Simultaneous Multislice Spiral and EPI Chemical Shift Imaging
By
Obaidah Anees Abuhashem
B.S. Electrical Engineering and Computer Science
Massachusetts Institute of Technology, 2013
Submitted to the Department of Electrical Engineering and Computer Science in
partial fulfillment of the requirements for the degree of
ARHtU
Master of Engineering in Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE
at the
OF TECHNOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
OCT 2 9 2013
September 2013
@MassachusettsJIstitute 9f Technology 2013. All rights reserved
LIBRARIES
...........................................................
A u th o r
Department of Electrical Engineering and Computer Science
September 3, 2013
C ertified b y ..........................................................
,.......................................
. ..... .......................................
Elfar Adalsteinsson
Associate Professor of Electrical Engineering and Computer Science
Associate Professor of Harvard-MIT Division of Health, Science and Technology
Institute for Medical Engineering and Science
Thesis Supervisor
Accepted by .......................................
.................................
Prof. Albert R. Meyer
Chairman, Masters of Engineering Thesis Committee
Simultaneous Multislice Spiral and EPI Chemical Shift Imaging
By
Obaidah Anees Abuhashem
Submitted to the Department of Electrical Engineering and Computer Science on
September 3, 2013 in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
The current prominent excitation methods of 3D slabs used for MR Spectroscopy
Imaging (MRSI) include long dead times in each TR. This dead time is necessary for
magnetization moments' longitudinal relaxation, and so a good SNR efficiency. The
fact that SNR is one of the dominant challenges that MRSI faces makes it an
expensive tradeoff to decrease the dead time in the 3D excitation methods.
Therefore, I investigate the possibility of using 2D slice excitation methods with
simultaneous multi-slice excitation and the associated reconstructions to address
some of the limitations of 3D excitations with the goal of increasing the overall
efficiency of the MRSI acquisition and acquiring one average in a much faster time.
The benefit of using simultaneous multislice for 2D spectroscopy is explained, and it
is shown how this method can accelerate the 2D acquisition with quantifications of
the involved tradeoffs. Then the details of designing RF pulses for SMS spectroscopy
imaging are explained, and a pulse to be used is proposed. I discuss the sequences
used to encode the k-space, where spiral trajectories are an efficient choice.
Therefore, a method that can be used for SMS spiral spectroscopy acquired data is
proposed. The proposed method accelerates the minimum scan time of 2D
excitation for a single average by a factor of four. This acceleration is possible with
SNR degradation less than 7% of the ordinary 2D excitation method.
Thesis Supervisor: Elfar Adalsteinsson
Title: Associate Professor of Electrical Engineering and Computer Science
Associate Professor of Harvard-MIT Division of Health, Science and Technology
iii
iv
Acknowledgements
Here, I would like to acknowledge the people who made this work possible.
First, and most importantly, I would like to express my sincere appreciation
to my advisor, Professor Elfar Adalsteinsson for his guidance, supervision,
assistance and encouragement during the times I worked with him. I have been
grateful and honored to be a part of his group and to take the classes he teaches
during my study time at MIT. Our discussions has taught me a lot, not only about
research and MRI, but how to be a creative thinker and successful in life, THANK
YOU.
I feel very fortunate that I had the chance to work with Dr. Berkin Bilgic. He
was always there to help me whenever I needed assistance since my first day at the
MIT MRI lab. The work in this thesis would have not been possible without his
assistance.
I am very thankful to Professor Kawin Setsompop for his help in the work
done in this thesis. I feel very honored that I had the chance to get advice and
discuss my project with the man who founded the novel approach that I am building
my thesis on. I am also grateful to Dr. Borjan Gagoski for teaching me about the
foundations for some of the methods I am using in this thesis. He was a great source
whenever I needed more information.
It has been a pleasure to work and interact with students in the MRI lab. I
learned a lot from Audrey about MR. Itthi's help was there whenever I needed it. I
am glad to know Filiz, Jeff, Trina and Shao Ying.
This thesis marks the end of the five years I spent at MIT for my
undergraduate and masters educations. Thanks to my friends who were around me
all the time, MIT became the most memorable and enjoyable place. I will not forget
the joy of pulling all-nighters working on p-sets and projects, or just to chill out. It
v
will be one of the many unique things that remind me of MIT. I will not forget
walking in the Infinite Corridor, Stata Center, RLE, or in libraries. It always made me
feel home away from home.
Finally, I would like to deeply thank my Mom and Dad for instilling in me the
love of knowledge. This work would have not possible without their support,
encouragement and love during my education years at MIT.
Obaidah A. Abuhashem
Cambridge, Massachusetts, USA
vi
Table of Content
Abstract.....................................................................................................................................
Acknow ledgem ent...................................................................................................................
Table of Content....................................................................................................................
iii
v
vii
List of Figures...........................................................................................................................ix
List of Tables ...........................................................................................................................
Chapter 1
Introduction and M otivation....................................................................
xv
1
1.1
The Im portance of M R Spectroscopic Im aging ........................................................
1
1.2
M R Spectroscopic Im aging Challenges ......................................................................
2
1.3
Excitation Sequences and Possible Solutions ...........................................................
2
1.4
Scan Tim e and SNR...................................................................................................................
5
1.5
Simultaneous Multislice Imaging for 2D Spectroscopy.......................................
7
Chapter 2
Simultaneous Multislice Imaging..........................................................
9
2.1
SM S Concept ..............................................................................................................................
10
2.2
SM S Theory ................................................................................................................................
12
2.2.1
Blipped-W ideband Approach...............................................................................
12
2.2.2
Blipped-CAIPI Approach........................................................................................
14
2.3
SM S Im age Reconstruction ...........................................................................................
19
2.4
SM S for spectroscopy ......................................................................................................
21
SM S RF Pulse Design.................................................................................
25
3.1
Excitation Background....................................................................................................
25
3.2
Design Param eters..................................................................................................................26
Chapter 3
Vii
3.2.1
Slice Thickness.................................................................................................................27
3.2.2
Separation Between Slices Excited Simultaneously ....................................
27
3.2.3
Spectral Bandw idth..................................................................................................
27
3.3
Design M ethod..........................................................................................................................29
3.4
RF Pulse Design........................................................................................................................30
3.4.1
RF Sinc Pulse.....................................................................................................................32
3.4.2
Parks-M cClellan Pulse Design Algorithm .........................................................
3.5
Bloch Simulator........................................................................................................................45
3.6
Sum m ary.....................................................................................................................................46
Chapter 4
SMS Spectroscopic Imaging: Data Acquisition ................................
39
47
4.1
Spiral acquisition for CS1..................................................................................................
48
4.2
Spiral with SM S ........................................................................................................................
51
4.3
SM S acquired data synthesis.........................................................................................
53
4.4
Data Reconstruction ..............................................................................................................
54
4.5
Spiral SM S Simulations and Results..........................................................................
55
4.5.1
Structural Data w ith one frequency..................................................................
55
4.5.2
Spectroscopy Data..........................................................................................................59
4.6
EPI SM S Spectroscopy...........................................................................................................68
4.6.1
EPI SM S Spectroscopy Sim ulations.....................................................................
69
4.6.2
EPI SM S Spectroscopy Results .............................................................................
70
4.7
Sum m ary.....................................................................................................................................78
Conclusion ...................................................................................................
81
Appendix A ...............................................................................................................................
85
Appendix B ..............................................................................................................................
87
Appendix C ..............................................................................................................................
91
Appendix D ..............................................................................................................................
95
Bibliography...........................................................................................................................
99
Chapter 5
viii
List of Figures
Figure 1.1: Encoding scheme for conventional 3D excitation, phase-encoded MRSI
a cq u isitio n ....................................................................................................................................................
3
Figure 1.2: Simulated SNR efficiency for different main field strengths......................
4
Figure 1.3: Encoding scheme for conventional 2D excitation, phase-encoded MRSI
a cq u isitio n ....................................................................................................................................................
5
Figure 2.1: The phase seen by each slice in SMS excitation and the phase difference
in o n slice ....................................................................................................................................................
12
Figure 2.2: Phase switching in the excited slices in SMS between Gz blips, and phase
accumulation at the slice edge when blipped-wideband is used..................................
13
Figure 2.3: Phase switching in the excited slices in SMS between Gz blips, and phase
accumulation at the slice edge when blipped-CAIPI is used...........................................
15
Figure 2.4: The phase seen by each slice in SMS excitation when isocenter slice is not
ex cite d ..........................................................................................................................................................
16
Figure 2.5: Phase switching in the excited slices in SMS between Gz blips when
isocenter slice is not excited...............................................................................................................17
Figure 2.6: Gz blips when 2, 3 or 4 slices to be excited using SMS. In Addition to the
caused Phase switching in the excited slices between Gz blips, and phase
accumulation at the slice edge of each slice............................................................................18
Figure 2.7: The acquired data using SMS of two slices without recon for un-aliasing.
.........................................................................................................................................................................
19
Figure 2.8: The acquired slices using single slice excitation for each one................. 20
ix
Figure 2.9:
An illustration of the GRAPPA recon method used to un-alias the
acq u ired d ata.............................................................................................................................................2
1
Figure 3.1: RF excitation and signal generation..................................................................
26
Figure 3.2: Simulation of the metabolites spectrum ...........................................................
28
Figure 3.3: Shape of RF pulse used for 900 excitation of isocenter slice....................32
Figure 3.4: Shape of the z gradient used simultaneously with the RF pulse during
excitation for slice selective excitation .....................................................................................
33
Figure 3.5: A simulation of the excited slice location after 900 sinc RF pulse
excitation. The right axis is for the phase in rad in green, and the left axis is the
m agnitude in dB in blue........................................................................................................................34
Figure 3.6: 2D profile that relates the excited spectral frequency in each slice, to its
lo catio n in z-axis......................................................................................................................................3
5
Figure 3.7: A simulation of the excited spectral frequency at z=O after 900 sinc RF
pulse excitation. The right axis is for the phase in rad in green, and the left axis is the
m agnitude in dB in blue........................................................................................................................35
Figure 3.8: Shape of RF pulse used for 900 excitation of 2 different slices...............
37
Figure 3.9: A simulation of the excited slices locations after 900 sinc RF pulse
excitation. The simulation also shows how the non-excited slices are affected. The
right axis is for the phase in rad in green, and the left axis is the magnitude in dB in
blu e ................................................................................................................................................................
38
Figure 3.10: A simulation of the excited slices locations after 900 PM RF pulse
excitation. The simulation also shows how the non-excited slices are affected. The
right axis is for the phase in rad in green, and the left axis is the magnitude in dB in
b lue ................................................................................................................................................................
40
Figure 3.11: 2D profile showing the excitation in x-z plane at y=0 .............................
41
Figure 3.12: 2D profile that relates the excited spectral frequency in each slice, to its
location in z-axes after using a PM RF pulse for exciting 2 slices..................................42
x
Figure 3.13: A simulation of the excited spectral frequency at z=O after 90 o PM RF
pulse excitation for SMS of 2. The right axis is for the phase in rad in green, and the
left axis is the magnitude in dB in blue .....................................................................................
43
Figure 3.14: A simulation of the excited slices locations at echo time, when PM RF
pulse is used. The simulation also shows how the non-excited slices are affected. The
right axis is for the phase in rad in green, and the left axis is the magnitude in dB in
blu e ................................................................................................................................................................
44
Figure 3.15: 2D profile, generated using Bloch simulator, which relates the excited
spectral frequency in each slice, to its location in z-axes after using a PM RF pulse for
ex citin g 2 slices.........................................................................................................................................4
6
Figure 4.1: The x and y gradients used for a spiral trajectory, this is for FOV of 12cm
and resolution of 2.5m m ......................................................................................................................
49
Figure 4.2: A spiral trajectory in the Kx-Ky plane ..............................................................
49
Figure 4.3: A spiral trajectory in the Kx-Ky-Kf space when the gradients in figure 4.1
a re ap p lied ..................................................................................................................................................
50
Figure 4.4: Encoding scheme for conventional spiral-encoded MRSI acquisition......50
Figure 4.5: Proposed z gradient blips to be used for SMS acquisition .......................
51
Figure 4.6: The resulted k-space sampling locations when SMS of 2 is used with
sp iral trajecto ry........................................................................................................................................
52
Figure 4.7: The two different spirals used to sample the 2 Kx-Ky two planes in the ksp a ce ..............................................................................................................................................................
53
Figure 4.8: Two slices that are acquired with fully sampled k-space using spiral
acq uisitio n ..................................................................................................................................................
56
Figure 4.9: Two slices that are acquired with under sampled by a factor of two kspace using SMS spiral acquisition ..............................................................................................
56
Figure 4.10: The difference between the fully samples k-space and the half sampled.
.........................................................................................................................................................................
xi
57
Figure 4.11: Four slices that are acquired with fully sampled k-space using spiral
acq u isitio n..................................................................................................................................................
58
Figure 4.12: Four slices that are acquired with under sampled by a factor of four kspace using SMS spiral acquisition ...........................................................................................
58
Figure 4.13: The difference between the fully samples k-space and the quarter
sam p led ........................................................................................................................................................
58
Figure 4.14: Reconstructing the under sampled k-space without using the proposed
optim ization technique.........................................................................................................................59
Figure 4.15: The process of creating spectroscopy data similar to the one acquired
using spiral SM S for M R SI....................................................................................................................60
Figure 4.16: The simulated spectra for some voxels when fully sampled k-space
using spiral acquisition is used. This is the ground truth ................................................
61
Figure 4.17: The simulated spectra for some voxels when acquisition simulation of
spiral SMS spectroscopy excitation of four is used..............................................................
62
Figure 4.18: The difference between the fully sampled k-space and SMS acquired kspace after reconstruction...................................................................................................................63
Figure 4.19: The image resulted after summing all the non-metabolite values for
each v ox el. ..................................................................................................................................................
64
Figure 4.20: The simulated spectra for some voxels when fully sampled k-space of
noisy data using spiral acquisition is used .............................................................................
65
Figure 4.21: The simulated spectra for some voxels when acquisition simulation of
spiral SMS spectroscopy excitation is used for noisy data...............................................
67
Figure 4.22: The difference between the fully sampled k-space and SMS acquired kspace after reconstruction for the noisy data .......................................................................
68
Figure 4.23: An EPI trajectory in the Kx-Ky-Kf space that is used for EPI
sp ectro sco p y..............................................................................................................................................
Xii
69
Figure 4.24: The simulated spectra for some voxels when fully sampled k-space
using EPI acquisition is used. This is the ground truth......................................................
72
Figure 4.25: Collapsed K-space data resulted of EPI SMS acquisition of four slices..73
Figure 4.26: The simulated spectra for some voxels when acquisition simulation of
EPI SMS spectroscopy excitation of four is used .................................................................
73
Figure 4.27: The difference between the fully sampled k-space and SMS acquired kspace after reconstruction...................................................................................................................74
Figure 4.28: The simulated spectra for some voxels when fully sampled k-space
using EPI acquisition is used w ith noisy data .......................................................................
75
Figure 4.29: The simulated spectra for some voxels when acquisition simulation of
EPI SMS spectroscopy excitation of four is used with noisy data.................................77
Figure 4.30: The difference between the fully sampled k-space and SMS acquired kspace after reconstruction for the noisy data ......................................................................
xiii
78
xiv
List of Tables
Table 1-1: Ti values for the different Bo fields.........................................................................
4
Table 3-1: Spectral bandwidth for the different Bo scanners.........................................29
Table 4-1: The average amplitude for the different metabolites' signals and the
correspondent SNR values based on the added noise.....................................................
64
Table 4-2: The SNR values for the different metabolites after spiral full k-space
reconstruction and spiral SMS reconstruction...................................................................
66
Table 4-3: The average amplitude for the different metabolites' signals and the
correspondent SNR values based on the added noise........................................................75
Table 4-4: The SNR values for the different metabolites after EPI full k-space
reconstruction and EPI SMS reconstruction.........................................................................
xv
76
xvi
Chapter 1
Introduction and Motivation
1.1
The Importance of MR Spectroscopic Imaging
Magnetic resonance spectroscopic imaging (MRSI), also known as chemical shift
imaging (CSI), is an application of Magnetic resonance (MR). It is used in current
clinical MRI machines to detect the metabolites in the scanned spatial location of
interest, and gives a frequency spectrum of biochemical compounds, e.g. brain
metabolites present in each spatial voxel of tissue. Detection of these signals is
based on the MR phenomenon of chemical shift - a frequency shift in the spectrum
that depends
on the
chemical
structure
of particular
compound.
MRSI
measurements are used to understand the rule of the different metabolites,
including N-acetyl-L-aspartate (NAA) - a neuronal marker, creatine (Cr) - one of
brain's energy suppliers, choline (Cho) - an essential nutrient. For example,
1
deficiency in the amount of NAA is correlated to the presence
of a
neurodegenerative disease, like the Alzheimer's disease [1-4].
1.2
MR Spectroscopic Imaging Challenges
The major challenge for the current MRSI techniques is the low SNR of the
metabolites. It is due to low metabolite concentrations of the order 1-10 mM,
compared to water signal for MRI at concentrations -50M [5-7]. In structural MRI,
where water is the primary signal source, scan time is in the order of minutes and
resolution is within millimeters, however, MRSI scan times are in the order of tens
of minutes and resolution is within centimeters. The reason for long scan time and
low resolution is to help achieving a better SNR; relation is in the following equation
[8].
SNR oc (Ax) (Ay) (Az) Vtotal readout interval
1.3
Excitation Sequences and Possible Solutions
A time diagram of the excitation and encoding process illustrates how time is spent
during the scan time. An example of a simple phase encoding sequence with RF
excitation pulses for 3D slab is shown figure 1-1.
2
TR
RF
Gz
Z
z
0
0
Gy
Gx
----
Magnetization relaxation
X
DAC
-Acq. tim"
X
U
Figure 1.1: Encoding scheme for conventional 3D excitation, phase-encoded MRSI acquisition.
Linear gradients (Gx, Gy and Gz) are used to traverse to a particular location in the
(Kx, Ky, Kz) space prior to switching on the analog-to-digital converter (ADC), which
then acquires samples along the Kf axis. This RF pulses for 3D slab excites the entire
spatial location of interest before encoding any point in the (Kx, Ky, Kz) space.
However, time is wasted after the ADC finishes acquiring data for one period and
before starting the RF excitation for the second period. This time is needed to allow
the return of the magnetization moments along the main field (Bo) before the
following excitation.
The value of the magnetization moment along (Bo) before excitation is proportional
to the acquired data's SNR, and so waiting is required here. The process where
magnetization moments gain magnitude in the main field direction can be modeled as
an exponential decay with time constant T1 . In order to find the best TR to use, TR is
selected to be the time that maximizes SNR efficiency; plots of SNR efficiency are
3
shown in figure 1-2. SNR efficiency is a relation between the received signal SNR
time spent, and so it
and the time spent to acquire that signal; that is signal SNR/
is a good metric to relate the trade off between time spent and SNR. The optimal TR
depends on T1, which depends on the main field strength, that TR is usually around
1.25xTi. The used values for T, in the different main field values are shown in table
1 below [23]. The values are chosen based on the metabolites with the longest
longitudinal relaxation time constant for each Bo, which is NAA.
Scanner Bo field strength
T 1 value
1.5 Tesla
1.27 sec
3.0 Tesla
1.47 sec
1.73 sec
7.0 Tesla
Table 1-1: T, values for the different Bo fields.
SNR Efficiency Diagram
E
1lEeslu
*-
--
rrele
scmnnei
scam.e
. .1S.T,
S1.5
2
2
2
TR time (seconds)
.5
4
4.1
Figure 1.2: Simulated SNR efficiency for different main field strengths.
4
5
The key solution to avoid wasting time between two consecutive excitations is to
avoid exciting the same spatial location in both periods. This solution suggests
exciting one slice of the subject at the time and encodes one TR there, then excites a
different slice in the following pulse, where magnetization moments are already
relaxed "at equilibrium"; figure 1.3 illustrates the suggested scheme. If more than
one TR are needed to encode each slice, then one TR is done for each slice before
moving to the next one, so there is enough time for magnetization moments
relaxation at each slice.
TR
10
TR
RF
@1
@1
U
U
Gz
) g
z
Gy
0
0
z
0
is
-I-
I
Gx
V
I-
V
X
X
DAC
--
LU
F-Acq. time-*
L
IAcq. J
I
Figure 1.3: Encoding scheme for conventional 2D excitation, phase-encoded MRSI acquisition.
1.4
Scan Time and SNR
In this section, scanning parameters are proposed, then the scan time and SNR for
both 2D and 3D excitations are calculated. This helps understanding the difference
between the two scans and the advantages of each.
5
The timing proposed scan parameters are for isotropic voxel size of 1cc, ADC is on
for 0.32 seconds, another 0.1 seconds is for excitation, 6 TRs per slice, 8 slices in the
Z dimension. In 3D, TR is 1.8 seconds, 1.37 seconds are for relaxation. In 2D, TR is
0.42 seconds. A simple calculation shows that the total time needed for one full
acquisition in the 3D case is 84.6 seconds, however, a single full acquisition for 2D
needs 20.2 seconds. In addition to the time comparison, a SNR comparison in
required here.
Assume for the 3D acquisition SNR = X, then for the 2D case SNR = X/V.
The
reason behind the 1/V factor is the fact that 8 slices are acquired in the Z
dimension. Basically, in the 3D scans, any point in the k-space is coded by exciting
the entire space and so it has contributions from the 8 slices, where in 2D, the data
for any point in the k-space comes from one slice only. For any different number of
slices, the factor between 3D SNR and 2D SNR is 1/V# of slices in favor of the 3D.
If the same time that is used for a 3D scan is used for a 2D scan, four acquisitions of
the 2D scan can be averaged. This allows improving the 2D case SNR by a factor of
V4. Therefore, the final SNR result for spending the same time in both scans is better
for the 3D scan by a factor of VZ in the case illustrated here.
The calculation above show that 2D scans makes it possible to get one acquisition in
a faster time. However, the SNR will suffer degradation in compare to one
acquisition of 3D scan.
6
1.5
Simultaneous Multislice Imaging for 2D Spectroscopy
Simultaneous Multislice (SMS) is a technique that allows the excitation and
acquisition of more than one slice simultaneously, without any degradation in SNR.
SMS has been applied to many MR applications including structural imaging, and
Diffusion imaging. This thesis studies the extension of SMS applications to include
MRSI with excitation pulses 2D slices.
The addition of SMS to the tools that can be used for the 2D scans, makes it
necessary to revise the case analyzed in the section above. The 3D scan does not
benefit from SMS, and so the time and SNR will be the same there. However, assume
2 slices are excited at the same time in the 2D case. We can do 1 and 5, 2 and 6, 3
and 7, 4 and 8 at the same time. The required time for one full acquisition in reduced
to 10.1 seconds. The time between each two consecutive TRs for the same set of
slices is 1.68 seconds, so it is enough for the magnetization relaxation in each set of
slices. Looking at the SNR that can be achieved, 8 acquisitions of the 2D scan can be
performed in the time needed for one 3D scan. This allows the 2D scan to attain the
same SNR as the 3D scan.
The benefit that 2D SMS scan offers over the 3D is that it is possible to get one
average in a much faster time. In 2D SMS, if good SNR is the goal, then the same 3D
SNR can be achieved in the same time. If it is impossible to spend the long time that
one 3D acquisition requires, then one or more of the 2D acquisitions will be
possible, e.g. infants and fetus scanning. Also, the 2D SMS scan will not suffer the
7
same degree of motion effects, because less time will be spent for each acquisition.
Furthermore, different excitation pulses will be possible in shorter time, e.g. phase
cycling and 2D spectroscopy [21].
8
Chapter 2
Simultaneous Multislice Imaging
Over the last three decades, MRI has gained a high importance to clinicians and
researchers for its ability to produce high quality images non-invasively without the
side effects of ionizing radiations. This importance comes from the numerous
applications that MRI has in both medical diagnostic and research and perioperative
clinical imaging. Despite the great success of MRI in the clinic, it still faces many
challenges. The image encoding process is inherently time consuming, as different
variations of the applied fields are required throughout the acquisition procedure
for full sampling of the Fourier-space data. The scan time problem introduces
motion artifacts, limits the images resolution, reduces the patient throughput, and
can be a challenge for non-compliant or ill subjects who have problems to remain
still for minutes at a time. Therefore, there is a great interest among scientists,
radiologists, and manufacturers to achieve the fastest possible scan time while
9
maintaining image quality. Many techniques have been investigated to accelerate
imaging process including parallel imaging, compressed sensing and simultaneous
multislice (SMS) Imaging.
In this chapter, the background behind SMS will be illustrated. Some of the
techniques used for SMS imaging will be explained. In addition, the extension of SMS
to include MR spectroscopy imaging is justified here.
2.1
SMS Concept
Simultaneous multislice imaging accelerates the data acquisition by exciting and
acquiring multiple slices simultaneously. This process is possible by manipulating
the phase of the simultaneously excited slices differently, and so the received signal
from all the excited slices can be separated into the contributions from each slice.
Additionally, the fact that a larger number of coils are used in receiving the signal,
with a better special sensitivity makes it possible to encode data that make the
reconstruction possible. For example, if three imaging slices are excited and
acquired per shot instead of one, the total acquisition time decreases directly by a
factor of 3. Additionally, unlike standard parallel imaging techniques, simultaneous
multislice acquisition methods do not neglect some k-space samples, or shorten the
readout period. Therefore, they are not subject to a V\iY penalty on SNR (where R is
the acceleration factor) faced in parallel imaging acceleration.
Various methods have been proposed for single-shot simultaneous multislice
methods using slice selection to excite multiple slices simultaneously including the
10
"wideband" imaging [9-11], simultaneous echo refocusing (SER) [12] sometimes
referred to as simultaneous image refocused [13] and parallel image reconstruction
based multislice imaging [14]. However, each of these methods has its limitations.
They can be found in more details in the references. In this chapter, a method of
class of simultaneous multislice methods that uses parallel imaging concepts to un-alias
the pixels from slices excited and encoded simultaneously will be illustrated [14-15].
The "controlled aliasing in parallel imaging results in higher acceleration"
(CAIPIRINHA) technique [14] introduces an in plane image shift between the
simultaneously acquired slices to increase the distance between aliasing voxels and
thus make them easier to separate. The technique uses a different radio frequency
(RF) pulse for every other k-space line. The multiband pulse modulates the phase of
the magnetization excited in the individual slices for each k- space line. For example,
alternating the phase of every other k-space line's excitation by wr will result in a
spatial shift of FOV/2 in the phase-encoded (PE) direction for that slice.
Unfortunately, this technique is not applicable to where all the PE lines are read out
after a single RF excitation, or spiral readout when no uniform sampling of the kspace is used. Alternatively, another approach was proposed on the basis of the
wideband method, where a shift in the PE direction is also applied to introduce
further distance between aliasing pixels. The shift is achieved by applying Gz blips.
This chapter illustrates Kawin et al. [15] technique termed as "blipped-CAIPI", which
is an extension of Nunes et al. [16] wideband method.
11
2.2
SMS Theory
A brief explanation of the blipped-wideband approach will be given here for EPI
scan, with its associated tilted voxel artifact. And then blipped-CAIPI scheme will
then be explained as a solution to the blipped-wideband approach problem. The
basic method of causing a FOV/2 shift in the PE direction will be described using an
example of a two simultaneously excited slices, one of which is at isocenter, as
shown in figure 2-1.
z
e
2----- ^-
Slice 1
Figure 2.1: The phase seen by each slice in SMS excitation and the phase difference
2.2.1
in on slice.
Blipped-Wideband Approach
The blipped-wideband gradient scheme is shown in Figure 2-2. A train of constant
gradient blips in the slice-select gradient (Gz) is applied simultaneously with the
conventional y gradient (Gy) PE blips of the EPI readout. This creates an interslice
image shift in the PE direction for the off-isocenter slice.
12
The amount of the shift depends on the distance of the slice from isocenter and the
area of the blips. For a FOV/ 2 interslice image shift, yAblipZgap
=
IT, where AbIip
= fbl 1p Gz dt is the area of each blip. So each Gz blip increases the phase of the spins
at the center of slice 2 by 11, the outcome is FOV/2 image shift relative to slice one,
where there is no shift at that isocenter slice. The voxel tilting artifact associated
with this technique occurs as a result of the finite thickness of the slice (AZ). Each
blip of Gz introduces a ±8 phase variation across the slice where 6 = y Abip A , that
ultimately cases signal degradation, due to the accumulation after each blip. The
prewind lobe, with the area of Aprewind is chosen to ensure minimum through- plane
dephasing.
GzAAAA
center slice
top slice
--------- 0 0
----------
Of
/
------------------- 0
-------M0
I
Phase at edge of slice
Sir '~
86
------ 6
Figure 2.2: Phase switching in the excited slices in SMS between Gz blips, and phase
accumulation at the slice edge when blipped-wideband is used.
13
2.2.2
Blipped-CAIPI Approach
The goal of blipped-CAIPI [15] technique is to achieve a similar interslice image shift
to the one blipped-wideband method does, but without the undesirable voxel-tilting
artifact that causes signal degradation. This is accomplished using a modified
"phase-cycled" Gz blips that applies the desired phase modulation along PE, but
without causing significant phase accumulation over the full readout. Figure 2-3
shows the phase-cycled Gz blips, a sign reversal is now applied on every PE line to
cause a phase difference of w between every two adjacent PE lines. Additionally,
balancing blip (shown in red), of area
Aprewind = - Abiip
/2 replaces the prewinding
lobe. As shown in Figure 2-3, the desired w phase difference is still achieved at the
center of the top slice. However, the sign modulation of the Gz blips solves the phase
accumulation problem during readout at the slices' edges. The new Gz blips make
the edges' phase switch back and forth between two small states ±6/2, which are
centered at zero. Therefore, the signal degradation due to phase acclamation does
not exist here.
14
A
Gz
A
V
Balancing
V
center slice
top slice
o -11/2
kL-------- o n/2
o -n/2
\Wt/t
o n/2
/p
A/
o -n/2
Phase at edge of slice
kx
6/2
-6/2
8/2
-6/2
6/2
V
w
-6/2
Figure 2.3: Phase switching in the excited slices in SMS between Gz blips, and phase
accumulation at the slice edge when blipped-CAIPI is used.
The case shown in Figure 2-1 illustrates a special case where on of the excited slice
slices is at the isocenter position. However, for the rest of the scans neither slice is at
isocenter, for example case 2 in Figure 2-4. For all the non- isocenter cases, Gz blip
adds a phase of q5 to one of the slice close to center, and q + 7r phase to the farther
slice. Where q5 =
yAblipZoffset
and Zoffset is the offset of the closer to the center slice
from isocenter.
15
z
case2
case1
01
40
iTr+4)
Figure 2.4: The phase seen by each slice in SMS excitation when isocenter slice is not excited.
The prewind blip continues to have the same area as in the isocenter case; it adds a
phase of - q/2 to one of the slice close to center, and - (4 + n)/2 phase to the
farther slice. The phase switch in the same slice after the consecutive blips is shown
in figure 2-5. As the figure shows, the odd and even phase encoding lines don't have
the same phase for the bottom slice, or exact difference of 7r phase for the top slice.
This problem is addressed by adding 4/2 to the even PE lines and - q/2 to the odd
PE lines, and then we have the same k-space configuration as the isocenter case.
16
Center slice
Top slice
Case 1
J2R/
-0/2
0/2
It
Case 2
+
Case 2 with
added phase
(T
0
\/
+ 0) 2
JI
Figure 2.5: Phase switching in the excited slices in SMS between Gz blips when isocenter slice
is not excited.
The last point to be addressed here is extending the SMS excitation of two slices into
SMS excitation of more than two. The case of exciting three or fours slices
simultaneously requires achieving multiples of FOV/3 or FOV/4 for the different
slices based on the blipped-CAIPI technique. For example, a SMS of three introduces
no shift to one of the three slices, a shift of FOV/3 to another one, and a shift of
2FOV/3 to the last slice. A SMS of four introduces no shift to one of the three slices, a
shift of FOV/4 to another one, a shift of FOV/2 to the third slice, and a shift of
3FOV/4 to the last slice. This could be achieved by having PE line phase correction
cycle in periods of 3 or 4 instead of 2. So instead of a phase difference of w between
any two consecutive PE lines, there is 21/3 phase difference for any two consecutive
17
PE lines in SMS of 3, and r/2 phase difference for any two consecutive PE lines in
SMS of 3. Figure 2-6 presents the Gz blips scheme for interslice shift of FOV/2,
multiples of FOV/3 and multiples of FOV/4. It also shows the phase accumulation at
the edges excited slice. The blipped-CAIPI technique does not allow significant
phase acclamation to build up at the edges, and so avoiding voxel-tilting artifact that
is caused by phase accumulation. Additionally, the balancing blips center the phase
variations around zero.
VA
A
FOV/2
btw slices
A A
btw slices
A AA
F3
V/3
OV/3btw slices
/4
VA
in slices
A A
In slices
A A
in slices
4
Figure 2.6: Gz blips when 2, 3 or 4 slices to be excited using SMS. In Addition to the caused
Phase switching in the excited slices between Gz blips, and phase accumulation at the slice
edge of each slice.
18
2.3
SMS Image Reconstruction
The next step after exciting a number of slices simultaneously is to separate the data
and reconstruct the slices. In SMS explained earlier, the acquired data is the
summation of the k-space for the slices. For example, the acquired data for the
excitation in figure 2-1 is the summation of k-space of slice1 and the k-space of slice
2 after the FOV/2 shift that is due to the applied Gz blips. Applying the inverse
Fourier transform for the acquired k-space gives an image similar to figure 2-7.
Figure 2.7: The acquired data using SMS of two slices without recon for un-aliasing.
Using single slice excitation acquisition for the same slices results in the images
shown in figure 2-8. The reconstruction step aims to separate the images from the
collapsed data. The applied FOV/2 shift helps to minimize the error when
separating the images.
19
Figure 2.8: The acquired slices using single slice excitation for each one.
A method similar to parallel imaging GRAPPA reconstruction [17] is used to
separate the collapsed images. The slice-GRAPPA uses kernels that are optimized
using data acquired from separately excited single slice acquisition. For the SMS of
two, two kernels iterate over the collapsed k-space after the SMS acquisition and
each kernel builds the k-space for one of the two slices. Finally the inverse Fourier
transform is applied to the two new reconstructed k-spaces. And the one of the
image that has FOV/2 shift get shifted again to have it in the right orientation. Figure
2-9 illustrates the process. More details could be found in [15].
20
Slice 1
Slice 2
Figure 2.9: An illustration of the GRAPPA recon method used to un-alias the acquired data.
2.4
SMS for spectroscopy
The goal of this thesis is to accelerate Spectroscopy Imaging using image. In order to
understand how that is possible to extend the technique explained above to
spectroscopy, the MR signal equation is below.
M1 (kx(t), ky(t), kf(t)) =
f f fm(x, y)e-i2n[kx(t)x+ky(t)y+ftdx dy df
Where
kx(t) =
271 0
21
Gx(T) dr
ky(t) = 2
f Gy (c) d-c
kf(t) = t
In the ordinary SMS case for MRI discussed in this chapter so far, Gz blips are
applied to achieve the phase difference in the PE lines in the k-space. Adding the
applied Gz blips to the signal equation adds the factor Kz to the exponent in the
equation:
Msms(t)
=
f f fm(x,
y)e-i27[kx(t)x+ky(t)y+kz(t)z+ftdx
dy df
Where
M(t) = M(kx(t),ky(t),kf(t))
Based on the description of the 2 slices excitation case, the following applied:
Y ft Gz(-r) d kz (t)=-- 27r
Now assume an odd number of blips
kz(t)
kz = 2)-1AiP
w (A blip
22
-
2 Ablip)
nt
kz-(t)
y'
4
-
T
=
7 y zgap
1
4 zgap
The equation becomes
Msms (t) =
f f fm(x,y)e-i2z[kx(t)x+kY(t)y+ft] e-inr/2df dy dx
So for odd PE lines
Msms (kx(t), ky(t), kf(t)) = M1 (kx(t), ky(t), kf(t))e
-i/ 2
And for even PE lines
Msms (kx (t), ky t(t))
= M1 (kx (t),yky (t),Ykf (t)) e in/2
In conclusion, the phase modulation for SMS achieved in the basic case could be
extended to the MR Spectroscopy without problems theoretically. The acquired
signal's phase modulation does not depend on the value of kf, and so the phase
modulation is applied to all values of kf. The result here is that the FOV/2 shift in
the image will be applied to all frequencies. Therefore, after applying the 3D inverse
Fourier transform to get m (x, y, f), the same process that is used in the ordinary
case should be applied to all frequencies f
One advantage here is that the kernel
should be trained only on onefand then it could be applied to the other frequencies.
23
24
Chapter 3
SMS RF Pulse Design
In this chapter, the work to design the excitation pulse for MR spectroscopy SMS is
presented. The discussion includes the excitation specifications that are required for
the MR spectroscopy SMS. In addition, methods used in the design are explained.
The outcome of the different methods is examined using simulation tools.
3.1
Excitation Background
In MRI machines, the main constant Bo field is always present and applied in the
positive z-direction. Applying this field affects a part of the magnetic moments in
atoms in the body causing them to align with it; this is called the equilibrium state.
In order to generate a signal in the spatial location of interest, radiofrequency (RF)
magnetic pulse B1 is applied in the x-y (transverse) plane using RF transmission
coils. The result of B1 field is rotating the atoms' magnetic moment 900 away from
25
the equilibrium state (parallel to Bo). The field strength is usually a small fraction of
a Gauss and has duration of few milliseconds. This filed is usually used to excite a
slice of the scanned object. After applying the RF pulse, the magnetic moments start
moving back toward equilibrium in the z-direction. These magnetic moments emit a
signal that is used for the imaging of the excited spatial location of interest. More
details can be found in [8].
Transmit coil(s)
RF pL4se
z
Excitation of magnetic
RF pulse
moments via
Receiver coil(s)
BO
Figure 3.1: RF excitation and signal generation.
3.2
Design Parameters
The parameters for the RF pulse design should be chosen such that the purpose of
the experiment is possible to achieve. This design aims for SMS excitation of two for
MR spectroscopic imaging.
26
3.2.1
Slice Thickness
The first parameter to decide here is the thickness of each slice. This variable
depends highly of the nature of the scan we are running. For a structural imaging
case, it can be 1mm. However, the main challenge in spectroscopy is low SNR. Based
on the equation below, the slice thickness (Az) is proportional to SNR, and so 1mm
slice thickness is not appropriate for spectroscopy. In this design, I decided to go
with a slice thickness of 1cm. This thickness is used in some spectroscopy scan. In
addition, it allows running fast spectroscopy with small number of averages to build
a good SNR.
SNR oc (Ax) (Ay) (Az) Vtotal readout interval
3.2.2
Separation Between Slices Excited Simultaneously
The second parameter is the separation between the two simultaneously excited
slices. This parameter is simply calculated by dividing the z dimension of the FOV
over 2. In this design I propose a FOV of 10cm in z, and so separation between
centers of simultaneously excited slices is 5cm.
3.2.3
Spectral Bandwidth
The third parameter is the spectral bandwidth (BW) of the RF pulse. In
spectroscopy, this depends on the metabolites the scan aims to detect.
The
frequency axis in spectroscopy is given in units of "parts per million", or PPM,
relative to the frequency defined by the main field. PPM is a unit less entity and if
one wants to convert the ppm axis in the units of Hertz (Hz), then 1ppm = (y
27
/2T)-Bo-10- 6 Hz. Here, (y/2n) is the gyromagnetic ratio and is equal to 42.576 MHz.
Figure 3-2 shows a simulation of the spectrum of the three important brain
metabolites: N-acetyl-L-aspartate (NAA), creatine (Cr), and choline (Cho).
H20
Cr+Cho
NAA
Lac
5
4
4.5
~
3.5
Cho CrLa
3
2.5
2
1.5
1
Chemical shift (ppm)
Figure 3.2: Simulation of the metabolites spectrum.
From the metabolites spectrum above, we can see that NAA signal is concentrated in
the singlet peak at 2.01 ppm. Creatine has a distinct peaks located at 3.03 ppm, and
Choline has a distinct peak at 3.21 ppm. A peak at 3.91 ppm is caused but both
Creatine and Choline. Here, 10ppm of the spectrum will be excited, this value is used
to ensure exciting the metabolites shown above even when there is a slight shift of
the excited band for reason that is explained later. Table 3-1 below shows the
spectral bandwidth of the RF pulse that should be used for the different Bo scanners,
so 10 ppm are excited.
28
Scanner Bo field strength
Spectral bandwidth in Hz
1.5 Tesla
600 Hz
3.0 Tesla
1200 Hz
7.0 Tesla
3000 Hz
Table 3-1: Spectral bandwidth for the different Bo scanners.
The last design decision here is the type of pulse to be used. Two different designs
will be examined here; one uses the simple sinc pulse, and the other uses ParksMcClellan Pulse Design Algorithm [18]. The used pulses are 900 excitation; they
excite the two slices simultaneously for SMS. A spin echo pulse sequence is used
here, and so a 1800 pulse will follow the 90 0 pulse. The time between the 90 pulse
and 180 pulses is in tens of milliseconds; it is called a half echo time (TE/2).
3.3
Design Method
In general, the magnetization vector behavior is descried by so-called Bloch
equation given by the following if relaxation is neglected [8].
m(
My
Mz
y
0
G
-B1,y
-G
0
B1,,
Bly
-Bl,x
0
MX\
(Mi
MZ
B1 is the RF pulse, and Mx, My, and Mz are the magnetization moments in the x, y
and z directions respectively, and G is the gradient amplitude.
In designing the pulses here, Shinnar-Le Roux (SLR) algorithm [18] is used. SLR is
an approximation of the Bloch equation. The method simplifies the solution of Bloch
29
equation to a design of two polynomials, and so RF pulse can be solved using known
digital filter design problem, for example, Parks-McClellan algorithm.
The SLR transform maps the RF field Bl(t) and the Cayley-Klein parameters
a and P. The Cayley-Klein parameters are used to map the magnetization moments
before and after the excitation based on the following equation.
2
M
(a*)
M') = y
MZ+
(f#*) 2
-a*f#*
_fl2
2a*#
0
-a#
2a#f*
aa*-ft*
MXY*
Mz
where Mx,y = M + iMy
Therefore, running the SLR transform, and using the equation above allows to
simulate the excited slice profile after applying any RF excitation pulse.
3.4
RF Pulse Design
The first excitation pulse to be examined is the windowed sinc pulse. When a simple
sinc pulse is used, it adds an undesired in slice and out of slice ripples as a result of
Gibbs phenomenon. Therefore, a sinc pulse that is windowed using a hamming
window to overcome this problem. All equations in this section can be found in [8].
In order to achieve the desired 900 flip in the magnetization moments, the following
equations explain the relation between the flip angle 0 and the RF excitation pulse
B1(t).
30
fw(s)
f=
ds
Where
w (t)
=
y B1 (t)
The last step to be considered before applying and RF pulse is setting the z gradient
(Gz) value during the excitation process. The Gz value is what decides the excited
slice thickness based on the following equation
Gz =
27* BW
y *AZ
Therefore, the applied Gz value during the excitation = 7.05 mT/m; here the used
BW is 3000 Hz.
At this point, a hamming windowed sinc pulse can be designed, however, this pulse
excites the slice at the isocenter only. In order to excite a different slice, the designed
pulse should be modulated. The following equation explains the relation between
the modulation frequency Wd, and the slice offset d.
d =
*
y * Gz
31
Thus, the modulation frequency cod = 94247.8 rad/s. when d = 5cm.
3.4.1
RF Sinc Pulse
The first step is to test the design above on a single slice excitation. This slice is
isocenter and the design parameters are the same ones indicated above. The pulse
to be used is shown in figure 3-3.
90 degrees excitation puls
2X 10
-
-
I
--.-
10 - --.... -
- -
- --
15 .............---.......--
-
Real(RF)
Imag(RF)
.-
-
-
- --
-- -
- -
0
0
05
1
1s
3
2
2.
Trme(mrSeconds)
35
4
45
5
x 104
Figure 3.3: Shape of RF pulse used for 900 excitation of isocenter slice.
The z gradient is applied simultaneously with the RF pulse for a selective excitation.
After the RF pulse in turned off, a negative z gradient is applied to unwind the linear
phase that has occurred over the slice width during the excitation; the process of
applying this negative gradient is called the refocusing. For a maximum refocusing,
the negative gradient must undo the phase shift by having the same magnitude as
32
the positive gradient and for half the time. The Gz for this case is shown in figure 34.
GZefor
1Z
10
10
_0
.
*.
1
3
2
mtcafte OxWOM
4
Time(milliSeconds)
5
6
.
7
.
6
x 10,
Figure 3.4: Shape of the z gradient used simultaneously with the RF pulse during excitation for
slice selective excitation.
The SLR transform is used in order to evaluate the excitation profile for this RF sinc
pulse and z gradient. Assuming that before excitation, the magnetization vector is
only in the z direction, that is M- = (0, 0, Mo). Solving the equation using a and j,
that gives the following output of magnetization after excitation
M'
= 2a*I3Mo
Figure 3-5 illustrates the excited slice after applying the RF pulse. The excited slice
is the one with log (IM,yI)
~ 0.
33
Mxy profile after refocus
.....
s.....t..n
................................
Figure
the
T
x i.
.e
s...e ..
.
xcited....
-5.C
C
..
5. ..U..
-60 .. . .. . .$
>t x..........
.......
. ..
. . ..
......
MM
01
..
..
I.
......
......
) ....
Figure 3.5: A simulation of the excited slice location after 900 sinc RF pulse excitation. The
right axis is for the phase in rad in green, and the left axis is the magnitude in dB in blue.
We can see that the figure achieves the parameters chosen earlier, but we still
cannot see the spectral bandwidth in this plot. Figure 3-6 demonstrates 2D plot of
the excited bandwidth based on the z location of the magnetization moments.
34
Mo
I.
I
I
Z-Om
Figure 3.6: 2D profile that relates the excited spectral frequency in each slice, to its location in
z-axis.
Looking at the center of the excited slice at z=O (blue line), the excited frequencies
are between -1.5 kHz and 1.5 kHz to give the bandwidth of 3 kHz. This can be seen
clearer in figure 3-7.
1
Mxy Bandwidth profile after refocus at z=
-2
Sd
-2
.... .. .. . .
.. -.. . ..
. .. .
. .. .. .. .... . . ...2
Ia
.. ..... . .. ..
-3
...
. . ..
.. ..
. ... . ..
-..
4.
-..
.. .. .
.. . . . .. . . . .
...
-2
.. .
..
.. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.. .. . .
.. . . .
. .. . . . . . . . . . . . . . . .
-4
4
a
16I
Pr1q (IC d
Figure 3.7: A simulation of the excited spectral frequency at z=0 after 900 sinc RF pulse
excitation. The right axis is for the phase in rad in green, and the left axis is the magnitude in
dB in blue.
35
The more interesting observation here is the excited frequencies at the edges of the
slice, that is at z = -0.5 and z = 0.5. The bandwidth is still 3 kHz, however, it is not
centered around zero. This effect is a result of the field modulation that is a function
of the point in the space based on the applied gradients. This modulation has
different effects for different points along z due to Gz. The z gradient is designed
such that the resonance frequency for the magnetization at the center of the excited
slice is what taken in account for the bandwidth. However, the changes of the
applied magnetic fiend at the edges of the slice, as a result of Gz, causes a slight
change in the resonance frequency for the magnetizations moments there, and so a
shift in the excited frequencies. This phenomenon is known in MR spectroscopy and
it is still possible to encode all the metabolites, since their spectrum is within the
excited bandwidth over the slice. This is another reason for choosing a bandwidth
that covers 10ppm, which is more than what is needed based on the spectrum in
figure 3-2, and selecting a thick slice.
One more observation is the phase at the excited slice. Based on the assumption
used to add the refocusing lobe to Gz, the phase should be the same over the entire
slice. However, it is linear with a slope here. The reason is the non-linearity of the
Bloch equation, which approximated to be linear the approximations made earlier.
Trying a refocusing lobe that is a bit more than 50% of the area of the positive z
gradients could solve this problem. The problem solution will be implemented in
the design of the RF pulses to be used in SMS excitation.
36
The next step toward our goal of simultaneous multislice excitation is to design the
RF pulse that will excite the slice centered at z = 5cm based on the parameter choice
earlier. The modulation frequency for the pulse to do that task was calculated
earlier. After modulating the RF pulse shown in figure 3-3 with this modulation, it is
added another RF pulse that is identical to the one in figure 3-3. The result pulse is
presented in figure 3-8.
90 degrees SMS excitation pulse
IMS
S..
11-1.1
...
.... ......
........ ..
iii
I
..........
...........
............
........ ...........
5.51 . ..............
-~F
......
........I
E
.. . . .. . . . .. .
... . .. . .
I
0
.6.5
.................
.............
-
...................
.11
I
___
j
1.5s
2
1111111
I
U
Tbm.(OONSeconds)
2
U3
4
4S
x1
Xl
Figure 3.8: Shape of RF pulse used for 900 excitation of 2 different slices.
The used Gz is similar to the one that was used before, shown in figure 3-4.
However, the non-constant phase problem, descried earlier, is solved by extending
the refocusing lobe to be 50.6% of the area of the positive z gradients.
The same SLR process that is used before is used here to examine the excitation
profile. Figure 3-9 illustrates the profile of M+ along z.
37
Mxy proile afer for SMS excitndon refocu
I
-P..ft(Xv
i Md
s5 ..
.
4e -
... . .0..
-
..
4
-3
-
-1
0
1
3
..
. . ..
. ..
5
Z-ex13 (cm)
of the excited slices locations after 900 sinc RF pulse excitation. The
simulation also shows how the non-excited slices are affected. The right axis is for the phase
in rad in green, and the left axis is the magnitude in dB in blue.
Figure 3.9: A simulation
The first observation is the constant phase along the excited slice, the solution
proposed earlier by extending the refocusing lobe works well here. Also the second
excited slice is centered at z = 5cm, and so it meet the specifications. The method of
adding 2 sinc RF pulse to excite the 2 slices of interest is proven to work here,
however some problems can be noticed. In the figure above, the out of slice ripples
(circled in red) witness amplification when compared to the case of one slice only.
When only 1 slice is exited, the out of slices ripples goes to -30 dB at 5mm away
from the excited slice, on the other hand, for excitation of 2 slices, the out of slice
ripples goes to -22 dB 5mm away from excited slice boundaries. The reason of this
is the out of slice ripples from each slice is having an affect on the other one. This
ripple interference is obvious, and could be problematic in some cases, in the
transition regions. This problem gets worse if the slices get closer, due to the nature
38
of the sinc pulse ripples. That is, the out of slice ripples have a larger magnitude, as
they get closer to the excited band. Therefore, the out of slice ripples will have more
severe affect, as excited slices get closer. Also this effect will be much worse when
the same method is used for more than two slices simultaneous excitation.
3.4.2
Parks-McClellan Pulse Design Algorithm
The problem faced when more than one sinc pulse are summed for SMS excitation
suggest that we need a design that can control the out of slice ripples everywhere.
The benefit of SLR is it makes it possible to design the RF excitation pulse using any
digital filter design algorithm. The Algorithm that is used when more than one band
are to be excited and can control the in slice and out of slice ripples is ParksMcClellan Algorithm. In addition to the freedom the Algorithm gives in specifying
the ripples, it makes it possible to specify the number of slices to be excited, the
separation between them and their thickness.
An RF pulse is designed using the Parks-McClellan Algorithm to satisfy the
parameters specified earlier and used in the RF sinc pulse. The same z gradient that
is used in the two added RF sinc pulses is used here. Finally, the same SLR process
that is used before is used here to examine the excitation profile. Figure 3-10
illustrates the profile of M+7y along z.
39
Mxy profile after for SMS excitation refocuw
a
le
.......
Fgr
i.
'6
.
(..
Axs
asshshof
Fiue1:Asimulation
the nexcited slices
90TMhple
aion afeted.
excitaxtsison hse
in rad in green, and the left axis is the magnitude in dB in blue.
It can be observed that the ripples problem is solved here. Although that the out of
slice ripples value here is -23 dB, which is similar to the previous design, it is not a
problem here. The out of slice ripples value is chosen in the design of the filter to be
less than .5% of the in-slice signal. The out of slice ripples are the same no matter
how far they are from the excited slice, and so there is no problem if slices number
increase or they get closer. When looking at the phase, we can see that it is almost
constant, however, it has small ripples. These ripples are due to the Parks-McClellan
Algorithm which trade off the in slice ripple, out of slice ripple and transition region
width, to output the optimal design based on the specified parameters. The in slice
ripples are set to 1% of the in-slice signal.
40
Based on the simulated excitation for the two examined designs here, I decided to
use Parks-McClellan Algorithm to design the pulses. More simulation for the RF
Parks-McClellan pulse is illustrated below to assure that it meets the specifications.
Figure 3-11 illustrates a 2D (x-z) profile of the resulting Mxy profile after excitation.
E
I
x-cmI
Figure 3.11: 2D profile showing the excitation in x-z plane at y=O.
The simulation presented in the figure above shows that for any x, the two slices
centered at z = 0 and z = 5 are excited. A cross section of this figure at x = 0 gives
figure 3-10 shown before. Figure 3-12 illustrates a 2D (z-f) profile of the resulting
Mxy profile after excitation.
41
50
44
-4
44
-4
4
4
45
4
-3
42
1
-1
2
3
4
U
Figure 3.12: 2D profile that relates the excited spectral frequency in each slice, to its location
in z-axes after using a PM RF pulse for exciting 2 slices.
The simulation results in the figure above illustrates that the spectral bandwidth of
3 kHz is excited in both slices. The spectral bandwidth from -1.5 kHz to 1.5 kHz can
be seen at the center of both slices (blue lines). Also it shows that the same excited
spectral band shift, which is witnessed when only one slice is excited at z = 0, is
happening here in both slices. The red lines show the excited slices boundaries. For
a better visualization, a cross section of the simulation above at z = 0 is taken and
plotted it in figure 3-13.
42
Mxy Bandwidth profile after refocus at z=O
o -
......
.....-E....
..........
.......-........
.....
_
X O Ind O
I
S
A W.
dBinbl
magnitudeOPX
in
Freq (kfW
Figure 3.13: A simulation of the excited spectral frequency at z=O after 90 0 PM RF pulse
excitation for SMS of 2. The right axis is for the phase in rad in green, and the left axis is the
magnitude in dB in blue.
The cross section plot demonstrated the phase in the excited spectral band; it is
constant here with some ripples, the in-slice ripples are set to be .5% of the signal.
The simulation that have been done so far are after the 900 pulse, however, we
decided to use spin echo technique for a better signal acquisition. The technique
applies a 1800 pulse at time
T
after the 900 excitation, and so it rotates the
magnetizations around the x-axis. More details about the technique can be found in
[8]. The task now is to find how this excitation profiles will change after the 1800
pulse. The pulse is similar to the one used for the 900 excitation, but has to be
multiplied by 2, so it gives 6 = 180. The Gz has the same value that is used with the
900 RF pulse, the positive part is applied simultaneously with the 1800 pulse, but
43
there is no refocusing lobe here. Instead, crusher gradients are applied before and
after the pulse by Gz for a better in slice signal and preserving the transition region.
The excitation profile can be anticipated using SLR transform, same process used
for the 900 pulse. Here crusher gradients are taken in account to get
= -fl2Mxy*
M+
Figure 3-14 illustrates the profile of log (IM' yI) along z. This is the profile at 2T
(echo time).
Mxy profile at echc
0-IMXY1I U1
-
bM d ......
inepx
.....
I
I
... ........
1 5
I
I
.......
......
2......0.....................
...........
0...
-2
......
..............
. . . . . . .. . .
. . . . .. .
....
. . . . ... . . .
. . . . ..
..
. . ..
. . . .. .
. . . .
I
.
...
..
...
..
..
...
..
..
...
..
........ ..... ....
x..
............. ..
..... .. ....
Figure 3.14: A simulation of the excited slices locations at echo time, when PM RF pulse is
used. The simulation also shows how the non-excited slices are affected. The right axis is for
the phase in rad in green, and the left axis is the magnitude in dB in blue.
We can see that the profile shares same shape when looking at slices thickness and
position as the one after 900. The similarity between "after 900" and "echo time" is
44
the same for the other simulations, and so the figures and not shown here. One
observation of the profile simulation at echo time is the magnitude of the
magnetization moment Mx,y for the out of slice ripples. Here, they are significantly
better than the case after the 900 pulse. Mx, is about -50dB here, where it is about 23dB after the 900. This means we go from out of slice ripples to be .05% of the inslice signal, to .0001% out of slice ripples at echo time. This is one reason why it is
better to start the acquisition around the echo time. The extra degradation of the out
of slice ripples is due to the crusher gradient used in Gz.
3.5
Bloch Simulator
In addition to the simulations done using the SLR tools. Bloch simulator was used to
confirm that the SLR approximations were valid. The 2D (z-f) profile of the Bloch
simulator output is illustrated in figure 3-15. The figure shows that the result indeed
meets what SLR tools have expected.
45
4
4
4
a.
4
a
IL
4
a
2
4
6
Z - cm
Figure 3.15: 2D profile, generated using Bloch simulator, which relates the excited spectral
frequency in each slice, to its location in z-axes after using a PM RF pulse for exciting 2 slices.
3.6
Summary
The group of simulations that is used to examine the RF pulse designed using ParksMcClellan algorithm assures that the pulse is good to be used in real scans. The
pulse meets all the specification that were set earlier for MR spectroscopic imaging,
and it is easy to be extended for more than SMS excitation of two, without creating
any problem. Unlike the sinc RF pulse, which starts to face challenges, even with
SMS of two. On the way to reach this conclusion, SLR pulse design technique is used;
in addition, Bloch simulator is used to examine the designed RF pulse.
46
Chapter 4
SMS Spectroscopic Imaging: Data
Acquisition
The RF pulse excitation of the spatial location of interest is the first step to scan a
subject in the MRI machine. In spectroscopy SMS, same as other scan techniques, the
next step is to acquire the data. In the 2D excitation process, both x gradient (Gx)
and y gradient (Gy) are used in the data acquisition process. The x and y gradients
traverse the kx-ky plane in the k-space. After the gradients specify the k-space index
to sample, the machine's analog to digital converted (ADC) is turned on to acquire
the signal. Usually the z gradient (Gz) is not used during the 2D acquisition process
of a slice, however, Gz in necessary in SMS acquisition. Chapter 2 illustrates the
necessity of Gz during the acquisition process, and the use of blipped-CAIPI
47
technique to apply the required phase modulation. Many methods are used in the
acquisition of K-space, some use uniform sampling -Echo planner Imaging (EPI)-,
and others use non-uniform one, -Spiral Trajectories-. In this chapter, I propose a
method that can use spiral trajectory acquisition for spectroscopy SMS. The
acquisition steps and how to use the different gradients are explained. Also,
simulations that produce the acquired data are implemented. Finally, I propose
algorithm to be used in the reconstruction process of the collapsed acquired data,
and the performance of this algorithm is tested on the simulated data.
4.1
Spiral acquisition for CSI
Spiral acquisition is an efficient method for MR spectroscopy scans. It uses time
varying gradients during the long acquisition readout periods, so that in addition to
sampling the time (Kf) axis, samples are simultaneously acquired along Kx and Ky.
The spiral CSI algorithm, introduced by Adalsteinsson et al [19], uses 2D spiralshaped k-space trajectories that are frequently played during the long acquisition
window, to simultaneously collect samples in the 3D (Kx, Ky, Kf) space in each
repetition period (TR).
The spiral-shaped trajectories are formed by playing sinusoidal gradient waveforms
simultaneously along Gx and Gy. Figure 4-1 shows the played gradients in Gx and
Gy.
48
Gx and Gy during acquisition
-Gx
II
...............
10
. .........
U.
........
...........
...........
...... .......
.............
..........
........................
. ... .. . .. ... ..
...............
S
0
. .............
-16
....
-15
a
L
......
z
.....
....................
4
...
....
..........
...... .........
.....
4
....
.........
.......
.............
U
is
9U
...
...
U
12
U
14
EU
go
Nm.Cm.1
W
I
Figure 4.1: The x and y gradients used for a spiral trajectory, this is for FOV of 12cm and
resolution of 2.5mm.
Figure 4-2 illustrates the resulted spiral k-space trajectory that samples k-space
here.
Spiral K-Space Trajectory
Kx
Figure 4.2: A spiral trajectory in the Kx-Ky plane.
Figure 4-3 shows the acquisitions of one spiral interleave in the 3D space of (Kx, Ky,
Kf).
49
Kx
II
I]')
nfl
n((llnr(Yfl
I Kf
Ky
Figure 4.3: A spiral trajectory in the Kx-Ky-Kf space when the gradients in figure 4.1 are
applied.
Figure 4-4 shows an encoding sequence scheme when spiral is used.
TR
RF
Gz
z
0
Gy
Gx
DAC
'U
L~1
Acq. time
Figure 4.4: Encoding scheme for conventional spiral-encoded MRSI acquisition.
50
P
4.2
Spiral with SMS
Applying the SMS technique to spiral acquisition requires using a reconstruction
method other than the one explained in chapter 2 for EPI acquisition. In EPI, SMS
makes use of the phase difference of ir between every two consecutive Ky lines in
the k-space. Due to the nature of the acquisition of spiral trajectories, this is not
possible. Also, it will not be possible to apply the Gz blips when moving between PE
(Ky) lines in spiral trajectories, since there are no PE lines here. Another method has
been proposed to apply blipped-CAIPI SMS for spiral in [20]. A similar design for
the blips used that method is proposed below.
The phasing blips for SMS of 2 will be inserted in Gz as shown in figure 4-5. This will
result in an under sampling by a factor of 2 of the k-space if we encode each phase in
a different plane as shown in figure 4-6.
Ox, Gy and Gz duung acquisition
4
U.
1
6
5
435
7
Figure 4.5: Proposed z gradient blips to be used for SMS acquisition.
51
The result will be k-space that is sampled in a spiral trajectory, and there is a phase
difference of 7r between every two consecutive rotations. Therefore, the switching
between -7/2 and 7/2 in every two consecutive full cycles could be represented as
in figure 4-6.
-0.
...-..
Ky
e
-2
x
-
Figure 4.6: The resulted k-space sampling locations when SMS of 2 is used with spiral
trajectory.
The acquired data in the scanner are one 2D k-space. However, it is possible to
represent these data as an under sampled 3D k-space as shown in figure 4-6. The
reason is that for any two consecutive full cycles in the acquired k-space, each cycle
belongs to a different kx-ky plan. The goal after applying the acquisition with this
similar blipped-CAIPI technique is to reconstruct the full 3D k-space. Here, the 3D kspace could be considered as an under sampled 3D k-space by a factor of two that
was acquired using spiral trajectory. Therefore, it is possible to use a similar
technique as the ones used in reconstructing under sampled spiral k-space.
52
4.3
SMS acquired data synthesis
In order to demonstrate how the acquired data can be processed and un-aliased to
reconstruct the real image, simulations of the acquired data are carried out here.
The data sets used here are acquired on a machine of 64 coils. It is part of a
structural scan. Two slices were taken in order to simulate the artifacts of the spiral
acquisition proposed here. The spiral-acquired data are simulated by applying
data acq = C mU.S S(FSi * img)
Where, Si is the i coil sensitivity. C -+.s S is the process of converting the Cartesian
data that we have, to under sampled data resulted using a spiral trajectory. Two
different under sampled k-space trajectories are used in each k-space plane, so the
fact that a different cycle is sampled for each plans is reflected. Figure 4-7 shows the
two spirals that are used.
Figure 4.7: The two different spirals used to sample the 2 Kx-Ky two planes in the k-space.
53
An important factor to consider when looking at the data we are using is the number
of acquisition coils. A higher number of coils make it easier to reconstruct an image
since there are more data to be used. That is, in multi coils systems, the data come
from smaller coils in an array with special sensitivity profiles that are better
separated than for arrays with smaller number of larger coils. In the simulations for
spiral SMS here, the data were acquired using 64 coils.
4.4
Data Reconstruction
In order to solve the data reconstruction problem for spiral SMS, I will implement
optimization techniques that are used to solve under sampled k-space problems.
The problem we are trying to solve can be represented in this cost function
11C --+.
S(FSi * x) - data i
112
+ A * TV(x)
Where x is the image we are trying to find, C -
., S is the operation of
transforming a fully sampled Cartesian k-space into an under sampled spiral kspace.
F is the Fourier transform, TV is the total variation transform, A is a
coefficient and Si is the i coil sensitivity. data i is the image data coming from coil i,
and it is simulated in this work using the equation shown in the previous section.
The solution of this optimization problem is implemented using the conjugate
gradient method.
54
4.5
Spiral SMS Simulations and Results
In this section, the results of applying spiral SMS reconstruction on simulated data
are presented.
4.5.1
Structural Data with one frequency
Figure 4.8 shows two slices that are acquired with fully sampled k-space using spiral
acquisition. It is reconstructed using the optimization equation explained before.
Figure 4.9 shows two slices that are acquired with under sampled k-space by a
factor of two using spiral acquisition, that is spiral SMS acquisition. Each plane in the
k-space was under sampled using a different spiral trajectory (as shown in figure
4.7) to simulate the SMS acquisition. It is reconstructed using the optimization
equation explained before. Figure 4.10 shows the difference between the two
different acquisitions after reconstruction. It is obvious that the difference is not in
the brain details. It is just making some areas darker and others brighter randomly.
The RMSE between the two reconstructions is 16% in the masked region of the
head. This RMSE value is a result of the difference shown in figure 4.10, and so the
reconstruction output is good since a large part of the difference is not in the image
details.
55
Figure 4.8: Two slices that are acquired with fully sampled k-space using spiral acquisition.
Figure 4.9: Two slices that are acquired with under sampled by a factor of two k-space using
SMS spiral acquisition.
56
Figure 4.10: The difference between the fully samples k-space and the half sampled.
The same method that is applied for two slices is applied for four slices now. The
simulation is done by under sampling every plane in k-space by 4. Figure 4.11
shows four slices that are acquired using fully sampled k-space using spiral
acquisition. It is reconstructed using the optimization equation explained before.
Figure 4.12 shows four slices that are acquired using under sampled k-space by a
factor of four using spiral acquisition. Each plane in the k-space was under sampled
using a different spiral trajectory to simulate the SMS acquisition. It is reconstructed
using the optimization equation explained before. Figure 4.13 shows the difference
between the two different acquisitions after reconstruction. It is obvious that the
difference is not in the brain details. It is just making some areas darker and others
brighter randomly. The RMSE between the two reconstructions is 27%. Since this
RMSE value is a result of the difference shown in figure 4.13, I can consider the
output to be good because a large part of the difference is not in the image details.
Finally, to appreciate the output of the proposed optimization method, figure 4.14
57
illustrates the simulated data for SMS of four if only standard gridding is used in
solving k-space from the SMS spiral trajectories without optimizing the cost
function in solving for the image.
Figure 4.11: Four slices that are acquired with fully sampled k-space using spiral acquisition.
Figure 4.12: Four slices that are acquired with under sampled by a factor of four k-space using
SMS spiral acquisition.
Figure 4.13: The difference between the fully samples k-space and the quarter sampled.
58
Figure 4.14: Reconstructing the under sampled k-space without using the proposed
optimization technique.
4.5.2
Spectroscopy Data
The motivation for the work done in this thesis is to apply SMS for spiral
spectroscopy in MRSI. The benefit of using spiral SMS spectroscopy is accelerating
the process of data acquisition for 2D spectroscopy excitations without causing
degradation in SNR when voxel size and imaging time are kept constant. In the
simulations in this chapter, SMS of four is used for spiral spectroscopy. This SMS
excitation achieves an acceleration factor of four, and so it is possible to finish one
acquisition in quarter the time that the normal 2D excitation takes, the associated
SNR with this acceleration is quantified. The FOV for the 2D slice is 240mm. The
slice size is 96x96 pixels, and so resolution is 2.5mm. The scanner gradients are
specified to have maximum amplitude of 40 mT/m, and maximum slew rate of 150
T/m/s.
In chapter 2, I have proved using the signal equation that in EPI, there is no problem
in extending SMS to include the frequency (Kf) dimension. The same concept is used
to prove that the spiral SMS could be used to acquire spectroscopy data using spiral
trajectories. In order to show that this is possible, a simulation similar to the one
59
applied in the structural data in the sub-section above is used. However, the
simulation is extended to include a number of frequencies, instead of one frequency
only. Figure 4.15 explains the computations used to simulate the SMS spiral
spectroscopy data from the scanner.
I- 4D data (x, y, z, f) from 64 coils with estimated coil
sensitivities
2- Use Fourier transform to get K-Space (Kx, Ky, Kz, KO)
Ir
3- Under sample in every Kx-Ky plane by a factor of 4 using spiral trajectory
(This data is to be used as the acquired data)
4) The output data now is 3D (Kxy, Kz, Kf)
where Kxy is the acquired vector for a spiral in Kx-Ky plane
Figure 4.15: The process of creating spectroscopy data similar to the one acquired using spiral
SMS for MRSI.
The same 3D data set acquired using 64 coils is used here. The addition of the
frequency access is simulated by adding another 63 frequency points to each voxel.
Therefore, there is 64 frequency points for each voxel now, one is used for H2 0, one
for Cr, one for Cho and one for NAA. The decision of using 64 frequency points was
made so that it is possible to run a large number of simulations in short time. The
method has the same performance regardless the number of the frequencies used if
run time is ignored. The simulation values for the metabolites are a function of the
60
index in x, y and the value of that voxel in acquired 3D data set. The values at the
other 60 frequencies are set to zero for now.
Figure 4.16 shows the ground truth for the spectroscopy simulations. It illustrates
the four slices after reconstructing the acquired data using a fully sampled k-space
spiral trajectory. The presented slices are the values of the H2 0 from each voxel
spectrum. The spectrum of two voxel from each slice is presented under that slice.
$AAA
LAAL
an
~
A
U
*
Il
AA
A
Figure 4.16: The simulated spectra for some voxels when fully sampled k-space using spiral
acquisition is used. This is the ground truth.
Figure 4.17 illustrates the four slices after reconstructing the acquired data using
the proposed spectroscopy SMS spiral acquisition. The same technique that was
used for a single frequency in the previous sub-section is used here for all
frequencies to simulate spiral SMS spectroscopy. The main observation that can be
made here is that there is no aliasing between the different frequencies; that is non61
metabolite frequencies, where the signal is zero with fully sampled k-space in 4.16,
continue to have a signal of zero after the spiral SMS spectroscopy.
AA
em-
A
'AAt
j
hAA
A., A
Figure 4.17: The simulated spectra for some voxels when acquisition simulation of spiral SMS
spectroscopy excitation of four is used.
The RMSE between the ground truth in figure 4.16, and the SMS acquisition in figure
4.17 is 27%. It is important to notice that this RMSE is the same as the one when
only one frequency was for structural data; that means using spiral SMS for
spectroscopy without the addition of noise does not change the RMSE. Similar to the
case of structural imaging, I look at the difference between the two acquisitions
before judging if this RMSE is good or bad.
Figure 4.18 illustrates the difference between the two spectroscopy-data sets is
shown in figure 4.16 and 4.17. It is obvious that the difference here is only where
there are metabolites. Also, the value of the difference is proportional to the value of
62
the signal at that metabolite. Finally, the difference does not affect the details of the
data, in a similar trend to the structural case where it only makes some areas darker
and others brighter. Therefore, I can consider the reconstruction performance to be
good here.
B-
-
em I
am
0
I
se
IB
m
am
$AD
0"nI
.. aI
amI*
a-s
.
B-*
an
enI
.a
*
T1
1,
AA JAA
Figure 4.18: The difference between the fully sampled k-space and SMS acquired k-space after
reconstruction.
In order to confirm that aliasing is not happening between difference adjacent
frequencies, the values from all the non-metabolite frequencies are summed to
create the image shown in figure 4.19.
63
X 104
4
3.5
3
2.5
2
1.5
1
0.5
0
Figure 4.19: The image resulted after summing all the non-metabolite values for each voxel.
In the figure above, it is obvious that the signal is not zero, which means that it
might be affected by the metabolites signal. However, this effect is completely
negligible. In the data set used to compute figure 4.19, the max value for a
metabolite signal in the (x, y, z, t) space is 1, however, the value in this summed nonmetabolite signal is less than 10-8.
The next step of simulations, that examine the proposed method for solving spiral
SMS spectroscopy, involves adding zero mean Gaussian to the entire data set. In
order to decide the standard deviation of the noise, the average amplitude of each
metabolite signal is calculated from the ground truth data as shown in table 4-1
below. The standard deviation is chosen to be 0.003 so it gives the SNR shown in the
table.
Metabolite
Average Amplitude
SNR
H20
0.148
0.148/0.003=49.3
Cho
0.033
0.033/0.003=11.0
Cr
0.024
0.024/0.003=8.0
NAA
0.037
0.037/0.003=12.3
Table 4-1: The average amplitude for the different metabolites' signals and the correspondent
SNR values based on the added noise.
64
Figure 4.20 shows the four slices of the noisy data acquired using a spiral trajectory
for fully sampled k-space. The presented slices are the values of the H20 peak from
each voxel spectrum. The spectrum of two voxel from each slices is presented under
each slice. We can see that it is easy to recognize the four peaks for H20, Cr, Cho,
NAA, and noise in other frequencies.
saI
ofa
a
so
I
aEM
am
am
*A-
*1
E
I a"
toI
a
am
as
Ia
voxels when fully sampled k-space of noisy data
using spiral acquisition is used.
Figure 4.20: The simulated spectra for some
Figure 4.21 shows the four slices after reconstructing the noisy acquired data using
the proposed spectroscopy SMS spiral acquisition. It is obvious here that the
metabolites signals can be recognized without difficulties in the reconstruction of
spiral SMS spectroscopy with added noise. In order to quantify the changes in the
SNR between the fully sampled spiral k-space and the spiral SMS sampling, SNR
65
calculations are done on both data sets illustrated in figures 4.20 and 4.21. The SNR
results are presented in the table below.
Metabolite
SNR for figure 4.20=
SNR for figure 4.2 1=
Average Amplitude/noise std
Average Amplitude/noise std
H2 0
0.150/0.0037 = 40.5
0.142/0.0037 = 38.4
Cho
0.035/0.0037 = 9.5
0.033/0.0037 = 8.9
Cr
0.026/0.0037 = 7
0.024/0.0037 = 6.5
NAA
0.038/0.0037= 10.2
0.036/0.0037 = 9.7
Table 4-2: The SNR values for the different metabolites after spiral full k-space reconstruction
and spiral SMS reconstruction.
The table above includes the information needed to understand how the
reconstruction method proposed here affects the SNR in the acquired data. First,
one can see that the noise is the same weather spiral SMS acquisition is used, or fully
sampled k-space spiral acquisition is used, it comes to be zero mean Gaussian with
standard deviation of 0.0037 after the reconstruction in both cases. The average
value for the signal amplitude changes within a range of 7% of its original value. For
SNR, the degradation after spiral SMS spectroscopy is less that 7% in the worst case.
The results above give the conclusion that the spiral SMS spectroscopy proposed
algorithm could still function in the presence of noise without problems.
In order to have a better understanding of the difference between the two
acquisitions, fully sampled and SMS under sampled for spiral spectroscopy. The
difference is computed, that is (Ifully sampled - SMS sampledi), and is shown in
figure 4.22. Looking at the figure, the first conclusion is that the difference is
proportional to the signal amplitude at a given frequency. We can see that whenever
66
the difference has a higher value at a specific frequency, then that is the H2 0
frequency. Also it is obvious that the signal difference at H20 frequency is higher,
when the H20 amplitude is initially high, as seen in the 2 slices to the right. The
conclusion is that the difference between the two acquisitions at any frequency is
affected by the metabolite amplitude there.
*A
MS
SM
MS
62
WA
an
IA.
MS
an
MS
j
Figure 4.21: The simulated spectra for some voxels when acquisition simulation of spiral SMS
spectroscopy excitation is used for noisy data.
67
m
e
am
Figure 4.22: The difference between the fully sampled k-space and SMS acquired k-space after
reconstruction for the noisy data.
4.6
EPI SMS Spectroscopy
In the previous section, the performance of Spiral SMS for spectroscopy was tested
by comparing it to the performance of ordinary spiral spectroscopy. Same data set
was used in both computations, and spiral SMS for spectroscopy has shown
promising results. In order to have another demonstration that SMS is possible for
spectroscopy data, simulations of using EPI sequence for spectroscopy data are
implemented here. The trajectory of EPI acquisition in the 3D space of (Kx, Ky, KfO is
shown in figure 4.23.
68
Kx
Ky
Figure 4.23: An EPI trajectory in the Kx-Ky-Kf space that is used for EPI spectroscopy.
4.6.1
EPI SMS Spectroscopy Simulations
The implementation process here requires the simulation of spectroscopy data, and
then extension of the slice-GRAPPA algorithm introduced in [15] to work for
spectroscopy data.
In chapter 2, it has been proven using the signal equation that the phase modulation
blips has the same effect in all Kx-Ky planes that are in the (Kx, Ky, Kf) space.
Therefore, the process of simulating spectroscopy data needs only applying the
appropriate FOV shift for all the x-y planes that are in the (x, y, f) space. In a method
similar to the one used to create SMS spiral spectroscopy data, SMS EPI
spectroscopy data will be created.
In order to solve the aliased data, the same slice-GRAPPA algorithm is extended to
include the frequency dimension. In this process, to build each slice (Kx, Ky, Kf) from
69
the collapsed data, it is enough to train the kernel at one Kf value, and then use this
trained kernel in the reconstruction for all the other Kf point, of course only for the
slice k-space where the kernel is trained. For example if the collapsed data set for 4
slices has Kx = 64, Ky = 64, Kf = 320, then only 4 kernels need to be trained, one for
each slice.
4.6.2
EPI SMS Spectroscopy Results
3D data set with simulated coil sensitivity to create 8 coils acquisition is used here.
The addition of the frequency access is simulated by adding another 63 frequency
points for each voxel. For each voxel there is 64 frequency points now, one is used
for H2 0, one for Cr, one for Cho and one for NAA. The simulation values for the
metabolites are a function of the index in x, y and the value of that voxel in acquired
3D data set. The values at the other 60 frequencies are set to zero for now. SMS
acquisition of four is used here; and so one slice is not affected, one is shifted by
FOV/4, another by FOV/2 and the last by 3FOV/4. This shift is applied in all
frequencies. Then the after shift slices are collapsed into one slice. Image
reconstruction has been done on the simulated data to demonstrate how possibility
of SMS with EPI spectroscopy.
The benefit of using EPI SMS spectroscopy is accelerating the process of data
acquisition for 2D spectroscopy excitations without causing degradation in SNR. In
the simulations in this chapter, SMS of four is used for EPI spectroscopy. This SMS
excitation achieves an acceleration factor of four, and so it is possible to finish one
70
acquisition in quarter the time that the normal 2D excitation takes, the associated
SNR with this acceleration is quantified.
It is important to notice here that SMS with EPI require a pre-acquired data set
using single shot and SMS so that it can be used to train the kernel. In the work done
here, the same set that is used to train the kernel is tested with this kernel.
Therefore, the results might not be possible to achieve in real time, but the point
here is to demonstrate the possibility of extracting the un-aliased data if the right
kernel is used for EPI SMS spectroscopy.
Figure 4.24 shows the ground truth for the EPI spectroscopy simulations. It
illustrates the four slices after using a fully sampled k-space for each. The presented
slices are the values of the H2 0 from each voxel spectrum. The spectrum of two
voxel from each slice is presented under that slice.
71
$A
*a
as
Wa
02
a."
*'a.
6A
M
*1
0.06
0.1
00
AA
MI
L
AA
I
0.
'531
A
Figure 4.24: The simulated spectra for some voxels when fully sampled k-space using EPI
acquisition is used. This is the ground truth.
Figure 4.25 shows the collapsed four slices when EPI SMS (blipped-CAIPI) is used to
excite four slices simultaneously. Figure 4.26 illustrates the four slices after
reconstructing the acquired collapsed data using the extended slice-grappa
algorithm. The main observation that can be made here is that there is no aliasing
between the different frequencies; that is non-metabolite frequencies, where the
signal is zero with fully sampled k-space in 4.24, continue to have a signal of zero
after the spiral SMS spectroscopy.
72
Figure 4.25: Collapsed K-space data resulted of EPI SMS acquisition of four slices.
an
AA
an
*
IM
a"]
ILA
Ma~L~K
Ilj
Figure 4.26: The simulated spectra for some voxels when acquisition simulation of EPI SMS
spectroscopy excitation of four is used.
The RMSE between the ground truth in figure 4.24, and the SMS acquisition in figure
4.26 is 2%, which means a very good reconstruction. Similar to the case of spiral
SMS spectroscopy imaging, I look at the difference between the two acquisitions.
Figure 4.27 illustrates the difference between the two EPI spectroscopy-data sets is
73
shown in figure 4.24 and 4.26. It is obvious that the difference here is only where
there are metabolites, and it is less than 2% of the signal value.
.me
A
-A
.
*a
M"
a-
I
. ---
A.
I
Figure 4.27: The difference between the fully sampled k-space and SMS acquired k-space after
reconstruction.
Similar to the spiral SMS spectroscopy, the EPI spectroscopy slice-grappa algorithm
will be tested after the addition of zero mean Gaussian noise to the signal. In order
to decide the standard deviation of the noise, the average amplitude of each
metabolite signal is calculated from the ground truth data as shown in the table
below. The standard deviation is chosen to be 0.006 so it gives the SNR shown in the
table
74
Metabolite
Average Amplitude
SNR
H2 0
0.317
0.317/0.006 = 52
Cho
0.037
0.037/0.006= 6
Cr
0.023
0.023/0.006 =4
NAA
0.111
0.111/0.006= 18
Table 4-3: The average amplitude for the different metabolites' signals and the correspondent
SNR values based on the added noise.
Figure 2.28 shows the four slices using a fully sampled k-space for each after
addition of the zero mean Gaussian noise. The presented slices are the values of the
H2 0 peak from each voxel spectrum. The spectrum of two voxel from each slices is
presented under each slice. We can see that it is easy to recognize the four peaks for
H20, Cr, Cho, NAA, and noise in the other frequencies.
*
an
0*
01
0a
A*
0*
0.1
0*
an
&Is
U
Is
I
5I
a
V
Figure 4.28: The simulated spectra for some voxels when fully sampled k-space using EPI
acquisition is used with noisy data.
75
Figure 4.29 shows the four slices after reconstructing the noisy acquired data using
the proposed EPI SMS spectroscopy acquisition. It is obvious here that the
metabolites signals can be recognized without difficulties in the reconstruction of
EPI SMS spectroscopy with added noise. In order to quantify the changes in the SNR
between the fully sampled k-space and the EPI SMS sampling, SNR calculations are
done on both data sets illustrated in figures 4.28 and 4.29. The SNR results are
presented in the table below.
Metabolite
SNR for figure 4.28=
Average Amplitude/noise std
SNR for figure 4.29=
Average Amplitude/noise std
H2 0
0.317/0.006=52.8
0.317/0.006=52.8
Cho
0.037/0.006=6.2
0.037/0.006=6.2
Cr
0.03/0.006=5.0
0.03/0.006=5.0
NAA
0.111/0.006=18.5
0.111/0.006=18.5
Table 4-4: The SNR values for the different metabolites after EPI full k-space reconstruction
and EPI SMS reconstruction.
The table above includes the information needed to understand how the
reconstruction method proposed here affects the SNR in acquired data. We can see
that the SNR in the same id EPI SMS spectroscopy is used, or full sampling is used.
This means the extended to spectroscopy slice-grappa functions without any
problem in the presence of noise.
In order to have a better understanding of the small difference between the two
acquisitions, fully sampled and SMS sampled for EPI spectroscopy. The difference is
computed; that is (Ifully sampled - SMS sampledi), and is shown in figure 4.30.
76
Looking at the figure, the first conclusion is that the difference is proportional to the
signal amplitude at a given frequency. We can see that whenever the difference has
a higher value at a specific frequency, then that is the H20 frequency. The same
conclusion reached in the spiral SMS spectroscopy can be confirmed here; that is the
difference between the two acquisitions, fully sampled and SMS sampled, at any
frequency is affected by the metabolite amplitude there.
an
C.,
em
em
n
8
*4
*
U
U
U
U
S.
~
*
*
--j
a
-.
.
u
a
Figure 4.29: The simulated spectra for some voxels when acquisition simulation of EPI SMS
spectroscopy excitation of four is used with noisy data.
77
an
am
am
am
T*at
9"W
GM
04"
am
em
440*a
'~
-w
OM'
saw
em
OM
6*0
s
am
Figure 4.30: The difference between the fully sampled k-space and SMS acquired k-space after
reconstruction for the noisy data.
4.7
Summary
In this chapter, I have demonstrated the technique that can be used to apply
Simultaneous multislice acquisition technique to MR spectroscopy using spiral
trajectory acquisition. After that, simulations have been done to implement the
expected effect of the spiral SMS acquisitions on the data. The simulations were
implemented first on collapsed two slices of structural imaging only. Then it was
extended to include four slices, and finally to function with more than one
frequency, so MR spectroscopy can be used. The data were un-aliased using an
optimization method that is usually used to solve under sampled k-space data in
MRI. The reconstructions of both EPI and spiral SMS spectroscopy has been
quantified and compared to the ground truth data. The SNR for all reconstructions
78
was computed, and it is shown how spiral SMS spectroscopy degrades SNR by less
than 7% of it initial value in the worst case, where EPI SMS spectroscopy does not
change SNR.
79
80
Chapter 5
Conclusion
The work in this thesis investigates the possibility of using simultaneous multislice
acquisition with spiral trajectory for MR Spectroscopy. The SMS spectroscopy
advantages include the acceleration of 2D excitation spectroscopy and the
possibility of acquiring one average in a faster time than the current 3D excitations.
The potential of using these advantages was the motivation behind implementing
the work presented here. Signal equation was studied first to guarantee the
possibility of applying SMS on spectroscopy acquisitions.
The thesis has addressed the two important stages in using the MR scanner for MR
spectroscopy SMS with spiral trajectory. In chapter 3, the first stage discusses the
right method to be used for RF pulse design. Two methods, windowed Sinc and
Parks-McClellan Algorithm, were used to design the RF pulse. Extensive simulations
have been used to ensure that the better method is used, and the right pulse is
81
designed. The choice of RF excitation pulse parameter has justified for the goal of
exciting multislice simultaneously for spectroscopy. The results have shown that
Parks-McClellan Algorithm has advantages over the other designs for a more
selective excitation. It made it possible to decide the maximum value for the out-ofslice ripples in the design unlike the windowed sinc pulse method. Also a multi band
pulse is possible to design instead of adding modulated pulses to excite more than
one slice.
The second stage in using MR scanner is the data acquisition stage. This stage
includes the design of the x, y and z gradients that are used to encode the k-space
when the signal is received. In chapter 4, the discussion explains the basics of the
spiral trajectory acquisition. Spiral with SMS is explained, and the design for the
blips in the z gradient is justified. A demonstration of how the data are affected
when Spiral SMS acquisition used is presented. Also, I have explained how the data
can be treated as an under sampled k-space, and so optimization technique to be
used in solving the problem. The algorithm used to solve such a problem was
implemented here.
Simulations have been used to implement the effects that SMS excitation and
acquisition imposes on the data. Finally, a number of reconstruction examples have
been presented to confirm that the proposed method is working. The used examples
included spiral acquisition for structural scan, I illustrated that the design works for
two simultaneously excited slices, then this was extended to four slices. After that,
82
the method is extended to included spectroscopy data for both spiral SMS and EPI
SMS chemical shift scans.
Overall, the possibility of using SMS with spiral trajectory for MR spectroscopy
imaging has been investigated here. All simulations have confirmed that the
proposed methods make it possible. Also the EPI spectroscopy sequence is proven
to work with SMS excitation here. The future work that can be a follow up the work
presented here is implementing the designed RF pulse and gradients on the scanner.
Then use the technique explained in chapter 4.4 for data reconstruction.
The implemented code used in running the simulations of this thesis is included in
the appendix. The Parks-McClellan Algorithm to design the RF pulse for two slices
excitation is in appendix A. The code implemented to simulate the SMS spiral
spectroscopy data is in appendix B. The code used in solving spiral SMS
spectroscopy data is in appendix C. Finally the extension to the current slice-grappa
algorithm to work for EPI SMS spectroscopy is included in appendix D. Any prerequirements for the code to be used is stated in the appendix.
83
84
Appendix A
The function below is the one implemented to design the Parks-McClellan RF pulse
for excitation of 2 slices simultaneously. The code is an extension of John Pauly's
rftools [18]. The installation of rftools is required to be able to run this code. The
code is included in sms-spectroscopy/rf design.
function [ rf ] = dz2rfpm( n,tb,ptype,dis,dl,d2)
There are a lot of options, most of
%
%
Designs an rf pulse.
which have defaults.
%
%
%
%
%
%
%
%
%
%
%
%
%
Inputs are:
(required)
np -- number of points.
(required)
tb -- time-bandwidth product
ptype -- pulse type. Options are:
(default)
st -- small tip angle
ex -- pi/2 excitation pulse
se -- pi spin-echo pulse
sat -- pi/2 saturation pulse
inv -- inversion pulse
dis - for seperation between slices
(default = 0.01)
dl -- Passband ripple
(default = 0.01)
d2 -- Stopband ripple
(default = 1.5)
pclsfrac -- pcls tolerance
if nargin < 5, dl = 0.01; d2 = 0.01; end;
end;
if nargin < 3, ptype = 'st';
if nargin < 4, dis = 0 ; end;
if strcmp(ptype, 'st'),
bsf = 1;
elseif strcmp(ptype, 'ex'),
bsf = sqrt(1/2);
dl = sqrt(dl/2);
d2 = d2/sqrt(2);
elseif strcmp(ptype,'se'),
bsf = 1;
dl = dl/4;
d2 = sqrt(d2);
elseif strcmp(ptype, 'inv'),
85
bsf = 1;
dl = dl/8;
d2 = sqrt(d2/2);
elseif strcmp(ptype,'sat'),
bsf = sqrt(1/2);
dl = dl/2;
d2 = sqrt(d2);
else
',ptype]);
disp(['Unrecognized Pulse Type -disp('Recognized types are st, ex, se, inv, and sat');
return;
end;
di = dinf(dl,d2);
w = di/tb;
shift = dis*n/2;
(1+w)*(tb/2)
trans =
-
(1-w)*(tb/2);
band = 2*(1-w)*(tb/2);
f = [0 shift shift+trans shift+trans+band shift+2*trans+band
(n/2)]/(n/2);
m = [0 0 1 1 0 0];
w
b
[dl/d2 1 dl/d2];
firpm(n-1,f,m,w);
=
=
if strcmp(ptype,'st'),
rf = b;
else
b = bsf*b;
rf = b2rf(b);
end;
end
86
Appendix B
The output of this code is ZI, the simulation of the acquired data using spiral SMS
spectroscopic imaging. It contains the data for the 4 slices, each slice has 64
frequencies in this simulation and each one of this comes from 64 different channels
in this simulation. The code here creates spiral trajectories, then samples Cartesian
x-y-z-f based of those trajectories. The code and data are included in
sms.spectroscopy/spiralsim&recon.
% 150 T/m/s
% G/cm
% Sampling time on GE scanners [Seconds]
% No of Interleaves
smax = 15000;
gmax = 4;
T = 4e-6;
N = 1;
Fcoeff =
[6,
-0];
% FOV
im size = 96;
FOV max = 240;
res = FOV max / im size;
% resolution in mm
kmax = 5 / res;
% cm^(-l), corresponds to
rad = [0:.1:kmax];
FOV = zeros(size(rad));
for t = 1:length(Fcoeff)
FOV = FOV + Fcoeff(t)
.*(rad/kmax)
1mm resolution.
(t-1);
end
figure(2), plot(rad, FOV) ,axis([O kmax 0 max(FOV)])
kvds = k-space trajectory (kx+iky) in cm-1.
%
[k-vds, g, s, time, r,theta] = vds(smax, gmax, T, N, Fcoeff, kmax);
87
figure(1), plot(real(kvds) / 1, imag(kvds) /
.5]
1),
axis([-.5,
.5, -. 5,
* kmax * 2)
numchan = 64;
%% Import data and build the x-y-z-f matrix
%
load SENS_96x96x64x57.mat;
sens = imresize(SENS, [im size,imsize]);
sensl = sens ./ max(abs(sens(:)));
sensitivity maps
%
x-y-z data for different channels
load img_96x96x64x57.mat
imgs = imresize(img, [im size,imsize]);
imgs = imgs / max(imgs(:));
mask = ones(96,96,64,2);
y = 48.5:.5:96;
for j = 1:64
for i=1:96
mask (:,i,j,1) = y;
mask (i,:,j,2) = y.^2;
end
end
mask(:,:,:,,1)
= mask(:,:,:,1)/max(max(mask(:,:,1,1)));
mask(:,:, : ,2) = mask(:,:,:,2)/max(max(mask(:,:,1,2)));
num slices = 4;
spectrumsize = 64;
img-sens = zeros(imsize,imsize,numslices,spectrumsize,numchan);
%zero noise case
k_space sens =
zeros(im-size,imsize,numslices,spectrumsize,num-chan);
%add water
img-sens(:,:,1,12,:)
imgsens(:,:,2,12,:)
imgsens(:,:,3,12,:)
imgsens(:,:,4,12,:)
%add Cho
img-sens(:,:,1,40,:)
imgsens(:,:,2,40,:)
imgsens(:,:,3,40,:)
imgsens(:,:,4,40,:)
%add Cr
imgsens(:,:,1,44,:)
imgsens(:,:,2,44,:)
imgsens(:,:,3,44,:)
imgsens(:,:,4,44,:)
%add NAA
imgsens(:,:,1,56,:)
img-sens(:,:,2,56,:)
img-sens(:,:,3,56,:)
imgs(:,:,:,18);
imgs(:,:,:,28);
imgs(:,:,:,38);
imgs(:,:,:,48);
0.1*imgs(:,:,:,18).*mask(:,:,:,1);
0.1*imgs(:,:,:,28).*mask(:,:,:,1);
0.1*imgs(:,:,:,38).*mask(:,:,:,1);
0.1*imgs(:,:,:,48).*mask(:,:,:,1);
0.1*imgs(:,:,:,18).*mask(:,:,:,2);
0.1*imgs(:,:,:,28).*mask(:,:,:,2);
0.1*imgs(:,:,:,38).*mask(:,:,:,2);
0.1*imgs(:,:,:,48).*mask(:,:,:,2);
0.25*imgs(:,:,:,18);
0.25*imgs(:,:,:,28);
0.25*imgs(:,:,:,38);
88
img-sens(:,:,4,56,:) =
0.25*imgs(:,:,:,48);
for c = 1:size(kspacesens,5)
k_spacesens(:,:,:,:,c) =
(1/sqrt(length(imgsens(:,:,:,:,c))))*fftshift(fftn(fftshift(imgsens(:
end
%% find k-space on the spiral points using interp2
ZI = zeros(length(k vds), numslices,spectrumsize,numchan);
[Y,X] = meshgrid(linspace(-kmax,kmax,imsize), linspace(kmax,kmax,im size));
XI = real(kvds);
YI = imag(kvds);
%different trajectories for the different Kz planes to simulate SMS
XI_1 = real(exp(2i*pi/4)*k vds);
YI_1 = imag(exp(2i*pi/4)*k-vds);
XI_2 = real(exp(4i*pi/4)*k-vds);
YI_2 = imag(exp(4i*pi/4)*k-vds);
XI_3 = real(exp(6i*pi/4)*k-vds);
YI_3 = imag(exp(6i*pi/4)*k-vds);
XI_4 = real(exp(8i*pi/4)*k-vds);
YI_4 = imag(exp(8i*pi/4)*k-vds);
for f=l:spectrumsize
for c = 1:size(kspacesens,5)
ZI(:,1,f,c) = interp2(Y, X,
kspace sens(:,:,1,f,c),
YI_1, XI-1).';
ZI(:,2,f,c) = interp2(Y, X, kspace sens(:,:,2,f,c), YI_2, XI_2).';
ZI(:,3,f,c) = interp2(Y, X, kspace sens(:,:,3,f,c), YI_3, XI_3).';
ZI(:,4,f,c) = interp2(Y, X, kspace-sens(:,:,4,f,c), YI_4, XI_4).';
end
end
89
90
Appendix C
The function fnlCg_SENSE2DSMS()
implemented here to solve for the image
using spiral SMS spectroscopy data as input. The code below modifies the algorithm
implanted by Lustig et al. [22] to solve image from sparse under-sampled k-space
using conjugate gradient with line search method. The code and data are included in
sms.spectroscopy/spiralsim&recon.
%% spiral-sense recon for SMS
oversamplingfactor = 1;
param = init;
param.FT
k_vds.');
=
Resampling2D(kmax, oversamplingfactor*imsize, num slices,
param.XFM = 1;
param.Itnlim = 5;
param.numcoils = numchan;
Res = zeros([oversamplingfactor*imsize, oversamplingfactor*imsize,
numslices,spectrumsize]);
param.data = [I;
param.sens =
temp = zeros(im size, im-size, num slices ,spectrum-size)
for c = 1:numchan
param.data{c} = ZI(:,:,:,c);
for f=1:spectrumsize
%add coil sensitivites for the diffetn slices
,1,f) = sensl(:,:,c,18);
temp (:,:
,2,f) = sensl(:,:,c,28);
temp (:,:
,3,f) = sensl(:,:,c,38);
temp (:,:
,4,f) = sensl(:,:,c,48);
temp (:,:
end
param.sens{c} = temp;
91
end
for
1:1
Res = fnlCgSENSE2D_SMS(Res, param); %solve for the image
n =
end
function x = fnlCg_SENSE2DSMS(xO,params)
%----------------------------------------------------------------------
% implementation of a Li penalized non linear conjugate gradient
% reconstruction The function solves the following problem:
% given k-space measurments y, a fourier operator F, and Cartesian to
spiral operator C-->S, the function
% finds the image x that minimizes:
% Phi(x) =
II(C-->S)F* C *x - y||^2
% the optimization method used is non linear conjugate gradient with
fast&cheap backtracking
% line-search.
%
params.sens : sensitivity maps, with size [im sizex, im-size-y,
num coils]
%---------------------------------------------------------------------x = xO;
% line search parameters
maxlsiter = params.lineSearchItnlim
gradToll = params.gradToll ;
alpha = params.lineSearchAlpha;
beta = params.lineSearchBeta;
to = params.lineSearchTO;
k =
t =
0;
1;
% copmute gO
=
grad(Phi(x))
gO = wGradient(x,params);
dx = -go;
% iterations
while(1)
% backtracking line-search
% pre-calculate values, such that it would be cheap to compute the
objective
% many times for efficient line-search
[FTXFMtx, FTXFMtdx] = preobjective(x, dx, params);
fO = objective(FTXFMtx, FTXFMtdx, DXFMtdx, 0, params);
t = to;
92
[fl,
ERRobj]
=
objective(FTXFMtx, FTXFMtdx, t, params);
isiter = 0;
while (fl > fO - alpha*t*abs(gO(:)'*dx(:)))^2 & (lsiter<maxlsiter)
lsiter = isiter + 1;
t = t * beta;
[fl, ERRobj]
=
objective(FTXFMtx, FTXFMtdx, t, params);
end
if
lsiter == maxlsiter
disp('Reached max line search..... not so good... might have a
bug in operators. exiting... ');
return;
end
% control the number of line searches by adapting the initial step
search
if lsiter > 2
tO = tO
*
beta;
end
if lsiter<1
tO = tO / beta;
end
x =
(x
+ t*dx);
%--------uncomment for debug purposes -----------------------if mod(k,1) == 0 && (k > 0)
disp(sprintf('%d
, obj: %f, L-S: %d', k,fl,lsiter));
end
%---------------------------------------------------------------
%conjugate gradient calculation
gl = wGradient(x,params);
bk = gl(:)'*gl(:)/(gO(:)'*gO(:)+eps);
gO = gl;
dx = - gl + bk* dx;
k = k + 1;
%TODO: need to "think" of a "better" stopping criteria
if (k > params.Itnlim) I (norm(dx(:)) < gradToll)
break;
end
;-)
end
return;
% ---------------------------------------------------------------------
function [FCtx, FCtdx] = preobjective(x, dx, params)
% precalculates transforms to make line search cheap
tx = params.XFM'*x;
93
tdx = params.XFM'*dx;
FCtx =
FCtdx =
[];
for coil = 1:params.numcoils
FCtx{coil} = params.FT * (params.sens{coil} .* tx);
FCtdx{coil} = params.FT * (params.sens{coil} .* tdx);
end
% --------------------------------------------------------------------function [res, obj] = objective(FCtx, FCtdx,t, params)
%calculates the objective function
p = params.pNorm;
obj
=
0;
for coil = 1:params.num coils
temp = FCtx{coil} + t * FCtdx{coil} - params.data{coil};
obj = obj + norm(temp(:))^2;
end
res = obj
function grad = wGradient(x,params)
gradObj = gOBJ(x,params);
grad = (gradObj );
% ---------------------------------------------------------------------
function gradObj = gOBJ(x,params)
% computes the gradient of the data consistency
tx = params.XFM'*x;
temp = 0;
for coil = 1:params.numcoils
Res = params.FT * (params.sens{coil}
temp = temp + conj(params.sens{coil})
end
gradObj = params.XFM*temp;
gradObj = 2*gradObj;
94
.*
.*
tx) - params.data{coil};
(params.FT'*Res);
Appendix D
The code below is an extension of slice-grappa algorithm implemented in [15] to
solve for EPI SMS. The implementation here makes it possible to apply slice-grappa
for
EPI
SMS
spectroscopy
data. The
code
and
data
are
included
sms.spectroscopy/EPIsim&recon.
load tse_27msdicom;
img = tse_27msdicom;
imagesize = 96;
img = imresize(img, [imagesize,imagesize]);
img = img / max(img(:));
mosaic(img(:,:,20:end),3,5,1,
sens = (bl);
load bl_8chan;
sens = imresize(sens, [imagesize,imagesize]);
sens = sens ./ max(abs(sens(:)));
numchan = size(sens,3);
spectrumsize = 64;
mask = ones(imagesize,imagesize,38,2);
y = image size/2+.5:.5:imagesize;
for j=1:38
for i=1:64
mask (:,i,j,1) = y;
mask (i,:,j,2) = y.^2;
end
end
mask(:,:,:,1) = mask(:,:,:,1)/max(max(max(mask(:,:,:,1))));
mask(:,:,:,2) = mask(:,:,:,2)/max(max(max(mask(:,:,:,2))));
%csimg = normrnd(O, .006, [dim,dim,38,spectrum size]); % Gaussian
noise
% zero
cs_img = zeros([imagesize,imagesize,38,spectrumsize]);
noise
%add
cs img(:,:,:,1 2 )+ img;
csimg(:,:,:,12) =
water
95
in
%add
csimg(:,:,:,40) =csimg(:,:,:,40)+ .2*img.*mask(:,:,:,1);
Cho
cs img(:,:,:,44) = cs img(:,:,:,44)+.2*img.*mask(:,:,:,2);
cs img(:,:,:,56) =csimg(:,:,:,56)+ 0.35*img;
NAA
training-slices = 10:6:30;
% location of training slices
num_slices = length(trainingslices);
sliceshift = round(linspace(0, size(img,1) *
,
%add Cr
%add
(num slices-1)/num-slices
numslices));
imgslices = [];
for s = 1:numslices
for f = 1:spectrumsize
imgslices(:,:,:,s,f) = circshift(
repmat(csimg(:,:,trainingslices(s),f),
[sliceshift(s),O,O]
[1,1,num-chan])
.*
sens,
);
end
end
kspaceslices = zeros(size(img slices));
% generate collapsed and individual k-space
for s = 1:numslices
for c = 1:numchan
kspaceslices(:,:,c,s,:) =
fftshift(fftn(fftshift(imgslices(:,:,c,s,:))));
end
end
kspacecollapse = (sum(kspaceslices,4));
% collapsed k-space%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% kernel training using individual k-space data
%% train one set of kernels for each collapsed slice using onr
frequency only
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
kernel sizeX = 5;
kernelleftSizeX = round((kernelsizeX-1.1) / 2);
kernel rightSizeX = round((kernel-sizeX-.9) / 2);
kernel sizeY = 5;
kernel leftSizeY = round((kernelsizeY-1.1) / 2);
kernel rightSizeY = round((kernel-sizeY-.9) / 2);
kxlimits = kernelleftSizeX + 1 : size(csimg,1) - kernel_rightSizeX;
kylimits = kernel leftSizeY + 1 : size(csimg,2) - kernel_rightSizeY;
96
kspacetraining = zeros(length(kx limits)*length(ky limits),
kernelsizeX*kernelsizeY*num-chan);
kspace targets = zeros(length(kx limits)*length(kylimits),
size(sens,3), num slices);
ind =
1;
for kx = kx limits(1) : kx limits(end)
kylimits(end)
for ky = kylimits(1)
for s = 1:numslices
kspacetargets(ind, :, s) = kspaceslices(kx, ky,
1:size(sens,3), s,spectrum-size/2);
end
temp = kspace_collapse(kxkernelleftSizeX:kx+kernel_rightSizeX, kykernelleftSizeY:ky+kernelrightSizeY, :,1,spectrum-size/2);
kspacetraining(ind, :) = temp(:).';
ind = ind + 1;
end
end
kernels = [];
%% least squares kernel fit
for s = 1:numslices
kernels(:,:,s) = kspacetraining \ kspacetargets(:,:,s);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% recon collapsed slices with the trained kernels
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
testslices = training-slices;
imslices = [];
for f= 1:size(csimg,4)
for s =
1:numslices
imIslices(:,:,:,s,f)
circshift(repmat(csimg(:,:,test-slices(s),f), [1,1,size(sens,3)])
sens, [sliceshift(s),O,O]);
end
end
kspcslices
=
[1;
% generate collapsed and individual k-space
for s =
1:numslices
for c = 1:size(sens,3)
kspcslices(:,:,c,s,:) = fftshift(fftn(fftshift(
im-slices(:,:,c,s,:) )));
end
end
97
.*
kspccollapse =
(sum(kspcslices,4));
% Recon with convolution in k-space
kspcrecons = zeros([size(kspccollapse(:,:,1)), size(sens,3),
numslices,size(csimg,4)] );
ind =
1;
for kx = kx limits(1) : kx-limits(end)
for ky = ky-limits(1) : kylimits(end)
for f= 1:size(csimg,4)
temp = kspccollapse(kx-kernelleftSizeX:kx+kernelrightSizeX,
ky-kernelleftSizeY:ky+kernel-rightSizeY, :,1,f);
for s = 1:num slices
kspcrecons(kx,ky,:,s,f) =
(temp(:).')
*
kernels(:,:,s);
end
end
ind = ind + 1;
end
end
= zeros(size(im slices));
img true = [];
[];
imslicesreal = zeros(size(im slices));
im recons
im-res =
rmse =
[I;
for s = 1:numslices
for c = 1:size(sens,3)
templ = fftshift(ifftn(fftshift(kspc_recons(:,:,c,s,:))));
1,1,:);
im recons(:,:,c,s,:) = templ(:, :,
temp2 = fftshift(ifftn(fftshift(kspace_slices(:,:,c,s,:))));
im-slicesreal(:,:,c,s,:) = temp2(:, :, 1,1,:);
end
for f= 1:size(csimg,4)
im res(:,:,s,f) = adaptivecombine(
circshift(imrecons(:,:,1:size(sens,3),s,f), [-sliceshift(s),0,0]),
sens );
img-true(:,:,s,f) = adaptive combine(
circshift(imslices real(:,:,:,s,f), [-slice shift(s),O,O]), sens );
end
t2 = imgtrue(:,:,s,12);
tl = im res(:,:,s,12);
rmse(s) = 100 * norm(tl(:)-t2(:)) / norm(t2(:));
end
98
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