

2.945

0.355

φF

(◦ )

3.082

2.962

0.238

Table 7.13 shows the RMS standard deviations (SD) of the ADTI parameters for the
ROI voxels when using MF30 and OPT30 schemes on a single healthy subject. In general, the SD values for all the parameters are less for the OPT30 case than the MF30
case indicating reduction in the parameter estimation uncertainty due to optimization.
However, for the diffusivities estimation, the reduction is more statistically significant
(especially for D⊥ ) than the angular parameters estimation which is expected since the
optimization cost function specifically optimized for the partitioned Rician CRLB for the
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Table 7.14. Comparison of ADTI parameter mean values in the ROI voxels for data based
on MF30 and OPT30 gradient schemes and one healthy subject
Model parameters

MF30

OPT30 (b opt.)

D × 10−3 mm2 s−1

2.311

2.246

D⊥ × 10−3 mm2 s−1

0.212

0.233

0.912

0.904

FA

MD ×

10−3

mm2 s−1

0.886

0.873

(◦ )

88.413

88.545

φ F (◦ )

0.294

0.868

θF

diffusivities. Table 7.14 shows the mean values of the estimated ADTI parameters and FA
and MD obtained by averaging over the ROI voxels for the MF30 and OPT30 schemes.
The parameters values are similar under the two protocols indicating no significant bias
due to the gradient scheme and b-factor optimization. Note that the coordinate axis is
rotated towards the X-axis so that the mean fiber angles are close to (θF , φF ) = (90◦ , 0◦ ).
This is done to avoid the discontinuity in φF at (0◦ , 0◦ ) which affects the calculation of
mean and SD.

7.4
7.4.1

Discussion
Justification for the use of axisymmetric diffusion model
for cervical spinal cord

In this section, the pertinence of the choice of the ADTI model is examined instead of the
use of the general DTI model by comparing the estimated values of the diffusion model
parameters, FA and MD. To estimate these quantities, the five subject data collected
using the MF30 scheme only and the same ROI voxels from the C1-C2 region of the
cervical spinal cord as described in the procedure in the “Experimental Protocol” section
were used.
For transverse diffusivities, the following group statistics (mean and standard deviation
(SD)) were obtained: D⊥ = (0.345 ± 0.131) × 10−3 mm2 s−1 , λ2 = (0.468 ± 0.147) ×

10−3 mm2 s−1 and λ3 = (0.288 ± 0.133) × 10−3 mm2 s−1 . The radial diffusivity (λr )
146

is defined as the mean of the secondary and tertiary eigenvalues of the diffusion tensor
(λr = (λ2 + λ3 )/2) and find that λr = (0.378 ± 0.13) × 10−3 mm2 s−1 . Fig. 7.4(a)
shows the distribution of the relative difference between λr and D⊥ , defined as 100 ×
(λr − D⊥ )/((λr + D⊥ )/2), for which the mean and SD are (11.1 ± 12.6) %. Also, it is
observed that about 70% of the voxels have less than 10% difference between D⊥ and λr
(90% voxels less than 21% difference). These results indicate that D⊥ is approximately
equal to the λr with a relatively small bias. With regard to longitudinal diffusivities,
D = (1.84 ± 0.34) × 10−3 mm2 s−1 and λ1 = (1.70 ± 0.29) × 10−3 mm2 s−1 . In Fig.
7.4(b), the group mean and SD for the relative difference between λ1 and D , defined
as 100 × (λ1 − D )/((λ1 + D )/2), is (-7.4 5.9)% and about 80% of the voxels have the
relative difference less than 10% between D and λ1 , thereby revealing that D obtained
with ADTI is equivalent to the primary eigenvalue λ1 obtained with DTI.
Figs. 7.4(c)–(f) show the distributions of the relative differences in FA, MD and difference in angular parameters (θF , φF ) between the ADTI and the DTI models. The group
means and SDs for the relative difference for FA (defined as 100 × (FADTI − FAADTI ))
is (-3.6 ± 3.3) % and for MD (defined as 100 × (MDDTI − MDADTI )/MDDTI ) is (-2.57
2.58) %. Similarly, for the angular parameters, the group mean and SDs for the difference
is (-0.08 2.13)◦ for θF and (0.09 ± 2.04)◦ for φF . Thus, both models lead to similar
estimations of FA, MD and the angular parameters (θF , φF ). Hence, any diagnosis based
on FA, MD or fiber tractography should not be affected by the ADTI assumption. Moreover, the ADTI model reduces the number of diffusion model parameters from six to
four, and thus allows a shorter computation time for the optimization framework. This
is advantageous for routine clinical studies.
Thus, in terms of diffusivities, the ADTI and DTI models result in analogous distributions in the cervical spinal cord white matter tracts (C1-C2 region). Hence, this
framework based on the ADTI model can be used in applications where only the transverse diffusivity is used to characterize the tissue properties. To deal with the inherent
limitation of the ADTI model in cases where the secondary and tertiary eigenvalues of the
diffusion tensor need to be characterized, a future study based on the non-axisymmetric

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DTI model will be performed.
0

0.7
0.5

50

θ

θ

0

0.7
0.5

50

0.3
−100
(a)

0
φ sin θ

100

−100
(b)

0

0.5

50

0
φ sin θ

100

0

0.7
θ

θ

0.3

0.7
0.5

50

0.3
−100
(c)

0
φ sin θ

0.3

100

−100
(d)

0
φ sin θ

100

Figure 7.10. Distributions of optimized diffusion gradient directions (indicated by white
circles) on an opened unit hemisphere. Underlying grayscale image shows the normalized
MR signal levels over the hemisphere changing with respect to the diffusion gradient
direction (θF , φF ). Optimized gradients are shown for the ADTI model in (a) and the
non-axisymmetric DTI model in (b) for healthy white matter tracts with FA = 0.745, MD
= 0.818 ×10−3 mm2 s−1 . For the pathological case (cervical spinal cord in ALS patients),
optimized gradients are shown for the ADTI model in (c) and the non-axisymmetric DTI
model in (d) with FA = 0.45, MD = 0.96 ×10−3 mm2 s−1 . For all cases, the fiber
orientation angle is (θF , φF ) = (0◦ , 0◦ ).

7.4.2

Optimized distribution of gradient directions

The optimized gradient schemes computed based on the ADTI and DTI models lead to
different characteristics in the spatial distribution of gradient directions. Figs. 7.10(a) and
7.10(b) represent the optimized gradient distributions from these two model assuming FA
= 0.745 and MD = 0.818 ×10−3 mm2 s−1 obtained from the group averages of the healthy
subjects. Let us then consider a pathological condition (amyotrophic lateral sclerosis
(ALS) case reported by Nair et al. [128]), for which the DTI diffusivities were calculated
directly from the definition of FA, MD and λr : λ1 = 1.44 × 10−3 mm2 s−1 , λ2 = 0.927 ×
10−3 mm2 s−1 and λ3 = 0.513 × 10−3 mm2 s−1 , resulting in λr = 0.72 × 10−3 mm2 s−1 .
The ADTI diffusivities are calculated by fitting the ADTI model to 30 diffusion-weighted
normalized MR signals, which are simulated based on the eigenvalues of the diffusion
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tensor values above, b-factor = 1000 s mm−2 , MF30 gradient scheme and fiber oriented
along +Z direction. The estimated ADTI parameters are: D = 1.46 × 10−3 mm2 s−1
and D⊥ = 0.712 × 10−3 mm2 s−1 . Figs. 7.10(c) and 7.10(d) show the corresponding
optimized gradient distributions using DTI and ADTI, respectively, with FA = 0.45, MD
= 0.96 ×10−3 mm2 s−1 for the pathological condition defined above. It is observed that
the optimized gradients are distributed more uniformly in the transverse orientation in
the ADTI case than in the DTI case for both the healthy and the pathological case.
This is expected given the axisymmetry assumption in the ADTI model, which results in
the axisymmetry of the MR signal with respect to the fiber orientation as shown in the
underlay images in Figs. 7.10(a) and 7.10(c).
The optimized distribution of the gradient direction on the unit sphere largely depends
on the symmetry in the transverse diffusivity, the overall diffusion anisotropy (the difference in the longitudinal and transverse diffusivity) and the cone angle within which
the optimal performance is designed. The higher the overall diffusion anisotropy, the
greater the MR signal is in the transverse direction to the fiber orientation in comparison
to the MR signal in the longitudinal direction (as seen in the underlay images in Fig.
7.10). Since D-optimality based framework uses the square of sensitivity matrices (which
are also a function of normalized MR signal), regions of higher sensitivities are along the
transverse direction rather than along the direction of the fiber orientation when diffusion
anisotropy is high. The axisymmetry assumption of the diffusion tensor results in the
normalized MR signal and diffusivity sensitivities also being axisymmetric about the fiber
orientation (see Figs. 7.10(a) and 7.10(c)). Thus, for the ADTI model, the optimized
gradient directions tend to be uniformly distributed about the fiber orientation (also see
Fig. 5.12). This is an important effect of the gradient optimization framework on the
gradient distribution.
If the general DTI model is used, and if the secondary and tertiary eigenvalues of the
diffusion tensor are significantly different from each other, the optimized gradients will no
longer tend to be distributed uniformly about the axis along the fiber orientation (also see
Fig. 5.13). The gradients will sample more of the region corresponding to the direction

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of the tertiary eigenvector than the region corresponding to the secondary eigenvector
of the diffusion tensor (shown in Fig. 7.10(d)). In this case, the diffusion tensor model
without the axisymmetric assumption has to be employed in the signal formulation since
the use of the axisymmetric DTI formulation would result in additional uncertainty in the
estimation of diffusion model parameters. The case when the overall diffusion anisotropy
is low (low FA cases), the signal level difference between the longitudinal and transverse
directions will also become low. Under such circumstance, the optimization framework
would distribute the gradients more evenly on the unit sphere as compared to more
transversely in case of high anisotropy (as seen in Fig. 7.10(c)).

7.4.3

Clinical relevance

FA threshold of 0.6 has been used to extract white matter tracts specifically and minimize
the selection of voxels with gray matter of CSF contamination. Since the validation study
was only on healthy subjects, this high FA thresholding worked well. Also, use of FA
threshold for ROI voxel extraction has been previously reported by Wheeler-Kingshott
et al. [129]. For low FA cases (pathology), the choice of the FA threshold will depend
on the application. If only the pathological voxels need to be extracted, a specific range
of FA values (corresponding to the pathology) can be used instead of a high FA cutoff.
A manual segmentation based on the anatomical structure would be more appropriate
(for example, using T1-weighted or T2*-weighed anatomical images). But the assistance
from the FA image has been quite valuable based on my experience in identifying the
white matter regions without gray matter or CSF contamination.
Cardiac and respiratory motions were not factored in this protocol optimization. The
parameter estimation will likely be improved if the effect of the cardiac and respiratory
motion is reduced through the use of gated sequences. However, these procedures require
more acquisition time and were not suitable in the present study. Both the bulk motions
mentioned above have been minimized by applying image co-registration techniques in the
pre-processing steps. In addition, physiological motion would affect both the optimized
schemes and MF30 equally, without changing the outcome of the comparisons.

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A limitation in the proposed work is that the optimization procedure needs a preliminary scan (2 min for the OPT30 protocol). An extra 20 min is required to generate the
optimized gradient scheme with four 3.3 GHz CPUs. The computation time can be shortened by parallelizing more CPUs which will allow the data acquisition with the optimized
gradient scheme in the same scan session as the preliminary scan. Most clinical spinal
cord MR protocols consist of routine anatomical scans. These scans can be conducted
while the computation of the optimized gradient scheme is being performed.
The use of the determinant of the CRLB matrix results in a global optimal state with
respect to the uncertainty in the estimation of the diffusion model parameters. But
the precisions of all the individual diffusion model parameters might not be improved
simultaneously. If one aims to optimize the estimation of only a subset of the diffusion
model parameters (for example, only the diffusivity parameters and not the angular
parameters), then the diffusion gradients can be optimized by minimizing the determinant
of a partition of the CRLB matrix for D-optimality [120].
Finally, it is noted that the use of the gradient optimization can be extended to regions
outside the central nervous system wherever the white matter tracts are directed in a
particular orientation and over a range of distance, such as the median nerves in human
wrists. Work done by Meek et al. [29] who demonstrated the feasibility of in vivo threedimensional reconstruction of the median nerves in human wrists by DTI-based fiber
tracking can be further improved by this gradient optimization. The ADTI can also
be applied to non-nervous tissues that exhibit organized and orientated structure, such
as the skeletal muscle tissues. DTI has been reported to track muscle fibers in skeletal
muscles and to create biomechanical models of the muscles based on the estimate of local
orientation of the fibers (pennation angle), fiber length and cross-section of the fiber
bundles in humans [32]. An optimized gradient scheme will improve the estimation of
such metrics.

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7.4.4

Fiber tracking in cervical spinal cord

The effect of the gradient scheme optimization on fiber tracking metrics (TF, AF and
AL) can be shown by a simple demonstration of the FACT algorithm given in Fig. 7.11.
In this case, a 2D FACT algorithm is implemented using brute-force method [103] for an
image of size 5 × 3. The boxes represent the image pixels, the arrows indicate the fiber
orientations within the pixels and the orange lines are the fiber tracks. The true fiber
orientation is along the vertical direction.
Going from left to right column, the number of pixels with uncertain fiber orientation
increases. It is observed that the left column with the least uncertain fiber orientations
has the most number of tracks reaching the seed voxels. Also, the number of tracks per
voxel is higher for this column of fibers and the fibers track the length of the column. For
the middle column, there is a pixel in the middle with higher uncertain fiber orientation
which results in reduced track density although the number of fibers reaching the seed
voxels is same as in the left column. Finally, at the right column, there are more pixels
with uncertain fiber orientations and this affects both the number of fibers reaching the
seed voxels as well as the track density.
This simple demonstration clearly explains the fiber tracking results from the simulation experiment as well as ADTI experiment based on the 5 subject ADTI data where
the tracking using the OPT30 protocol gave higher TF and AF metrics on an average for
each subject than the MF30 protocol. The average length of the fiber tracks were only
moderately higher for the OPT30 protocol indicating that MF30 could also track similar
distances as OPT30 but only with fewer tracks. The larger the number of tracks generated, the better is the confidence in the estimated tract bundle which indicates reduced
uncertainty in the overall fiber tracking. This shows that gradient scheme optimization
to reduce the uncertainty of the diffusion model parameters also reduces the uncertainty
in the results of a secondary post-processing such as fiber tracking.
Fiber tracking results can be sensitive to the choice of FACT thresholds (FA and track
curvature) and could result in the omission of certain tracks by minor change in the
threshold values as shown by Brecheisen et al. in the brain [106]. However, FA ∼ 0.6 has
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Figure 7.11. An illustration of the FACT algorithm applied to a 2D distribution of fiber
orientations. Tracks are shown in orange. Each box represents a pixel with the fiber
orientation shown by the arrow. True fiber orientation is along the vertical. Light red
pixels are ones with more uncertain fiber orientation. The green pixels are the seed (ROI)
pixels. Only tracks penetrating the seed pixels are retained.
been widely reported for the healthy cervical spinal cord white matter in other studies
[28, 69, 129] and its selection as a threshold should reduce spurious fiber tracks passing
through gray matter and CSF. The curvature threshold is to prevent sharp turns in the
fiber track within the locality of the voxels and use of a 40◦ threshold is also commonly
used [67, 103] and is not a restrictive value. Moreover, by generating more tracks because
of the gradient scheme optimization, the sensitivity of these thresholds to fiber tracking
could potentially reduce.

7.4.5

Partitioned CRLB-based b-factor and gradient scheme optimization

This validation experiment aimed to verify two aspects of the CRLB-based optimization,
namely, the use of the partitioned CRLB to selectively improve the precision of a subset
of the diffusion model parameters and the use of a combined b-factor and gradient scheme
153

optimization. The study can be considered to be a more general case of the study for
only gradients directions described in section 7.1.
Simultaneous optimization of b-factor and gradient scheme optimization has been recently reported by Gao et al. [39] where they demonstrated a unified optimization approach for the selection of the b-factor, the diffusion gradient directions and the timing
parameters (such as ∆, δ, TE and readout time, R) using a stochastic optimization framework based on the simulated annealing algorithm [122]. They also used a cone of fiber
angle information as a prior knowledge for the optimization. Their work aimed to minimize the trace of the covariance of the estimated parameters assuming an additive white
Gaussian noise model. While their work is similar to the work presented in this section,
there are certain limitations of their work that have also been considered here. They did
not use a Rician noise model which is relevant for magnitude MR signals at low SNR,
such as in DTI of the cervical spinal cord. Also, the selective improvement in precision
of subset of diffusion parameters was not used by them.
In this work, the b-factor has been considered as a single lumped experimental parameter. However, this assumes that TE and the readout time for the b = 0 and b = 0
images are kept equal and are not optimized. The b-factor and the gradient directions are
independent of each other and can be simultaneously optimized (as done in this work)
since the b-factor can be decoupled from the b matrix definition as shown by Basser et
al. [12] (for trapezoidal diffusion gradient pulses in traditional spin-echo sequence) and
Finsterbusch [94] (for square gradient pulses in dual-spin echo sequence).
This work presents a generalized framework for the selective improvement in the precision of a subset of diffusion parameters based on CRLB. For the validation, the precision of the diffusivities are minimized. A somewhat similar work has been reported
by Brihuega-Moreno et al. [24] where b-factor was optimized for reducing the Gaussian
CRLB of the estimated ADC values either for a single ADC measurement or a range of
ADC measurements. The full CRLB matrix consisted of the S0 (non-diffusion weighted
MR signal) as well as ADC as the model parameter and the b-factor optimization focused
on the element of the CRLB matrix corresponding to ADC only. However, their frame-

154

work was much simplified due to the use of scalar ADC instead of the diffusion tensor
and also the use of Gaussian noise model. This work and work on Chapter 5 section 5.3
provides a more generalized framework for DTI/ADTI model and for both Rician and
Gaussian noise model.
Increase in the b-factor affects the SNR of the DWIs adversely (as shown by the simulations in Chapter 6, section 6.4) since higher b-factors are achieved by higher gradient
strength and/or longer TE and timing parameters all of which result in lower MR signal.
In this section, the optimized b-factor at an FA = 0.9 was 2062 s mm−2 which is higher
than the generally used b = 1000 s mm−2 . The eSNR was correspondingly lower (∼ 3) for
this study as compared to section 7.1 (∼ 5). This variation in the SNR necessitates the
use of the Rician noise model especially when b-factor is optimized in a DTI experiment.

7.4.6

Justification for the use of prior information

Use of prior information in the optimization of gradient scheme can have potential drawbacks and is debatable. Firstly, using prior knowledge results in improved parameter
estimation only towards estimating certain specific model parameter values. Secondly, if
prior knowledge is available with certainty, there is no need for imaging. The optimization framework developed in this work intends to address these issues and thereby justify
the use of such optimized scheme as compared to un-optimized schemes.
In the developed optimization technique, preliminary short DTI scans are performed
on each subject to collect subject-specific information, such as the cone angle (Λ) and
mean diffusivities (D , D⊥ ). The cone angle used as an input for solving the robust
optimization problem not only results in improved overall precision of estimates within the
cone but also beyond it, upto certain safety margin, as shown in the performance curves
in Section 5.5.2. Similarly, for diffusivities, ranges of FA and MD can also be used to
constrain the robust optimization problem along with the use of cone angle, as described in
Section 5.4.5. Use of subject-specific information and presence of safety margin indicates
that the optimization framework improves the parameter estimation for subject-specific
parameter values, which is relevant, and allows for a range or margin beyond the subject-

155

specific parameter values, making it less biased. Even for pathological cases, the proper
subject-specific information would help improve the precision of parameter estimation
since pathological condition does not necessarily mean completely uncertain scenario.
For example, DTI of pathological cases (e.g., multiple sclerosis [130]) report pathological
ranges of FA or MD values and degeneration of nerve fibers can result in larger cone angle
values. This subject-specific information can be used to develop optimized scheme for
pathological cases also. Also, the prior knowledge is only approximate and not known with
certainty since it is based on a short DTI scan (with fewer diffusion gradient directions),
thus there is a need for a full scan with the optimized gradient scheme and the prescribed
number of gradient directions.
A practical scenario for the use of the optimization framework can be outlined as
follows: a preliminary scan on the subject provides subject-specific information on the
diffusion model parameter values. While the program for computing the optimized gradient scheme is running, the subject goes through routine scans, such as T1 -weighted
volumetric and T2 -weighted. These routine scans take about 20 minutes in total which
is similar to the computation time to obtain the optimized gradients. Finally, data is
acquired using the optimized gradient scheme. By using multi-core computers, the optimized gradients can be computed faster than 20 minutes. And thus the total procedure
can be “online” without having the patient exit the scanner.

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CHAPTER 8
Conclusions
8.1

Summary

Diffusion-weighted imaging provides a non-invasive technique to delineate the nerve fiber
tracts at a macroscopic level and to perform in vivo fiber tracking. Compared to other
neuroanatomical imaging methods which are generally microscopy-based methods (for
example, scanning electron microscope [52], MR microscopy [49]) which require special
hardware and high field gradients (MR microscopy [49]) and cannot image in vivo or
are invasive (tract tracing using fluorescent dyes [4]), DWI-based methods can be easily implemented on a conventional MRI scanner and does not require special hardware
requirements for the scanner. This technique is applicable to humans in vivo and a
considerable coverage of the imaged target is achievable (for example, full brain DTI).
While neurophysiology-based methods, such as EEG and fMRI can provide effective and
functional connectivity information in neural networks respectively, the neuroanatomical
information can be used to validate and augment such connectivity information, thus
providing a complete picture of the neuronal connectivity. Owing to these advantages,
DWI-based methods (especially, diffusion tensor imaging, DTI [7]) is being used extensively to explore nerve fiber tracks.
The focus of this research work was to develop a diffusion gradient optimization framework for DTI to improve the precision or reduce the uncertainty in the diffusion model

157

parameter estimation. A detailed evaluation of the effect of experimental settings (such
as the b-factor, number of diffusion gradients (N)), effects of selection of noise models
and effects of selection of parameter estimators on the overall estimation uncertainty has
been demonstrated. Finally, the estimation performance of the gradient optimization is
validated by computer simulations and a five-subject human study for cervical spinal cord
DTI.

8.1.1

Gradient scheme optimization

The diffusion gradient optimization framework is developed for the DTI and the ADTI
diffusion models. These are investigated under both Rician and Gaussian noise. The
diffusion gradient optimization framework is based on D-optimality ([33, 34]) which minimizes the CRLB on the estimation variances of the diffusion model parameters. The
framework utilizes the prior structural knowledge of the imaged organ. For the spinal
cord, the prior structural information of the nerve fibers can be represented by a cone
model with known mean fiber orientation and the spread of the fibers from the mean
as defined by the cone angle (Λ). The gradients are optimized to perform within the
cone angle and thus for a range of fiber orientations. The gradient optimization can be
performed for improved estimation of either all the diffusion model parameters or for
selected model parameters, such as only for diffusivities or angular parameters for fiber
orientation. Depending on the user’s choice of the model parameters to have improved
estimation, appropriate optimization cost function can be selected.
The optimization framework was extended to include b-factor optimization along with
the optimization of the diffusion gradient directions. Due to limitation of scanner hardware, b-factor cannot be set any value and hence its optimization requires more constraints
(such as limits on the gradient strength and the timing parameters) than that of the diffusion gradient directions. In this work, the gradient scheme optimization was performed
at an assumed fixed value of the diffusion model parameters (i.e., diffusivities and the
mean fiber direction are fixed) which are obtained from preliminary DTI experiments.
While the cone angle (Λ) incorporates a range of fiber orientation for which the diffusion

158

gradients are optimized by the optimization framework, a range of diffusivity values can
also be used as prior information in the optimization problem, thus, generalizing the prior
knowledge to both diffusivity and fiber direction information.

8.1.2

Simulation experiments

An evaluation of the noise models used for the normalized MR signal is performed by
simulations. The normalized MR signal being a ratio of two magnitude MR signals is a
ratio of two Rician distributed random variables. However, in this work, the ratio pdf has
been approximated by a Rician pdf assuming that the normalizing signal has a negligible
variance as compared to its magnitude (hence high SNR). This approximation allows
using a fixed value of σ in the Rician CRLB definition. The simulations indicated that
when the Rician approximation is not used, σ of the Rician fit of the ratio pdf shows
dependence on the normalized MR signal. However, this dependence can be minimized
by increasing the number of MR signal acquisitions (NEX). The DTI experiments are
conducted at NEX = 2 for which I assume the approximation of fixed σ holds good.
Next, a comparison of the estimation performance of three estimators, namely, LS,
LSC and ML estimators, is done. While LS estimator is applicable when noise model is
not known, LSC is more applicable when systematic error (bias) is needed to be removed
from the measurements. ML estimator is used when noise model is known a priori .
From the performance comparison of the estimators for both the optimized gradient and
MF30 gradient scheme, it was shown that while LS estimator obtained marginally lower
hypervolume values than ML estimator, ML provided the most unbiased estimates of
the ADTI model parameters while having comparatively low hypervolume values. Thus,
ML estimator was used for the data analysis in the spinal cord ADTI experiment. Also,
the performance curves which shows the variation of the normalized hypervolume with
respect to angular deviation from the mean fiber orientation was validated by Monte
Carlo simulations with all the three estimators (for the Rician noise) and LS estimator
(for the Gaussian noise). The simulations indicated that MLE was the most appropriate
estimator in the case of Rician noise model and ADTI diffusion model.

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For the study on the effect of b-factor, it was shown that for both the optimized
and MF30 schemes, the performance (based on performance indices µr1 and PS ) only
improves within a certain range of b-factor. However, for the optimized case, the range
of high performance is broader than the MF30 case. Thus, when a cone angle of fiber
distribution is used for optimizing the diffusion gradient, the b-factor range increased
as compared to an un-optimized (MF-based) scheme. Increase in b-factor reduces the
eSNR. This shows that a limit on allowable eSNR can also be applied while doing the
b-factor optimization. For the effect of the number of gradient directions, it was seen
that increasing the number of gradients reduced the estimation uncertainty which was
expected since more measurements result in more extracted information from the data.
For the effect of cone angle, smaller the cone angle, better is the estimation performance
(i.e., more reduced is the uncertainty). Smaller cone angle indicates less uncertainty in
the fiber orientations.
Monte Carlo simulations for the ADTI and DTI experiments were conducted to assess
the estimation performance of the optimized gradient scheme compared with the MF30
scheme when either all model parameters or diffusivities only or the angular parameters
for fiber orientation only were selected. While the hypervolumes of uncertainty were
reduced in all the cases as indicated by the performance indices (PS and µr1 ), the effect on
individual parameter estimation varied. For example, for ADTI, when all parameters are
selected, most of the improvement in precision was visible in the angular parameters. But,
when the diffusivity only parameters are selected, uncertainties of both the diffusivities
(D , D⊥ ) as well as uncertainty in FA was reduced. Also, for the angular parameters
case, uncertainty of both the angle parameters (θF and φF ) and angular deviation (α)
was reduced. Results for the DTI model indicated that while overall improvement in
the hypervolumes was achieved, individual parameters not always improved in precision,
especially D which showed no improvement for the all parameters and the diffusivities
only cases. This was since the majority of the high square sensitivity regions for the
diffusion model parameters were transverse to the fiber direction, which was not the
case with D . Improving the precision of D would require the framework to exclusively

160

optimize for D and not for other diffusivities.
Selecting all parameters during optimization of diffusion gradients results in a global
optimal state and an overall improvement in precision. However, this does not mean improvement in the precision of all the model parameter estimates simultaneously. Selection
of model parameters for diffusion gradient optimization finally depends on the application. For diagnosis based on diffusivity-related metrics, such as FA or MD, it would
be desirable to improve the precision of diffusivities only. While for fiber tractography
applications which rely on the precise estimation of fiber direction angles, the angular
parameters should be selected during gradient optimization.

8.1.3

Spinal cord ADTI experiments

This work included a five-subject human study on the cervical spinal cord DTI for the
validation of the diffusion gradient optimization for reduced uncertainty in the diffusion
model parameter estimation. Step-by-step procedure including the ADTI protocol has
been described for conducting the study and analyzing the data.
Based on the spinal cord and brain stem imaging results, it is conclusively demonstrated
that the diffusion-encoding gradient optimization has performed significantly well in the
region of interest (spinal cord/brain stem white matter tracts). These regions contain
fibers oriented within certain range (or cone) of directions and the gradient optimization takes into account this prior knowledge and has performed better in this cone of
fiber directions. The new gradient optimization scheme does not bias the estimation as
differently as the standard gradient scheme (MF30). This is crucial since a bias in the
estimates will affect the secondary metrics, such as FA, MD and would cause erroneous
diagnosis of the imaged tissue. Another aspect of the CRLB-based gradient optimization
is that performance improvements can be predicted prior to conducting the DTI experiment. This is because the CRLB provides an analytical lower bound on the expected
variance. The expected performance can be predicted from the analytical formulation of
the CRLB. Such a prediction capability is convenient and essential for the evaluation of
the new gradient scheme before human subject scanning is performed.

161

This work assumes axisymmetry in the diffusion tensor model, i.e., the secondary and
the tertiary eigenvalues of the diffusion tensor are assumed equal. The comparison of
the estimates of the ADTI and the DTI parameters indicated that D and D⊥ of the
ADTI model are equivalent to λ1 (primary eigenvalue of diffusion tensor) and λr (radial
diffusivity defined as the mean of secondary and tertiary eigenvalues of diffusion tensor)
of the DTI model, respectively. Also, FA, MD and the fiber angles (θF , φF ) estimated for
both the ADTI and DTI models were approximately equal indicating that the use of the
ADTI model will not affect any diagnosis based on these metrics. Also, the transverse
diffusivity (D⊥ ) estimated from the ADTI model could be potentially used for diagnosing
spinal cord pathology.
By reducing the uncertainty in parameter estimation, more confidence is ensured in
the estimates of diffusivities and fiber orientation. This implies that any secondary processing, such as fiber tracking, performed based on these estimates would also have lower
uncertainty. In the fiber tractography analysis, FACT-based [16] fiber tracking was performed on the ADTI datasets in the cervical spinal cord region. For all the five subjects,
the average of number of fibers tracked and the average fiber density were higher for
the optimized ADTI protocol than MF30 protocol. This shows that the quality of fibers
tracked in the spinal cord region improved upon the optimization of the gradient scheme.
Since the optimization has so far used a narrow cone angle representing higher certainty
in the fiber direction, a case where the cone angle of 90◦ indicating completely uncertain
fiber orientation was used for gradient optimization and its performance was validated
by a one-subject ADTI experiment. The estimation results indicate that the optimized
gradient scheme still outperformed the MF-based technique, albeit marginally.
In this work, a combined b-factor and gradient scheme optimization method was also developed and a preliminary experimental validation was presented by a one-subject ADTI
experiment. The technique is applied in the cervical spinal cord/brain stem region near
C1-C2 vertebral levels and aimed at reducing uncertainty of only the diffusivities while
using prior knowledge of the fiber orientation. The optimization was performed at a high
FA value (FA = 0.9). From the validation experiment, the model parameter estimation

162

results indicate that the uncertainty of the diffusivities was reduced as expected from
the CRLB formulation. Also, the estimates were not biased. It was observed that at
high b-factor, the SNR of the DW image was low and only at higher NEX (= 4) could
a discernible diffusion-weighted image be scanned. Also, the TR and TE of the sequence
was changed to accommodate the high b-factor. This indicates that changes in the bfactor requires additional consideration and these could be formulated as constraints in
the optimization problem.
Use of prior information can be justified since by using subject-specific information
based on a preliminary scan, each optimized gradient scheme is made specifically optimized for the subject. Also, safety margin on the performance extends the range of
expected performance. Finally, development of an “online” method where the subject
does not exit the scanner, will make the optimization more practical.

8.2

Contributions

The research work done in this thesis contributes to different aspects in optimal design
of DTI/ADTI experiments. The following are some of the contributions of this research
work:
1. The diffusion gradient scheme for DTI/ADTI was improved so as to achieve more
precise estimates of tissues properties (diffusivities and fiber orientation). Practical
application to spinal cord ADTI experiments was demonstrated. Improvements in
the fiber tracking of white matter tracts was also verified from the experiments.
2. A five-subject human study on the spinal cord as well as detailed simulations to
validate the optimization procedure was presented. There are no stringent requirements on the MRI scanner in terms of the RF pulse strength and timing, gradient
strengths or use of custom-built pulse-sequence. Hence, this technique is readily
usable by other research groups in further studies and the results can be crossvalidated easily.

163

3. Application of advanced techniques in the design of optimal experiments were proposed, such as the use of CRLB, injection of prior knowledge into the optimization
problem, use of various signal and noise models, partitioned-CRLB methods and
use of various estimators. The theoretical framework for the optimization can be
used for the design of any experiment that fits similar requirements of the DTI
experiment. This shows an immense possibility of the use of techniques developed
in this research for other fields, either within MRI or beyond.
4. The equivalence of the 4-parameter ADTI model and general 6-parameter DTI
model is demonstrated in terms of the estimates of longitudinal and transverse
diffusivities. Also, the FA, MD and fiber angle orientations are similar in these two
models. Since only four diffusion-weighted MR images are sufficient to estimate the
ADTI diffusion parameters, ADTI experiments can be shorter in scan time than a
general DTI experiment which indicates potential clinical application.
5. ADTI diffusion gradients can be optimized to precisely estimate the transverse
diffusivity, which has recently shown high sensitivity to the detection of spinal cord
diseases, such as ALS or MS. ADTI optimized gradient schemes can be used in
place of current protocols to detect such pathologies.

8.3

Future Work

In this research, a comprehensive diffusion-encoding gradient optimization framework has
been developed and analyzed for the ADTI/DTI model with Rician or Gaussian noise.
The framework uses subject-specific information and performs within the prescribed range
of diffusion parameter values including certain safety margin, thus making this technique
less biased and more applicable to clinical DTI studies. An important future work would
be the development of an online optimization technique where the subject does not exit
the scanner and the total examination time is within an acceptable time limit.
Although the work has mainly focussed on ADTI/DTI, the framework can be extended
to any diffusion model, such as the CHARMED model ([9, 131]) and any cost function
164

can also be used (for example, determinant of CRLB matrix can be replaced by variance
of FA). In methods such as HARDI or q-ball imaging [86, 88], the post-processing does
not use any assumption of the diffusion model although the data acquisition is done
using the similar DWI pulse sequences as DTI, such as Stejskal-Tanner PGSE sequence.
These non-parametric techniques estimate the orientation distribution function directly
from the diffusion-weighted MRI data, but require a large amount of data due to their
non-parametric nature. In the study of the effect of number of gradient directions on the
estimation performance, it was seen that equivalent estimation performance (in terms
of performance indices) was predicted with the use of fewer diffusion gradient directions
when optimization of the gradient directions was performed based on prior structural
knowledge. This observation can be exploited in the case of non-parametric DWI (such as
q-ball imaging) to reduce the number of gradient directions used in the DWI experiment.
This could potentially reduce the overall acquisition time and make these non-parametric
techniques more clinically usable.
Instead of using a cone angle of the fiber orientation, information of the range of
diffusivities (or FA and MD) can also be used for the optimization of the gradient scheme.
One application that seems relevant in this regard is the tract-based spatial statistics
(TBSS) [14]. TBSS is a voxel-based multiple subject analysis of diffusion-weighted data
where FA maps from a group of subjects are co-registered using non-linear techniques and
from the group mean of the aligned FA maps a “group mean FA skeleton” is created. The
FA skeleton is obtained by thinning the FA maps and applying thresholding to remove
low mean FA and/or high inter-subject variability. Essentially, the mean FA skeleton
corresponds to the centers of all fiber bundles that are common to the subjects and
generally the range of FA in the mean skeleton is narrow. If a priori knowledge of the
FA range in the mean skeleton FA is available about a subject group, this optimization
framework could potentially be used to estimate the FA values in the range more precisely
and hence extract the FA skeleton more precisely as well.
Going beyond neuroimaging, DTI has also been applied to track muscle fibers in skeletal
muscles to study the three-dimensional (3D) architecture of skeletal muscles in mice ([63])

165

and in humans [64]. DTI has also been reported to be used to create biomechanical
models of the quadriceps mechanism in humans [32]. These studies show feasibility of
fiber tracking using DTI of skeletal muscles. The fiber tracking provides information
on the local orientation of the fibers (pennation angle), fiber length and cross-section
of the fiber bundles which correlate with muscles physiological cross sectional area, an
indicator of force that the muscle can exert [32]. Muscle fibers are fairly organized and
oriented in a particular direction. This prior knowledge of the fiber orientation can
be incorporated in the DTI gradient scheme optimization framework to more precisely
estimate the pennation angle.
By the above instances, the potential future uses of the gradient scheme optimization
framework are demonstrated for applications in neuroimaging and imaging of non-nervous
tissues, such as skeletal muscles.

166

APPENDICES

167

APPENDIX A
Flowcharts and instructions for the
overall optimization procedure
In this section, a detailed step-by-step description along with flowcharts of the overall
procedure is given. The steps should provide necessary information for any third party
to optimize the gradients, conduct the optimized DTI experiment and perform the postprocessing necessary. Although the described steps are specifically for spinal cord ADTI
experiment, the gradient optimization algorithm can be extended to other regions, such
as the brain or muscles. I begin with some information on the selection of subjects and
other formalities, such as Institutional Review Board (IRB) approval.
1. Obtain IRB approval to conduct human research: A proper certificate of approval
from the Institution’s review board (for Michigan State University, this is the
Biomedical and Health IRB under the Human Research Protection Program) must
be obtained if DTI experiments are to be carried out on human subjects. For
animal subjects, certificate of approval needs to be obtained from appropriate authority (for Michigan State University, this is the Institutional Animal Care and
Use Committee).
2. Subject recruitment: For the optimization study, no specific subject recruitment
criterion was used. Only criterion was that subject should be normal with no
history of spinal cord disease or injury. Age matching is not necessary, but was
168

used for the experiments.
3. Preliminary DTI scan: This step is to collect subject-specific structural information
of the spinal cord region. A standard MF15 based DTI protocol is used. Typical
protocol is given by: T2 and diffusion-weighted images acquired using a dual spinecho EPI sequence on a 3T GE Signa HDx scanner (GE Healthcare, Waukesha, WI),
equipped with an 8-channel head coil with the following parameters: 22 contiguous
3-mm axial interleaved slices, TR = 7000 ms, TE = 77.4 ms, matrix size =128 x 128,
FOV = 16 cm x 16 cm, number of excitations = 1, parallel imaging acceleration
factor = 2, b = 1000 s mm−2 , 15 diffusion gradient directions (MF15) and scan
time = 1 min 52 sec.
4. Preliminary DTI data processing: The flowchart of the preliminary data processing
is shown in Fig. A.1. DTI images from the scanner are stored in DICOM (Digital
Imaging and Communications in Medicine) format which are converted to NIfTI
(Neuroimaging Informatics Technology Initiative) 4D volume data format. The
data ordering in the NifTI file is converted from neurological (RAS) to radiological
(LAS) order so as to use this data in FSL software package. Eddy current correction
is performed to reduce the image distortion (contraction, shift and shear) caused
due to eddy currents. This is performed using FSL’s FDT toolbox “eddy correct”
routine. Here the diffusion-weighted images are registered with the T2 -weighted
image (i.e., the image with no diffusion weighting) using affine transformation (12
degrees of freedom). After the preprocessing, spinal cord region is identified visually
and DTI/ADTI fit is performed to estimate the model parameters. Also, FA, MD
are calculated. Spinal cord is extracted from the region by applying FA > 0.35 and
MD < 2.5 ×10−3 mm2 s−1 thresholds. Now, distribution of fiber angular deviation
(α) is computed and the cone angle is determined by selecting the α at 80% cumulative distribution. Next, spinal cord tracts voxels are extracted in the spinal cord
region by putting α < Λ and FA > 0.6 thresholds. Finally, the mean diffusivities
and the fiber angle in the spinal cord tract voxels are calculated and along with the
cone angle information are saved for the gradient scheme optimization.
169

Figure A.1. Flowchart for the processing of the preliminary DTI data to compute the
cone angle (Λ) and the mean of ADTI model parameters in the spinal cord tract region.
5. Gradient scheme optimization: The optimization is performed off-line and the subject does not need to be in the scanner during the calculation of the optimized
gradient scheme. The flowcharts for the optimization scheme including b-factor
optimization are shown the following figures (Fig. A.2 to Fig. A.6). At the
beginning, the optimization problem is initialized by selecting the signal model
(ADTI/DTI) parameters, noise model(Rician/Gaussian), cost function (full CRLB
or partial CRLB), prior knowledge (cone angle and mean values of diffusion model
parameters) and finally to include b-factor optimization or not. Depending on

170

whether b-factor optimization is used or not, there are two options for the optimization. One is for only gradient direction optimization which uses a user input
b-factor and the other is the optimization of both b-factor and gradient directions.
For the optimization of the gradient directions only, the procedure asks the user for
a b-factor and then proceeds in a two-stage gradient directions optimization. In the
first stage, a sub-optimal solution of the optimization problem (i.e., a set of gradient
directions, Ω ) is obtained by simulated annealing method (see Fig. A.3). At this
stage of the optimization, the cost function is minimized without using the cone
angle (Λ) (uses only the mean values of the model parameters). Basic flowchart of
the simulated annealing method is shown in Fig. A.3 and the stopping criteria is
expanded in Fig. A.4. Simulated annealing (SA) is a stochastic method [73] used
here for the minimization of the cost function. The method is inspired from the
annealing technique in material science which involves heating the material to a
temperature and then cooling is slowly in a controlled fashion so that the crystals
in the material are formed without defects. In SA, at each iteration, the algorithm
allows for an additional case where the intermediate solution is accepted even if the
cost function is not reduced. The criteria is based on a probabilistic thresholding
method shown in the flowchart (Fig. A.3). This helps to prevent local minima
problems which are particularly rampant in gradient-based methods.
The next step in the optimization is the robust optimization of the gradient directions which uses the cone angle information (see Fig. A.5). The algorithm starts
by using the sub-optimal solution. Then, it discretizes the cone of angles into a set
of angular locations (grid) and looks for the worst case of cost function (maximum)
within the discretized cone of angles. After the worst case location is identified,
gradient of the cost function is calculated by finite difference method at the worst
case location and the solution is updated along the negative gradient direction (for
minimization of cost function). The process is iterated till generally the relative difference in the cost function is within tolerance levels or number of iterations have
exceeded or if the solution starts to oscillate. The oscillation issue arises due to

171

fixed step size issues. At each iteration, the algorithm finds a different worst case
location within the cone of angles and minimizes the cost function at that location.
Thus, this algorithm literally flattens the cost function curve within the cone of
angles and thus brings a uniform cost performance for a range of angles in the cone,
making the gradient scheme robust within the cone. Another important aspect of
this algorithm is the reformulation of the gradient scheme into a ring-based distribution of gradient directions. This greatly reduces the number of optimization
parameters (from the original set of gradient directions angles to a few parameters
such as number of rings, number of points on the rings and there zenith angle and
azimuthal offset angles). Algorithm uses a predefined set of number of rings and
number of points on the rings and tries a number of such configurations to choose
the best case.
For the b-factor optimization step, a simulated annealing method is preferred since
both the b-factor as well as the gradient directions are varied simultaneously and
local minima problems are significant. Gradient-based methods are bound to be
stuck at the local minima instead of the global minimum. The algorithm merges
the SA methods with the robust optimization methods described previously except
for the reformulation of the gradient scheme part. The flowchart is shown in Fig.
A.6.

172

Optimization settings initialization: select
signal (ADTI/DTI),
noise (Rician/Gaussian),
cost function (Full CRLB or partial CRLB),
prior information (cone angle (Λ), mean values of
diffusivities, (D||, D⊥) for ADTI)

User input: bfactor
No

Do b-factor
optimization?

Sub-optimal solution
using Simulated
annealing (without
using cone angle)

Yes

Robust optimal solution
for b-factor and gradient
directions using Simulated
Annealing (using cone
angle)

Robust optimal
solution with gradient
descent algorithm
(using cone angle)

Performance check
via simulations

Performance check
via simulations

Figure A.2. General overview of the overall gradient and b-factor optimization scheme.
Flowcharts for simulated annealing, robust optimization and b-factor optimization are
shown in Figs. A.3–A.6.

173

Set initial solution Ω0 and calculate cost function, S0

Set the simulated annealing parameters: start and stop temperature,
cooling rate, starting step size, success count (Nsucc) = 0, reject
count (Nrej) = 0, iterations at fixed temperature (NtryT) = 0

Exit algorithm
and set Ω as
the final
solution

Yes

Is stopping
criteria
satisfied?
No

Update gradient direction angles, Ω, by random increment
and calculate new cost function, S

Accept Ω as new
solution,
++Nsucc, Nrej=0,
S0=S

Accept Ω as
new solution,
++Nsucc,
S0=S

Yes

Yes
Is S < S0 ?
No

Is rand <
exp[-0.5 (S-S0)/S0T]?

No

Reject Ω
as new
solution,
++Nrej

Figure A.3. Flowchart describing the simulated annealing algorithm to find the suboptimal solution (not using the cone angle information). Ω = {g i ; i ∈ [1, N]}, N is
the number of gradient directions (N = 30). g i is the ith gradient direction vector:
g i ≡ [gxi , gyi , gzi] ≡ [θi , φi ]. Cost function, S = detΣCR, f ull or detΣCR, partial . S is a
function of Ω . S0 is the initial value of cost function for gradient scheme Ω0 and it is later
updated in every iteration. ‘rand’ is a function that generates random numbers between
0 and 1 with uniform probability. The flowchart for the decision box for stopping criteria
is shown in Fig. A.4.

174

Stopping
criteria
Satisfied

Yes

Is step size
too small?
No
Is NtryT >
max_NtryT or
Nsucc >
max_Nsucc?

Stopping
criteria
NOT
Satisfied

No

Yes
Is Nrej >
max_Nrej?

No

Is T >
min_T?

No

Stopping
criteria
Satisfied

Yes
Stopping
criteria
Satisfied

No

Yes
new T = cool
(T), reset
NtryT = 1,
Nsucc = 1

Is T > min_T?

Stopping
criteria
NOT
Satisfied

Yes
Reduce step size, new T =
cool (T), reset NtryT = 1,
Nsucc = 1, Nrej = Nrej/2

Stopping
criteria
NOT
Satisfied

Figure A.4. Flowchart for the stopping criteria in the simulated annealing algorithm.
Here, NtryT = number of trials at a fixed temperature T, max NtryT = maximum limit of
NtryT, Nsucc = success count, max Nsucc = maximum limit of Nsucc, Nrej = consecutive
rejection count, max Nrej = maximum limit of Nrej, T = temperature(simulated), min T
= minimum limit of T, cool(T) = cooling method for T.

175

Set sub-optimal solution as
initial solution, Ω0

User input:
ring
configurations

Reformulate the gradient
scheme in ring-based
parameters

Initialize optimization parameters: iter,
iterMax, del, delMin, osc, oscMax,
stepsize

Update
iter, del,
osc

Is iter >
iterMax or del
< delMin or
osc >
oscMax?

Yes

Exit algorithm
and set Ω as
final solution

No
Discretize cone region into finite number
of angular locations

Find worst case (max) cost function within cone
and calculate gradient of cost function at worst
case location

Change solution, Ω, towards negative gradient
direction (gradient descent)

Figure A.5. Flowchart for the robust optimization procedure for the gradient directions
utilizing the cone angle information of fiber orientations. Ω = {g i ; i ∈ [1, N]}, N is the
number of gradient directions (N = 30). g i is the ith gradient direction vector. Cost
function, S = detΣCR, f ull or detΣCR, partial . iter = number of iterations, iterMax
= maximum limit of iter, del = absolute relative change in the cost function (abs(S −
S0 /S0 )), delMin = minimum limit of del, osc = number of oscillations in the cost function,
oscMax = maximum limit of osc.

176

Set initial solution Ω0 and preset b-factor and calculate
cost function, S0

Set the simulated annealing parameters: start and stop temperature,
cooling rate, starting step size, success count (Nsucc) = 0, reject
count (Nrej) = 0, iterations at fixed temperature (NtryT) = 0

Exit algorithm
and set Ω, b
as the final
solution

Yes

Is stopping
criteria
satisfied?
No

Update Ω and b by random
increment

Find worst case (max) cost function within discretized
cone and set it as the new value of S
Accept Ω, b as
new solution,
++Nsucc, Nrej=0,
S0=S

Yes

Is S < S0 ?
No

Accept Ω, b
as new
solution,
++Nsucc,
S0=S

Yes

Is rand <
exp[-0.5 (S-S0)/S0T]?

No

Reject Ω, b
as new
solution,
++Nrej

Figure A.6. Flowchart for the robust optimization of gradient directions and b-factor using
simulated annealing algorithm utilizing the cone angle information of fiber orientations.
Here, Ω = {g i ; i ∈ [1, N]}, N is the number of gradient directions (N = 30). g i is
the ith gradient direction vector: g i ≡ [gxi , gyi , gzi] ≡ [θi , φi ]. Cost function, S =
detΣCR, f ull or detΣCR, partial . S is a function of Ω . S0 is the initial value of cost
function for gradient scheme Ω0 and it is later updated in every iteration. The flowchart
for the decision box for stopping criteria is shown in Fig. A.4.

177

6. The performance check consists of calculating the 1D plot the cost function w.r.t.
angular deviation, α, or 2D plot of cost function w.r.t. (θF , φF ). Next, the cost
functions are calculated using the CRLB formulation for a range of diffusivity as
well as fiber directions. Finally, performance indices, such as mean ratio of cost
functions of OPT30 versus MF30 under successful voxels (µr1 ) and the percentage
success (PS ) is calculated from simulations.
7. Performing the DTI/ADTI experiment with optimized gradient and/or b-factor:
Multiple sets of data are collected using the optimized protocol and a standard
MF30 protocol. Typical protocol remains the same as the preliminary DTI scan
except that number of excitations = 2 (for additional SNR improvement). The
scan time per dataset was typically 7 min 21 sec. For b-factor optimization, the
optimized b-factor is used and number of excitations = 4. Since a different b-factor
is used, this results in different TE .
8. Processing of the data from the optimized ADTI/DTI scan: The flowchart for
the processing is shown in Fig. A.7. The flowchart is similar to the preliminary
DTI data processing except that since multiple datasets are involved, a 4D volume
registration technique is incorporated in the work flow. After the spinal cord tract
voxels are extracted in each of the 4D volume, a common set of voxels are identified
amongst all the datasets. This common voxels data is then extrapolated using
repetition bootstrapping method to obtain 6000 data samples. The extrapolated
dataset is then used for model parameter estimations and calculation of covariance
matrix and the variances of various quantities.

178

DTI Datasets

Data conversion (DICOM to NifTI)

Eddy Current Correction

Spinal cord region mask generation

Linear rigid-body registration of different data volumes

DTI/ADTI model parameter estimation of co-registered
DTI data in the spinal cord region

ROI voxel extraction

Repetition bootstrapping to extrapolate 6
original datasets to 6000 datasets

Statistical analysis of estimated ADTI parameters
in ROI voxels
Compare estimation performance of OPT30
and MF30

Figure A.7. Flowchart for the processing of the multiple DTI 4D volume datasets obtained
by using the optimized protocol and compared to the MF30 protocol.

179

APPENDIX B
Gradient tables for the spinal cord
imaging study
0
θ

0.6
50

0.4
−100

0
φ sin θ

100

0.2

Figure B.1. Diffusion gradient directions (white circles) for Subject 1 optimized using
prior structural information. The underlay shows the normalized MR signal w.r.t. gradient direction angles (θ, φ).

180

Table B.1. DTI Gradient Table for Subject 1
gx

gy

gz

gx

gy

gz

1.000

-0.000

0.001

0.215

-0.923

0.318

-1.000

-0.000

0.001

-0.692

-0.648

0.318

0.800

-0.600

0.001

-0.907

0.275

0.318

-0.800

0.600

0.001

-0.215

0.923

0.318

0.674

0.712

0.198

0.692

0.648

0.318

-0.674

-0.712

0.198

0.580

-0.111

0.807

0.896

-0.011

0.445

0.404

-0.430

0.807

-0.011

-0.896

0.445

0.074

-0.586

0.807

-0.896

0.011

0.445

-0.284

-0.517

0.807

0.011

0.896

0.445

-0.534

-0.251

0.807

0.843

-0.039

0.536

-0.580

0.111

0.807

-0.039

-0.843

0.536

-0.404

0.430

0.807

-0.843

0.039

0.536

-0.074

0.586

0.807

0.039

0.843

0.536

0.284

0.517

0.807

0.907

-0.275

0.318

0.534

0.251

0.807

181

θ

0

0.8
0.6

50

0.4
−100

0
φ sin θ

100

Figure B.2. Diffusion gradient directions (white circles) for Subject 2 optimized using
prior structural information. The underlay shows the normalized MR signal w.r.t. gradient direction angles (θ, φ).
Table B.2. DTI Gradient Table for Subject 2
gx

gy

gz

gx

gy

gz

-1.000

-0.000

0.008

-0.227

-0.915

0.334

1.000

-0.000

0.008

0.679

-0.654

0.334

-0.787

-0.617

0.011

0.906

0.261

0.334

0.787

0.617

0.011

0.227

0.915

0.334

-0.742

0.656

0.139

-0.679

0.654

0.334

0.742

-0.656

0.139

-0.578

-0.110

0.809

-0.879

-0.004

0.477

-0.403

-0.429

0.809

0.004

-0.879

0.477

-0.074

-0.584

0.809

0.879

0.004

0.477

0.283

-0.516

0.809

-0.004

0.879

0.477

0.532

-0.251

0.809

-0.823

-0.023

0.567

0.578

0.110

0.809

0.023

-0.823

0.567

0.403

0.429

0.809

0.823

0.023

0.567

0.074

0.584

0.809

-0.023

0.823

0.567

-0.283

0.516

0.809

-0.906

-0.261

0.334

-0.532

0.251

0.809

182

θ

0

0.8
0.6

50

0.4
−100

0
φ sin θ

100

Figure B.3. Diffusion gradient directions (white circles) for Subject 3 optimized using
prior structural information. The underlay shows the normalized MR signal w.r.t. gradient direction angles (θ, φ).
Table B.3. DTI Gradient Table for Subject 3
gx

gy

gz

gx

gy

gz

-1.000

-0.000

-0.000

-0.229

-0.915

0.332

1.000

-0.000

-0.000

0.678

-0.656

0.332

-0.789

-0.615

0.010

0.907

0.259

0.332

0.789

0.615

0.010

0.229

0.915

0.332

-0.740

0.656

0.144

-0.678

0.656

0.332

0.740

-0.656

0.144

-0.580

-0.110

0.807

-0.879

-0.004

0.477

-0.404

-0.430

0.807

0.004

-0.879

0.477

-0.075

-0.586

0.807

0.879

0.004

0.477

0.284

-0.518

0.807

-0.004

0.879

0.477

0.534

-0.252

0.807

-0.822

-0.022

0.569

0.580

0.110

0.807

0.022

-0.822

0.569

0.404

0.430

0.807

0.822

0.022

0.569

0.075

0.586

0.807

-0.022

0.822

0.569

-0.284

0.518

0.807

-0.907

-0.259

0.332

-0.534

0.252

0.807

183

0

0.8

θ

0.6
50

0.4
−100

0
φ sin θ

0.2

100

Figure B.4. Diffusion gradient directions (white circles) for Subject 4 optimized using
prior structural information. The underlay shows the normalized MR signal w.r.t. gradient direction angles (θ, φ).
Table B.4. DTI Gradient Table for Subject 4
gx

gy

gz

gx

gy

gz

-1.000

-0.000

0.003

-0.184

-0.933

0.310

1.000

0.000

0.003

0.716

-0.626

0.310

-0.774

-0.634

0.005

0.900

0.307

0.310

0.774

0.634

0.005

0.184

0.933

0.310

-0.643

0.743

0.187

-0.716

0.626

0.310

0.643

-0.743

0.187

-0.579

-0.112

0.807

-0.910

-0.004

0.414

-0.403

-0.431

0.807

0.004

-0.910

0.414

-0.072

-0.585

0.807

0.910

0.004

0.414

0.286

-0.516

0.807

-0.004

0.910

0.414

0.534

-0.250

0.807

-0.856

-0.059

0.513

0.579

0.112

0.807

0.059

-0.856

0.513

0.403

0.431

0.807

0.856

0.059

0.513

0.072

0.585

0.807

-0.059

0.856

0.513

-0.286

0.516

0.807

-0.900

-0.307

0.310

-0.534

0.250

0.807

184

0
θ

0.6
50

0.4
0.2
−100

0
φ sin θ

100

Figure B.5. Diffusion gradient directions (white circles) for Subject 5 optimized using
prior structural information. The underlay shows the normalized MR signal w.r.t. gradient direction angles (θ, φ).
Table B.5. DTI Gradient Table for Subject 5
gx

gy

gz

gx

gy

gz

-1.000

-0.000

0.000

-0.251

-0.884

0.395

1.000

0.000

0.000

0.640

-0.659

0.395

-0.728

-0.686

0.001

0.891

0.225

0.395

0.728

0.686

0.001

0.251

0.884

0.395

-0.688

0.721

0.082

-0.640

0.659

0.395

0.688

-0.721

0.082

-0.564

-0.118

0.817

-0.721

0.624

0.301

-0.387

-0.427

0.817

-0.624

-0.721

0.301

-0.062

-0.573

0.817

0.721

-0.624

0.301

0.287

-0.500

0.817

0.624

0.721

0.301

0.526

-0.236

0.817

-0.868

-0.087

0.489

0.564

0.118

0.817

0.087

-0.868

0.489

0.387

0.427

0.817

0.868

0.087

0.489

0.062

0.573

0.817

-0.087

0.868

0.489

-0.287

0.500

0.817

-0.891

-0.225

0.395

-0.526

0.236

0.817

185

0
θ

0.6
50

0.4
−100

0
φ sin θ

0.2

100

Figure B.6. Diffusion gradient directions (white circles) for Subject 1 optimized using
completely uncertain fiber orientation. The underlay shows the normalized MR signal
w.r.t. gradient direction angles (θ, φ).
Table B.6. DTI Gradient Table for Subject 1 with Λ = 90◦
gx

gy

gz

gx

gy

gz

-0.718

-0.000

0.696

0.908

0.065

0.414

0.000

-0.718

0.696

0.398

0.819

0.414

0.718

0.000

0.696

-0.510

0.754

0.414

0.000

0.718

0.696

-0.508

-0.258

0.822

-0.707

-0.707

0.000

0.478

-0.311

0.822

0.707

-0.707

0.000

0.030

0.569

0.822

0.707

0.707

0.000

-0.008

-0.533

0.846

-0.707

0.707

0.000

0.008

0.533

0.846

-0.447

-0.391

0.804

-0.834

-0.473

0.284

0.391

-0.447

0.804

-0.150

-0.947

0.284

0.447

0.391

0.804

0.647

-0.708

0.284

-0.391

0.447

0.804

0.957

0.065

0.284

-0.908

-0.065

0.414

0.546

0.788

0.284

-0.398

-0.819

0.414

-0.276

0.918

0.284

0.510

-0.754

0.414

-0.890

0.357

0.284

186

BIBLIOGRAPHY

187

BIBLIOGRAPHY

[1] B. Horwitz. The elusive concept of brain connectivity. NeuroImage, 19:466–470,
2003.
[2] L. Lee, L. M. Harrison, and A. Mechelli. A report of the functional connectivity
workshop, Dusseldorf 2002. NeuroImage, 19:457–465, 2003.
[3] S. A. Huettel, A. W. Song, and G. McCarthy. Functional Magnetic Resonance
Imaging. Sinauer Associates, Sunderland, MA, USA, 2004.
[4] D. L. Sparks, L.-F. Lue, T. A. Martin, and J. Rogers. Neural tract tracing using
Di-I: a review and a new method to make fast Di-I faster in human brain. J.
Neuroscience Methods, 103:3–10, 2000.
[5] E. O. Stejskal and J. E. Tanner. Spin diffusion measurements: Spin Echoes in the
Presence of Time-Dependent Field Gradients. J. Chem. Phys., 42:288–292, 1965.
[6] E. O. Stejskal. Use of spin echoes in a pulsed magnetic-field gradient to study
anisotropic, restricted diffusion and flow. J. Chem. Phys., 43:3597–3603, 1965.
[7] P. J. Basser, J. Matiello, and D. Le Bihan. MR diffusion tensor spectroscopy and
imaging. Biophys. J., 66(1):259–267, 1994.
[8] L. G. Raguin, D. Hernando, D. C. Karampinos, L. Ciobanu, B. P. Sutton, Z.-P.
Liang, and J. G. Georgiadis. Quantitative Analysis of q-Space MRI Data. In
IFMBE Proc. 3rd European Medical and Biological Engineering Conference, vol.
11, 2005.
[9] Y. Assaf, R. Z. Freidlin, G. K. Rohde, and P. J. Basser. New modeling and experimental framework to characterize hindered and restricted water diffusion in brain
white matter. Magn. Reson. Med., 52(5):965–978, 2004.
[10] V. J. Wedeen, T. J. Reese, D. S. Tuch, and et al. Mapping fiber orientation spectra
in cerebral white matter with Fourier-transform diffusion MRI. In Proc. Int. Soc.
Magn. Reson. Med. 7th Scientific meeting ISMRM99, page 321, 1999.
[11] D. S. Tuch. Q-Ball Imaging. Magn. Reson. Med., 52:1358–1372, 2004.
[12] P. J. Basser, J. Matiello, and D. Le Bihan. Estimation of the effective self-diffusion
tensor from the NMR spin echo. J. Magn. Reson. B, 103:247–254, 1994.
[13] P. B. Kingsley. Introduction to diffusion tensor imaging mathematics part II:
Anisotropy, diffusion weighting factors and gradient encoding schemes. Concept
Magnetic Res. A, 28:101–122, 2006.

188

[14] S. M. Smith, M. Jenkinson, H. Johansen-Berg, D. Rueckert, T. E. Nichols, C. E.
Mackay, K. E. Watkins, O. Ciccarelli, M. Z. Cader, P. M. Matthews, and T. E. J.
Behrens. Tract-based spatial statistics: Voxelwise analysis of multi-subject diffusion
data. NeuroImage, 31:1487–1505, 2006.
[15] P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi. In vivo fiber
tractography using DT-MRI data. Magn. Reson. Med., 44:625–632, 2000.
[16] S. Mori, B. J. Crain, V. P. Chacko, and P. C. van Zijl. Three-dimensional tracking
of axonal projections in the brain by magnetic resonance imaging. Ann. Neurol.,
45:265–269, 1999.
[17] K. M. Hasan, D. L. Parker, and A. L. Alexander. Comparison of gradient encoding
schemes for diffusion-tensor MRI. J. Magn. Reson. Imag., 13:769–780, 2001.
[18] N. G. Papadakis, D. Xing, C. L.-H. Huang, L. D. Hall, and T. A. Carpenter. A
comparative study of acquisition schemes for diffusion tensor imaging using MRI.
J. Magn. Reson., 137(1):67–82, 1999.
[19] S. Skare, M. Hedehus, M. E. Moseley, and T.-Q. Li. Condition number as a measure
of noise performance of diffusion tensor data acquisition schemes with MRI. J.
Magn. Reson., 147(2):340–352, 2000.
[20] P. G. Batchelor, D. Atkinson, D. L. G. Hill, F. Calamante, and A. Connelly.
Anisotropic noise propagation in diffusion tensor MRI sampling schemes. Magn.
Reson. Med., 49(6):1143–1151, 2003.
[21] H. Peng and K. Arfanakis. Diffusion tensor encoding schemes optimized for white
matter fibers with selected orientations. Magn. Reson. Imaging, 25:147–153, 2007.
[22] D. K. Jones, M. A. Horsfield, and A. Simmons. Optimal strategies for measuring
diffusion in anisotropic systems by magnetic resonance imaging. Magn. Reson.
Med., 42(3):515–525, 1999.
[23] D. C. Alexander. A general framework for experiment design in diffusion MRI and
its application in measuring direct tissue-microstructure features. Mag. Res. Med.,
60(2):439–448, 2008.
[24] O. Brihuega-Moreno, F. P. Heese, and L. D. Hall. Optimization of diffusion measurements using Cramer-Rao lower bound theory and its application to articular
cartilage. Mag. Res. Med., 50(5):1069–1076, 2003.
[25] S. Standring. Gray’s Anatomy: The Anatomical Basis of Clinical Practice.
Churchill-Livingstone, London, UK, 40th edition, 2008.
[26] J. C. E. Underwood and S. S. Cross. General and systematic pathology. ChurchillLivingstone, London, UK, 5th edition, 2009.

189

[27] D. Facon, A. Ozanne, P. Fillard, J. F. Lepeintre, C. Tournoux-Facon, and
D. Ducreux. MR diffusion tensor imaging and fiber tracking in spinal cord compression. Am J Neuroradiol., 26(6):1587–1594, 2005.
[28] K. Shanmuganathan, R. P. Gullapalli, J. Zhuo, and S. E. Mirvis. Diffusion tensor
MR imaging in cervical spine trauma. American Journal of Neuroradiology, 29:655–
659, 2008.
[29] M. F. Meek, M. W. Stenekes, H. M. Hoogduin, and J. P. Nicolai. In vivo threedimensional reconstruction of human median nerves by diffusion tensor imaging.
Experimental Neurology, 198:479–482, 2006.
[30] U. Techavipoo, A. F. Okai, J. Lackey, J. Shi, M. A. Dresner, T. P. Leist, and
S. Lai. Toward a practical protocol for human optic nerve DTI with EPI geometric
distortion correction. J Magn Reson Imaging, 30:699–707, 2009.
[31] J. Hiltunen, T. Suortti, S. Arvela, M. Seppa, R. Joensuu, and R. Hari. Diffusion
tensor imaging and tractography of distal peripheral nerves at 3 T. Clin Neurophysiol, 116:2315–2323, 2005.
[32] J. H. Kan, A. M. Heemskerk, Z. Ding, A. Gregory, G. Mencio, K. Spindler, and
B. M. Damon. DTI-based muscle fiber tracking of the quadriceps mechanism in
lateral patellar dislocation. Journal of Magnetic Resonance Imaging, 29:663–670,
2009.
[33] J. V. Beck and K. J. Arnold. Parameter Estimation in Engineering and Science.
John Wiley & Sons, New York, NY, USA, 1977.
[34] J. V. Beck and K. A Woodbury. Inverse problems and parameter estimation: integration of measurements and analysis . Meas. Sci. Technol., 9:839–847, 1998.
[35] P. G. Lindberg, A. Feydy, and M. A. Maier. White matter organization in cervical
spinal cord relates differently to age and control of grip force in healthy subjects.
J. Neurosci., 30:4102–4109, 2010.
[36] S. Morisaki, Y. Kawai, M. Umeda, M. Nishi, R. Oda, H. Fujiwara, K. Yamada,
T. Higuchi, C. Tanaka, M. Kawata, and T. Kubo. In vivo assessment of peripheral
nerve regeneration by diffusion tensor imaging. J Magn Reson Imaging, 33:535–542,
2011.
[37] S. M. Hesseltine, M. Law, J. Babb, M. Rad, S. Lopez, Y. Ge, G. Johnson, and R. I.
Grossman. Diffusion tensor imaging in multiple sclerosis: assessment of regional
differences in the axial plane within normal-appearing cervical spinal cord. AJNR
Am J Neuroradiol, 27:1189–1193, 2006.
[38] J. Renoux, D. Facon, P. Fillard, I. Huynh, P. Lasjaunias, and D. Ducreux. MR
diffusion tensor imaging and fiber tracking in inflammatory diseases of the spinal
cord. AJNR Am J Neuroradiol, 27:1947–1951, 2006.
190

[39] W. Gao, H. Zhu, and W. Lin. A unified optimization approach for diffusion tensor
imaging technique. NeuroImage, 44:729–741, 2009.
[40] K. J. Friston. Functional and effective connectivity in neuroimaging: A synthesis.
Human Brain Mapping, 2:56–78, 1994.
[41] A. McLntosh and F. Gonzalez-Lima. Structural equation modeling and its application to network analysis in functional brain imaging. Human Brain Mapping,
2:2–22, 1994.
[42] K. Friston, L. Harrison, and W. Penny. Dynamic causal modelling. NeuroImage,
19:1273–1302, 2003.
[43] R. Goebel, A. Roebroeck, D. Kim, and E. Formisano. Investigating directed cortical
interactions in time-resolved fMRI data using vector autoregressive modeling and
Granger causality mapping. Magnetic Resonance Imaging, 21:1251–1261, 2003.
[44] L. Harrison, W. Penny, and K. Friston. Multivariate autoregressive modeling of
fMRI time series. NeuroImage, 19:1477–1491, 2003.
[45] S. Ogawa, R. S. Menon, D. W. Tank, S. G. Kim, H. Merkle, J. M. Ellermann,
and K. Ugurbil. Functional brain mapping by blood oxygenation level-dependent
contrast magnetic resonance imaging. A comparison of signal characteristics with
a biophysical model. Biophysical journal, 64:803–812, 1993.
[46] J. A. Brunberg, K. A. Frey, J. A. Horton, J. P. Deveikis, D. A. Ross, and R. A.
Koeppe. H2 O positron emission tomography determination of cerebral blood flow
during balloon test occlusion of the internal carotid artery. American Journal of
Neuroradiology, 15:725–732, 1994.
[47] E. Niedermeyer and F. H. L. Silva. Electroencephalography: basic principles, clinical
applications, and related fields. Lippincott Williams & Wilkins, Philadelphia, PA,
USA, 2005.
[48] M. H¨m¨l¨inen, R. Hari, R. J. Ilmoniemi, J. Knuutila, and O. V. Lounasmaa.
a aa
Magnetoencephalography–theory, instrumentation, and applications to noninvasive
studies of the working human brain. Rev. Mod. Phys., 65(2):413–497, 1993.
[49] G. A. Johnson, H. Benveniste, R. D. Black, L. W. Hedlund, R. R. Maronpot, and
B. R. Smith. Histology by magnetic resonance microscopy. Magnetic Resonance
Quarterly, 9:1–30, 1993.
[50] G. M. Fatterpekar, T. P. Naidich, B. N. Delman, J. G. Aguinaldo, S. H. Gultekin,
C. C. Sherwood, P. R. Hof, B. P. Drayer, and Z. A. Fayad. Cytoarchitecture of the
human cerebral cortex: MR microscopy of excised specimens at 9.4 Tesla. American
Journal of Neuroradiology, 23:1313–1321, 2002.

191

[51] R. Mizutani, A. Takeuchi, K. Uesugi, S. Takekoshi, R. Y. Osamura, and Y. Suzuki.
Microtomographic analysis of neuronal circuits of human brain. Cerebral Cortex,
20:1739–1748, 2010.
[52] K. L. Briggman and W. Denk. Towards neural circuit reconstruction with volume electron microscopy techniques. Current Opinion in Neurobiology, 16:562–570,
2006.
[53] M. D. Greicius, K. Supekar, V. Menon, and R. F. Dougherty. Resting-State Functional Connectivity Reflects Structural Connectivity in the Default Mode Network.
Cerebral Cortex, 19:72–78, 2009.
[54] M. Smits, M. W. Vernooij, P. A. Wielopolski, A. J. Vincent, G. C. Houston, and
A. van der Lugt. Incorporating functional MR imaging into diffusion tensor tractography in the preoperative assessment of the corticospinal tract in patients with
brain tumors. American Journal of Neuroradiology, 28:1354–1361, 2007.
[55] D. Le Bihan, J. F. Mangin, C. Poupon, C. A. Clark, S. Pappata, N. Molko, and
H. Chabriat. Diffusion tensor imaging: concepts and applications. J. Magn. Reson.
Imaging, 13:534–546, 2001.
[56] D. J. Werring, C. A. Clark, G. J. Barker, A. J. Thompson, and D. H. Miller.
Diffusion tensor imaging of lesions and normal-appearing white matter in multiple
sclerosis. Neurology, 52(8):1626–1632, 1999.
[57] P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi. In vivo fiber
tractography using DT-MRI data. Magnetic Resonance in Medicine, 44:625–632,
2000.
[58] T. E. Conturo, N. F. Lori, T. S. Cull, E. Akbudak, A. Z. Snyder, J. S. Shimony,
R. C. McKinstry, H. Burton, and M. E. Raichle. Tracking neuronal fiber pathways
in the living human brain. Proceedings of National Academy of Science, USA,
96:10422–10427, 1999.
[59] T. E. Behrens, H. J. Berg, S. Jbabdi, M. F. Rushworth, and M. W. Woolrich.
Probabilistic diffusion tractography with multiple fibre orientations: What can we
gain? Neuroimage, 34(1:144–155, 2007.
[60] G. J. M. Parker, H. A. Haroon, and C. A. M. Wheeler-Kingshott. A framework for
a streamline-based Probabilistic Index of Connectivity (PICo) using a structural
interpretation of MRI diffusion measurements. J. Magn. Reson. Imaging, 18:242–
254, 2003.
[61] C. M. Ellis, A. Simmons, D. K. Jones, J. Bland, J. M. Dawson, M. A. Horsfield,
S. C. R. Williams, and P. N. Leigh. Diffusion tensor MRI assesses corticospinal
tract damage in ALS. Neurology, 53(5):1051–1058, 1999.

192

[62] H. Mamata, F. A. Jolesz, and S. E. Maier. Apparent diffusion coefficient and
fractional anisotropy in spinal cord: age and cervical spondylosis-related changes.
J Magn Reson Imaging, 22(1):38–43, 2005.
[63] A. M. Heemskerk, G. J. Strijkers, A. Vilanova, M. R. Drost, and K. Nicolay. Determination of mouse skeletal muscle architecture using three-dimensional diffusion
tensor imaging. Magnetic Resonance in Medicine, 53:1333–1340, 2005.
[64] D. A. Lansdown, Z. Ding, M. Wadington, J. L. Hornberger, and B. M. Damon.
Quantitative diffusion tensor MRI-based fiber tracking of human skeletal muscle.
Journal of Applied Physiology, 103:673–81, 2007.
[65] M. Ries, R. A. Jones, V. Dousset, and C. T. Moonen. Diffusion tensor MRI of the
spinal cord. Magn. Reson. Med., 44:884–892, 2000.
[66] J. P. Mottershead, K. Schmierer, M. Clemence, J. S. Thornton, F. Scaravilli, G. J.
Barker, P. S. Tofts, J. Newcombe, M. L. Cuzner, R. J. Ordidge, W. I. McDonald,
and D. H. Miller. High field mri correlates of myelin content and axonal density in
multiple sclerosis: A post-mortem study of the spinal cord. J. Neurol., 250:1293–
1301, 2003.
[67] D. Ducreux, J. F. Lepeintre, P. Fillard, C. Loureiro, M. Tadi´, and P. Lasjaue
nias. MR diffusion tensor imaging and fiber tracking in 5 spinal cord astrocytomas.
American J. Neuroradiology, 27:214–216, 2006.
[68] A. W. Anderson. Measurement of fiber orientation distributions using high angular
resolution diffusion imaging. Magnetic Resonance in Medicine, 54:1194–1206, 2005.
[69] M. Onu, P. Gervai, J. Cohen-Adad, J. Lawrence, J. Kornelsen, B. Tomanek, and
U. N. Sboto-Frankenstein. Human cervical spinal cord funiculi: investigation with
magnetic resonance diffusion tensor imaging. Journal of Magnetic Resonance Imaging, 31:829–837, 2010.
[70] J. Gullapalli, J. Krejza, and E. D. Schwartz. In vivo DTI evaluation of white matter
tracts in rat spinal cord. J. Magn. Reson. Imaging, 24(1):231–234, 2006.
[71] V. Gulani, G. A. Iwamoto, H. Jiang, J. S. Shimony, A. G. Webb, and P. C. Lauterbur. A multiple echo pulse sequence for diffusion tensor imaging and its application
in excised rat spinal cords. Mag. Res. Med., 38(6):868–873, 1997.
[72] N. Yanasak, J. D. Allison, Q. Zhao, T. C.-C. Hu, and K. Dhandapani. Non-uniform
gradient prescription for precise angular measurements using DTI. Med. Image.
Comput. Comput. Assist. Interv., 5241:866–873, 2008.
[73] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller.
Equations of State Calculations by Fast Computing Machines. J. Chem. Phys.,
21:1087–1092, 1953.

193

[74] J. L. R. Andersson. Maximum a posteriori estimation of diffusion tensor parameters
using a Rician noise model: Why, how and but. NeuroImage, 42(4):1340–1356, 2008.
[75] E. M. Haacke, R. W. Brown, M. R. Thompson, and R. Venkatesan. Magnetic
Resonance Imaging: Physical Principles and Sequence Design. Wiley–LISS, New
York, NY, USA, 1999.
[76] D. G. Nishimura. Principles of Magnetic Resonance Imaging. Stanford University,
Stanford, CA, USA, 1996.
[77] P. Woodward. MRI for Technologists. McGraw–Hill, New York, NY, USA, 2nd
edition, 2001.
¨
[78] A. Einstein. Uber die von der molekularkinetischen Theorie der W¨rme geforderte
a
Bewegung von in ruhenden Fl¨ ssigkeiten suspendierten Teilchen.(English: On the
u
movement of small particles suspended in a stationary liquid demanded by the
molecular-kinetic theory of heat.). Annalen der Physik, 322:549–560, 1905.
[79] P. T. Callaghan. How Two Pairs of Gradient Pulses Give Access to New Information
about Molecular Dynamics. The Open-Access Journal for the Basic Principles of
Diffusion Theory, Experiment and Application: Diffusion Fundamentals 2, 64:1–18,
2005.
[80] H. C. Torrey. Bloch equations with diffusion terms. Physical Review, 104:563–565,
1956.
[81] J. Mattiello, P. J. Basser, and D. Le Bihan. Analytical expressions for the b matrix
in nmr diffusion imaging and spectroscopy. JMRA, 108:131–141, 1994.
[82] J. E. Tanner and E. O. Stejskal. Restricted self-diffusion of protons in colloidal
systems by the pulsed-gradient, spin-echo method. J. Chem. Phys., 49(4):1768–
1777, 1968.
[83] P. T. Callaghan. Pulsed-gradient spin-echo NMR for planar, cylindrical, and spherical pores under conditions of wall relaxation. J. Magn. Reson. A, 113:53–59, 1995.
[84] S. Majumdar, S. S. Udpa, and L. G. Raguin. Robust optimization of diffusionweighted MRI protocols used for fiber reconstruction. J. Phys: Conf. Series,
135:012069, 2008.
[85] S. Majumdar, D. C. Zhu, G. Raguin, and S. S. Udpa. Optimization of diffusion
encoding gradients in axisymmetric diffusion tensor imaging using a priori structure information. In Proc. Int. Soc. Magn. Reson. Med. 17th Scientific meeting
ISMRM09, 2009.
[86] D. S. Tuch, T. G. Reese, M. R. Wiegell, and V. J. Wedeen. Diffusion MRI of
complex neural architecture. Neuron, 40:885–895, 2003.

194

[87] D. C. Alexander. Multiple-Fiber Reconstruction Algorithms for Diffusion MRI.
Ann. N.Y. Acad. Sci., 1064:113–133, 2005.
[88] D. S Tuch, T. G. Reese, M.R. Wiegell, N. Makris, J.W. Belliveau, and V. J. Wedeen.
High angular resolution diffusion imaging reveals intravoxel white matter fiber heterogeneity. Mag. Res. Med., 48:577–582, 2002.
[89] C. P. Hess, P. Mukherjee, E. T. Han, D. Xu, and D. B. Vigneron. Q-ball reconstruction of multimodal fiber orientations using the spherical harmonic basis. Magnetic
Resonance in Medicine, 56:104–17, 2006.
[90] C. Lenglet, J. S. Campbell, M. Descoteaux, G. Haro, P. Savadjiev, D. Wassermann,
A. Anwander, R. Deriche, G. B. Pike, G. Sapiro, K. Siddiqi, and P. M. Thompson.
Mathematical methods for diffusion MRI processing. NeuroImage, 45:S111–S122,
2009.
[91] P. Mansfield. Multi-planar image formation by using NMR spin-echoes. J Phys C,
10:L55–L58, 1977.
[92] F. Schick, J. Forster, J. Machann, R. Kuntz, and C. D. Claussen. Improved Clinical
Echo-Planar MRI Using Spatial-Spectral Excitation. JMRI, 8:960–967, 1998.
[93] T. G. Reese, O. Heid, R. M. Weisskoff, and V. J. Wedeen. Reduction of EddyCurrent-Induced Distortion in Diffusion MRI Using a Twice-Refocused Spin Echo.
Mag. Res. Med., 49:177–182, 2003.
[94] J. Finsterbusch. Double-spin-echo diffusion weighting with a modified eddy current
adjustment. Mag. Res. Imag., 28:434–440, 2010.
[95] B. J. Mock and C. R. Michelich. B-value calculation and correction using a linear
segment gradient waveform model. United States Patent, 2003. Patent no. US 6
670 812 B1.
[96] P. J. Basser, S. Pajevic, C. Pierpaoli, J. Duda, and A. Aldroubi. In vivo fiber
tractography using DT MRI data. Magn. Reson. Med., 44(4):625–632, 2000.
[97] C. Poupon, C. A. Clark, V. Frouin, J. Regis, I. Bloch, D. Le Bihan, and J. Mangin.
Regularization of diffusion-based direction maps for the tracking of brain white
matter fascicles. Neuroimage, 12:184–195, 2000.
[98] G. J. Parker and D. C. Alexander. Probabilistic anatomical connectivity derived
from the microscopic persistent angular structure of cerebral tissue. Philos. Trans.
R. Soc. Lond., B, Biol. Sci., 360:893–902, 2005.
[99] B. Chen and A. W. Song. Diffusion tensor imaging fiber tracking with local tissue
property sensitivity: phantom and in vivo validation. Magn Reson Imaging, 26:103–
108, 2008.

195

[100] M. W. Woolrich, S. Jbabdi, B. Patenaude, M. Chappell, S. Makni, T. Behrens,
C. Beckmann, M. Jenkinson, and S. M. Smith. Bayesian analysis of neuroimaging
data in FSL. Neuroimage, 45:S173–186, 2009.
[101] T. E. J. Behrens, M. W. Woolrich, M. Jenkinson, H. Johansen-Berg, R. G. Nunes,
S. Clare, P. M. Matthews, J. M. Brady, and S. M. Smith. Characterization and
propagation of uncertainty in diffusion-weighted MR imaging. Mag. Res. Med.,
50(5):1077–1088, 2003.
[102] S. Mori and P. C. M. van Zijl. Fiber tracking: principles and strategies – a technical
review. NMR in Biomedicine, 15:468–480, 2002.
[103] H. Jiang, P. C. van Zijl, J. Kim, G. D. Pearlson, and S. Mori. DtiStudio: resource
program for diffusion tensor computation and fiber bundle tracking. Comput Methods Programs Biomed, 81:106–116, 2006.
[104] S. Mori, W. E. Kaufmann, C. Davatzikos, B. Stieltjes, L. Amodei, K. Fredericksen,
G. D. Pearlson, E. R. Melhem, M. Solaiyappan, G. V. Raymond, H. W. Moser,
and P. C. van Zijl. Imaging cortical association tracts in the human brain using
diffusion-tensor-based axonal tracking. Magn Reson Med, 47:215–223, 2002.
[105] S. Hofer and J. Frahm. Topography of the human corpus callosum revisited–
comprehensive fiber tractography using diffusion tensor magnetic resonance imaging. Neuroimage, 32:989–994, 2006.
[106] R. Brecheisen, B. Platel, A. Vilanova, and B. ter Haar Romeny. Parameter sensitivity visualization for DTI fiber tracking. IEEE Trans Vis Comput Graph, 15:1441–
1448, 2009.
[107] S. Correia, S. Y. Lee, T. Voorn, D. F. Tate, R. H. Paul, S. Zhang, S. P. Salloway,
P. F. Malloy, and D. H. Laidlaw. Quantitative tractography metrics of white matter
integrity in diffusion-tensor MRI. Neuroimage, 42:568–581, 2008.
[108] A. Stadlbauer, E. Salomonowitz, G. Strunk, T. Hammen, and O. Ganslandt. Agerelated degradation in the central nervous system: assessment with diffusion-tensor
imaging and quantitative fiber tracking. Radiology, 247:179–188, 2008.
[109] S. M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, New Jersey, USA, 1 edition, 1993. pp. 47–49.
[110] R. M. Henkelman. Measurement of signal intensities in the presence of noise in MR
images. Med. Phys., 12(2):232–233, 1985.
[111] S. O. Rice. Mathematical analysis of random noise. J. Bell System Tech., 23:282–
332, 1944.
[112] S. O. Rice. Mathematical Analysis of Random Noise. J. Bell System Tech., 24:46–
156, 1945.

196

[113] H. Gudbjartsson and S. Patz. The Rician distribution of noisy MRI data. Magn.
Reson. Med., 34(6):910–914, 1995.
[114] C. G. Koay and P. J. Basser. Analytically exact correction scheme for signal extraction from noisy magnitude MR signals. J. Mag. Res., 179(2):317–322, 2006.
[115] K. Levenberg. A Method for the solution of certain non-linear problems in least
squares. The Quarterly of Applied Mathematics, 2:164–168, 1944.
[116] D. Marquardt. An algorithm for least-squares estimation of nonlinear parameters.
SIAM Journal on Applied Mathematics, 11:431–441, 1963.
[117] J Sijbers, A. J. den Dekker, P. Scheunders, and D. Van Dyck. Maximum-likelihood
estimation of Rician distribution parameters. IEEE Trans. Med. Imag., 17(3):357–
361, 1998.
[118] D. C. Alexander. Axon radius measurements in vivo from diffusion MRI: a feasibility
study. In Proc. Eleventh IEEE International Conference on Computer Vision,
Workshop on Mathematical Methods in Biomedical Image Analysis, pages 1–8, 2007.
[119] L. G. Raguin, S. Majumdar, and S. S. Udpa. Design of optimal experimental parameters for diffusion-weighted MRI fiber-tracking protocols. Int. J. Appl. Electromagn.
Mech., 28(1–2):61–67, 2008.
[120] L. L. Scharf and L. T. McWhorter. Geometry of the Cramer-Rao bound. In Proc.
IEEE Sixth SP Workshop on Statistical Signal and Array Processing, pages 5–8,
1992.
[121] M. A. Woodbury. Inverting modified matrices. Memorandum Rept., Statistical
Research Group, Princeton University, 42:4, 1950.
[122] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by Simulated Annealing. Science, 220(4598):671–680, 1983.
[123] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. Numerical
recipes in C: the art of scientific computing. Cambridge University Press, Cambridge, MA, USA, 2 edition, 1992.
[124] National Electrical Manufacturers Association. Determination of signa-to-noise ratio (SNR) in diagnostic magnetic resonance images. Technical Report No. MS 1,
NEMA, 2008.
[125] A. H. Poonawalla and X. J. Zhou. Analytical error propagation in diffusion
anisotropy calculations. J. Mag. Res. Imag., 19:489–498, 2004.
[126] M. Jenkinson, P. R. Bannister, J. M. Brady, and S. M. Smith. Improved optimization for the robust and accurate linear registration and motion correction of brain
images. NeuroImage, 17(2):825–841, 2002.

197

[127] S. Pajevic and P. J. Basser. Parametric and non-parametric statistical analysis of
DT-MRI data. JMR, 161:1–14, 2003.
[128] G. Nair, J. D. Carew, S. Usher, D. Lu, X. P. Hu, and M. Benatar. Diffusion tensor
imaging reveals regional differences in the cervical spinal cord in amyotrophic lateral
sclerosis. NeuroImage, 53:576–583, 2010.
[129] C. A. Wheeler-Kingshott, S. J. Hickman, G. J. Parker, O. Ciccarelli, M. R. Symms,
D. H. Miller, and G. J. Barker. Investigating cervical spinal cord structure using
axial diffusion tensor imaging. Neuroimage, 16:93–102, 2002.
[130] M. Filippi, M. Cercignani, M. Inglese, M. A. Horsfield, and G. Comi. Diffusion
tensor magnetic resonance imaging in multiple sclerosis. Neurology, 56:304–311,
2001.
[131] Y. Assaf and P. J. Basser. Composite hindered and restricted model of diffusion(charmed) mr imaging of human brain. J. Neuroimage, 27:48–58, 2005.

198