Echo Planar Spectroscopic Imaging and
31
P In Vivo Spectroscopy
ECHO PLANAR SPECTROSCOPIC IMAGING AND
31
P IN VIVO
MAGNETIC RESONANCE SPECTROSCOPY
By
SERGEI I. OBRUCHKOV, M.Sc.
McMaster University
©Copyright by Sergei I. Obruchkov, March 2010
DOCTOR OF PHILOSOPHY (2010)
McMaster University
(Medical Physics and Applied Radiation Sciences)
TITLE:
Echo Planar Spectroscopic Imaging and
Hamilton, Ontario
31
P In Vivo Magnetic Resonance
Spectroscopy
AUTHOR:
Sergei I. Obruchkov, B.Sc.(Dalhousie University), M.Sc.(McMaster Uni-
versity)
SUPERVISOR:
Professor M.D. Noseworthy
NUMBER OF PAGES:
xv, 178
ii
Abstract
The work in this thesis deals with pre–clinical development of rapid in vivo 31 P magnetic resonance spectroscopy (MRS) techniques. Current MRI literature of 31 P spectroscopy presents evidence of increased concentrations of phosphomonoesters (PME),
and phosphodiester (PDE) as well as inorganic phosphate concentrations in tumor
tissue. Human breast cancer studies have demonstrated correlation between disease
progression and both PME and PDE peaks. Furthermore, 31 P MRS can be used to
detect, grade tumours and monitor response to chemo and radiation therapy.
Tumor measurements are typically static (i.e. single measurement per scan). In other
experiments, on muscle for example, dynamic measures are required the purpose of
which is to assess temporal function and recovery. In all 31 P acquisitions there are
problems surrounding RF coil design, pulse sequence speed, localization and system
calibration. The work presented here focused on improving all these aspects and
provide easy and reliable work flow to use 31 P MRS in a clinical setting.
One of the aspects of this thesis lies in designing and construction of an RF coil
that is well suited for integration with a clinical MRI breast imaging and biopsy
system. The designed coil was tuned for simultaneous operation at 31 P (51.73 MHz)
and 1 H (127.88MHz) Larmor frequencies. This design has advantages in the fact that
complex pulse sequences with heteronuclear decoupling could be performed easily.
The additional features of the coil design is that it is possible to swap it into the
breast imaging system without moving the patient. Along with the designed coil,
custom software was written to assist with transmit gain calibration of 31 P RF pulses,
to ensure maximum MR signal. The automated prescan ensures easy work flow and
minimizes the operator variability and patient time inside the MR scanner.
Another aspect of this thesis deals with rapid pulse sequence development, to further
speed up the 31 P MRS data acquisition. Echo planar spectroscopic imaging (EPSI)
with a fly–back gradient trajectory is currently one of the most reliable and robust
techniques for speeding up chemical shift imaging (CSI) acquisitions. A 31 P EPSI
sequence was written to acquire spectroscopic imaging data at 1, 2 and 2.6 cm spatial
resolution and spectral bandwidth of 3125 Hz. The sequence showed an ability to
speed up data acquisition up to 16 times, where SNR permits.
Phantom studies were used to verify the double tuned coil and EPSI sequence ensuring proper and safe operation. In vivo measurements of an exercising muscle
demonstrated the ability of 31 P EPSI to play an important role in rapidly acquiring
spatially localized 31 P spectroscopic data.
With these preclinical developments in place a clinical trial is possible using 31 P MRS
rapidly and efficiently. Furthermore the increased usability of 31 P MRS provided by
the tools developed in this thesis can prove to be beneficial by integrating 31 P MRS
into existing clinical protocols.
iii
Acknowledgements
I wish to thank my supervisor professor Michael D. Noseworthy, Ph.D for all his guidance,
support and encouragement throughout this project. I also would like to thank my supervisory committee, Dr. Alex Bain, Dr. Charles Cunningham and Dr. Michael Patterson it
is an honor to be guided by such bright people.
There is an incredible number of people who helped me with this project. I would like to
thank the entire Diagnostic Imaging Department of St. Joseph’s Healthcare. Thank you
for taking me in as one of your own, I hope I gave back as much as I took. Toni Cormier
thank you very much for all your help. A big thanks goes to Norm Konyer, who has shared
his wisdom and allowed his calm personality to rub off on me.
A very special thank you goes to Ken Bradshwaw from Sentinelle Medical who provided
much needed direction in MR hardware development. Thanks to John Snider from GE
Healthcare, for spending so much time with me and trusting me to take appart the scanner,
you are an incredible resource that I will greatly miss.
I would like to acknowledge the department of Medical Physics and Applied Radiation
Sciences. I could not complete this thesis without their support.
A special thanks goes out to the Dalhousie University Physics class of 2003, my development
as a student and a physicist could not have had a better start. Thank you very much for
still being there for me.
And of course I would like to thank the most important people in my life, my parents Igor
and Galina I couldn’t be here without your help. Thank you for all of your sacrifices and
your unwavering support. My beautiful and incredible wife Heather I simply don’t know
how I could do anything without you, thank you for your love, your support and your
confidence in me. And lastly to my children Hali and Anya, thank you for enriching my
life, you are my angels.
iv
Dedicated to the memory of
Dr. Masayoshi Senba
everyone can teach but
not everyone can
teach like you...
v
Table of Contents
Abstract
iii
Acknowledgements
iv
List of Figures
ix
List of Tables
xii
List of Notations and Abbreviations
xiii
Chapter 1
1.1
1.2
1
General Overview And Introduction to in vivo Spectroscopy . . . . .
1
1.1.1
Clinical MRS . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.2
Clinical MRS Techniques and Experimental Considerations . .
4
1.1.3
Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Multinuclear spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . .
11
1.2.1
On the nature of spin . . . . . . . . . . . . . . . . . . . . . . .
11
1.2.2
Phosphorus NMR . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.2.3
Decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.2.4
Adiabatic RF . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
1.2.5
Echo Planar Spectroscopic Imaging (EPSI) . . . . . . . . . . .
20
1.2.6
The Problem and the Approach to a Solution . . . . . . . . .
22
Chapter 2
2.1
Introduction to In Vivo Spectroscopy
27
Theory
Basic NMR Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.1
Quantum Description . . . . . . . . . . . . . . . . . . . . . . .
27
2.1.2
Classical Description . . . . . . . . . . . . . . . . . . . . . . .
30
2.1.3
Bulk Magnetization . . . . . . . . . . . . . . . . . . . . . . . .
31
2.1.4
NMR Signal Detection . . . . . . . . . . . . . . . . . . . . . .
33
2.1.5
Longitudinal and Transverse Relaxation . . . . . . . . . . . .
35
2.2
Chemical Shift
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.3
J–Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
2.4
Decoupling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5
Introduction to Imaging: Hard vs Soft RF excitation . . . . . . . . .
48
2.6
Adiabatic experiments . . . . . . . . . . . . . . . . . . . . . . . . . .
53
2.6.1
64
Spatial Encoding . . . . . . . . . . . . . . . . . . . . . . . . .
vi
Chapter 3
jectory
Echo Planar Spectroscopic Imaging of
31 P
using Flyback Tra-
69
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
3.2
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.2.1
Chemical Shift Imaging signal equation . . . . . . . . . . . . .
71
3.2.2
Optimization of Echo–Planar Flyback Readout Trajectory Gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3.3
Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . .
82
3.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.4.1
Calibration Results . . . . . . . . . . . . . . . . . . . . . . . .
86
3.4.2
EPSI Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.2.3
3.5
Chapter 4
Imaging
4.1
Design and Construction of a Dual Tuned RF Coil for MR Breast
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
Motivation for Dual Tuned Coil for Breast 31 P Spectroscopy
With Two Port Input . . . . . . . . . . . . . . . . . . . . . . .
95
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
4.1.1
4.2
4.2.1
4.3
97
Resonating RLC Circuits . . . . . . . . . . . . . . . . . . . . . 101
4.2.3
Reciprocity Principle and Coil Quality Factor . . . . . . . . . 106
4.2.4
Transmit–Receive Switch . . . . . . . . . . . . . . . . . . . . . 108
4.2.5
Methods for Dual–Tuning Coils . . . . . . . . . . . . . . . . . 111
4.2.6
Dual Tuned Coil Design for Breast Imaging and Spectroscopy
116
Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 119
31
P Solenoid Coil For System Stability and QA Assurance Tests120
4.3.2
Dual Tuned Surface Coil Design and Construction . . . . . . . 123
4.3.3
B1 and B0 Field Calibration . . . . . . . . . . . . . . . . . . . 125
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
4.4.1
4.5
Propagation of Electromagnetic Waves and Magnetic Field Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.2
4.3.1
4.4
93
Coil Performance Imaging, Spectroscopy and Decoupling . . . 128
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
vii
Chapter 5 Design and Construction of an inexpensive Polyvinyl Alcohol
Breast Like Phantom for use with MR Imaging
135
5.1
Introducion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2
Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Chapter 6 Dynamic Measurements of 31 P Metabolites in a Posterior Compartment of the Leg During Plantar Flexion Exercise Using Surface Coil or
Echo Planar Spectroscopic Imaging for Localization
143
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.2
Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Chapter 7
Conclusions and Future Directions
155
7.1
Future Applications of the Constructed Double Tuned 1 H – 31 P Breast
RF Coil and the Designed flyback 31 P EPSI Sequence . . . . . . . . . 155
7.2
Future Directions and other Applications of ESPI sequence and Multi
Tuned MR Coils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Bibliography
161
Appendix A
169
Appendix B
175
Appendix C
177
viii
List of Figures
1.1
RF Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.2
Nuclear structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.3
31P Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
1.4
EPSI PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.1
Zeeman Splitting of Energy Levels . . . . . . . . . . . . . . . . . . . . .
29
2.2
Spin Precession of 1 H Nuclei in Magnetic Field Bo . . . . . . . . . . . .
30
2.3
Contribution of individual 1 H Nuclei Magnetic Moments to Bulk Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.4
Magnetization in Lab vs Rotating Frames of Refference . . . . . . . . . .
34
2.5
Longitudinal and Transverse Relaxation . . . . . . . . . . . . . . . . . .
39
2.6
Spin–Echo Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.7
Chemical Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.8
J–Coupling between two heteronuclear systems energy levels. . . . . . . .
43
2.9
J–coupling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.10 Slice Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
2.11 RF Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2.13 NMR a reversible adiabatic experiment . . . . . . . . . . . . . . . . . . .
56
2.14 Effective Field created by the Ofresonance effects . . . . . . . . . . . . .
61
2.15 Adiabatic Pulses and Their Profile . . . . . . . . . . . . . . . . . . . . .
63
2.16 Spatial Encoding of 2D Frequency Space and Fourier Transformation . .
67
3.1
2D CSI Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.2
2D CSI data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
3.3
Comparing Point Spread Functions . . . . . . . . . . . . . . . . . . . . .
75
3.4
EPSI Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.5
Segmented Flyback EPSI sequence . . . . . . . . . . . . . . . . . . . . .
77
3.6
Phase offset of the EPSI acquisition . . . . . . . . . . . . . . . . . . . . .
79
3.7
3.0 Tesla MRI Scanner . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3.8
Linear Readout with Dead Time . . . . . . . . . . . . . . . . . . . . . .
84
3.9
Synchronising Gradients and Data Sampling . . . . . . . . . . . . . . . .
85
3.10 Interleaving Gradient Waveforms . . . . . . . . . . . . . . . . . . . . . .
86
ix
3.12 Overlay of Echoes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.13 CSI and EPSI Data of a Spherical Phantom . . . . . . . . . . . . . . . .
89
3.14 High Resolution CSI and EPSI Data . . . . . . . . . . . . . . . . . . . .
90
3.15 SNR Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.1
Vanguard Breast Biopsy Table . . . . . . . . . . . . . . . . . . . . . . . .
96
4.2
Field generated from a circular loop . . . . . . . . . . . . . . . . . . . . .
98
4.3
Field Profile of a Surface Coil . . . . . . . . . . . . . . . . . . . . . . . . 100
4.4
Resonating RLC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5
Tuning and Matching Capacitors in MR coil . . . . . . . . . . . . . . . . 106
4.6
Reactance and Resistance of the RF Coil . . . . . . . . . . . . . . . . . . 108
4.7
Linear and Hybrid TR switch schematics . . . . . . . . . . . . . . . . . . 110
4.8
Circuits for dual tune approaches . . . . . . . . . . . . . . . . . . . . . . 111
4.9
Two Coil Approach to Dual Tuning . . . . . . . . . . . . . . . . . . . . . 113
4.10 Crossed Elements Dual–Tuned Coils . . . . . . . . . . . . . . . . . . . . 115
4.11 Layout of Orthogonal Dual Tuned Birdcage Coils . . . . . . . . . . . . . 115
4.12 Circuit for a Dual Tuned Coil . . . . . . . . . . . . . . . . . . . . . . . . 116
4.13 Reactances of RLC, Tank and dual tuned circuits . . . . . . . . . . . . . 117
4.14 Decoupler Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.15 RF Coil Dock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.16 Simulated Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.17 Physical Layout of the Coil . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.18 Varying value of L2 inductor . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.19 Dual Tuned Coil and Sentinelle Breast Biopsy Table . . . . . . . . . . . 125
4.20 TG Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.21 High Order Shimming data . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.22 Bench Testing Results of a Dual Tuned Coil . . . . . . . . . . . . . . . . 129
4.23 B1 Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.24 Gradient Echo Excitation Profile . . . . . . . . . . . . . . . . . . . . . . 130
4.25 Spin Echo Excitation Profile . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.26 Decoupling of Methylphosphonic Acid . . . . . . . . . . . . . . . . . . . 132
5.1
Polyvinyl alcohol molecule . . . . . . . . . . . . . . . . . . . . . . . . . . 137
x
5.3
T1 and T2 Weighted Phantom MR Images . . . . . . . . . . . . . . . . . 141
6.1
Schematic of the biochemical reaction in muscle bioenergetics . . . . . . 146
6.2
CSI and EPSI Data Overlaid Over 1 H Image of a Leg . . . . . . . . . . . 149
6.3
Dynamic Study of Plantar Flexion Exercise Amplitude Change . . . . . . 150
6.4
PCr and Pi in an Exercising Leg . . . . . . . . . . . . . . . . . . . . . . 151
6.5
Dynamic Study of Plantar Flexion Exercise Spectral Frequency Change . 152
6.6
pH Calculation of 31 P MRS in an Exercising Leg Without Spatial Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
xi
List of Tables
1.1
Phosphorus chemical shift . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.1
In vivo concentration of phosphorous metabolites in human brain tissue.
71
3.2
SNR is proportional to square root of total acquisition time . . . . . . .
73
3.3
1.5 T and 3.0 T Magnet Parameters
. . . . . . . . . . . . . . . . . . . .
82
3.4
List of Gradient Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.1
T1 and T2 of 10% and 15% PVA cryogel . . . . . . . . . . . . . . . . . . 141
6.1
Relaxation Times of
31
P Metabolites and Their Chemical Shifts . . . . . 146
xii
List of Notations and Abbreviations
R1
Longitudinal relaxation constant of a contrast agent
T1
Longitudinal relaxation time
T2
Transverse relaxation time
χ
Magnetic susceptibility
CEMRA
Contrast enhanced magnetic resonance angiography
CNR
Contrast to noise ratio
CT
Computed Tomography
DC
Direct current
ETL
Echo train length
FID
Free induction decay
FOV
Field of view
FSE
Fast spin echo
GE
General Electric
GRE
Gradient recalled echo
Gd-DTPA-BMA
Gadolinium complex of diethylenetriaminepentaacetic
acid bismethylamide.
Gd
Gadolinium
IR
Inversion recovery
LV
Levenberg-Marquardt
MIP
Maximum intensity projection
MRA
Magnetic resonance angiography
MRI
Magnetic resonance imaging
MTT
Mean transit time
MVA
Minimal variance algorithm
xiii
NEX
Number of excitations
NMR
Nuclear magnetic resonance
PE
Phase encoding
PR
Projection reconstruction
RF
Radio Frequency
ROI
Region of interest
SE
Spin echo
SNR
Signal to noise ratio
HOS
High Order Shimming
SPGR Spoiled gradient recalled
SoS
Sum of squares
TE
Time to echo
TR
Repetition time
TTP
Time to peak
T
Tesla
bpm
Beats per minute
cc
Cubic centimeter
ml
Milliliter
mM
Millimolar
mm
Millimeter
ms
Milliseconds
s
Second
xiv
E
Electric Field Intensity (volts/meter)
D
Electric Flux Density (coulombs/meter2 )
H
Magnetic Field Intensity (amperes/meter)
B
Magnetic Flux Density (webers/meter2 )
J
Electric Current Density (amperes/meter2 )
ρ
Electric Charge Density (coulombs/meter3 )
0
Free space permitivity (farads/meter)
µ0
Free Space Permeability (henrys/meter)
σ
Conductivity (siemens/meter)
xv
xvi
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Chapter 1
Introduction to In Vivo Spectroscopy
1.1 General Overview And Introduction to in vivo
Spectroscopy
Nuclear magnetic resonance theory describes the physical interaction of particles with
a nuclear magnetic dipole moment µ̂ in a magnetic field Bo . In vivo NMR refers to
a technique which enables identification and quantification of a large number of biologically important molecules in the intact tissue of a living organism (some good
references to in vivo NMR are de Certaines et al. (1992), de Graaf (2005a) and
de Graaf (1998)). In vivo NMR also known as magnetic resonance spectroscopy
(MRS) is used as a diagnostic tool on a daily basis world wide. The first in vivo
spectra were obtained from rat brain in 1983 by Behar at Yale University, using a
modified NMR apparatus. Soon after, papers by Lauterbur and Mansfield appeared
describing in vivo NMR imaging concepts. Applying position dependent magnetic
field gradients in addition to the static field allowed Lauterbur and Mansfield to spatially localize NMR signal and construct an image. Soon after the first successful
experiments, apparatus for clinical applications became available. The first hardware
for producing images was big and bulky, and imaging was slow as only a few pulse
sequences were available. First clinical diagnosis were done largely by measuring T1
1
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
and T2 times of tissues. Today in vivo NMR got renamed as simply MRI, allowing
images to be acquired with high soft tissue contrast and in vivo spectroscopy can
be performed from any particular volume within the tissue using accurate localization techniques. Clinical diagnosis today can be performed by much more advanced
techniques than was first thought possible: for example, assessment of diffusion of
water inside the tissues using diffusion tensor and diffusion weighted imaging (DTI
and DWI, respectfully), functional brain imaging using fMRI, angiography using contrast agents, susceptibility weighted imaging (SWI) allowing for venous mapping and
observing calcium and iron deposits, and spectral editing techniques for studying
metabolic concentrations of lactate and gamma-aminobutyric acid (GABA) in vivo,
just to name a few.
The following sections will describe in more detail clinically available magnetic resonance spectroscopy (MRS) techniques, current approaches and challenges.
1.1.1 Clinical MRS
Proton magnetic resonance spectroscopy or hydrogen spectroscopy (1 H-MRS) is the
most clinically applied spectroscopy technique today. The proton is one of the most
NMR sensitive nuclei not only in terms of high gyromagnetic ratio but also of its natural abundance (99.9%) and concentration (i.e. since virtually all metabolites contain
protons). Proton spectroscopy has been most successfully applied in identifying and
quantifying cerebral metabolites and is considered a valuable diagnostic tool in diag2
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
nosing and grading many diseases including metabolic disorders, and cancer. Even
though the sensitivity of in vivo 1 H NMR is limited to the detection of metabolites
with concentrations above 0.5 mM (de Graaf 2005a) it has still proven invaluable in
diagnosing aberrant metabolic processes.
The success of proton spectroscopy, in many parts of the body, has been limited due
to technical difficulties and limited information content e.g. liver and muscle spectra
are relatively simple in comparison to brain spectra due to large dominating lipid
resonances. In addition, liver spectroscopy is challenging due to high iron concentrations and respiratory motion which both contribute to broadening of resonance
lines. Muscle 1 H spectroscopy is of interest as there are significant differences in
metabolic content between muscle groups and metabolic states (e.g. rest/exercise).
In addition, metabolite resonances such as carnosine 2 and 4 exhibit chemical shift
dependence on pH enabling the use of 1 H spectroscopy as an intracellular pH meter
(Moon and Richards 1973; Pan et al. 1988). With the introduction of commercially
available pulse sequences optimised for prostate and breast cancer studies 1 H MRS
is undoubtedly a growingly useful technique that can be applied to all parts of the
body.
Other NMR visible nuclei (see section 1.2.1) such as
31
P,
23
Na,
13
C, 7 Li,
19
F and
39
K
are either used rarely or almost never used for clinical spectroscopy. There are many
reasons for this, the first being that most commercial clinical MR systems are designed
to work only with 1 H nuclei (i.e. what is only required for clinical imaging). Hardware
for “multi” nuclear work is available as an extension to most existing clinical systems
3
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
however rough estimate puts less then 5% of all MRI systems in Canada capable of
this functionality∗ . Section 1.2 will deal with multinuclear spectroscopy in particular,
for now I will focus on MRS in general bringing examples from the clinical application
of proton spectroscopy.
1.1.2 Clinical MRS Techniques and Experimental Considerations
Obtaining MR images and MR spectra depend on selective excitation of a certain
volume from which signal is recorded. This is accomplished by manipulating magnetic
gradients and RF pulses in order to excite a particular volume of interest. The signal
from the excited volume is acquired, digitised and processed via Fourier transform.
Following acquisition, data reconstruction results get processed via a semi-automatic
analysis and the end result is diplayed. State of the art MR systems today routinely
present only a simple analysis, and are often incapable of automatically correcting
for errors if the spectra are deemed of poor quality.
From the clinical perspective the technical process is transparent to the end user.
Clinical users are given only an ability to choose the volume of interest on the anatomy
and specify the most basic NMR related parameters. At first sight clinical MRS seems
straight forward. In reality, however, clinical MRS acquisition is incredibly complex.
Even if all the technical problems could be solved, challenges of patient positioning,
patient cooperation, voxel placement and diagnosis are still great. Thus, even though
∗
Currently there are approximate;y 200 MRI systems in Canada, therefore around 10 have MNS
capabilities.
4
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
the data from MRS is so rich it is simply not as widely clinically accepted as one
would imagine.
There are a few reasons as to why wide use of clinical spectroscopy is limited. First,
the acquisition spectral artifacts can be hard to identify and often require a physical
scientist specialising in spectroscopy to perform analysis. Secondly achieving absolute
metabolite concentrations (and how these can be related to clinical conditions) is not
established, or even completely agreed upon, which results in many between-study,
and even between-patient variations.
Spectral Localization
There are two types of single voxel spectral localisation approaches: multi shot techniques, such as image selected in vivo spectroscopy (ISIS) and single shot techniques
such as stimulated echo acquisition mode (STEAM) or point resolved spectroscopy
(PRESS). In addition, an array of other techniques have previously been used, for
example volume selective spectroscopy (VOSY) and volume restriction using RF surface coils. In current clinical practice however, STEAM and PRESS are considered
the “tools of the trade”. Both techniques utilise three slice selective RF pulses and
gradient combinations exciting three orthogonal slabs in such a way as to generate
an echo (MR signal) from an intersecting volume.
The PRESS technique utilises a combination of 90o -180o -180o (slice selective pulses)
to produce a spin-echo while the STEAM sequence use a 90o -90o -90o combination to
5
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
produce a stimulated echo. Over a single TR the PRESS pulse sequence is usually
longer (i.e. longer TE) than STEAM and thus has difficulty visualizing metabolites with short relaxation times. The PRESS sequence is known to better visualize
metabolites with longer T2 relaxation times, is less susceptible to motion, diffusion,
and quantum effects, and has a theoretically better SNR than STEAM over the same
voxel.
All of the above described techniques are known as single voxel spectroscopy as they
are used to acquire data from a single volume of interest (VOI). For more information
on localisation techniques please see the paper by Keevil (2006). A comparison between STEAM and PRESS, in terms of variability in metabolite measurements can
be seen in Kim et al. (2005).
Although a single voxel acquisition is tremendously useful for understanding in vivo
metabolic processes, often comparison between diseased and healthy tissues, or a
study of regional metabolite variation is needed. With single voxel scan times often
approaching 10 minutes it is not clinically feasible to perform multiple single voxel
acquisitions. Hence, multi-voxel, or chemical shift imaging (CSI) was developed.
Chemical shift imaging is an extension of the single voxel MRS technique used to
obtain data from multiple voxels. This may be achieved essentially by sequentially
acquiring data from adjacent voxels to produce spectral maps in 2D and 3D. Details
of the CSI approach is discussed further in section 1.2.5.
6
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
RF Coils
Radiofrequency (RF) coils are used to obtain MR signals. These are high Q resonating
circuits tuned to a specific resonating frequency, dependent on the nucleus (more
specifically the nuclear gyromagnetic ratio), and the strength of magnetic field. There
are two types of coils: surface coils and volume coils. Volume coils are typically large
assemblies in a Helmholtz, solenoid, or birdcage geometry, that can produce a uniform
B1 RF field which is used to excite the spins. Surface coils are much more efficient
at receiving a signal close to the coil but their effectiveness significantly decreases
at about half the coil radius away. They are rarely used for excitation as they do
not produce a uniform RF B1 field (see Figure 2.11). Therefore current approaches
in MR utilize an array of receive-only surface coils that can be positioned close to
the patient’s body (i.e. ensuring good coupling) to achieve a large volume coverage
to receive MR signal. Volume coils are used for excitation purposes. Volume coils
and arrays are superior for imaging applications. However, the use of coil arrays for
spectroscopy adds a layer of complexity as it becomes non-trivial to add multiple
broadband spectroscopic signals together. Specifically, phase correction and scaling
factors during the adding process may affect quantification during the data analysis.
Relaxation, Diffusion, Coupling Effects and Other Distortions.
MR spectra are affected by user parameter selections: repetition time (TR), echo
time (TE), location of the voxel within the anatomy, and other factors such as mag7
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 1.1: B1 field plots of a birdcage coil (right) and surface coil cross–section
(left). The flip angle iso-lines clearly show a region of uniform excitation that can be
achieved using bird cage geometry.
netic field uniformity (∆B0 , ∆B1 ), patient motion, and temperature to name a few.
Nevertheless MR spectroscopy is clinically extremely useful, as long as the effects
and artifacts are understood and identified during data analysis to ensure correct
diagnosis.
TR and TE times set by the user affect spectral appearance as different species relax
with different T1 and T2 relaxation times. Ideally a very short TE and a very long
TR, are best for capturing all of the metabolites on a spectrum. However this is often
disadvantageous as the spectrum would be extremely crowded and hard to interpret
plus patient scan time would be unreasonable long. Often longer TE times (e.g. 144,
288 ms) are used in order to let species with shorter T2 times relax and produce
spectra with only certain metabolites.
Molecular diffusion during the delays between RF pulses and application of magnetic field gradients results in signal contamination from spins that are not properly
“tagged”. This produces a spectrum which is difficult to quantify, especially when
comparing molecules with large diffusion coefficients. Minimizing TE, using smaller
8
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
gradients, or using a pulse sequence that is insensitive to diffusion e.g. utilising
Carr-Purcell-Meiboom-Gill (CPMG) echoes instead of Hahn echoes can be used to
eliminate this effect (de Graaf 2005a).
Other factors that affect spectroscopy include scalar coupling between different nuclei
e.g.
1
H–13 C, and the chosen water/fat suppression techniques used. To add more
to the complexities, angular orientation of structured tissue in the magnetic field,
truncated sampling causing “voxel bleeding”, blood pulsatility near the voxel and
many more factors not mentioned here tend to complicate the life of an in vivo NMR
scientist.
1.1.3 Data Analysis
Fourier transformed NMR, in principle, can be used as a quantitative technique to
derive absolute metabolite concentrations in tissues. The area under the curve of the
FFT data is related to the longitudinal thermal equilibrium magnetization vector,
which in turn is proportional to the number of spins in the sample. True quantification of the data however is challenging as often the RF field (B1 ) varies spatially
throughout the sample. In addition, in vivo NMR lacks a true internal or external
reference for quantification: internally all metabolites can potentially vary, while external references (e.g. vials of chemical solutions placed on the subject’s skin) do not
experience the same B1 and B0 fields as the voxel of interest. Proton spectroscopy
often utilises concentration of the endogenous metabolite creatine in the brain. The
9
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
reported concentrations of other metabolites are then reported relative to the creatine
peak. But absolute concentration cannot truly be calculated this way due to large
variability of creatine within people, and also between ages and diseases. The external
reference technique utilises a small vial with a known concentration of metabolites.
Thus this approach requires acquisition of at least two spectra; one from the VOI
another from the reference sample. The RF inhomogeneity between the two locations
can be accounted for if the coil’s RF field distribution is known.
The analysis of spectra often requires input from the user, as automated techniques
either do a poor job (i.e. do not work on all spectra due to poor SNR, artifacts, mysteriously varying baselines, etc.) or require previous knowledge such as a database
of standards. There are a few approaches to automated spectral analysis, the most
successful of these techniques utilising a least squares fitting algorithms. The general
principle of these methods is that the fitted “theoretical” spectrum should resemble the experimental spectrum as closely as possible by varying parameters Mo , T2 ,
∆ω, φo for each resonance. The two most frequently used algorithms are VARPO
(Vanhamme et al. 2001), (Mierisova and Ala-Korpela 2001) a time domain algorithm, and LCmodel (Provencher 1993), a frequency domain algorithm. The time
domain method is flexible enough to quantify frequency shifts and can avoid problems with intense baselines, while frequency domain analysis is most often utilised
for more complex spectra, such that are observed with short echo times. However,
the LCmodel technique can perform analysis only for specific echo times TE=30, 35,
50 and 144ms. Although if other echo times are desired new basis sets can be devel10
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
oped (a time consuming process). Automated spectral analysis approaches, without
a priori knowledge are quite poor as results have to fit within 95% of a certainty calculation otherwise the convergence will fail. Further, to avoid baseline issues, some
methods, instead of integrating peaks, use convolution and deconvolution, to collapse
the peaks into a series of delta like functions the heights of which are proportional to
peak “areas under the curve”. This approach reports relative heights of the peaks as
the relative concentrations.
1.2 Multinuclear spectroscopy.
Clinical implementation of Nuclear Magnetic Resonance (NMR) is largely focused on
1
H nuclei mostly due to their abundance and biological importance. There is a vast
number of other NMR visible nuclei. The visibility of a nuclear isotope is dependent
on its atomic structure. This section will focus on
31
P in vivo spectroscopy.
1.2.1 On the nature of spin
Spin I is an intrinsic property of a fundamental particle. In order to understand why
some atoms have spin and other do not, a brief review of subatomic particles and
their properties is necessary (see Figure 1.2). The classical view of an atom consists
of a positively charged nucleus surrounded by a cloud of orbiting negatively charged
electrons. The nucleus of an atom always has a positive net charge, and is made up
from neutrons and protons. However the neutron and proton are not fundamental
11
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
atomic particles, as if either is broken apart smaller subatomic particles, called quarks,
will be released. Quarks come in six different flavors and three colors. Both neutron
and proton each consist of three quarks that are held together by gluons. From Figure
1.2 it is noted that the neutron consists of two down quarks and one up quark, adding
up the charges of these three quarks −1/3e − 1/3e + 2/3e = 0 gives a neutron net
zero charge, while adding the charges of three quarks that make up a proton we end
up with a net charge of −1/3e + 2/3e + 2/3e = +1e, where e is electron charge.
Performing additions with quark spins results in a net spin of the proton and neutron
calculated as I = 21 , however a proton has a positive spin and neutron negative spin.
Since most nuclei are made up from multiple neutrons and protons, it is possible to
divide them into three categories:
• Even-even nuclei: Having an even number of protons and neutrons, the spins
cancel out producing a net spin of zero.
• Even-odd and odd-even nuclei: The final spin of nuclei with uneven numbers of
protons and neutron is determined by the single unpaired nucleon.
• Odd-odd: This type of nucleus is harder to describe as the unpaired protons and
neutrons produce a net spin that is determined by the parallel and anti-parallel
alignment of the proton and neutron spins.
Nuclei with spin have net positive or net negative spin depending on the excess of
protons or neutrons. The sign of the spin has only one effect on a nucleus in a
magnetic field: positive spins will align with the magnetic field and will precess in
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 1.2: Modern view of a nucleus.
a clockwise direction, while a nucleus with negative spin will align anti-parallel with
the magnetic field and precess counter clockwise around the magnetic field.
1.2.2 Phosphorus NMR
The success of proton NMR in clinical applications has only been matched by phosphorus
31
P NMR. A high natural abundance and sensitivity of around 7% that of
proton, allows acquisition of a good quality spectrum within clinically relevant/acceptable time scales (i.e. minutes). A large chemical shift dispersion of 30 ppm,
and considerably fewer metabolites than 1 H-NMR, produces “good” peak resolution,
see table 1.1. Most importantly
31
P spectroscopy is capable of detecting all of the
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
key compounds involved in energy metabolism. Furthermore the chemical shift of
some phosphorus containing compounds is sensitive to the concentrations of magnesium and pH. This property can be explained by the chemical exchange phenomenon,
where during the protonation with magnesium, changes in the local magnetic environment of
31
P nucleus occur. Chemical exchange creates two species of the compound
e.g. protonated and unprotonated, with two distinct chemical shifts but because this
exchange process is rapid in contrast to the NMR time scale only an average chemical
shift of the two species is observed. Clinically relevant pH measurements are performed on the inorganic phosphate peak using a Henderson-Hasselbach relationship:
pH = pKA + log
δ − δHA
δA − δ
(1.1)
were δ is the observed chemical shift, δHA and δA are protonated and unprotonated
chemical shifts, and pKA is a equilibrium constant of a compound, (pK=6.77 for inorganic phosphate) (Hoult et al. 1974; Moon and Richards 1973; Pan et al. 1988).
The calculations of free magnesium concentrations is often deduced similarly by using
equation 1.1 from the chemical shift of the β-ATP dominant peak. Phosphorus spectroscopy has been also applied for non invasive measurements of the creatine kinase
enzymatic reaction (de Graaf 2005a).
P Cr2− + M gADP − + H + ←→ M gAT P 2− + Cr
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Table 1.1: Phosphorus chemical shifts for low molecular weight metabolites, at pH 7.2
and fully complexed with magnesium (Shulman et. al. 2005). Typically spectral width
on a clinical 1.5T spectrometer of 500Hz is enough to acquire all of the 1 H resonances,
phosphorus on the other hand requires about 3 kHz.
31
Compound
Adenosine monophospate (AMP)
Adenosine diphosphate (ADP)
α
β
α
β
γ
Adenosine triphospate (ATP)
Dihydroxyacetone phosphate
Fructose-6-phosphate
Glucose-1-phosphate
Glucose-6-phosphate
Glycerol-1-phospate
Glycerol-3-phosphorycholine
Glycerol-3-phosphorylethanolamino
Inorganic Phosphate
Nicotinamide adenine dinucleotide (NADH)
Phosphocreatine
Phosphoenolpyruvate
Phosphorylcholine
31
P chemical shift (ppm)
6.33
-7.05
-3.09
-7.52
-16.26
-2.48
7.56
6.64
5.15
7.20
7.02
2.76
3.20
5.02
-8.30
0.00
2.06
5.88
P NMR spectra are comparatively simple (relative to proton spectra) with reso-
nances from phosphocreatine (PCr), adenosine triphosphate (ATP), inorganic phosphate (Pi) and phosphomono and phosphodiesters. In vivo
31
P NMR can be used
to probe dynamic changes of rapid and slow high energy phosphate reactions. For
example, the chemical shift of the Pi and the β–ATP resonances are dependent on the
pH and magnesium concentration. These exchange reactions are typically very fast as
they are not catalyzed by enzymes. Many of the other compounds that are typically
observed by
31
P NMR are very important in enzyme–catalyzed reactions, such as:
ATP synthesis and the interconversion of PCr and ATP by creatine kinase. Combin15
PhD Thesis
ing
31
Sergei Obruchkov
Dept. of Medical Physics
P MRS with magnetization transfer (MT) experiments have been extensively
used to quantitatively study metabolic pathways (Forsen and Hoffman 1963; de Graaf
2007; Brown et al. 1977). An extensive amount of literature exists describing how
31
P MRS has been used to assess phosphorous containing metabolites in healthy and
diseased brain, heart, liver and muscle tissues (see section 6).
Adult brain is highly (and exclusively) aerobic, which results in high rates of oxygen
consumption. Glucose is the fuel of choice, in the healthy brain. Glucose is metabolized via glycolysis and subsequent tricarboxylic acid cycle to form a maximal
amount of ATP. The resultant energy is stored in the high energy phosphate bonds of
ATP. The energy is used (i.e. hydrolysis of high energy phosphate bonds) to maintain
ongoing electrophysiological activities in the resting brain, and to supply additional
energy for increased neuronal activity during brain activation and task performance.
31
P MRS offers a non–invasive neuroimaging technique that enables quantitative as-
sessment of the cerebral metabolic rates of ATP in both resting and activated brain
states.
MR imaging and spectroscopy has an important role in helping understand muscle
pathologies. More importantly, 31 P MRS has been used to assist in drug development
and evaluation. Recent research in development of gene therapy treatments are being
introduced (such as gene and stem cell therapies) for which MR methods may be very
useful to monitor the effect. MR biomarkers have been proposed that can provide
feedback information on the treatment, one of which uses arginine kinase as a reporter
16
PhD Thesis
gene and the
Sergei Obruchkov
31
Dept. of Medical Physics
P signal of phospho-arginine as biomarker (Walter et al. 2000; Leroy-
Willig et al. 2003).
Several
31
P MRS detectable metabolic changes have been observed following anti-
cancer treatment. Both radiotherapy and DNA-damaging chemotherapy have shown
elevation in the phospholipid precursor phosphocholine (PC) in tumors compared to
normal tissue. Furthermore, this lowers following treatment, frequently prior to tumor shrinkage (Gillies and Morse 2005). This observation can be explained in part
by inhibition of cellular proliferation (Ronen et al. 1992).
31
P MRS is also frequently
used to assess tumor energy metabolism reporting on parameters such as the levels
of ATP, inorganic phosphate and sugar phosphate intermediates of glycolysis, and
tissue pH. Typically, a large reduction in ATP and intracellular pH, as well as increased levels of inorganic phosphate, are observed with successful drug therapies
(Beloueche-Babari et al. 2010). Therefore
13
31
P and MRS of other nuclei such as 1 H,
C and 17 O are highly valuable as noninvasive longitudinal biomarkers that can pro-
vide information on success of drug delivery and efficacy of drug-target treatments.
These unquestionably provide unique non invasive feedback that aid in therapeutic
guidance and allow rapid identification of resistance to therapies.
1.2.3 Decoupling
Coupled nuclear spins exchange energy in the same manner as two pendulums attached by a spring do: the spring causes them to swing intermittently. In NMR this
17
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
effect causes spectral splitting. This often causes an additional complexity for in vivo
NMR as the peaks become overlapped and quantification becomes virtually impossible. This is classically seen as a problem in in vivo 1 H-NMR . For
31
P-NMR there is
clear separation in resonances observed in muscle while in brain there are overlapping
resonances, particularly in the phosphate mono and diesters (figure 1.3).
The process of removing the splitting between spins is called decoupling. Decoupling
is achieved with the aid of a RF selective saturation pulse. Often compound pulses
i.e. a set of 90, 180, 270 pulses with varied phase cycling approaches are used to
saturate hydrogen spins, resulting in collapse of the splitting, thus eliminating the
added complexity.
1.2.4 Adiabatic RF
In vivo NMR uses radio frequency pulses that are used simultaneously for localisation
and excitation. When surface coils are used for excitation it becomes difficult to
excite a sample uniformly due to inhomogeneous distribution of the B1 RF field.
Adiabatic RF pulses however achieve uniform excitation by sweeping the amplitude of
the pulse sinusoidally while sweeping the frequency as a cosine function. Two types of
adiabatic pulses exist; selective and non-selective. Non-selective pulses are the easiest
to design and implement, while selective adiabatic pulses are much more sophisticated
and require one to simultaneously modulate the RF phase and amplitude, and the
amplitude of the magnetic field gradients.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 1.3: 31 P spectra from different parts of the body. Adenosine triphosphate
(ATP) is the energy currency in living systems. Phosphocreatine (PCr), another high
energy phosphate, is used for storing energy and converting ADP to ATP. It is absent
in liver, kidney and mammalian red blood cells. Inorganic Phosphate (Pi ) is
generated from hydrolysis of ATP and often seen increased in compromised tissues.
Phosphomonoester (PME) signal contains contribution from membrane phospholipid
constituents, glucose-6-phosphate and glycerol-3 phosphate. Phosphodiester (PDE)
signal contains contribution from membrane constituents.
A good review of adiabatic full passage pulses for spectroscopic imaging has been
written by Sacolick et al. 2007. There are many different variations of selective
excitation from Carr-Purcell adiabatic excitation, localization by adiabatic selective
refocusing (LASER) (Garwood and DelaBarre 2001), B1 insensitive universal rotator
BIR-4 (Staewen et al. 1990), B1 insensitive slice selection using 8 adiabatic segments
BISS-8 (de Graaf et al. 1996), and other varieties each with their own advantages
and disadvantages.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
1.2.5 Echo Planar Spectroscopic Imaging (EPSI)
Chemical shift imaging (CSI), is a process of acquiring 2D or 3D adjacent multiple
voxels, often using standard PRESS or STEAM localization. The general idea is to use
a train of 3 slice selective RF pulses to localise a volume, during which time spectral
information is lost due to T2 and J-coupling effects. It takes a long time to obtain
multiple voxels using phase encoding acquisition of individual voxels using PRESS CSI
or STEAM CSI pulse sequences. In order to speed up the process faster techniques
can be used. Echoplanar spectroscopic imaging (EPSI) was originally proposed by
Mansfield in 1984 motivated by the need for an in vivo spectroscopy method with a
short TE time. Posse et al. (1994) successfully demonstrated acceleration of spectral
acquisition of a 3D data set using EPSI. EPSI unlike PRESS/STEAM CSI utilises
only one slice selective RF pulse, and the localization in the spatial dimension is
performed by a sweeping magnetic gradient i.e. kx see figure 1.4.
Gradient reversal during the echo planar readout creates “odd” and “even” echoes,
as the data is sampled during the forward and during the fly-back readout. In theory,
odd and even echoes should be equivalent after a phase reversal and correction. However, technical limitations make odd and even echoes incompatible with each other
often due to timing errors, low spectral bandwidth, eddy currents, susceptibility and
other artifacts (Chen et al. 2007; Cunningham et al. 2005). Often spectral spatial oversampling is performed in order to compensate, with an added requirement
20
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 1.4: Basic Design of the EPSI pulse sequence, slice selection and refocusing
can be in any form: e.g. gradient echo, Hahn echo, or CPMG echo, each with their
own advantages and disadvantages. Variations of EPSI utilise additional 1 H nuclear
Overhauser effect (NOE) pre-pulses to increase SNR. Furthremore, decoupling for 31 P
and 13 C are often performed as well.
of extra encoding steps during the EPI readout. Therefore fly-back techniques are
considered more reliable and do not increase total scan time.
EPSI essentially accelerates the acquisition of spectral and spatial information. Consider a scan requiring 32x32 spatial points and 512 spectral points, EPSI will acquire
all of the data in 32 TR steps. In comparison, conventional CSI techniques used
clinically will acquire the same amount of data in 32x32 TR steps, or 32 times longer.
For excitation purposes a single channel volume coil is typically used, while the acquisition is performed using an array of small surface coils. Depending on geometric distribution of the surface coils parallel acquisition techniques such as sensitivity encoding (SENSE) and generalized autocalibrating partially parallel acquisitions
(GRAPPA) can be used for accelerating spatial encoding. These techniques have
21
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
been significantly explored for accelerating MR imaging. 1 H and multinuclear spectroscopy can benefit from this speed-up, and has recently been demonstrated with
proton spectroscopy (Lin et al. 2007; Zhu et al. 2007). Currently there is very little
acceleration with SENSE or GRAPPA done with in vivo multinuclear NMR.
1.2.6 The Problem and the Approach to a Solution
Magnetic resonance spectroscopy is a valuable method for evaluation of tissue metabolism.
Current clinical spectroscopy techniques acquire static snapshots of metabolite concentrations in tissues. This “static” data is limited in information and is not very
representative of changes that occur on a physiological time scale (i.e. less then a second). While some single voxel spectroscopy can be performed as fast as within 10-20
seconds, current multivoxel acquisitions are not similarly as fast. Thus single voxel
placement is relied upon, which demands accurate voxel placement (i.e. to properly
sample the diseased area). This requires previous knowledge about the diseased location, which would necessarily be obtained [possibly] from other imaging modalities.
Therefore a multivoxel approach is more desirable as it would allow simultaneous
comparison between healthy and diseased tissue regions, and it would display localisation and differentiation of the diseased tissue. While 1 H is an important nucleus
the project will focus on
31
P as it is extremely useful to observe dynamic changes in
high energy metabolism and also intracellular pH.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
EPSI is a fast spectroscopic technique which allows acquisition of localised in vivo
MR spectra, and is ideal for dynamic studies. Development of this pulse sequence
was an important step in the success of this project. Excitation was performed
using a dual tuned surface coil. Readout was performed using a trapezoidal fly-back
echo planar encoding (Cunningham et al. 2005). EPSI unlike PRESS/STEAM CSI
utilises only one slice selective RF pulse, and the localization in the spatial dimension
is performed by a sweeping magnetic gradient i.e. kx . EPSI essentially accelerates
the acquisition of spectral and spatial information. Consider a scan requiring 16x16
spatial points and 512 spectral points, EPSI will acquire all of the data in 16x1 TR
steps, conventional chemical shift imaging techniques used clinically will acquire the
same amount of data in 16x16 TR
Automatic prescan is not available for multinuclear spectroscopy (on clinical MRI
scanners). Auto prescan performs automated setting of appropriate recieve/transmit
gains, tuning to a specific resonance peak and magnetic field shimming. Therefore
the software to automate prescan was developed for a clinical MR scanner.
This project required design of RF coils, and the multi tuned coil approach was chosen
as it allows for simultaneous data acquisition and decoupling. Dynamic MRS studies
were performed on human volunteers and standardized phantoms to calibrate and
observe changes during pulse sequence development.
Proton decoupling of
31
P data is critical for obtaining good quality brain spectra.
This was done using a WALTZ-4 decoupling system during 31 P spectroscopy. However
23
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
much manual calibration is still required in order to perform decoupling experiments.
Since the motivation behind EPSI was to reduce scan time then ideally it should not
take more time setting up the patient and calibrating the equipment.
Data analysis must be performed on the acquired spectra. While there are a number of
software packages for analysing spectroscopy data (e.g. VARPRO, LCmodel, SAGE,
MRUI SiTools, etc.), collected data must be imported from raw format into these
packages first. A combination of SAGE and MATLAB software was used to read,
reformat and process the data.
This project was designed towards pre–clinical technical development of software
and hardware as clinical scanners are ill equipped to perform
31
P MRS “out of the
box”. Following technical developments tests were performed on a series of specially
designed phantoms. The first in vivo human tests were performed on leg muscles
of healthy volunteers during plantar flexion exercise inside the MRI. The technical
developments can be translated to clinical settings including the studying of muscle
diseases and breast cancer.
St. Joseph’s Healthcare is a leader in MRI assessment of breast cancer and currently
owns a state of the art MRI compatible breast imaging and biopsy system (Sentinelle
Medical, Toronto Ontario Canada). Coupling rapid spectroscopy with breast biopsy
data is ideal for studying breast cancer. Current MRI literature of
31
P spectroscopy
presents clear evidence of increased concentrations of phosphomonoesters (PMEs),
24
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
and phosphodiester (PDEs) as well as inorganic phosphate peaks in breast tumour
tissue (Leach et al. 1998; Arias-Mendoza et al. 2004).
25
PhD Thesis
Sergei Obruchkov
26
Dept. of Medical Physics
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Chapter 2
Theory
2.1 Basic NMR Physics
Nuclear magnetic resonance is a science of understanding the behavior of spin in
a static magnetic field and its interactions with surrounding chemical and physical
environments. The physical nature of spin however is a complex question that lies
in the realm of theoretical physics which, along with other phenomena such as the
nature of mass, inertia and gravity are still not very well understood. Uncovering the
true nature of spin is not the subject of this thesis. However one point that must be
left with the reader is that spin can be regarded as a fundamental property of the
particle like mass or charge.
In order to understand NMR principles an understanding of nuclear interactions
within a magnetic field is required. First the energy of a proton in a magnetic field
is considered, then the classical picture of a proton as a charged spinning particle is
presented.
2.1.1 Quantum Description
Once a proton with spin I, is placed in a uniform magnetic field B~a it possesses a
~ Magnetic moment and angular momentum
moment µ
~ and an angular momentum ~I.
27
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
quantities are parallel vectors and they may be written as:
µ
~ = γ~I~
(2.1)
with γ as the magnetogyric ratio, a nuclear specific constant present to make the two
quantities equal. For the hydrogen nucleus, or the proton, the magnetic moment is
tiny 1.4102 × 10−26 JT−1 , and magnetogyric ratio is
γ
2π
= 42.575 MHzT−1 .
In the applied magnetic field B~a , the energy of interaction for a particle with moment
µ
~ is:
~ a.
E = −~µ · B
(2.2)
The applied magnetic field is usually a vector in a single direction B~a = Bo ẑ then
equation 2.2 may be presented as:
E = −µz · Bo = −γ~Bo Iz
(2.3)
Where the allowed values of spin Iz are mI = I, I − 1, ..., −I which correspond to
quantized energy levels EI = −mI γ~Bo . The number of quantized energy levels in
the magnetic field is 2I + 1. This means that once a nucleus with non zero spin is
placed in a magnetic field it can take on one of 2I + 1 energy states. This is known
as Zeeman splitting and is illustrated in Figure 2.1 for hydrogen nuclei.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.1: Zeeman splitting of energy levels in the presence of the magnetic field
Bo . Higher energy state corresponds to those nuclei that are aligned antiparallel to Bo .
Energy difference between two states is described by Eq. 2.4.
The 1 H and 31 P nuclei in a magnetic field Bo have two possible energy states mI = ± 12 .
The energy difference between the two states in terms of ~ωo can be calculated from
equation 2.3 to be:
∆E = 2µBo = γ~Bo = ~ωo
(2.4)
NMR is a quantum mechanical phenomenon and the observed signal can be viewed
as the energy measured between the transition of the proton from the higher energy
state m− to the lower m+ . With a sufficiently large magnetic field, ∆E becomes
larger than the thermal energy kb T at room temperature so there are no spontaneous
transitions from lower to higher energy state. Most commonly this energy is in the
form of electromagnetic radiation. Calorimetric techniques had also been attempted
in order to measure magnetic resonance by Gorter (1936) however these methods did
not succeed.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.2: A spin placed in magnetic field Bo will precess around the field with
frequency ωo .
2.1.2 Classical Description
In a classical picture, rotation of the proton generates a magnetic moment µ
~ . In
the absence of a magnetic field it is oriented randomly but when placed in a static
~o it aligns with it, either parallel or anti–parallel. Once aligned it
magnetic field B
~o as shown in Figure 2.2; this is similar to the spin top
will start to precess about B
precessing in earth’s gravitational field. This is readily observed from the rate of
change of angular momentum of the system, which is equal to the torque ~τ that acts
on the system. Mathematically this is a cross product between the two vectors:
~τ = ~
dI~
~o
=µ
~ ×B
dt
(2.5)
substituting from Eq. 2.1 this becomes:
d~µ
~o .
= γ~µ × B
dt
30
(2.6)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Combining constants from the above equation into a single constant,
ωo = γBo
(2.7)
where ωo is a natural frequency of the precession also known as the Larmor frequency.
The solution to the differential equation Eq. 2.6 is the motion of the proton around
the magnetic field, written in terms of ωo as:
µ = µo e−iωo t
(2.8)
This precession is field dependent and further it dictates the conditions at which
the occurrence of resonance is observed. If a proton precesses in a magnetic field
with frequency ωo , and is irradiated with electromagnetic radiation field of the same
frequency perpendicular to the Bo , the two fields couple, energy is absorbed and the
spin is tipped.
2.1.3 Bulk Magnetization
NMR experiments are performed on large numbers of spins; usually on the order of
~ can be viewed as the
Avogadro’s number (6.0223x1023 ). Then bulk magnetization M
sum of all the nuclei moments µ
~ in the unit volume:
~ =
M
X
i
31
~i
M
(2.9)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
It is possible to rewrite the equation of motion Eq. 2.6 using Eq. 2.9
~
dM
~o
= γ~µ × B
dt
(2.10)
In thermal equilibrium at temperature T the magnetization will be along the ẑ direction:
Mxy = 0; Mz = Mo
(2.11)
where Mxy is transverse and Mz is longitudinal magnetization.
If nuclei are placed in a static magnetic field, and are in thermal equilibrium, the
population ratio of spins in different energy states is determined by the Boltzmann
factor for the energy difference of 2µBo :
N+
− 2µBo
= e kB T
N−
(2.12)
where kB is Boltzmann’s constant (1.38065x1024 JK−1 ) and T is absolute temperature
in Kelvin. The magnetization of a system is related to the population difference
N+ − N− of the spins. The excess of spins aligned with the magnetic field contribute
~ z = (N+ − N− )~µ as shown in
to the total nuclear magnetization of the sample M
Figure 2.3.
32
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.3: In a static magnetic field the population of nuclei aligned with Bo is
larger then those that are antiparallel to the field N+ > N− . This excess of spins with
individual magnetic moments µ contribute to the longitudinal magnetization Mo of the
entire sample.
2.1.4 NMR Signal Detection
In order to measure the NMR signal, the longitudinal magnetization Mz must be
tipped into the transverse x–y plane, which gives rise to transverse magnetization
Mxy . This is achieved by applying a pulsed electromagnetic field B~1 perpendicular
~o at the resonant frequency ωo . In effect the B1 applies
to the main magnetic field B
~ by a prescribed angle α. The torque
torque onto the nuclear spins which rotates the M
is dependent on the strength of the applied field B1 and the pulse duration. This can
be represented on a Bloch sphere in laboratory coordinates, or in rotating frame of
reference where the x–y plane rotates at the Larmor frequency (Figure 2.4). The B1
field is usually generated by a coil that is perpendicular to Bo . That coil can also
be used to detect a time varying magnetic field created by transverse magnetization
Mxy rotating at the Larmor frequency.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
~ z is flipped by angle alpha using the field B1 . In the
Figure 2.4: Magnetization M
lab frame the motion of the nuclei is described by a nutation around the z axis. In the
rotating frame of reference the x–y plane is rotating at ωo and magnetization, in this
example, is seen as tipping along the y’–axis. The rate of tipping is determined by the
frequency and amplitude of the applied field B1 .
The nuclei will absorb energy most efficiently at the Larmor frequency which from
Eq. 2.7 is magnetic field dependent. When a sample is irradiated with RF satisfying
~ will tip away from
the resonance condition in the rotating frame of reference, the M
the z axis at a precessional frequency ω1 (Figure 2.4) that is dependent on the B1 RF
field as described by:
ω1 = γB1
(2.13)
The magnetization will continue tipping as long as B1 is applied. Usually B1 is
~ by a specific angle α. Because the receiver
applied in pulses of RF energy, tipping M
is sensitive to magnetization components in the x–y plane the signal will be directly
related to Mxy . The maximum signal is then obtained when a tip angle α = 90o .
In order to detect NMR signals most efficiently a series of pulses must be applied
(discussed bellow).
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Ignoring relaxation (i.e. return to equilibrium) effects , the measured signal S(t) from
a 3D sample located in a static magnetic field Bo , following a B1 excitation pulse,
with the distribution of transverse magnetization Mxy (~r) would be:
Z
S(t) =
Mxy (~r)e−iωo t d3~r.
(2.14)
vol
In this case it is not possible to distinguish signals originating from different locations
within the volume. Instead, NMR averages signal from the entire sample.
2.1.5 Longitudinal and Transverse Relaxation
Following the tipping of magnetization, by angle α, the resultant initial vector can be
decomposed into two magnetization components: Mxy transverse and Mz longitudinal
magnetization. Following excitation, the transverse component of the magnetization
decays away while the longitudinal component returns to its thermal equilibrium
state. The magnetization component Mz of the system approaches equilibrium magnetization at a rate proportional to the departure from equilibrium Mo . Mathematically this can be described by:
dMz
Mo − Mz
=
dt
T1
35
(2.15)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
which is the differential equation for T1 relaxation. The solution to this is:
Mz = Mo + (Mz (0) − Mo )e−t/T1 ,
(2.16)
where T1 is called the longitudinal or spin–lattice relaxation time constant, and characterizes the return of magnetization to equilibrium along the z direction (Figure 2.5).
This relaxation is done predominantly via exchange of energy between the nucleus
and surrounding environment.
The T1 values lengthen with increasing field strength, because the amount of energy
nuclei have to release, in order to return to equilibrium, increases with Bo . At the
atomic level the nuclei are surrounded by randomly fluctuating magnetic fields. These
fields result from moving molecules and other nuclei all possessing magnetic moments.
These fluctuating fields tend to enhance the energy exchange between the nuclei and
the lattice, shortening T1 . This effect is the basic principle behind MR contrast agents
such as Gd–DTPA–BMA.
The behavior of the transverse magnetization component is dominated by the de–
phasing of individual magnetic moments leading to magnetization loss (incoherence).
This behaviour can be described by:
dMxy
Mxy
=−
dt
T2
36
(2.17)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
which is the differential equation describing T2 relaxation. The solution to this is
simply:
Mxy = Mxy (0)e−t/T2 ,
(2.18)
where T2 is known as the spin–spin or transverse relaxation time constant. The
interaction of nuclear dipoles with each other causes some spins to precess faster or
slower then others. This brings a uniform distribution of spins that causes loss of
spin coherence and results in decay of transverse magnetization, as shown in Figure
2.5.
Transverse relaxation is determined by molecular mobility where spins bound to big,
slow–turning macromolecules experience rapid T2 decay while smaller more mobile
spins exhibit much slower decay. Compared to longitudinal relaxation, transverse
magnetization always decays faster, generally, in biological systems,
T2 6 T1
In addition to T2 another source of loss of transverse magnetization is Bo inhomogeneities which give rise to a time constant T2∗ . Mathematically this is expressed
as:
1
1
=
+ γ∆Bo
∗
T2
T2
(2.19)
where ∆Bo defines the inhomogeneity of the external magnetic field (γ∆ Bo is sometimes called 1/T 2 0 ). In a biological tissue with nonzero ∆Bo transverse magnetization
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
is largely described by T2∗ . In most cases T2∗ ≤ T2 causing a shorter free induction
decay (FID) and accelerated loss of NMR signal. However using special techniques
such as the spin echo, the signal loss can be recovered.
While T1 relaxation is an irreversible process, while transverse relaxation, T2 (arising
from spin-spin interaction) on the other hand can be partially reversed to produce an
echo. The phase coherence loss due to spin-spin relaxation is essentially irreversible,
whereas the coherence loss due to magnetic field inhomogeneity,T2∗ , can be reversed by
generating a so-called Hahn-echo (or spin-echo). This can be achieved by application
of RF energy in order to re–phase the spins back into coherence. For example, consider
a set of spins which at t=0 are all aligned in phase (Figure 2.6). The existing field
inhomogeneities between the spins will create slightly different rates of precession
thus the phase between the spins will accrue at different rates. For a set of spins the
phase difference at location ~r and time t can be calculated using:
φ(~r) = γ∆B0 (r)t
(2.20)
where ∆B0 (~r) is the magnetic field inhomogeneity at ~r. This phase accrual can be
reversed by applying a 180o RF pulse at time t = T E/2. This will cause the spins to
be mirrored in z such that the phase +φ(~r) will become −φ(~r). Since they are still
located at their respective location ~r they will preccess at the same rates as before
and at time t = T E will re–phase forming an echo, see Figure 2.6.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.5: Longitudinal relaxation must recover to the initial state along z, as described by Eq. 2.16. The energy during the longitudinal recovery is absorbed by the
surrounding lattice. Transverse relaxation occurs due to spin–spin interaction and acts
to disperse the distribution of precessional frequency ωo . Some spins slow down while
others precess more rapidly. The end result is loss of spin precessional coherence.
Transverse relaxation is governed by Eq. 2.18.
39
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.6: Echo formation following a spin echo experiment. The transverse magnetization is flipped around the axis, while still traveling at the same rate all the spins
reffocus to produce a spin echo at time TE.
40
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
2.2 Chemical Shift
It is often impractical to look at a single, or even a small group of nuclei, alone. In the
real world these events are too quick to measure or are in very hostile environments.
Nuclei typically are surrounded by electrons and form atoms which in turn often form
complex molecules. By taking a look at a single nucleus and an electron one quickly
realizes that one of the most prominent sources of the field inhomogeneities around
the nucleus is the surrounding electron shell. When both are placed in a magnetic
field B0 , the electron will oppose the applied field resulting in an effective field B:
B = B0 (1 − σ)
(2.21)
where σ is called the shielding constant, and is dependent on the chemical environment i.e. surrounding electron shell. The nuclei will resonate at slightly different
frequencies depending on their physical position on the molecule itself (Figure 2.7).
Using equation 2.21 and equation 2.7 it is possible to quantify the expected frequency
in radians per second to be:
ω = γB0 (1 − σ)
(2.22)
However it is often more practical to use a unit–less scale to describe chemical shift
delta such that it is independent of the applied magnetic field B0 :
δ=
ω − ωref 6
10
ωref
41
(2.23)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.7: Electrons surrounding the nucleus give rise to the chemical shift effect.
This is demonstrated using the 1 H spectrum of lactate where the electronegative oxygen
atoms shift electrons away from the protons thus giving rise to greater chemical shift.
Thus the CH quartet is more de–shielded then the CH3 doublet in the above spectrum.
where ωref is a Larmor frequency of a reference chemical. Therefore chemical shift of
the nuclei dictates the peak position on the frequency axis. Using this information
and the fact that the area under the peaks is proportional to the number of chemically
equivalent nuclei residing at each location it is possible to describe chemical structures.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.8: The SI spin system has four possible energy states which dictate possible
transition states that could be detected by magnetic resonance experiment a) In the case
of non–interacting spins there will be four possible transitions, with only two different
energy differences. b) Interacting spins will split individual levels producing unique
transisition energies.
2.3 J–Coupling
Peak splitting is a phenomena that is usually the product of the interaction between
two or more nuclei. The nuclei with magnetic moments generate magnetic fields
which tend to influence the neighbouring spins directly through space in the form of
dipolar coupling. In liquids molecular tumbling averages out these effects and no peak
splitting is observed due to dipolar coupling. Peak splitting in liquids, however, can
still be observed due to indirect coupling through the electron cloud of the chemical
bonds this is called scalar coupling or J–coupling.
Take spin system IS where both I and S possess magnetic moment for example
13
C
and 1 H atoms and the precessional frequencies can be written down as:
ωI = γI B0
(2.24)
ωS = γS B0 .
(2.25)
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Adopting |↑i as the low energy state and |↓i as the high energy state of the spin I
and S it is possible to write down four possible combinations that will result in four
possible energy states: |↑↑i, |↑↓i, |↓↑i, |↓↓i. If the spins are not interacting with each
other then the transition energy can be calculated using eq. 2.4. The energy level
diagram for such a system is shown in Figure 2.8 a). The transitions between levels
in the non interacting nuclei will be symmetric:
∆E = ~ωS = hfS
(2.26)
∆E = ~ωI = hfI .
(2.27)
In an NMR spectrum this will simply appear as single peaks occurring at the Larmor
frequencies of fS and fI .
However if scalar–coupling is present this will perturb the energy levels of the system
as shown in Figure 2.8 b). This perturbation can be explained by the fact that
due to Pauli exclusion principle the electrons occupying the same electron shell must
be anti–parallel to each other, and thus through the hyperfine interaction between
electrons and the nucleus the energy levels will be slightly altered from the case were
no scalar–coupling effect is present. The energy levels will be altered proportional to
J/4, where J is called a scalar coupling constant. The four possible transitions will
no longer appear symmetric instead four unique transitions are observed:
44
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
∆E = h(fS + J/2)
∆E = h(fS − J/2)
∆E = h(fI + J/2)
∆E = h(fI − J/2).
This appears as peak spitting on the NMR spectrum (see Figure 2.9) with the peaks
separated by J(Hz). The scalar–coupling interaction between heteronuclear nuclei can
generate complex splitting in the spectra that can be determined by the method of
successive splitting or mathematically using Pascal’s triangle. The relative peak areas
of split peaks can also be determined from the Pascal’s triangle where the relationship
between the areas is directly proportional to the number of the successive splittings
that need to occur in order to produce that peak (see Figure 2.9).
J–coupling provides another extremely valuable source of information for properly
identifying the chemical compound, however for the in-vivo applications, where many
metabolites can reside in the same location with overlapping peaks J–coupling can
decrease the sensitivity of detection by decreasing the amplitude of the peaks, spreading the frequency information and contaminating information from other metabolites.
This is often undesirable and is often thought to be controlled or removed if possible.
A technique called decoupling is often used in high resolution NMR spectroscopy in
45
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.9: Simulated spectra of the J–coupling effects on the 13 C spectra, a carbon
nuclei instead of producing a singlet produces a doublet at 0Hz when one 1 H is chemically bonded to it, a triplet at 225 Hz appears for the CH2 group and a quartet at 500
Hz for a CH3
order to simplify the spectra and increase the signal to noise via Nuclear Overhauser
effect.
2.4 Decoupling
In order to decrease the interaction between two or more magnetic dipoles, experiments that require simultaneous control of both nuclei are required. By affecting
certain energy level transitions it is possible to eliminate coupling effect which are
observed as peak splitting in the NMR spectra. These types of experiments are referred to as decoupling. It has been observed from experiments involving irradiation
of nuclei at two frequencies simultaneosly, such that when the magnetic moments of
the two nuclei diverge (become orthogonal), the effects of the scalar coupling disap46
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
pear. Thus keeping two magnetic moments orthogonal to each other it is possible
to reduce the J–Coupling effect and eliminate the splitting of the spectra. There are
many different decoupling methods, all of which utilise a second magnetic field B2 .
This field is produced at RF frequency of the nucleus that is spin-spin coupled to
the nucleus of interest. For example we are interested in acquiring
31
P spectra from
methylphospohic acid P (OH)2 O(CH3 ), due to coupling between hydrogens in the
hydroxyl group and the
31
P nucleus acquired spectra appears as a quadruplet. One
type of decoupling method applies a continuous B2 field. As long as the field applied
is at the resonant frequency of 1 H in the hydroxyl group the resulting
31
P spectra
will collapse into a singlet, however if B2 is even slightly off resonance the decoupling
effect will be greatly diminished. This type of decoupling belongs to the narrow band
decoupling methods they are typically very inefficient, and have a very low figure of
merit Ξ ≈ 0.01:
Ξ=
2π∆F
,
γB2
(2.28)
where ∆F is the effective decoupling bandwidth. Narrow band decoupling methods
require large amounts of RF power that can cause sample and tissue heating.
Another type of decoupling method utilises what is called cycles and super cycles in
order to improve the efficiency while minimizing the RF power required. WALTZ-4
is a composite decoupling sequence consisting of multiple flip angles with different
phase cycling that when experimentally applied have broad bandwidth that they
could perform decoupling, the Ξ ≈ 1.0 for WALTZ–4.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
WALTZ–4 sequence applies 4 cycles of RF pulses in increments of π, one WALTZ
cycle would be (1 π2 , 2 π2 , 3 π2 ) eq. 2.29, at the end of the four cycles the spins are brought
back into the Mz orientation, by proper phase cycling eq. 2.30.
For
31
R = 90x 180−x 270x
(2.29)
W ALT Z − 4 = RRR̄R̄
(2.30)
P–1 H decoupling WALTZ-4 has been extensively used as the decoupling se-
quence of choice ((Barker et al. 2001; de Graaf 2005b; Widmaier et al. 1998)).
2.5 Introduction to Imaging: Hard vs Soft RF excitation
Slice selective excitation is one of the most basic principles behind Magnetic Resonance Imaging (MRI). It is possible to excite a specific volume of spins and spatially
encode it through the combined use of magnetic gradients and RF pulses. For example: a gradient Gz can be superimposed on top of the main magnetic field Bo , this
creates a gradient of spins with spatial distribution of Larmor frequencies along the
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
z axis. In order to excite a slice of thickness ∆z an RF pulse with bandwidth of ∆ω
must be applied to the system where:
∆ω = γGz ∆z
(2.31)
The only spins excited will be those falling within the bandwidth of the applied RF
pulse. Spins residing outside of the RF pulse bandwidth will remain at equilibrium.
A summary of this is shown in Figure 2.10.
SINC or soft pulses have been a natural choice for slice selective excitation as they
have narrow bandwidth and the Fourier transform of a SINC pulse is a rectangular
function, an ideal slice profile. A typical SINC function is given by:
B1 =


)
sin( πt

t0

ASIN C πt
≡
At
0 πt
t0
−NL t0 ≤ t ≤ NR to



0
everywhere else
(2.32)
with A as the peak amplitude at time t = 0, and t0 being one half width of the largest
central lobe of the sync function, NL and NR are number of zero crossings or one less
number of side lobes of the SINC function. The bandwidth of the SINC function or
the FWHM of its Fourier transform, can be approximated as
∆ω ≈
49
1
t0
(2.33)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Figure 2.10: A magnetic field gradient Gz distributes spin precessional frequencies
ω(z) linearly throughout a sample, along z. A Fourier transform of a SINC pulse is a
square function in the frequency domain. Applying a SINC pulse of bandwidth ∆ωo
will excite only those spins that fall within ∆z.
Unlike soft pulses, hard or rectangular pulses have much broader bandwidth and are
applied without magnetic field gradients. Thus hard pulses will excite many spins
simultaneously, assuming a homogeneous B0 magnetic field. Generally the bandwidth
of a rectangular pulse will depend on its length. When considering a rectangular pulse,




1 if |t| >
t
)=
RECT (

T /2


0 if |t| <
T
2
T
2
the Fourier transform is naturally:
F(RECT ) = T SIN C(πf T )
50
(2.34)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
(i.e. a SIN C function) with
F W HM ≈ 1.21/T
where T is the pulse length. Thus the shorter the RF pulse the broader its excitation
profile. Short pulses have many other benefits, as they excite the entire spin system
rapidly, and cleanly (i.e. excited spins will be in phase in the transverse plane).
Imaging can be performed using either of the pulses, however hardware limitations,
safety, and many other factors have to be carefully considered for the type of imaging
and experiments needed. Hard pulses can be used for performing inversion, and
3D readouts, while soft pulses are used for 2D readout, slice selection and spatial
localization. The flip angle is easy to control and is dependent on the B1 RF field
strength and pulse duration T as
Z
T
θ=γ
B1 (t)dt.
(2.35)
0
MRI utilizes many types of RF pulses for different types of purposes, in the following
sections, amplitude and frequency modulated RF pulses, and their uses in cases where
B1 inhomogeneity can not produce desirable flip angles, are described.
Consideration of Using Surface Coils
RF coils are used to obtain MR signal, these are high Q resonating circuits tuned
to a specific frequency, where the resonating frequency depends on the nuclear gyro51
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
magnetic ratio and the magnetic field strength. There are two types of coils: surface
coils and volume coils. Volume coils are typically large assemblies in a Helmholtz or
birdcage geometry, that can produce a uniform B1 RF field which is used to excite
the spins. Surface coils are much more efficient at picking up signal close to the coil,
but their effectiveness significantly decreases at about half the radius away from the
coil. They are rarely used for excitation as they do not produce a uniform RF field
(see Figure 2.11). Therefore current approaches in MR utilize an array of receiveonly surface coils that can be positioned close to the patients body (i.e. to ensure
good coupling) to achieve a large volume coverage to receive MR signal. Volume
coils are used for excitation purposes. When using arrays of surface coils for RF
receiving a large volume coil (body coil) performs RF transmission. These types of
coils are superior for imaging applications but for spectroscopy, surface arrays add
unnecessary complexity as it becomes non–trivial to add multiple signals together.
Specifically, phase correction and scaling factors during the adding process may affect
quantification during the data analysis.
Figure 2.11: RF Excitation profiles of a surface coil (left) and volume coil (right).
The flip angle iso–lines clearly show a region of uniform excitation that can be
achieved using birdcage geometry. In MRI volume coils are typically used for
excitation purposes, and surface coils are used for signal recieving.
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PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Off–Resonance Effects and RF pulses
The RF pulses described in the above section and the flip angle (eq: 2.35) assume a
uniform B1 field acting on spin isochromats exactly on resonance:
ωRF = ω0 = −γB0
There are a few ways of thinking about the off–resonance case when ωRF 6= ω0 ; in
the ‘swing’ analogy the ‘pushes’ can be very small but if they are timed right they
can really get the swing going. Or one can think of the transition probability of spins
|+i → |−i in the case of a 180o pulse: the probability is highest when the RF is on
resonance and diminishes as the |∆ω| increases.
It is often useful to think about the off–resonance as a vector component
∆ω
γ
which
will effectively tilt the RF B1 field out of the xy plane. This is known as the effective
field Bef f (see figure 2.12) with magnitude:
|Bef f | =
q
B12 + (∆ω/γ)2
2.6 Adiabatic experiments
Thermodynamics of NMR
53
(2.36)
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Dept. of Medical Physics
Figure 2.12: The effective field tips the rotation axis and therefore the expected flip
angle will be different than expected.
What does NMR have to do with Thermodynamics? Spin temperature Ts probed
by a magnetic field is slightly above the temperature of an experiment and is one of
the single most important reasons why we can detect NMR signal. The temperature
difference contributes to the separation in populations which we are probing. At
equilibrium this population is governed by the Boltzmann distribution as:
γ~B0
P−
−
= e kB Ts
P+
(2.37)
where P+ and P− are the populations of spins in higher |+i and lower |−i states.
In thermodynamics the definition of an adiabatic process is one that keeps entropy
constant. This is often experimentally achieved by changing one of the physical conditions very slowly. NMR experiments were first done using continuous RF radiation
and the magnetic field was swept until the resonance condition was met. This in54
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Dept. of Medical Physics
duced absorption of energy from the RF field was detected as a change in the coil
induction. Earlier NMR experiments were attempted using mangetocaloric measurements. In these experiments it was thought that spin temperature could be measured
directly. However, those experiments were spectacularly unsuccessful. With later advancements in technology, physicists cooling atoms below µK started thinking about
cooling spins further using magnetocalorimetry.
The basic principle behind magnetic spin cooling relies on the fact that sweeping
a magnetic field can excite or saturate a system. If the sweeping process is very
slow it does not matter about system initial equilibrium, the final equilibrium will
be dependent only on the field and type of system. This allows one to ‘dial in’ spin
temperature. Careful experiments have been performed which showed this cooling
effect using adiabatic sweep of a magnetic field, a good example of an adiabatic reversible process. If a sample at temperature Ti and field Hi is placed in an apparatus
in which the magnetic field is slowly ramped down, the system will end up at a new
position, Tf and field Hf see figure 2.13. This effect is often used to cool systems
to temperatures approaching zero degrees kelvin, and is often used in low temperature superconductors, in masers, to achieve Einstein–Bose condensate, and other low
temperature physics.
Adiabatic RF Pulses
From the above sections, we can see the effect of Bef f on flip angle θ is quite pronounced when using surface coils, where B1 tends to vary as much as a factor of 10
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Dept. of Medical Physics
Figure 2.13: By slowly varying the magnetic field it is possible to reduce spin temperature. This is a reversible adiabatic process.
over the volume of interest. Adiabatic pulses are often used in these situations, as
they do not require complex setup of using separate coils for transmit and receive.
Typically an adiabatic RF pulse is decribed in the form:
B1 (t) = A(t)e−iω1 (t)t
(2.38)
where A(t) is the envelope and ω1 (t) is the frequency sweep function. One can think of
these pulses as composite pulses containing many hard pulses with varying amplitude
and frequency.
There are a few interesting properties of adiabatic RF pulses, that make them interesting for surface coil imaging. First of all they do not require multiple coil set
up (figure 2.11), allowing surface coils to be used alone for transmitting and receiving signal. Furthermore, they can act as inversion pulses that are RF amplitude
independent, and also be useful in frequency selection as well.
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It is best to introduce a reference frame
∗
Dept. of Medical Physics
called the frequency rotating frame which
will rotate at the frequency of the RF field, i.e. modulated by ω1 (t). Starting with
~ is the product of the
the Bloch equations, in the laboratory frame, magnetization M
~ and
interaction between two magnetic moments: magnetization vector from spins M,
~
the magnetic field be it the static magnetic or magnetic component of the RF B:
~
dM
dt
!
~ ×M
~
= −γ B
(2.39)
Lab
where
~ = B1 cos(ω0 t)î0 + B1 sin(ω0 t)î0 + B0 k̂0
B
(2.40)
and î0 , ĵ0 , k̂0 are fixed unit vectors in the lab frame. For convenience we’ll define the
RF field only in the x direction:
B1 = B1,x
Magnetization in the laboratory frame rotates at ω0 and
î = î0 cos(ω0 t) + ĵ0 sin(ω0 t)
ĵ = −î0 sin(ω0 t) + ĵ0 cos(ω0 t)
(2.41)
k̂ = k̂0
∗
Lab reference frame is the one that we would observe as bystanders. The spin would appear to
nutate or precess around the magnetic field. The rotating reference frame is one where we would
be rotating at frequency ω around the spin. In this frame spins would appear stationary until RF
pulses are applied.
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Dept. of Medical Physics
then
~ = B1 î + B0 k̂
B
(2.42)
~ = Mx î + My ĵ + Mz k̂
M
~
Where Mx , My , Mz and î, ĵ, k̂ are time varying in the lab frame. Differentiating M
with respect to time produces:
~
dM
dt
!
dMx
dMy
dMz
dî
dĵ
dk̂
î +
ĵ +
k̂ + Mx + My + Mz
dt
dt
dt
dt
dt
dt
!
~
dî
dĵ
dk̂
dM
+ Mx + My + Mz
=
dt
dt
dt
dt
=
Lab
(2.43)
Rot
In the laboratory reference frame vectors î, ĵ are rotating unit vectors at the Larmor
frequency ω0 and can be expressed as a cross product so
ω~0 = ω0 k̂
and the time derivatives would then be:
dî
dĵ
dk̂
= ω~0 × î;
= ω~0 × ĵ;
= ω~0 × k̂
dt
dt
dt
substituting above we get
~
dM
dt
!
=
~
dM
dt
=
~
dM
dt
Lab
!
+ ω~0 × (Mx î + My ĵ + Mz k̂)
!Rot
(2.44)
~.
+ ω~0 × M
Rot
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Dept. of Medical Physics
In the rotating frame then we can rearrange the equation such that
~
dM
dt
!
=
Rot
~
dM
dt
!
~
− ω~0 × M
Lab
~ ×M
~ − ω~0 × M
~
= −γ B
(2.45)
~ − ω~0 ) × M
~
= (−γ B
~
~ + ω~0 ) × M
= −γ(B
γ
where the quantity in brackets is known as the effective field
~ + ω~0
B~eff = B
γ
(2.46)
ω~0 = −γB0 k̂
(2.47)
At the Larmor Frequency of
the applied magnetic vector simplifies to
~ = B1 cos(ω0 t)î0 + B1 sin(ω0 t)ĵ0 + B0 k̂0
B
(2.48)
= B1 î + B0 k̂
and the effective field will be
−γB
~ ef f = B1 î + B0
0 k̂ = B1 î
B
k̂
+
γ
59
(2.49)
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Dept. of Medical Physics
The changing reference frame cancels the static B0 field in the rotating frame, which
is convenient, when working with frequency modulated RF pulses.
An RF pulse with a time dependent ω1 (t) component in the form of eq: 2.38, can
now be easily applied. The equation for the effective field can be calculated in the
rotating frame ω0 + ω1 (t):
~ = A(t)cosω1 (t)îω0 + A(t)sinω1 (t)ĵω0 + 0k̂
B
(2.50)
ω1 (t)
k̂
B~eff ω0 +ω1 (t) = A(t)îω0 +ω1 (t) +
γ
(2.51)
This will create a z magnetization component equal to
ω1 (t)
,
γ
creating an effective field
Bef f with the magnitude:
s
B
ef f =
A2 (t) +
ω1 (t)
γ
2
(2.52)
This effect is also known as spin–locking, because the magnetization is locked and
precesses around the effective field (see figure 2.14).
Adiabatic Condition
Now without the effect of B0 , the magnetization will precess around the effective field
~ to the effective
Bef f . It is possible by sweeping ω1 (t) to lock the magnetization M
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Dept. of Medical Physics
Figure 2.14: Left: With the cancellation of B0 in the rotating reference frame, the
spins precess about the effective field. Right: For a trajectory with a slow sweep rate the
magnetization follows the effective field vector in a full adiabatic passage experiment.
field, this is satisfied when the angle Φ of Bef f changes relatively slowly compared to
its magnitude:
dΦ γ Bef f dt
(2.53)
which is known as an adiabatic condition. It is often convenient to define an adiabatic
factor:
γ Bef f η = dΦ
dt (2.54)
To achieve the adiabatic condition one will have to satisfy η > 1.
In order to understand the effects of adiabatic RF pulses it is useful to plot amplitude
γA(t) vs the ω1 (t) frequency sweep function. This is best done considering the hyperbolic secant function as an example (amplitude and phase of B1 according the general
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form are presented in eq: 2.38). Amplitude and phase for the hyperbolic secant are
defined as:
γA(t) = γA0 sech(βt)
1
ω(t) = −µβtanh(βt)
γ
(2.55)
A plot of γA(t) vs γ1 ω(t) (see figure 2.15), provides visualization and understanding
of adiabatic pulses. The hyperbolic secant is known as an inversion pulse or adiabatic
full passage pulse as it will tip magnetization 180o . It is possible to design different
kinds of pulses, such as adiabatic half passage. This can tip magnetization by 900 and
thus be considered as an excitation pulse. It is even possible to create an arbitrary
angle rotation pulse. One example would be B1 insensitive rotation or BIR–4. (The
description of BIR–4 will follow at later date and will be in more detail).(Slichter
1991)
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Dept. of Medical Physics
Figure 2.15: Representation of a hyperbolic secant pulse (top left) and its frequency
~ ef f field rotating
modulating function (top right). Bottom left: with the effective B
clockwise, the rotation is controlled by the ω1 (t) function. Bottom right: Adiabatic
pulses are insensitive to B1 . The shaded contour describes the area where the adiabatic
condition is satisfied, i.e. the magnetization vector will follow the effective field when
the adiabatic condition is met. In this figure adiabacity is best in the middle but is
poor for the low and high B1 values.
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Dept. of Medical Physics
2.6.1 Spatial Encoding
~ Varying the magnetic
Spatial encoding of a selected slice is done using gradients G.
field in the spatial domain, distributes the Larmor frequency of the nuclei as a function
of their spatial position ~r according to:
~ · ~r
ω(~r) = ωo + γ G
(2.56)
By prescribing gradients at specific times, intervals, and magnitudes, in combination
with RF pulses, the MR signal S(t) can be spatially encoded. Thus, with the application of gradients, the measured NMR signal can be rewritten from Eq. 2.14 using
the spatially dependent Larmor frequency:
Z
S(t) =
Mxy (~r)e
−iωo t+γ
Rt
~ )·~
G(τ
rdτ
0
d3~r.
(2.57)
vol
The above signal equation with spatial encoding parameter ~k(t):
~k(t) = γ
2π
Zt
~ )dτ,
G(τ
(2.58)
0
can be further simplified by understanding that the signal S(t) is demodulated. Thus
the S(t) can be rewritten without e−iωo t as a function of the spatial encoding parameter
~k(t) as:
s(~k) =
Z
~
Mxy (~r)e−i2πk·~r d3~r
vol
64
(2.59)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Thus the signal is acquired over time, as the gradients are applied to encode the
position of the changing ~k parameter. In a 2D case then gradients Gx and Gy can be
applied such that to collect a two dimensional map of the spatial encoding parameters
kx (t), ky (t) we can rewrite these as:
γ
kx (t) =
2π
Zt
Gx (τ )dτ
0
and,
γ
ky (t) =
2π
Zt
Gy (τ )dτ
(2.60)
0
After time=t the collected signal can be represented as a function of kx and ky :
Z Z
s(kx , ky ) =
x
Mxy (x, y)e−i2π[kx x+ky y] dxdy
(2.61)
y
Therefore, from this equation the magnetization distribution Mxy (x, y) can be recovered via Fourier transformation.
In general, the Field of View (FOV) associated with other modalities, is directly associated with the area that is imaged, or sampled. In MR, sampling takes place in
the spatial frequency domain. In order to get the image the data is Fourier transformed: this transformation results in an inverse proportional relationship between
the sampling rates ∆kx and ∆ky and FOV:
F OVx =
1
1
, F OVy =
∆kx
∆ky
65
(2.62)
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Sergei Obruchkov
Dept. of Medical Physics
The signal FID or an echo is acquired digitally. A 2D array of s(kx , ky ) or k–space
data is produced by frequency and phase encoding. During frequency encoding a
gradient, Gx , is applied during the read–out of the echo signal. This fills one line of
k–space and resolves one spatial dimension x. To produce a second dimension a phase
encoding gradient, Gy ,is applied with different strength. The two steps are repeated
with varying amplitudes of Gy until all of the desired k–space is acquired (see Figure
2.16). The collected k–space data is digitized into real and imaginary components
which are acquired by mixing the signal with cos(ωo t) and sin(ωo t) functions. The
resultant real and imaginary k–space is converted to image space using a 2D Fourier
transform. Often the number of points in k–space is acquired with the dimensions as
a function of power of 2 to allow use of the Fast Fourier Transform (FFT) algorithm.
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Dept. of Medical Physics
Figure 2.16: By applying Gx and Gy gradients it is possible to acquire 2 dimensional
information about the distribution of Mxy (x, y). Steps 1 through 4 show the application
of gradients and their direct effect of traversal through k–space. Data is recorded only
during steps 2 and 4. Data collected in the frequency domain must be reconstructed
using a 2D FT algorithm to produce an anatomical image.
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PhD Thesis
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Dept. of Medical Physics
Chapter 3
Echo Planar Spectroscopic Imaging of
Phosphorus and Hydrogen using Flyback
Echo Planar Readout Trajectory.
This chapter deals with the development and design of a MR pulse sequence which
is used to accelerate Chemical Shift Imaging using a modified readout technique.
Theory, of the design of echo planar spectroscopic imaging with flyback trajectory
for use on
31
P nucleus is discussed. Results of EPSI sequence implementation on a
clinical GE 3T scanner are presented.
3.1 Introduction
1
H magnetic resonance spectroscopic imaging (MRSI) or chemical shift imaging (CSI)
is widely used as part of clinical protocols for measuring spatial distribution of different metabolites, (Kurhanewicz et al. 2000). However due to long acquisition times
CSI is rarely used in a clinical setting.
The commercial success of MR scanners with higher magnetic fields (i.e >3.0T), over
the past few years was largely driven by the fields of neuroimaging and angiography
(Krüger et al. 2001). The MR spectroscopy has also seen some gains by the increased
signal-to-noise ratio (SNR) and narrowing of spectral resolution (Hetherington et al.
1997). However, theoretical predictions of two fold increase in SNR with the doubling
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Dept. of Medical Physics
of the B0 , have not been observed (Gonen et al. 2001). In the field of spectroscopy
even a slight increase in SNR can be traded to reduce the long scan times (Dydak
and Schar 2006) according to the equation:
SN R ∝ V
√
t,
(3.1)
where V is voxel size and t is acquisition time.
The
31
P MRS on the other hand has always been of high interest to researchers and
clinicians giving the ability to measure energetic metabolism in vivo. However due to
the lack of clinical scanner capabilities, 31 P MRS is mainly used for research purpose.
In order to push the
31
P MRS into a clinical realm much technical development
needs to occur. In this chapter the design of rapid localization techniques for
31
P
spectroscopy is discussed.
While there are many multi shot fast chemical shift imaging methods available for
accelerating
31
P CSI (Pohmann et al. 1997), a recent study comparing fast spectro-
scopic imaging methods by Zierhut et al. 2009 found that the best approach was
EPSI, with a flyback gradient design, when compared to other k–space sampling
methods. In contrast to parallel imaging acceleration techniques, and other rapid
trajectories a flyback trajectory was found to be extremely robust in measuring and
assessing metabolite concentrations in comparison to the gold standard CSI technique
(Cunningham et al. 2005).
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Dept. of Medical Physics
Table 3.1: In vivo concentration of phosphorous metabolites in human brain tissue.
Metabolite
Adenosine triphosphate (ATP)
Phosphocreatine,
Inorganic phosphate (Pi)
Concentration (mM/L)
3.03
5.18
1.5
For MRS the limit of detection is about 0.5 mM for metabolic concentrations, and 31 P
containing metabolites have concentrations that are very close to this detection limit
(Table 3.1) (Bottomley and Hardy 1989). Therefore the main goal of rapid imaging
is to speed up the acquisition while keeping the signal to noise ratio for phosphorous
metabolites in the range of SNR = 10 – 50 for valid data analysis.
3.2 Theory
For gradient design hardware considerations, maximum amplitude and slew rate are
important. This is often limited by the amplifiers, physical dimensions of the coils and
the safety standards set by governing bodies to limit dB/dt exposure to the patients
for prevention of peripheral nerve stimulation.
3.2.1 Chemical Shift Imaging signal equation
As seen in chapter 2 magnetic resonance imaging usually deals with just relaxation
parameters T1 , T2 and proton density ρ to describe the magnetization distribution
m(x, y) and the effect of the imaging pulse sequence on the weighting it will produce.
A fourth parameter, chemical shift σ, is often ignored and even considered an artifact
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Dept. of Medical Physics
(chemical shift artifact), often requiring complex approaches trying to separate signal
from fat and water (Merkle and Dale 2006; Dietrich et al. 2008). In order to understand chemical shift imaging we need to extend the magnetization description with
frequency domain. For 2D CSI magnetization distribution can therefore be described
using
m(x, y, f )
. For example imaging muscle and fatty tissue in humans can be considered as
imaging a volume from water H2 O and fatty acids CH2 , producing two lines separated
by about 3.5 ppm. Most imaging methods integrate over the frequency domain,
producing images that have equal contribution from fat and water. If scan parameters
are not considered properly the signal from fat and water will appear overlapped
resulting in a chemical shift artifact (called chemical shift artifact of the 1st kind,
noted in the frequency encoding direction).
Spectroscopic imaging methods, on the other hand, can be thought of as a combination spatial–spectral imaging method, where the goal is to obtain NMR spectra for
each spatial position. This is often done at coarse spatial and high spectral resolution, due to low concentrations of metabolites. Signal from volume m(x, y, f ) can be
written down as:
Z Z Z
S(t) =
x
y
m(x, y, f )e−i2πkx (t)x e−i2πky (t)y e−i2πf t df dydx
f
72
(3.2)
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Sergei Obruchkov
Dept. of Medical Physics
Table 3.2: SNR is proportional to square root of total acquisition time
Acquisition Time
Single Voxel MRS Number of Acquisitions
2D CSI
NEX × X encodes × Y encodes
3D CSI
NEX × X encodes × Y encodes × Z encodes
which is essentially a 3D Fourier transform of the m(x, y, f ), where the kx (t) and ky (t)
are trajectories through k-space and are functions of applied magnetic gradients:
t
γ
kx (t) =
2π
Z
γ
ky (t) =
2π
Z
Gx (τ )dτ
0
t
Gy (τ )dτ
(3.3)
0
Signal to noise ratio of 1 H MRS varies with the square root of acquisition time.
Compared to single voxel spectroscopy, CSI has the same SNR as single voxel data
and provides spatial information. (Table 3.2) shows how time acquisition depends on
number of phase encoding steps.
2D CSI has 2 spatial and 1 spectral dimension. The simplest pulse sequence (i.e. a
CSI sequence) for this type of acquisition scheme is shown in Figure(3.1 & 3.2).
Due to the limited number of phase–encoding steps and large voxel sizes used in
31
P
CSI (>1 mL), it is therefore important to consider the point spread function (PSF)
of the pulse sequence. The shape of the PSF depends on the acquisition scheme, the
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Dept. of Medical Physics
Figure 3.1: 2D CSI sequence and its associated k-space trajectory
Figure 3.2: 2D CSI data sets showing both kx and ky spatial dimensions.
coarse resolution and large voxel sizes can cause the possible side lobes of the PSF to
cover substantial distances, thereby spreading incorrectly localized signal.
Furthermore, nominal voxel sizes usually differ substantially from effective voxel sizes,
as defined by the FWHM of the PSF. This becomes especially important in the design
of fast CSI sequences, where sometimes the SNR per voxel is artificially increased by
the large effective voxel sizes. Depending on the acquisition scheme it is often difficult
to calculate and can be even more difficult to measure the real PSF of the sequence
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Dept. of Medical Physics
(Robson et al. 1997). When using surface coils for excitation, measurement of the PSF
becomes even more challenging as one needs to compensate for B1 inhomogeneities
(Murphy-Boesch et al. 1998).
The PSF can be calculated using the Fourier transform of the acquisition pattern in
k–space, since EPSI reads out data in a Cartesian fashion the PSF of EPSI and the
conventional CSI are very similar Figure 3.3.
Figure 3.3: a) PSF of conventional CSI, which is the same as the flyback EPSI readout;
and b) PSF of a circular k-space readout trajectory.
The EPSI sequence with a flyback readout trajectory reads out data where the kx
direction is traversed in an echo planar manner with readout only in one direction.
This ensures that the phase of the spins always evolve in the same direction (Figure
3.4). This minimizes artifacts associated with the phase reversal in most echo planar
spectroscopic and imaging techniques (Cunningham et al. 2005).
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Dept. of Medical Physics
Figure 3.4: (Left) EPSI pulse sequence with a flyback readout trajectory. (Right)
The readout is performed using Gx gradient showing the resulting k-space trajectory.
3.2.2 Optimization of Echo–Planar Flyback Readout Trajectory Gradients
When designing an echo planar spectroscopic readout trajectory both spatial and
spectral bandwidths must be considered (Figure 3.4). Spectral bandwidth can be
defined as the repetition of individual segments of the flyback trajectory defined as:
BWspect =
1
Tf + Tr
(3.4)
where Tf and Tr are the periods of the trajectory as shown in Figure 3.5. While the
spatial bandwidth requirements are defined by the Nyquist limit in the field of view
(FOV) direction,
BWspatial =
1 γ
Gx Lx
2 2π
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(3.5)
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Dept. of Medical Physics
where Gx is gradient amplitude and Lx is the required FOV. The trajectory is further
constrained by the MRI system itself, in particular the slew rate (SR) and maximum
amplitude of the gradients and the maximum sampling rate. Therefore in order
to design an optimal trajectory five parameters need to be specified: spectral BW,
spatial resolution, maximum system slew rate, maximum gradient amplitude and the
sampling rate.
In order to calculate the optimal Gx gradient trajectory that will generate MR signal
(i.e. an echo), with the required spatial bandwidth, the area under the readout
gradient (|A|) must equal:
|A| =
1
,
γ
δ
2π x
(3.6)
where δx is the required spatial resolution for a CSI voxel.
Figure 3.5: Flyback EPSI trajectory broken apart into segments required for calculating needed parameters. The segment after the rewinder portion of the sequence is
repeated the required number times.
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Dept. of Medical Physics
If the desired spectral BW is evenly divisible by the sampling rate of the MR system,
the problem simplifies to a geometric one where the area of any triangle can be
defined using the system gradient slew rate as a constant (Figure3.5). Then defining
the problem as a system of equations becomes easy:
Tf + Tr =
|A|
+ 2r + 2r1
h
h
= SR
r
(3.7)
Tf + Tr = BWspectral
The solution to the above system of equations was found to be:
|A|
1
−r
−
2 ∗ BWspect (2 ∗ SR ∗ r)
2
−
|A|
− r2 = 0
SR
(3.8)
where the spectral bandwidth, BWspect , gradient area, |A| and slew rate, SR are
all known quantities. From the two solutions to the above quadratic equations the
shortest r is chosen such that it does not violate the maximum gradient amplitude of
the system. The above equation has no solution if the requirements for spectral and
spatial resolution are too high i.e. when:
(2.0 ∗ r − 1/BWspect ))2 − 4 ∗
|A|
) < 0.
SR
(3.9)
In order to design the gradient trajectory the exact solution is linearly interpolated
and converted to an even 16 bit signed integer vector with temporal resolution of 4
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Dept. of Medical Physics
Figure 3.6: A Cartesian dataset is acquired using flyback echo planar trajectory thus
maintaining an optimal point spread function, PSF. The EPSI data is slightly phase
shifted in the kx direction. This is easy to correct by phase shifting the data set
according to the acquisition parameters.
µs i.e. the system gradient update time. Due to integer conversion the gradient area
is verified to ensure read out and rewinder areas are exactly the same. Source code
used for system design can be found in Appendix A.
Efficiency of the EPSI readout is determined by its duty cycle (i.e. the time the
sequence spends in the linear readout trapezoid Tf ) where the effective SNR ESN R
can then be written down as:
s
ESN R =
Tf
Tf + Tr
(3.10)
The data that is acquired using an EPSI with a flyback trajectory does not exactly
fall onto the Cartesian grid. Instead each point is shifted by a certain amount, φ,
which depends on the spectral bandwidth and spatial resolution of the readout EPSI
gradients see Figure 3.6.
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Dept. of Medical Physics
3.2.3 Reconstruction
The reconstruction of EPSI data can be performed in either the time or frequency
domain however both methods first require reshaping the acquired time data into the
3D data set (i.e. 2 spatial, 1 spectral dimension) and then shifting each data point
by the proper phase amount. Each 1D FID was first reshaped to form a 2D data set
according to the acquisition parameters. For a 2D acquisition the result would be
a 3D data set and for 3D CSI a 4D data set would be generated. The transformed
data is Fourier transformed in the time domain creating a frequency domain data set
I(f )mn were m is the sample number in the nth gradient echo. Recovering a Cartesian
data set is then performed by applying a linear phase shift to each point according
to equation:
0
Imn
(f ) = I(f )mn ei2π
mn∆tBWspect
N
(3.11)
were m,n and ∆t are point location as described in figure 3.6 and N is the total
number of gradient echoes.
Equation 3.11 describes the phase shifting in the frequency domain, which in the time
domain is equivalent to a sinc interpolation (aka Whittaker–Shannon interpolation)
as described by ((Oppenheim et al. 1998)):
x(t) =
∞
X
x[n] · sinc
n=−∞
80
t − nT
T
(3.12)
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
where T = 1/fs is the sampling interval, fs is the sampling rate, and sinc(x) is the
normalized sinc function,
sinc(x) =
sin(πx)
.
πx
(3.13)
Following interpolation, the data is then Fourier transformed and is analyzed using
tools designed for CSI analysis. The source code for the reconstruction algorithm can
be found in Appendix B.
Figure 3.7: The GE Healthcarer 3.0 Tesla MRI scanner in the Imaging Research
Center of St Joseph’s Healthcare.
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3.3 Methods and Materials
A 3T Signa Excite HD MRI (GE Healthcarer Milwaukee, WI) equipped with the
multinuclear accessory was used to perform all experiments. A 3.0 Tesla scanner is
shown in Figure 3.7 with the specifications for the system provided in Table 3.3. The
scanner has capabilities to perform imaging using two different gradients, whole and
zoom, each with different slew rate and maximum amplitudes.
Table 3.3: Parameters of MR Magnets used in the experiments.
Magnetic Field Strength (T)
Water Center Frequency (Hz)
Maximum FOV (cm)
Shimmed area (cm)
Field Inhomogeneity (ppm)
Band Width (Hz)
3.0
127,799,074
60
45x48
0.349593
500
Narrow Band RF Power Amplifier(Body Coil)
Max Output Power (Watts)
35,000
Max Average Power (Watts)
1,250
Max Pulse Energy (Joules)
125
Max Pulse Width (ms)
50
Broad Band RF Power Amplifier(Body Coil)
Max Output Power (Watts)
4,000
Gradients (Whole)
Max Rise Time (µs)
Max Amplitude (Gauss/cm)
Slew rate (Tesla cm−1 s−1 )
288
2.3
80
Gradients (Zoom)
Max Rise Time (µs)
Max Amplitude (Gauss/cm)
Slew rate (Tesla cm−1 s−1 )
268
4.0
150
Software
OS version
12.0 M5B 0846.d
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Dept. of Medical Physics
The pulse sequence was designed to use zoom gradients in order to gain high spectral
and spatial resolution. Because the system has limited memory the trajectory was
designed as two segments: a half rewinder and a trapezoidal readout with a flyback.
When running this sequence it was important to ensure the designed segments were
executed ’back to back’ with dead time minimized, in a periodic/symmetric fashion
throughout the EPSI readout. This ensures symmetric k-space sampling and phase
accrual.
Design and optimization of the gradients was performed in MATLAB (The Mathworks, Natick, MA, USA). Gradient waveforms were then integrated into the scanner pulse sequence by modifying a CSI pulse sequence written in the EPIC (GE
Healthcarer Milwaukee, WI) programming language. Acquired data was processed
in MATLAB and formatted for use with the spectral analysis software SAGE (GE
Healthcarer Milwaukee, WI).
3.4 Results
Gradients were designed to ensure symmetrical data acquisition where dead time
effects were taken into consideration in the acquisition scheme. Originally gradient
files were designed to fill system machine memory limited to 16384 points and then
to repeat itself, unfortunately this resulted in an 8µs dead time which shifted the
k-space sampling every 16384 points. This occurred due to differences in the way
the MRI system memory was managed, and the original gradient waveform file was
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Dept. of Medical Physics
Figure 3.8: Instead of a linear readout the 8µs dead time caused phase accrual every
0.6ms resulting in a step like readout.
designed. Therefore in this schema the acquired k–space readout could not be properly
reconstructed since the points did not fall on a Cartesian grid, Figure 3.8.
Dividing the gradient waveform into a rewinder and readout chunks ensured that
gradients could be designed to be in sync with the data acquisition system, which
has a limited set of readout filters with required bandwidths. The maximum readout
bandwidth allowed by the system was BWreadout = 32.75 kHz. Therefore the designed
gradients required BWspectral to divide evenly with the BWreadout . The collected data
would then be acquired properly, Figure 3.9.
Flyback echo planar trajectories were optimised for the available MR hardware. In
Figure 3.10a), it is noted that data collected during the flat portion of the trapezoid
contributes to the SNR, thus the EPSI efficiency will be higher for gradients designed
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PhD Thesis
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Dept. of Medical Physics
Figure 3.9: (Left) Designed gradients need to be equally divisible into the allowed
system receiver bandwidth to ensure proper data acquisition. (Right) Gradients are
out of sync with the receiver.
for low spatial resolution and low spectral resolution, than those with higher resolution
requirements. While 1 H MRS spans only 4 ppm at 127.8 MHz or 512 Hz, sampling at
the rate of 1100 Hz is sufficient to satisfy the Nyquist criterion and as such is easy to
design EPSI gradients at 1 cm.
31
P MR spectra, on the other hand, spans 25 ppm at
51.7 MHz or 1300 Hz, thus in order to acquire
31
P spectrum EPSI gradients need to
have BWspectral = 2600 Hz. Designing gradients with high spatial resolution for
31
P
becomes a real challenge. While any waveform gradient could be designed as long as
they fall within the operating parameters of the MRI hardware, the gradients that
were optimised and tested for
31
P are listed in Table 3.4.
In order to have 1 cm and 2 cm spatial resolution, gradient waveforms were interleaved
to ensure full spectral coverage (Figure 3.10b).
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Dept. of Medical Physics
Figure 3.10: a) Waveform designed for 2cm spatial resolution and spectral bandwidth
of 1000 Hz. Only the first 2.5 ms of the waveforms are shown. The calculated k-space
trajectory and the points sampled at 31.25 kHz are shown as an overlay. b) In order
to ensure complete spectral coverage interleaved waveforms were used to ensure full
spectral coverage of the gradients waveforms.
Table 3.4: List of readout gradients created and optimised for the 3T MRI system.
31
P waveforms
Spatial Resolution
(cm)
1
2
2.6
4
1
Spectral Resolution
(Hz)
868
1302
1562.5
2083
ESN R
%
73.9
68.8
63.5
64.3
504
93.0
H waveforms
1
3.4.1 Calibration Results
To achieve the most accurate reconstruction possible the BWspect was measured directly from an EPSI sequence, for each optimised gradient, from a high concentration
phantom producing high SNR data. This was done by applying gradients in an off
resonance condition and observing phase of the FID signal (Figure 3.11), (Jung et al.
2007).
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Dept. of Medical Physics
Figure 3.11: Measuring FID in an off resonance condition. (Left) the phase of the FID
will reflect the k-space trajectory and its derivative will reflect the applied gradients
(Right).
After the excitation RF pulse, a gradient echo will form every time the gradients travel
through the center of k-space. With an echo planar flyback trajectory a gradient echo
is formed twice, once during the flat trapezoid gradient time Tf and once during the
flyback Tr . Only the echo generated during the Tf is used during the reconstruction
process. In order to verify that the sequence is operating properly and is recording
data symmetrically, high SNR phantom data was used (Figure 3.12). Multiple echoes
from raw data were superimposed on top of each other to display 256 gradient echoes.
In figure 3.12b) the first points (1–55) are produced during the trapezoidal readout
Tf , while the last points (55–63) are the echoes generated by the flyback gradients.
Ensuring that the flyback echoes are positioned in a tight grouping, at the end of the
dataset demonstrates quality of the readout gradient.
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Dept. of Medical Physics
Figure 3.12: 256 Gradient echoes generated by the Echo Planar Readout spanning 63
points across k–x a) Raw data showing the echoes generated by the trapezoid and the
flyback gradients b) After the data is properly reshaped to fit back on to the Cartesian
grid the rewinder echoes are tightly bunched and are properly positioned at the end of
the dataset.
3.4.2 EPSI Results
Flyback EPSI and CSI data was acquired using gradients with 1 cm spatial resolution
for both 31 P and 1 H. 31 P spectra were acquired using gradients designed with spectral
bandwidth of 868 Hz (interleaved 3 times) and effective SNR of 73.9% (Table 3.4).
1
H spectra were acquired using gradients with BWspect = 504 Hz and ESN R = 93.0%
as calculated per equation 3.10.
A spherical phantom, filled with 0.3% saline (NaCl), was used to acquire 1 H CSI and
EPSI data Figure 3.13. Chemical shift images were produced by integrating the area
under the water peak. The resulting EPSI images geometrically correlated well with
CSI data, with line widths (FWHM) of 30±5 and 25± 5 Hz respectively.
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Dept. of Medical Physics
Figure 3.13: CSI and EPSI data of a Spherical Phantom (Top) 1 H 12x12 CSI data 1
cm3 spatial resolution 504 Hz spectral resolution no water suppression, scan time 4.8
minutes. (Bottom) 18x12 EPSI 1 H reconstructed data 1 cm3 504 Hz spectral resolution
acquisition time 24 seconds.
SNR comparison test of
31
P CSI and EPSI data were acquired on phantoms con-
taining 1M concentration of KH2 P O4 and K2 HP O4 (Sigma-Aldrich, St. Louis, MO,
USA) (Figure 3.14). CSI images were produced from computing areas under the two
chemical peaks. EPSI offers acceleration times, for low bandwidth applications, 20
times faster then conventional CSI. Measured SNR varied according with theoretical
prediction and decreased as a square root of the acquisition time. For 8 NEX and 2
NEX, SNR was measured to be 35% and 21.8% respectively, compared to that of the
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PhD Thesis
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Dept. of Medical Physics
Figure 3.14: High Resolution CSI data from 1M mono and di–basic potassium phosphate buffer solution. Images produced from the 20x20 CSI spectra and interpolated
to 128x128 for display purposes.
CSI acquisition. The 1 cm gradients corresponded well with theoretical predictions
for the SNR Figure 3.15.
3.5 Discussion
Successful design and implementation of
31
P EPSI sequence with flyback trajectory
has been demonstrated. High SNR EPSI sequence with flyback trajectory can speed
up acquisition without significantly affecting image quality as assessed using peak line
width and geometric representation.
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PhD Thesis
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Dept. of Medical Physics
Figure 3.15: Relative SNR data as measured using ESN R = 73.9% gradients with 1
cm spatial resolution compares well with theoretical predictions.
Overall EPSI data showed good correspondence with CSI data. Since flyback trajectory does not contain reversed echoes the quantification of spectra is more robust and
is insensitive to acquisition errors. Thus this approach has great potential for in-vivo
applications (Ulrich et al. 2007; Posse et al. 1994).
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Dept. of Medical Physics
PhD Thesis
Sergei Obruchkov
Dept. of Medical Physics
Chapter 4
Design and Construction of a
Heteronuclear 1H and 31P Double Tuned
Coil for Breast Imaging and Spectroscopy
This chapter deals with the development and design of a multi–tuned RF coil for
simultaneous
31
P spectroscopy and 1 H WALTZ-4 decoupling. Background theory
and design of a double tuned RF coil for breast imaging and spectroscopy application
is presented. The results from RF calibration and imaging using the MRI coils are
discussed.
4.1 Introduction
The importance of RF coils was previously discussed in chapter 2. The MR signal is
in a form of RF waves that belongs to very high frequency (VHF) band 30–300 MHz
with typical bandwidth of less then 1 MHz. MR coils are designed to operate at a
very specific frequency which depend on the Larmor frequency of the nucleus being
imaged. Therefore a separate coil is needed for each nucleus of interest, e.g. one
coil for 1 H and one for
31
P, 3 He,
13
C etc. However there are a specific type of coils
called dual tuned or multituned coils that can operate at two or more frequencies
simultaneously. This enables the MR scanner to perform experiments which require
ability to interrogate multiple nuclei at the same time.
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Dept. of Medical Physics
The practice of using volume coils for transmit and using surface coils for receive
occurred in the last 15 years, and has played a large role in being able to increase
the strength of the magnets while optimizing the available SNR. There are two main
differences between coils that are used for transmission and reception of RF signal.
Transmission coils are typically much larger and must be able to handle large amounts
of power 15–35 kW is a typical strength of RF power amplifier that are used today
in clinical imaging. Most clinical scanners have a built in transmit coil tuned to 1 H
frequency called body coil, which is driven by a large narrow band power amplifier
designed to run at the same frequency. Therefore a separate transmit coil and a
broadband RF amplifier are needed for work with nuclei other than 1 H.
Reception RF coils on the other hand are much smaller in size and use low power
components. In order to use them safely the receive coils are detuned during the time
when the transmit coils are turned on thuss isolating the coil. This is accomplished
in two ways: passive and active blocking circuits. Both utilize diodes and capacitors
to detune the center frequency such that the transmit and receive coils will not be
interacting. T/R switches are typically used for these applications and often each
T/R switch is designed to operate only at one frequency at a time therefore two T/R
switches are needed to be used with a dual tuned coils.
Multinuclear work requires many extra components that are often not available with
most clinical scanners. When designing a dual tuned coil one must then consider
multiple factors such as; type of experiments the coil will be used in, target anatomy,
physical size, number of ports, transmit receive capabilities and many more.
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PhD Thesis
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Dept. of Medical Physics
4.1.1 Motivation for Dual Tuned Coil for Breast
31
P Spectroscopy With
Two Port Input
Normally if a scanner has multinuclear capabilities, in order to perform phosphorus
spectroscopy or imaging only require a coil tuned to 31 P would be required. However,
there are several common applications that utilize multiple–frequency excitation to
study the interactions and to enrich acquired data with extra information. For example, it is possible to increase SNR by transferring spin polarization from 1 H to 13 C
using the nuclear Overhauser effect (Bomsdorf et al. 1991). Proton decoupling is
an effective technique for mitigating the effects of spin–spin coupling which tends to
broaden and increase the complexity of the spectra (Ch 2). And the inverse detection
methods used in 1 H and 13 C spectroscopy and many other application all require coils
that can operate with high efficiency with two or more nuclei.
In addition, dual tuned coils, are very useful for producing shimming and localizer/scout data using the 1 H channel prior to doing
abundance of
31
P,
13
31
P spectroscopy. Because of the low
C and other nuclei, 1 H shimming and localization often helps to
increase the accuracy of spectral localization and the quality of field homogeneity.
Furthermore the simultaneous acquisitions of 1 H and
very useful, as it is possible to interleave 1 H and
31
31
P data has been shown to be
P acquisitions (Jesmanowicz et al.
2002).
Phosphorus spectroscopy presents clear evidence of increased concentrations of phosphomonoesters (PMEs), and phosphodiester (PDEs) as well as inorganic phosphate
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PhD Thesis
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Dept. of Medical Physics
Figure 4.1: a) Sentinelle Medical Vanguard breast biopsy table attached to a clinical
scanner. b) Breast coils are swapable for a grid to MR guide breast biopsy c) Vanguard
table with the MR coils.
peaks in breast tumour tissue (Leach et al. 1998; Arias-Mendoza et al. 2004). Therefore it is of great interest to create and design a coil that would be compatible with
clinical workflow and compliment the use of a specialized table that utilizes breast
compression for imaging and performing breast biopsy (Figure 4.1).
A double tuned coil, compatible with Sentinelle Vanguard breast biopsy table (Sentinelle Medical Inc., Toronto ON), was designed and built to efficiently operate at
resonant frequencies of
31
P and 1 H. Dual tuned RF coil allows performance of exper-
iments that simultaneously require RF transmission at both frequencies, as well as
better quantification of
1
31
P metabolites, by applying B1 corrections acquired using
H channel.
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Dept. of Medical Physics
4.2 Theory
This section will describe basic electromagnetic theory, from the creation of electromagnetic radiation to the basic principles behind RF circuitry and MRI RF system
in general. An overview of modern design of MRI transmit and receive RF chain
needed for implementation of dual tuned coils in MR experiments is provided.
4.2.1 Propagation of Electromagnetic Waves and Magnetic Field Generation
In chapter 2 B1 field which is responsible for the generation of transverse magnetization Mxy and its decomposition into circular components have been discussed. The
generation of the this time–varying magnetic field requires, like any electromagnetic
field, some arrangement of electric charge. This is achieved using accelerating charges
and changing currents. However unlike radio stations that use electrical dipole antennas to broadcast their signal with high efficiency over hundreds of kilometers,
MRI systems require the use of magnetic dipole antenna to produce electromagnetic
radiation, which are typically not very efficient. However they can produce good
directionality and coverage over short distance e.g. human body.
Take a wire loop of radius a, around which alternating current I is being driven as:
I(t) = I0 cos(ω0 t).
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(4.1)
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Dept. of Medical Physics
Figure 4.2: A circular loop with current I will generate magnetic field which could be
calculated using Biot–Savart law.
By driving the current I at resonant frequency of ω0 = γB0 two things occur: a B
magnetic and E electric fields will be generated they will be perpendicular to each
other and the coil and secondly they will radiate outward. The radiation carry energy
away from the coil, typically electric field is responsible for radiating more energy then
magnetic field, and the ratio of the amplitudes of the two components are E/B = c,
where c is the speed of light.
It is possible to calculate the field profile that is being radiated by an arbitrary shaped
source. This is typically done using a time–variant solutions to Maxwell equations
known as Jefimenko’s equations which describe generalized time–varying Coulomb
and Biot–Savart’s laws (Griffiths and Inglefield 1999).
Often the method of moments analysis subdivides a coil into segments from which,
first the electrical field that is produced by the coil can be calculated 4.2, and later
the magnetic field.
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~
~ = −∇V − ∂ A ,
E
∂t
Dept. of Medical Physics
~ =∇×A
~
B
(4.2)
~ and V~ are retarded potentials which can be calculated using:
Where the A
Z ~ ~0
J(r , tr )
)dτ 0
0
|r − r |
Z
1
ρ(r~0 , tr ) 0
dτ
V~ =
4π
|r − r0 |
~ r, t) = µ0
A(~
4π
(4.3)
(4.4)
~ r~0 , tr ) is the current density. A good review
where ρ(r~0 , tr ) is the charge density and J(
of method of moments analysis can be found at (Harrington 1993). This approach is
often used with a simplification for coils that are typically smaller then the wavelength
of the EM radiation emitted from them.
While the exact solution to the magnetic field is complicated, it is possible to estimate
it using Maxwell’s equations and static Biot–Savart law. Analytical solution can be
separated for the distribution of magnetic B1 field from a single loop coil in radial
and axial directions as:
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PhD Thesis
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Dept. of Medical Physics
Figure 4.3: Estimated magnetic field profile from a single loop coil, B1 profile of the
XY plane at z=0.
B1radial ∝ q
x2 + z 2 (r +
B1axial
y
√
r 2 + x2 + y 2 + z 2
2
−K(κ ) + (r − √x2 + z 2 )2 + y 2 E(κ )
x2 + z 2 )2 + y 2
(4.5)
2
r 2 − x2 − y 2 − z 2
2
2
√
∝q
K(κ ) +
E(κ )
√
(r
−
x2 + z 2 ) + y 2
2
2
2
2
(r + x + z ) + y
1
(4.6)
where K(κ2 ) and E(κ2 ) are Legendre’s elliptical integrals of the first and second kind,
κ is an elliptical modulus with values between 0 < κ2 < 1 and r = x2 + y 2 + z 2 . The
B1 field is visualized in Figure 4.3 (de Graaf 2007).
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Dept. of Medical Physics
RF coils can be made out of large loops of copper however using a long continuous
piece of copper or even multiple loops is not desirable. Large coils increase the size
of the inductor, and because inductors are electrically lossy the image quality will be
degraded. It is common practice to introduce breaks in the RF coil, thus separating
long inductor into several components and distributing capacitance throughout the
entire RF coil.
Image quality will also largely depend on the scan parameters used to perform imaging experiment. For example Gradient Echo (GE) and Spin Echo (SE) will produce
images with quite drastically different images. Typically when using GE based sequences such as spoiled gradient echo (SPGR) can produce images of better quality
with transmit–receive surface coils then SE counterparts. This is largely due to the
fact that transmit surface coils have hard time refocusing spins properly, and in cases
where the TR << T1 , smaller flip angles is more desirable to maximise the SNR of
the images (Evelhoch et al. 1984). Furthermore the Adiabatic RF pulses could be
used to maximize the sensitivity of the surface transmit/receive coil, see Chapter 2.
4.2.2 Resonating RLC Circuits
Any loop of wire can be used to transmit and receive RF, however the properties of
that loop will define the optimal alternating current that can easily travel through
that conductor. Typical MR coils can be simplified to RLC circuit which maximize
the energy transfer at a given frequency. These can be found by solving Kirchoff’s
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Dept. of Medical Physics
Figure 4.4: Resonating RLC circuit.
law for RLC circuit shown in Figure 4.4. The sum of potentials across the resistor,
inductor capacitor and source V must be equal to zero V :
−L
d2 Q
dQ Q
−R
− +V =0
2
dt
dt
C
(4.7)
One can then solve the above differential equation to find out the maximum current
that the circuit can support. However it is easier to understand the behaviour of the
circuit from the phasor notations. Because capacitor and inductor both have a similar
property such that current and voltage are π/2 out of phase with each other, it is
often preferred to introduce the impedance of an element Z = V /I and write it such
that the impedance for inductor and capacitor is in a complex plane i.e. orthogonal
to the resistance.
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ZR = R,
ZL = iXL = iωL,
ZC = iXC = −
(4.8)
i
ωC
Where the real impedance Real(Z) = R is called resistance and complex impedance
Complex(Z) = X is called reactance. The inverse of an impedance is an admitance:
Y =
1
.
Z
(4.9)
Now the Kirchoff’s law still applies and one can write down the total impedance of
the circuit shown in figure 4.4.
−RI +
i
I − iωLI + V = 0
ωC
(4.10)
where ω is the angular frequency at which V is oscillating. The maximum current
that one can generate can be calculated using
I=
Vo i(ωt−φ)
e
|Z|
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Dept. of Medical Physics
where maximum current will be
Vmax
Imax = p
R2 + (iωL − i/ωC)2
(4.11)
The maximum current traveling through the circuit will depend on the impedance of
the circuit Z and will be frequency dependent. The impedance can be modeled as:
Z 2 = R2 + (jXL + jXC )2
Z 2 = R2 + (jωL −
1
ωC)2
(4.12)
(4.13)
The current within the circuit will be maximized when the impedance is minimized.
This will occur when the reactance component of the impedance vanishes and this
corresponds to frequency ω:
1
=0
ωC
1
ω = ω0 = √
,
LC
ωL −
(4.14)
(4.15)
where ωo is known as natural resonant frequency of the circuit. Applying voltage
at resonant frequency ω0 will maximize the current and thus the magnetic field B1
that is generated by the coil. The power transfered from the source to the coil itself
will largely depend on impedances of the two systems. It is common to utilise 50 Ω
transmission lines for their power transfer at broad range of frequencies. Thus the
RLC circuit must be matched to 50 Ω.
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Tuning and Matching
It is often not enough to just build a resonating circuit that can be used for MR
imaging. The coil must be connected to the transmitter, receiver or the preamplifier.
This is typically done though coaxial lines with impedance of 50 Ω. Thus the coil that
is tuned and matched to 50Ω can theoretically achieve 100% power transfer and 100%
efficiency simultaneously in a perfect resistance free reactive circuit. In an imperfect
reactive circuit (i.e. one with both reactance and resistance), the efficiency η will be
the power dissipated in the load PL divided by the power provided by the source PS .
η=
PL
PS
When the resistance is a small fraction of the reactance, the efficiency will be very
high even at the maximum power transfer point. This is a key to designing high
efficiency MR coils, ensuring that the current generated in the coil from the transverse
magnetization of the spins will be detected with maximum efficiency.
An introduction of a matching capacitor CM is necessary to convert the loaded coil
impedance from usually a very high impedance to 50 Ohm, Figure 4.5. The natural
frequency of the circuit will shift and thus the value of a tuning capacitor CT needs
to be changed such that the coil will resonate at the needed frequency ω0 with the
impedance at that frequency being 50 Ohm.
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Dept. of Medical Physics
Figure 4.5: RLC circuit connected to the source ZS = 50Ω via coaxial cable Z0 = 50Ω.
In order to ensure efficient power transfer at the resonant frequency tunning CT and
matching CM capacitors are used to ensure that the coil acts as a 50Ω load at the
resonating frequency ω0 .
4.2.3 Reciprocity Principle and Coil Quality Factor
Reciprocity principle states that the B1 field generated by current It should be iden~ . In other words the
tical to current Ir generated in the coil by magnetic moment M
knowledge about the transmit field of the coil will give direct information about the
signal generated in the coil. This symmetry between received and transmitted fields
can be used to map the B1 , which in turn could be used for quantification of MR
signal, e.g. molar concentrations of metabolites. It is possible to measure B1 field of
the coil using several different methods, this information then can be used to compare
a known concentrations of a reference sample to unknown concentration in the tissue.
From section above we know that LCR circuit will resonate when reactance is zero,
while the resistance of the coil usually peaks, Figure 4.6. Thus energy will be dissipated in the circuit decreasing the quality of the received signal. A quality factor
Q of the built coil, is defined as the ratio of maximum energy stored in the circuit
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over the total energy dissipated. Quality factor can be measure from the resistance
of LCR circuit and it is related to the FWHM of its peak:
Q=
ω0
2∆ω
(4.16)
This quality factor depends on the design of the coil and its components. Quality
factor of a tuned and matched unloaded RF coil can be approximated by
Qunloaded =
ωL
,
Rc
where Rc is the resistance of the coil. RF coils ideally should have high Q so that they
will be very sensitive to the RF frequency, without damping or dissipating RF energy.
However as soon as the coil is placed on the sample or a patient that is magnetically
lossy (sample usually introduce high resistance in the coil), the coil will couple to the
sample and the Q will be reduced to:
Qunloaded =
ωL
,
Rc + Rs
where Rs is resistance introduced by the sample. The coil sensitivity, S is defined as
the ratio of quality factors of loaded to unloaded coil as
S
=
S0
s
1−
Qloaded
,
Qunloaded
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(4.17)
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Dept. of Medical Physics
Figure 4.6: Resistance and Reactance of the coil tuned and matched to 50 Ω. Resistance of the coil will change depending on the load that is placed onto the coil. Ideal
col will have very narrow and tall peak thus have a large Q.
where S0 is the theoretically maximum achievable sensitivity when Rc → 0.
It is common practice to use a saline phantom with loading properties similar to
human tissue for tuning and matching the coil. While the phantom can not mimic all
of the loading characteristics that the device will be subjected to in clinical imaging
it is often impractical to tune and match the coils for every single subject.
4.2.4 Transmit–Receive Switch
The purpose of a transmit–receive (T/R) switch is to redirect RF, protect the receiver
and the coil electronics. Additionally because most T/R switches use diodes, noise
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between transmit and the receiver port is uncoupled ensuring proper operation of
the low noise pre amplifier. The resonating circuits are very efficient at absorbing
energy at their resonant frequencies. It is often dangerous to operate an unplugged
coil in the bore of the magnet or have two transmit coils that resonate at the same
frequencies operate in the magnet at the same time.
During transmit the energy that is delivered to the coil often exceeds 2kW for small
birdcage and linear coils, the body coil on the other hand often requires more than
10kW of power to generate required B1 field (Hayes 2009). The receive only coils
are typically made of low power components that are not designed to handle large
amount of power. Therefore a number of different techniques are utilized to ensure
that the receive coil is detuned during the transmit phase.
Typically transmit receive switches utilize DC bias and diodes to direct RF. Figure
4.7 shows two types of TR switches, Fig. 4.7a that can be used for simple linear
coils, and Fig. 4.7b T/R switch for hybrid coils. During the transmit a forward DC
bias is applied to the two diodes turning them on. Diode D1 lets RF through, and
D2 shorts the λ/4 cable. From the coil view it becomes an open circuit, therefore no
power goes through, thus protecting the preamplifier and the receiver. During receive
phase, reverse bias is applied to the diodes, D1 effectively disconnects the transmit,
and the coil is connected directly to the receiver port through the transmission line.
During the whole process the matching of the coil remains the same to ensure most
efficient power transfer.
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Figure 4.7: a) Linear coils usually utilize a single port for transmission and reception.
b) Hybrid coils actually utilize four ports, two of which are a separate transmit and
receive ports. Because the quadrature hybrids are not always perfect, transmit and
receive ports are separated to ensure proper operation. This type of a TR switch, is
commonly used because it can handle both linear and quadrature coils.
Quadrature hybrid is a 4 port device that during transmit splits RF power from port
1 equally into port 2 and 3. The RF in port 2 and 3 are π/2 out of phase with
each. During receive mode the RF from port 2 and 3 are recombined into a single
channel on port 4. Ideally port 1 and port 4 are completely isolated from each other
so in principle no T/R switch is necessary. However, because it impossible to tune
and match for every patient exactly, the reflected power will be recombined by quad
hybrid into port 4, thus a T/R switch is still required, see Figure 4.7. During the
transmit forward DC bias is on, RF is transfered into the coil any reflected power
will be sinked into the load on port 4. During receive reverse biased is applied, the
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Figure 4.8: a) Typical trap circuits used to split the resonances in a dual tuned coil
design b) This approach uses DC bias and extra capacitors for dual tuned coil design.
transmit port is effectively terminated by a 50 Ω load, and the RF can enter the
receive port through the λ/4 transmission line (Chen, Hoult, and Sank 1983).
4.2.5 Methods for Dual–Tuning Coils
Currently there is a great demand on using magnetic resonance imaging and spectroscopy for multinuclear applications. These demands often require the ability to
perform 1 H localization, B0 shimming and proton decoupling which could be achieved
using dual tuned probes. The dual tuned probes could be subdivided into two categories: single coil and dual coil approaches. Single coil rely on introduction of extra
lumped components to the MR coil in order to increase the number of resonances.
The dual coil approach uses multiple coils and separates the two modes using either
geometric decoupling or using trap circuits. Generally the application in which the
coil will be used is the main constraint for the design of a dual tuned coil.
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Single–Coil Method
From section 4.2.2 we know that at the frequency when reactance of the coil reaches
zero the coil resonates. So it is possible to design a circuit such that reactance goes
through zero at multiple frequencies. There are a few ways of achieving this either
using trap circuits or having extra capacitors that are activated using DC bias. Trap
circuits are a type of tank circuits typically build out of capacitors and inductors
in parallel. A dual tuned circuit introduce by Schnall et al. 1985 uses these tank
circuits (Figure 4.8a) to create multiple resonance modes. These trap circuits are
purely reactive components made up of capacitors and inductors in parallel, they can
be used to design coils with any number of resonances. Typically, however, a single
coil is not tuned to more then three resonances. The trap circuits are often non ideal
and always have resistance associated with them. This often causes dual tuned coils to
have lower Q then the single tuned counterparts. As well, often it is desirable to have
multiple ports for the coil and each port must be tuned and matched to its particular
frequency, this is often is the most challenging part of building dual tuned coils. This
approach has been used to design surface or volume coils by Murphy-Boesch et al.
1994 and Matson et al. 1999.
Another approach uses diodes and DC bias to electrically connect a second capacitor
to the circuit (Figure 4.8) such that the resonant frequency of the coil will change.
This approach however can utilise only a single frequency at a time (Weyers 2009).
The advantages of this design is the fact that the coil can be optimized for two
frequencies but they cannot be used at the same time.
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Figure 4.9: a) Two coils are tuned and matched separately for the two nuclei of
interest while physically they are super positioned on top of each other. b) A schematic
diagram showing the approximation of the two coil approach. The inductive coupling
of the two coils acts similar to a trap circuit splitting the resonance in two.
Dual–Coil Method Crossed Elements Configuration
It is possible to use two surface coils tuned to two different frequencies of interest
and placed adjacent to each other, Figure 4.9. This approach is similar to using
trapped circuits as the coils will be inductively coupled to each other, shifting the
desired frequencies. Once the coils are placed next to each other and loaded with a
sample the coils are retuned and rematched to the final desired frequencies. Similar
to the trapped circuits when operating the system at the hydrogen frequency, the
phosphorus coil will act as an inductor; while at the phosphorus frequency, hydrogen
coil will act as a capacitor (Fitzsimmons et al. 1993; Fitzsimmons et al. 1987). This
approach is very useful as it is the easiest implementation of two port dual tuned coil.
It is often used when the design permits the position of two coils close together and
if the coupling effects can be geometrically eliminated.
Crossed element configuration of RF coils is often used as a form of geometric decoupling to fix the problem of counter–productive current generation. Generating
two magnetic fields that are perfectly perpendicular to each other stops the flux from
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inducing currents in the adjacent coil, Figure 4.10. Orthogonal configuration of the
coils is often the simplest way of producing this effect using surface coils. However
this approach is often limited by the imaging requirements and is always hard to
achieve. Often half volume coils (e.g. open birdcage coils or bent butterfly coils) are
used with this type of configuration in order to balance the volume coverage and the
geometric decoupling benefits offered by the orthogonal configuration (Adriany and
Gruetter 1997).
A figure–8 shaped, or butterfly coil, (Figure 4.10b) provides a magnetic field parallel to
the circular coil plane, thus significantly reducing the coupling between the two coils.
While these elements have shown to be able to produce good geometric decoupling
effect, figure–8 coil suffers from low depth penetration thus limiting its uses as a
transmit coil (Bottomley et al. 1989).
The ideal dual tuned coil that can be designed must have good excitation properties
and good efficiencies at two frequencies. Two coil birdcage system offers great benefits
over surface coil, such as uniform field excitation, and good geometric decoupling,
however birdcage coils are limited due to SNR, and often some more unique anatomy
can not be easily imaged using such a configuration. Figure 4.11, shows one possible
configuration of a dual tuned volume coil, such design operates at about 90% power
efficiency compared to the single tuned coils with the same geometric configuration
(Duan et al. 2009). The coupling effects between the two coils are very hard to
eliminate especially when they are run in quadrature.
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Figure 4.10: a) Two coils positioned such that the magnetic fields generated by them
are orthogonal and thus are naturally decoupled from each other. b) A figure-8 coil
(black) or a butterfly coil, generates magnetic field that is perpendicular to the magnetic
field generated by circular coil, which is pointing out of the page.
Figure 4.11: The layout of a typical dual tuned birdcage coil using crossed elements.
Each coil is tuned to a the desired frequency and placed within the other. The runs of
the birdcage coil are oriented such that minimal coupling is observed between the two
coils.
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4.2.6 Dual Tuned Coil Design for Breast Imaging and Spectroscopy
Figure 4.12: Simplified Circuit Diagram of a dual tuned coil.
While a number of different techniques can be used to create multiple resonances as
describe above, the challenge of accessing the anatomy and intrinsic power limitation
of broadband power amplifier guided the design of a dual tuned
31
P/1 H for breast
imagind and spectroscopy. A single element surface coil was decided to be best
solution to the problem. By integrating the coil within the breast biopsy table, the
coil is easily swappable with multichannel 1 H coil array or to perform a biopsy.
Dual tuning of a single coil can be analyzed as a LCR circuit and a reactance component made up of an inductor and a capacitor in parallel (Figure 4.12). The total
impedance can be written down as the sum of impedances of an LC circuit Z1 plus
impedance of an inductor and a capacitor in parallel Z2 .
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Figure 4.13: The RLC circuit has a single resonating frequency ω0 adding a tank
circuit to RLC circuit splitts the resonance to produce two resonant frequencies ω1 and
ω2 .
Z1 = R1 + iXL1 + iXC1
1
1
1
=
+
Z2
iXL1 iXC1
(4.18)
ZT = Z1 + Z2
From equation 4.13 the impedances for individual components can be combined to
form:
Z = R1 + i ωL1 (1 −
L2
1
)
+
ω
ω 2 L1 C1
1 − ω 2 C2 L2
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The circuit will have minimal impedance and will resonate when all reactance components are zero this will occur when:
C2 =
w 2 L1 C1 − 1 + w 2 L2 C1
w2 L2 (w2 L1 C1 − 1)
(4.20)
Solution to the above equation was performed explicitly which was used to calculate
the exact values for the capacitors needed to produced desired resonance frequencies.
Value L1 is fixed by the coil geometry, and value of second inductor L2 was chosen to
ensure high efficiency on the
31
P channel which is typically L1 /2, then the capacitors
values C1 and C2 are calculated for desired frequencies, (See Appendix C ).
Efficiency of the dual tuned coil has been shown to largely depend on the values of
the inductors used in construction of the coil (Schnall et al. 1985). This is largely
due to the fact that the inductors are the most lossy components present in the coil
and it is possible to ensure optimal efficiencies for the desired operation using:
r
E31P =
r
E1H =
L1
L1 + L2
(4.21)
L2
L1 + L2
(4.22)
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Figure 4.14: Decoupler hardware from the top: power meter, vector signal generator,
1 kW solid state power amplifier.
4.3 Methods and Materials
Several phosphorus coils were designed and built for a GE Signa (GE Medical Systems,
Milwaukee, WI) 3T whole body scanner. Broadband capabilities were provided by an
accessory consisting of a 4 kW amplifier, T/R switch and a single channel amplifier
for
31
P nucleus.
The decoupling hardware consisted of a stand alone system with 1 kW power amplifier
and is based on Agilent E4438C ESG Vector Signal Generator (Agilent Technologies,
Santa Clara, CA) (Figure 4.14). The system was preprogrammed to provide WALTZ–
4 pulse shape modulation, the calibration and control of the which was performed
using a PC with an graphical user interface written in Vee Pro (Agilent Technologies,
Santa Clara, CA).
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Figure 4.15: RF coil dock, a) TR switch for 31 P, b) 1 H connector, c) cable on the side
is the decoupler connection.
While the broad band system utilises the same receiver subsystem as the 1 H system, the signal pathway is different, this includes a separate T/R switch and a pre–
amplifier. Comparing the broadband pathway to the 1 H pathway is important to
insure optimal signal acquisition. This was performed once in order to establish a
known initial condition that were later monitored using Quality Assurance QA test
designed in house for the broadband system.
4.3.1
31
P Solenoid Coil For System Stability and QA Assurance Tests
When designing and building RF coils for multinuclear work, two things have to
be considered: a proper transmit receive switch must be used, and unlike 1 H coils
transmit and receive coils must be build. Typically coils for multinuclear work are
combined i.e. transmit and receive is performed using the same coil. While this usually simplifies the building process, B1 field homogeneity of a coil is limited by its size.
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However if the coil is too large the filling factors will decrease resulting in a lowered
SNR during receive. Thus transmit–receive coils must balance two parameters: field
homogeneity for uniform excitation and good SNR.
A 6 turn solenoid coil was tuned and matched for a 1 ml sample of 1M H3 P O4 .
The coil was first tuned and matched to the Larmor frequency of 1 H (127.8 MHz
at 3.0T). The sample was scanned using a spin echo spectroscopic sequence without
water suppression. In order to keep geometry the same the coil was then tuned and
matched at the Larmor frequency of
31
P nucleus (51.73 MHz at 3.0 Tesla). The spin
echo sequence was used to compare the relative peak height of 1 H and phosphorus, to
estimate the performance of the broad band receive chain. This data was later used
as a standard for QA tests of
31
P broadband system.
Broadband system quality was also verified using a two tone test, by injecting two
tones directly into the head–coil carriage coil connectors, and the TR switch using RF
signal generator (Model 2030; Marconi Instruments, St. Albans, Colchester). The
noise floor was measured and compared using the on board MR receiver system. The
noise floor, and peak height was used as a reference for the quality of the 1 H and
31
P
receive pathways.
Systems docking station shown in figure 4.15 was used in all the experiments. The
hydrogen channel of the coil was always connected to either the decouple or the 1 H TR
switch. The phosphorus coil was always connected through 31 P TR switch. Adaptors
were used to convert the proprietary connections to a female BNC connection.
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Figure 4.16: Simulating double tuned coil using QUCS software.
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4.3.2 Dual Tuned Surface Coil Design and Construction
Dual tuned surface coil was designed using the help of simulation software QUCS (
http://qucs.sourceforge.com ). Initial attempt of tuning and matching was performed
using optimization algorithm with a search for a global minimum of an S11 parameters at both resonant frequencies. However with the initial design of the coil it was
impossible to find values that tuned and matched both channels simultaneously, as the
value of matching capacitors on one channel would effect the second channel. A final
design of the dual tuned coil shown on figure 4.16, with the blocking circuit followed
by the matching network allowed independent matching of the two channels. The
tuning and matching was performed using line transformation methods. In practice,
parasitics create extra capacitances and inductances throughout the physical circuit
which are hard to include into simulations. Thus the simulated values and values
used during actual building of the coil differed slightly.
By providing the values of the inductors, L1 and L2 the values of the capacitors needed
to generate required resonances was calculated using in house software written in
Maple (Waterloo Maple Inc., Waterloo, ON). In order to understand the effect of
inductors on the circuit, simulation was run multiple times by varying the value of
the trap inductor L2 , since L1 is fixed and is determined by the geometry of the
coil (Figure 4.17). The results shown in figure 4.18, provide a good estimate of
balancing the inductors in order to achieve similar efficiencies on both channels. It
was found that using value of L2 = L1 /2 gave the best balanced efficiencies from
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Figure 4.17: Physical layout of the coil the capacitors C1 , C2 and inductors L1 and
L2 are responsible for tuning the circuit to the 31 P and 1 H resonant frequencies.
the S11 measurements. The final values of L1 and L2 , used to build the coil had
efficiencies of 87% and 48% for
31
P and 1 H channel respectively, which is confirmed
using equations 4.21 and 4.22.
The coil was put into a housing designed to fit into a commercial breast biopsy table,
Vanguard ( Sentinelle Medical, Toronto, ON Canada), Figure 4.19. It is possible
to swap the dual tuned coil during a clinical breast scan and replace it with the
multichannel coil or perform a biopsy. In order to eliminate the chance that the
blocking circuits would fail if too much power is fed into one of the ports, the power
allowed to enter the coil was limited by software.
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Figure 4.18: Circuit Response while varying the value of the inductor L2 = 100 −
−320nH inductor L1 = 300nH is fixed, the tuning and matching capacitors for the
circuit are recalculated for each value of L2 to ensure desired center frequencies.
Figure 4.19: Coil in the case and in the Sentinelle breast biopsy table.
4.3.3 B1 and B0 Field Calibration
Due to low intrinsic concentrations of of
31
P in–vivo, RF power calibration is chal-
lenging and unlike 1 H it is not automated on most clinical scanners. Therefore a
pulse sequence was written along with reconstruction software in order to calibrate
flip angle precisely to 90o .
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Figure 4.20: Transmit gain calibration results window, the ia rf1 value is used to
calibrate the power needed for a 90o flip angle.
Because it is non trivial to calibrate a slice profile, with a surface transmit coil a 1 ml
vial of H3 P O4 was used as a reference for power calibration. Reconstruction software
was written to run on the scanner console and present the user with a final power
setting value that should be used to produce the desired flip angle (Figure 4.20).
The user then sets the value manually and removes the vial from the coil. From
the presented user data it is also possible to perform a calculation that will provide
a 90o flip angle at a desired depth. This semi automated procedure of transmit
power calibration only takes 2–3 minutes and takes away the uncertainty of manual
calibration that varies from user to user.
Linear and high order shimming (HOS) are used to correct for the inhomogeneities in
the magnetic field that the sample introduces in the B0 field when it is placed inside
the magnet. Automated shimming can only be performed by collecting 1 H data, and
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Figure 4.21: High order shimming performed using double tuned coil.
applying known shim model data, it uses singular value decomposition method to
attempt and correct for the introduced inhomogeneities (de Graaf 2007). Shimming
the sample decreases the line width of the spectra, thus improving the quality of the
collected data. For
31
P spectroscopy, dual tuned coils is a practical option if HOS
is required. It is possible to streamline the scanning without moving the patient or
change the coils.
In house protocol for high order shimming was designed and implemented to use with
dual tuned coil. For this purpose 1 H coil port was connected to the scanner (Figure
4.15b) and the
31
P port connected to
31
P T/R switch. The HOS scan was run a
several times to ensure that the line width values are minimized (Figure 4.21).
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4.4 Results
The dual–tuned coil was first built using air–core high Q inductors (Coilcraft, Cary,
IL). After tuning and matching the coil was used, for imaging and spectroscopy
work, however, during decoupling tests the inductors have failed. They have been
later replaced by in house built inductors made of enamel covered magnet wire.
Bench top measurements of the dual tuned coil showed that capacitor and inductor values were very close to the values acquired through the simulations. Since
double–tuned coil is a two port device S11 measurements were performed on each
port separately. Figure 4.22, shows S11 measurements for
31
P and 1 H channel. The
blocking network worked as expected producing good blocking between the two channels. This can be observed from the fact that only a single resonant frequency was
present per channel. Dual–tuned coil operate normally with both ports plugged into
their respective channels on the scanner and the decoupler.
4.4.1 Coil Performance Imaging, Spectroscopy and Decoupling
Images and spectra were successfully acquired using both
31
P and 1 H channels. Be-
fore each measurement both ports were plugged into the scanner, the localization
and shimming was performed solely using 1 H channel. Then for
31
P channel a semi
automated transmit gain calibration and manual centering of the frequency was performed. For the decoupling purposes 1 H port was then disconnected from the scanner
and reconnected to the decoupler system. Manual calibration of decoupler power was
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Figure 4.22: Tuning and Matching top Tuning to a desired frequecny, 51.73MHz and
127.8 MHz. Matching to 50 Ω using Smith chart. The blocking circuit performs well,
no peaks from the second resonant frequency is observed.
performed before
31
P spectra was acquired. The total calibration procedure lasted
minimum of 10 minutes.
Because this is a transmit receive probe, several interesting challenges with excitation
profiles occur. In order to estimate the quality of the coil and the calibration protocol
B1 map profile was acquired using 1 H channel. Two different flip angles were used
to generate two images from which the B1 map was calculated. It was observed that
the flip angle calibration produced proper values as the highest intensity B1 is at the
center of the coil (Figure 4.23). Because this is a transmit receive coil, the depth of
penetration varies depending on the pulse sequence being used. Gradient and spin
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Figure 4.23: B1 map calculated using two different flip angles.
Figure 4.24: Gradient Echo Profile
echo sequences with same TE and TR times were used to collect images from a saline
phantom. It was observed that the gradient echo sequence produced images with
much greater coverage then the spin echo pulse sequence, (Figures 4.24,4.25).
The decoupling hardware was first tested for functionality using power attenuator
and an oscilloscope. WALTZ–4 sequence with parameters ton = 5ms tof f = 0.5ms,
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Figure 4.25: Spin Echo Profile
the nuclear Overhauser effect (NOE) attenuation varied between 0 and 20 dB was
used to decouple a 10 cm sphere of methylphosphonic acid (CH3 P (O)(OH)2 ). The
center frequency of the decoupler was centered about the hydrogen on the hydroxyl
group which resonates as a doublet at 1.2 ppm. The coupling between phosphorus
and hydrogen on the hydroxyl group, splits the phosphorous resonance into 4 peaks
which are separated by 17.24 Hz (Figure 4.26). After turning on the WALTZ–4 and
adjusting the power the peaks collapse into a singlet increasing the SNR 4 times.
Due to the chemical structure of methylphosphonic acid no NOE enhancement was
observed (Figure 4.26c). No noise injection was observed while running WALTZ–4
decoupling sequence on the proton channel.
4.5 Discussion
A dual tuned coil for breast imaging and spectroscopy was successfully designed and
built to integrate with Sentinelle’s Vanguard breast biopsy system. This coil allows
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Figure 4.26: a) 1 H imaging of polyvinyl alcohol phantom, b) 31 P spectra of
methylphosphonic acid c) 31 P spectra of methylphosphonic acid with WALTZ–4 decoupling on 1 H channel.
multinuclear data acquisition of
31
P while simultaneously performing decoupling ex-
periments on 1 H channel. It is also possible to use this system for further calibration
and quantification of
31
P metabolites.
Design and implementation of dual tuned RF coil largely depends on the application.
For example phosphorus spectroscopy of the breast with decoupling would require
a dual tuned coil in order to minimize SAR. However due to limits imposed by the
system and the difficulties integrating with the existing hardware, a single dual tuned
approach was the simplest to implement. While in general, a departure from single–
tuned to dual–tuned coils, will result in loss of performance for at least one of the
nuclei, regardless of method used. The 50% loss of sensitivity on hydrogen channel
is some what disappointing. This can become an issue from the patient safety point
as SAR limit as measured by the system may be reached faster during WALTZ–
4 decoupling. The RF power delivered to the coil is not being transmitted to the
patient and instead would be dissipated as heat in the RF components capacitors
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and inductors. While there are techniques that can move the coil sensitivities above
90% for both channels their application for breast spectroscopy and integration with
clinical work flow environment still needs to be explored (Dabirzadeh et al. 2008;
Duan et al. 2009).
Multi–nuclear work is slowly becoming popular, particularly as MRI field strengths
increase, making clinical spectroscopy possible. This will open new venues for high–
performance, versatile multi–frequency probes. In addition, as field strengths of
whole–body magnets increase, proton coil designs are becoming increasingly specialized, usually entailing the use of complex multi–channel array coils that cannot
easily be re–engineered for multi–frequency use. Therefore many vendors are exploring options for building array of dual tuned coils with parallel transmit and receive
capabilities.
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Chapter 5
Design and Construction of an
inexpensive Polyvinyl Alcohol Breast Like
Phantom for use with MR Imaging
This chapter covers construction of a breast–like phantom from polyvinyl alcohol.
An inexpensive method of producing PVA cryogels is discussed and the relaxation
parameters for 10 and 15 percent concentration of PVA are discussed.
5.1 Introducion
Various types of phantoms are used for various purposes in MR imaging and spectroscopy. Phantoms are typically designed to mimic tissue relaxation parameters T1
and T2 or the tissue biochemistry for spectroscopic studies. They also need to mimic
the electrical properties of the tissue in order to load the coil properly. Phantoms are
typically built to be used during pulse sequence design and optimization, developing
MR hardware like RF coils and for quality assurance purposes.
MR phantoms are usually designed using water solutions, or various types of hydrogels (e.g. agar, agarose) (Mazzara et al. 1996; Woo et al. 2007; Yoshimura et al.
2003) . While these types of phantoms are easily built and the materials are readily obtainable, they still have some disadvantages. Relaxation parameters T1 and
T2 often require addition of relaxation agents such MnCl2 , Aluminum or Gd-DTPA
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to change relaxation parameters, which can produce sediment or react and degrade
metabolites. Agar gels are susceptible to water loss, and degradation and typically
are very fragile. It is also impossible to build the phantoms to have multiple compartments without introducing a water impermeable barrier between different volumes.
As well, due to the low separation between agar setting and melting temperatures it
is difficult, if not impossible, to introduce layers.
Polyvinyl alcohol (PVA), Figure 5.1, is an easily obtainable chemical which is odourless and non-toxic. It is produced from hydrolysis of polyvinyl acetate (commonly
used a wood glue). It is known to have an interesting property that by freezing and
thawing an aqueous PVA solution promotes cross linking between polymer chains.
These type of gels are often called cryogels and have many uses in biotechnology
(Lozinsky et al. 2003; Tanabe and Nambu 1988). For example PVA has been used in
MRI, Ultrasound (Mano et al. 1986; Chu and Rutt 1997; Surry et al. 2004), CT and
X-ray phantoms and also in wound coverings for burn victims. The PVA cryogels can
be made mechanically equivalent to tissue (i.e. similar elastic moduli) for ultrasound
imaging. T1 and T2 for MR imaging can be tuned by using concentration and by
controlling the rate and number of freeze thaw cycles, by increasing the number of
cross links via hydrogen bonding in the polymer chains. Lately the PVA cryogels
have been extensively studied due to their unique ability to be used in temperature
and radiation dosimetry (Zhang and Yu 2004; Lukas et al. 2001).
Typically cryogels are hard to produce and require a complicated laboratory. They
often require long processing to produce the aqueous solution due to low solubility,
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Figure 5.1: Chemical structure of polyvinyl alcohol polymer.
and then typically freeze thaw cycles take 12 to 24 hours. However an inexpensive
method was developed to produce a breast–like phantom using 15% PVA cryogel,
which can be used for MR imaging and spectroscopy and for biopsy training. A
vacuum forming system was designed and built to produce a plastic mold to produce
phantoms of various types. The use of inexpensive equipment and quick production
could be used many times with PVA to make stable phantoms for a variety of imaging
experiments.
5.2 Methods and Materials
A casting compound (Activia, Marshall, TX) was used to duplicate a mold of a breast
from an expensive triple modality breast biopsy phantom (Supertech Elkhart, IN).
The cast was then used on a vacuum forming jig, Figure 5.2, to produce a polystyrene
negative mold of the breast phantom.
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Figure 5.2: Apparatus used for vacuum forming
Polystyrene has low glass transition temperature of 95o C and high tensile strength to
survive the freeze thaw cycles. A 1.5 mm sheet of polystyrene uniformly heated in an
oven and stretched over a cast on the vacuum table was used to produce the desired
shape after the styrene had cooled down.
Polyvinyl alcohol has low water solubility, and typically reflux condensers are used
to boil the PVA solution for two or more hours. Instead a pressure cooker was used
as an “autoclave” to simplify the process. A 1L Erlenmeyer flask with the desired
concentration of PVA solution was heated in a water bath inside for 30 minutes under
1.2 atmospheres and 125 o C. The flask was weighed before and after to ensure the
amount of water remained the same. Once the solution was prepared it needed to be
stored in a high humidity environment or used immediately in order to minimize the
exposure to air and prevent film formation on the surface.
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Aqueous solutions of PVA of 10% and 15% by weight were prepared, and poured
into 50 cc plastic vials then placed inside a -20o C freezer for 24 hours alongside a
1 Liter water container. Thawing was performed gradually by keeping the vials in
an insulated container alongside frozen water to increase the thermal mass and slow
down the thawing process. Vials were first kept in a fridge for 18 hours and then
brought to room temperature. Freeze thaw cycles were performed once or twice.
Another 15% batch was prepared and poured into the polystyrene breast mold. A 1
cc piece of previously frozen 15% cryogel was embedded into the breast mold before
addition of the PVA solution. This was done to mimic a tumour.
An inversion recovery pulse sequence was used to calculate T1 value of different samples. In order to speed up the measurements and minimize temperature fluctuations
during the scan, the FSE-IR pulse sequence was used.
The pulse sequence parameters were TE=16.3 ms, TR=10,000 ms and ETL = 12,
inversion time TI varied from 50 to 2500 ms (25 different TI measurement points in
total) with a 26cm FOV, slice thickness of 2 cm and acquisition matrix of 256x128.
Signal received from the FSE-IR pulse sequence as a function of TI can be written
mathematically as:
S(T I) = A|1 − 2βe−T I/T1 |
(5.1)
where A and β are constants. Parameter A ∝ Mo while β denotes the deviation from
a perfect 180o pulse. Mean and standard deviation data of the ROI was measured
using ImageJ software. Data was fitted using a three parameter least-squares fit
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based on the Levenberg-Marquardt (LV) method, to Eq. 5.1. Origin Lab (OriginLab,
Northampton, MA) software was used to fit the data using LV algorithm.
T2 measurements were done using a spin echo sequence producing 4 echoes from a
single excitation. The T2 values were fitted to the spin echo signal equation:
S(T E) = Ae
−T E
T2
.
(5.2)
5.3 Results and Discussion
The relaxation parameters for the vial samples are shown in Table 5.1. The values
compare well with those published in literature (Surry et al. 2004). The T1 and T2
images of 15% PVA breast phantom are shown in Figure 5.3. The simulated tumour
had clearly distinguishable contrast. The T2 weighted image shows the contrast
inhomogeneity of the phantom. This is probably due to the non-linear spatial freeze
thaw temperatures, which introduces realistic fibrous breast contrast. The mock
tumour is also easily distinguishable in a T2 weighted image.
The PVA phantom is cheap to manufacture and could be used for breast biopsy
training, as the mechanical properties of the cryogel are very similar to human tissue.
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Table 5.1: T1 and T2 relaxation parameters measured from 10% and 15% PVA samples. Values are mean plus or minus standard deviation, as taken from the ROI placed
over the entire 50 cc vial.
10%
15%
1 Freeze Thaw Cycle
T1
T2
(ms)
(ms)
1571±23 170±33
1450±20 108± 26
10%
15%
2 Freeze Thaw Cycles
1553±17 142±25
1442±12 80±16
PVA concentration
Figure 5.3: a) T1 weighted image acquired using 2D FSE-IR, ETL = 6, TR 2200 ms
5 mm slice thickness TE 9.26 ms TI 720 ms, 256x256 b) T2 weighted image acquired
using FLAIR , TR 9s TE 140 ms TI 2.25s 256x256
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Chapter 6
Dynamic Measurements of 31P
Metabolites in a Posterior Compartment
of the Leg During Plantar Flexion
Exercise Using Surface Coil or Echo
Planar Spectroscopic Imaging for
Localization
This chapter deals with the use of
31
P MRS for use in dynamic studies that re-
quire rapid measurements and spatial localization. In particular a human calf muscle
was studied during exercise inside an MRI scanner to dynamically measure relative
changes in metabolite concentrations. The EPSI sequence was evaluated for use in
dynamic studies.
6.1 Introduction
Phosphorus magnetic resonance spectroscopy is often used to access information on
the functionality and regulation of the central metabolic pathways of phosphate, i.e.
studying the cellular energy capacity. This has been extensively demonstrated to
provide a non-invasive in–vivo assessment of muscle metabolism (Carter et al. 2002;
Barker and Armstrong 2010). Another key feature of
31
P MRS lies in its ability
to provide intracellular pH and ionic magnesium (Mg2+ ) concentration. The pH
sensitivity has been extensively used to study muscle during exercise (Morikawa et al.
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1994), brain injury (Rango et al. 1990) and measure intracellular pHi in tumours
(Gillies et al. 1994).
31
P MRS is the only technique that can provide non–invasive measurements of pHi . To
measure extracellular pHe other techniques such as injection of exogenous compounds
like 6-Fluoropyridoxol (19 F MRS) (Mehta et al. 1994), and a 2-imidazole-1-yl-3ethoxycarbonyl propionic acid (IEPA) in 1 H MRS (van Sluis et al. 1999) have been
used in animal models.
The relative simplicity of a
31
P spectra, is due to the fact that only free or rapidly
exchanging metabolites are MR visible, i.e. highly mobile compounds in solution
have long transverse relaxation T2 times, (Table 6.1) and appear as sharp peaks.
Compounds that are bound to macromolecules or are located in a viscous medium
such as the mitochondrial matrix, have short T2 and are essentially MR invisible, i.e.
appear as an underlying broad variation of the spectral baseline.
The plantar flexion exercise is very simple to perform inside an MRI scanner, requiring
movement of the foot towards the shin. It has been shown to be an effective way to
exercise the gastrocnemius muscle and can be done to vary the level of work. The
bioenergetics pathway involved in energy consumption during muscle contraction is
shown in Figure 6.1. During muscle contraction myosine ATPase, utilizes the ATP
as energy needed for movement to produce ADP and inorganic phosphate. However
the amount of ATP in muscle is only enough to sustain the contractions for a very
short period. The back–up energy reserve comes from the phosphocreatine molecule
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in the cytoplasm, and creatine kinase catalyzes the reaction to transfer a phosphoryl
group from PCr to ADP to produce ATP. So the energy pathway that is typically
observed in the NMR time scale can be summarized as; once the energy requirement
increases beyond the capacity that can be provided by ATP, the PCr concentration
decreases and the amount of inorganic phosphate present in the cytoplasm increases.
Concurrent to this process is a decrease in pHi .
Under aerobic conditions mitochondrial respiration supplies most of the required energy as ATP, however once the enzyme system is saturated or oxygen is used up only
PCr and glycogenolysis can supply ATP. The later produces large amount of lactic
acid also decreasing pHi .
Dynamic studies demand good temporal resolution while maintaining high enough
SNR to achieve useful results. Temporal resolution of a standard Chemical Shift
Imaging sequence with a 5x5 CSI grid and TR of 4 seconds, NEX = 1 will only
achieve the temporal resolution of 100 seconds. The EPSI with a 5x10 grid and TR
of 4 seconds, NEX = 1 can achieve temporal resolution of 20 seconds. The speed
up is proportional to the number of phase encoding steps in the x-direction that are
encoded using echo planar readout. In contrast, the surface coil itself can be used for
localization with temporal resolution being the same as the TR. Size of a receive coil
is proportional to its B1 sensitivity profile. The study shows the ability of the EPSI
sequence to be useful in dynamic studies while providing spatial information.
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Figure 6.1: Schematic of the main biochemical reactions involved in energy consumption and production during muscle exertion. The rapid depletion of ATP by myosine
ATPase in muscle requires a reserve supply provided by phosphocreatine which is used
for a rapid ATP production directly in the cytosol of the cell via MM creatine kinase.
Table 6.1: Relaxation parameters of 31 P metabolites in a human calf muscle at 25.68
MHz/1.5T and their related chemical shifts (de Graaf 2007, de Certaines 1992).
ATP α
ATP β
ATP γ
PCr
Pi
T1
T2
Chemical Shift
(sec) (ms)
ppm
3.6
22
-7.52
4.3
8.1
-16.26
-2.48
4.75 16.1
5.517 424.1
0
4.017 204.7 5.02 (pH of 7.2)
6.2 Methods and Materials
An in–house built MR compatible ergometer was used for plantar flexion exercises.
Three subjects were scanned using the FID CSI or EPSI sequence while performing
plantar flexion using 8.3 kg resistance for a period of time followed by a period of
rest. The exercise paradigms varied with the experimental protocol (details below).
Subjects were placed supine feet first inside the 3T MRI scanner. In order to minimize
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motion during the exercise subjects held on to the provided rope handles to counter
the force exerted during the plantar flexion exercise.
The static
31
P MRS data was acquired using a 5 inch diameter transmit–receive
surface coil. The coil was manually tuned and matched to the subject inside the
MRI scanner using an MR compatible Network Analyzer, VIA Echo MRI (AEA
Technology, Inc. Carlsbad, CA, USA). An EPSI sequence was acquired and compared
to a CSI acquisition. A CSI acquisition was acquired using a 5x5 grid with a TR = 4
sec, 16 averages and a FOV = 13 cm, a spectral BW of 5000 Hz and 2048 points, with
a soft RF pulse was used for excitation with a slice thickness of 4 cm. The EPSI data
set was acquired using a 5x10 grid with a TR = 4 sec, 16 averages and a FOV = 13
cm with a 4 cm slice thickness. The EPSI gradient with a 2.6 cm spatial resolution,
spectral resolution of 3125 Hz (1562.5 Hz interleaved twice) and 16,384 points were
acquired per interleave. The RF pulse was calibrated using a 4 cc 1x1x4 cm vial of
1 M solution of phosphoric acid and a custom transmit calibration pulse sequence
(Section 4.3.3).
Data processing of static spectroscopic images was performed using SAGE (GE Healthcarer
Milwaukee, WI), while EPSI data was first processed in MATLAB (The Mathworks,
Natick, MA, USA) and then exported into SAGE.
The same coil was used to acquire dynamic
31
P MRS data. A FID CSI sequence,
with no spatial encoding, was used to acquire 16
31
P spectroscopic data points at a
temporal resolution of 16 seconds ( acquisition parameters were: 2048 points, spectral
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bandwidth 5000 Hz, TR = 8 sec and two averages per data point with π, −π phase
cycling). Plantar flexion exercise using 8.3 kg as resistance was performed for a period
of 1 minute followed by 3 minutes of rest. The scanner clock was used to guide the
subject during the exercise.
An EPSI sequence with spatial resolution of 2.6x2.6x4 cm was used to acquire 16
sequential 5x10x820, (5x10 spatial and 820 spectral points), spectroscopic images
every 20 seconds. EPSI gradients with spectral bandwidth of 1562.5 Hz, interleaved
once, was used to acquire 16,384 data points with TR = 4 sec. For EPSI acquisition
the exercise was performed using 8.3 kg resistance for 1 min 40 sec followed by 4 min
40 sec of rest. RF pulse excitation was calibrated using a transmit gain calibration
sequence to produce a 90o flip angle at the coil center, where a 1 M vial of phosphoric
acid was located. The vial was removed prior to the actual data acquisition.
In all cases data was collected after 4 [discarded] excitations in order to achieve steady
state prior to data acquisitions. The dynamic data was processed using a combination
of Matlab and jMRUI software (http://www.mrui.uab.es/mrui). Peak fitting was
performed using using a time-domain AMARES fitting algorithm (Vanhamme et al.
1997) to fit Lorentzian peaks using prior knowledge of peak positions. The pHi
calculations were performed using a modified Henderson–Hasselbach relation 1.1 with
pKa = 6.77.
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Figure 6.2: CSI and EPSI Data overlaid over the 1 H image of a leg a) 5x5 spectroscopic
image of a Leg 13 cm FOV 4cm slice thickness acquired with a TR = 4 seconds, 1 NEX,
spectral BW = 5000 Hz, 2048 points. Total scan time 26 minutes 40 seconds. b) 31 P
EPSI 5x10 CSI 2.6 cm resolution, spectral BW = 1562.5 Hz interleaved twice, TR = 4
seconds, slice thickness 4 cm. Total scan time 5 minutes 20 seconds.
6.3 Results and Discussion
The EPSI sequence was already verified in phantoms, however the short T2 values one
would find in vivo are hard to duplicate ex vivo. The acquired spectroscopic image
data overlaid over the 1 H image of the leg is shown in Figure 6.2. Overall the
31
P
EPSI data shows good correlation with 31 P CSI data. Both FIDs were apodized to 2.4
Hz and the real part of the spectra were manually phased. The SNR was measured
from the largest PCr peak in the spectroscopic image. The CSI dataset had SNR of
51.6 and EPSI SNR of 16.8. The PCr FWHM peak widths of both CSI and EPSI
were similar 9.4±0.8 Hz.
The dynamic data requires high SNR in order to observe peak amplitude and frequency variation. Thus a large voxel is required to produce high SNR when high
temporal resolution is needed. Using a surface coil for localization a volume of ap149
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Figure 6.3: 31 P spectra acquired during plantar flexion exercise using 8.3 kg of resistance. In total 15 spectra were acquired at temporal resolution of 16 seconds. Amplitude
variation of PCr and Pi can be clearly observed.
proximately 300 cc was acquired using the FID CSI sequence, without any spatial
encoding. The real part of the spectra was apodized to 1.8 Hz and manually phased
and is shown in Figure 6.3. The depletion of PCr and increase in inorganic phosphate
can be clearly seen from the acquired data. The ATP peaks remained unchanged
throughout the entire experiment, as expected. The measured SNR of the highest
PCr peak was 64.8, and the SNR = 29.2 of Pi was found to be the highest at the
end of the exercise, the SNR = 7.8 of ATP was unchanged during the entire dynamic
study. The PCr FWHM peak width of the dynamic data was comparable to the CSI
and EPSI of 8.12 ± 0.71 Hz.
From figure 6.5 the frequency shift of Pi can be clearly observed, this was used to
calculate the pHi (Figure 6.6).
The dynamic study using EPSI provided similar results but the voxel size of 27 cc
was the detrimental factor to the SNR and thus the quality of the spectra. Even after
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Figure 6.4: Dynamic Changes of phosphocreatine and inorganic phosphate peaks
during 60 seconds of plantar flexing followed by 3 minutes of rest.
apodization of 6.4 Hz the SNR of PCr was only 5.7. Furthermore, the ATP peaks were
not resolved and the Pi peak was only observable during the exercise (Figure 6.7).
The amplitude of Pi peak was maximal at 100 seconds when the exercise stopped
(Figure 6.8). The pHi was calculated using the AMARES algorithm, however due to
low SNR peak fitting was suboptimal (Figure 6.9).
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Figure 6.5: 31 P spectra acquired during plantar flexion exercise using 8.3 kg of resistance. Frequency variation of Pi can be clearly observed.
Figure 6.6: pHi Calculation of 31 P MRS in an exercising leg without spatial localization, frequency shifts of Pi with respect to PCr show the pH decrease to 6.52 at the
end of the exercise. Beginning of recovery was subsequently observed.
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Figure 6.7: Dynamic Study of planar flexion using EPSI 20 second temporal resolution
2.6 cm 5x10 ESPI CSI 1562.5 Hz TR 5 seconds without interleaving.
Figure 6.8: Area of Pi and PCr peaks as determined using AMARES algorithm.
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Figure 6.9: pH Calculation from frequency shifts of Pi with respect to PCr.
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Chapter 7
Conclusions and Future Directions
7.1 Future Applications of the Constructed Double
Tuned 1H –
flyback
31
31
P Breast RF Coil and the Designed
P EPSI Sequence
Proton magnetic resonance spectroscopy or hydrogen spectroscopy (1 H-MRS) is the
most clinically applied spectroscopy technique today. The success of proton spectroscopy in many parts of the body, however, has been limited due to technical difficulties and limited information content e.g. liver and muscle spectra are relatively
simple in comparison to brain spectra due to large dominating lipid resonances. Other
NMR visible nuclei such as
31
P,
23
Na,
13
C are either used rarely or almost never for
clinical spectroscopy. There are many reasons for this, the first being that most commercial clinical MR systems are designed to work only with 1 H nuclei. Hardware
for “multi” nuclear work is often available as an extension to most existing clinical
systems however a rough estimate puts less then 5% of all MRI systems in Canada
capable of this functionality.
The success of proton NMR in clinical applications has only been matched by phosphorus
31
P NMR. A high natural abundance and sensitivity of around 7% that of
proton, allows acquisition of a good quality spectrum within clinically relevant time
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scales i.e. minutes. A large chemical shift dispersion of 30 ppm, and considerably
fewer metabolites than 1 H-NMR, produces “good” peak resolution. Most importantly
31
P spectroscopy is capable of detecting the major compounds responsible for
energy metabolism. Furthermore the chemical shift of some phosphorus containing
compounds is sensitive to the concentrations of magnesium and pH.
The developed rapid chemical shift imaging pulse sequences,
31
P Echo Planar Spec-
troscopic Imaging, has been shown to be robust in acquiring 2D spectral images
using a 3T Signa Excite MR Scaner (GE Healthcare, Milwaukee IL USA) located at
St Joseph’s Healthcare in Hamilton Ontario, which is equipped with multi nuclear
capabilities.
Combining the
31
P EPSI sequence with the in–house developed auto prescan func-
tionality for multi nuclear spectroscopy enables the calibration to be done semi automatically, drastically reducing setup and prescan time while removing the operator
dependent inconsistencies. Furthermore, the ability to use dual tuned coil for higher
order shimming and decoupling minimizes the setup time, moving patients in order
to swap coils or alter the setup.
The project was oriented towards technical development of software and hardware
with the end goal to utilize this approach in a clinical population. The future work will
involve using the EPSI technique for studying breast cancer and muscle metabolism.
Using an MRI compatible breast imaging and biopsy system (Sentinelle Medical,
Toronto Ontario Canada), available at St. Joseph’s Healthcare to couple rapid spec156
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Dept. of Medical Physics
troscopy with breast biopsy is ideal for studying breast cancer. Current MRI literature
of
31
P spectroscopy presents clear evidence of increased concentrations of phospho-
monoesters (PMEs), and phosphodiesters (PDEs) as well as inorganic phosphate in
breast tumor tissue (Leach et al. 1998; Arias-Mendoza et al. 2004).
Acquisition of 31 P spectra is complex due to lack of developed hardware and software
for clinical scanners, this is further hampered with complex protocol development
required to complete clinical scans. However this thesis has successfully showed that
it is possible to acquire
31
P MRS in as little as 20 seconds with voxel size larger then
40 cc, and use this information for dynamic studies. Furthermore, the advantages
of
31
P EPSI sequence were observed in acquisition with high signal especially when
large number of voxel counts are required. The acquisition speed increase of up to 20
times could be achieved provided that the SNR can be sacrificed.
The fundamental limit of SNR remains the magnetic field strength and the optimization of the coils. Current state of the art MR systems all utilize volume transmit
and surface coil arrays to receive the signal. The smaller surface arrays drastically
increased the SNR for 1 H imaging purposes, by coupling less noise from the sample.
The new surface coil array technology utilises low noise amplifiers with noise figures
as low as 0.35 that can boost SNR by more than 40%. Utilizing the phased array
coil technology for
31
P MRS is challenging but not impossible, the difficulty lies in
the fact that the bore of the MR machine has limited size and must contain body
coil tuned to transmit only at 1 H frequencies, shim coils, gradient coils, insulation
and cooling. This drastically hampers the development of new devices optimized for
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other nuclei. One solution would be to have a large dual tuned transmit coil and
design surface receive arrays tuned for either 31 P or 1 H frequencies. Another solution
may come in the development of compact transmit/receive arrays that could provide
more B1 field homogeneity.
The dual tuned surface coil in this thesis has advantages that it remains compact
and mobile and it is easy to swap in and out with the existing 1 H hardware, while
still allowing for complex experiments require transmit at multiple frequencies. This
technology could easily be extended in designing small compact arrays for improving
spatial coverage and increasing SNR.
Using technology developed in this thesis a future study will be designed to collect
phosphorus spectra from the various clinical populations. First the protocols will be
designed and verified on phantoms. Subsequently clinical populations with breast
cancer, liver diseases and metabolic muscular diseases could be scanned using these
developed techniques. The studies could be easily designed such that they will be
performed along side of clinically diagnostic procedures during which the biopsy will
be performed by trained medical staff. The results from spectroscopy and biopsy can
be used to develop new non–invasive protocols with high sensitivity and specificity.
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7.2 Future Directions and other Applications of ESPI
sequence and Multi Tuned MR Coils.
Using RF coils to localize the signal is advantageous in dynamic studies, while EPSI
is better suited for speeding up high resolution CSI acquisition as shown in Chapter
3. Current CSI acceleration techniques have their challenges. It has been shown that
EPSI with a flyback trajectory is a strong contender enabling researchers to speed
up the acquisition process (Zierhut et al. 2009). However experimental design, SNR
requirements and artifact severity have to be considered.
The biggest challenge currently present for use of the
31
P EPSI sequence is the poor
SNR due to poor sensitivities and low metabolic concentrations. One key approach to
improving multinuclear SNR is to use phased array coils. In particular the design of
double tuned multinuclear coils is of current investigational interest (Lee et al. 2000;
Avdievich and Hetherington 2009; Avdievich and Hetherington 2007). The phased
array multinuclear coils require uniform B1 field for excitation, which presents an
extra layer of complication. Another point to consider is that as individual elements
of a phased array coil become minimized the coil thermal noise begins to dominate
over patient noise. At this point any further SNR improvement would require use
of cryogens to reduce the coil Johnson noise thereby increasing SNR (Ginefri et al.
2007; Darrasse and Ginefri 2003).
To allow for the CSI spatial encoding gradients 0.2 ms is required after the RF excitation. This is potentially a long time, especially when trying to resolve species with
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short T2 s, such as31 P and 23 Na. Alternative techniques could be used to acquire data
immediately following excitation thereby maximizing SNR of these species (Robson
et al. 2005).
31
P MRS requires a larger receiver bandwidth then 1 H MRS, demanding higher per-
formance from the gradients (i.e. faster slew rates and higher amplitudes). While
interleaving gradients is a robust option to eliminate the need for better gradients,
other techniques of sub sampling MRS data during dynamic acquisitions and using
prior knowledge for data reconstruction (e.g. a coil sensitivity profile) are also possible
to speed up the acquisition (Otazo et al. 2007; Lin et al. 2007).
For clinical work,
31
P EPSI is well suited as it requires minimal work flow modifi-
cations and can be utilized with commercially available hardware. For breast spectroscopic imaging (i.e. non-dynamic needs) the current setup is ready for a clinical
trial. To boost SNR over the slice of acquisition more averaging will be required while
EPSI will still keep the acquisition well within a clinically feasible time frame (30-40
minutes). The biggest future challenges will be in dynamic studies, where speed is
essential.
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Appendix A
function [ epsigrad , wave_form , my_hdr ] = epsigrad ( BW , RES , Sr )
% Generate EPSI g r a d i e n t s by g i v i n g p a r a m e t e r s
%
% I n p u t : Bandwidth R e s o l u t i o n , and S l e w r a t e
% BW i n Hz
% RES i n cm
% Sr i n Gaus/cm/ s
%
% Output :
% epsigrad . half
H a l f t r a p e z o i d s h o r t waveform
%
. full
F u l l waveform up t o max p u l s e w i d t h up t o 16384 p o i n t s
%
. c o m p l e t e two waveform s t i t c h e d t o g e t h e r
% wave form
Exact waveform
% my hdr
Header f o r t h e c r e a t e d g r a d i e n t s
%
% EPIC NOTE: MAKES SURE THAT THE FILES ARE EVEN XLATEBIN
% DOES NOT PRODUCE A PROPER HEADER/CHECKSUM OTHERWISE ! ! !
%
% Example :
% [ e p s i , wave , hdr ]= e p s i g r a d ( 2 0 0 0 , 2 , 1 5 0 0 0 ) ;
%
% Test t h e waves u s i n g e p s i s a m p l i n g
% Rewritten
% Jan 15 t h 2009 , Matlab 7 . 6 R. 2 0 0 8 a
%
% Mar 5 t h 2 0 0 9 :
%
A f t e r g r a d t e s t s found o u t t h a t a r e a i s s t i l l o n l y h a l f
%
o f what i t s u p p o s e d t o be . M u l t i p l i e d a l l a r e a c a l c s by 2 .
% S e r g e i Obruchkov
%% Args Test
i f nargin == 0
disp ( ' See ” h e l p e p s i g r a d ” f o r arguments ' )
end
i f nargin < 1
BW = input ( ' S p e c t r a l BandWidth [ 2 0 0 0 ] ( Hz ) : ' ) ;
i f isempty ( BW )
BW = 2 0 0 0 ;
end
end
i f nargin < 2
RES = input ( ' S p a t i a l R e s o l u t i o n [ 2 . 0 ] (cm) : ' ) ;
i f isempty ( RES )
RES = 2 . 0 ;
end
end
i f nargin < 3
Sr = input ( 'Max SlewRate o f g r a d i e n t s [ 1 5 0 0 0 ] ( Gauss /cm/ s ) : ' ) ;
i f isempty ( Sr )
Sr = 1 5 0 0 0 . 0 ;
end
end
%% MAIN
%G l o b a l v a r i a b l e s
global gamma;
%gamma = 4 2 . 5 7 5 e3 ;
gamma = 1 7 . 2 3 5 e3 ;
area = calcArea ( BW , RES , Sr ) ;
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[ hrewind , trap_rewind , my_hdr ] = calcParams ( BW , RES , Sr , area ) ;
my_hdr . Area_req = 1 0 / (gamma∗ RES ) ;
% Check t h a t a r e a s A r e q r e q matches Area k
wave_form = cat ( 1 , hrewind , trap_rewind ) ;
% I n t e r p o l a t i n g waveforms t o 4 us r e s o l u t i o n
hrewind_d = wavetoGradUP ( hrewind ) ;
trap_rewind_d = wavetoGradUP ( trap_rewind ) ;
half = convtoEvenINT ( hrewind_d ) ;
f u l l = convtoEvenINT ( trap_rewind_d ) ;
% c h e c k and c o r r e c t t h a t a r e a s match
i f sum( f u l l ) ˜= 0
[ C , I ] = max( f u l l ) ;
f u l l ( I −1) = f u l l ( I −1) − sum( f u l l ) ;
end
area_diff = sum( abs ( f u l l ) ) − sum( abs ( half ) ) ∗ 4 ;
i f area_diff ˜= 0
[ C , I ] = min( half ) ;
half ( I ) = half ( I ) − area_diff /4 + mod ( area_diff / 4 , 2 ) ;
end
% check t h a t
f u l l or h a l f do n o t have two z e r o e s a t t h e end
Lful = length ( f u l l ) ;
Lhalf = length ( half ) ;
i f f u l l ( Lful −1) == 0
f u l l ( Lful ) = [ ] ;
end
i f half ( Lhalf −1) == 0
half ( Lhalf ) = [ ] ;
end
Lful = length ( f u l l ) ; %Get new l e n g t h
% F i n a l i z e t h e waveforms
% h a l f i s a s h o r t waveform and d o e s n o t r e p e a t
% f u l l w i l l r e p e a t s u g e s t e d NumberofLobes t o max o u t t h e time p u l s e w i d t h o f 65536 ←us
% or 16384 p o i n t s w i t h 4 us time r e s
epsigrad . half=cat ( 2 , half , 1 ) ;
epsigrad . f u l l = cat ( 2 , f u l l , 1 ) ;
% F i n a l i z i n g header
my_hdr . half . Resolution = length ( epsigrad . half ) −1;
my_hdr . half . PulseWidth = ( length ( epsigrad . half ) −1) ∗ 4 ;
my_hdr . f u l l . Resolution = length ( epsigrad . f u l l ) −1;
my_hdr . f u l l . PulseWidth = ( length ( epsigrad . f u l l ) −1) ∗ 4 ;
my_hdr . SugestedminTR= ( 1 0 2 4 ∗ ( length ( epsigrad . f u l l ) −1)∗4+( length ( epsigrad . half ) −1)←∗ 4 ) / 1 0 0 0 ; % minTR t o u s e i n c v e v a l i n ms
% For s i m u l a t o r
% 1024 l o b e s
simulate = cat ( 2 , half , f u l l ( 2 : Lful ) ) ;
for n = 1:1024
simulate = cat ( 2 , simulate , f u l l ( 2 : Lful ) ) ;
end
epsigrad . simulate = cat ( 1 , 1 : 4 : 4 ∗ length ( simulate ) , double ( simulate ) ) ;% time i n us and←amplitude
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% MAIN END
% C a l c u l a t e needed p a r a m e t e r s
function Area_r = calcArea ( BW , RES , Sr ) ;
% Global Constants :
% Gamma o v e r 2∗ Pi f o r 31P i n Hz/ Gauss = 1 7 . 2 3 5 e3
% Gamma o v e r 2∗ Pi f o r 1H i n Hz/ Gauss = 4 2 . 5 7 5 e3
%gamma = 4 2 . 5 7 5 e3 ;
%gamma = 1 7 . 2 3 5 e3 ;
% Area
% i .e.
% AREA
global
needed t o f l y t h r o u g h t h e needed r e s o l u t i o n
Area needed t o t r a v e l k−s p a c e
= x r e s c s i ∗10/(2∗gamma∗FOV)
gamma;
area =10/(gamma∗ RES ) ;
eq1
eq2
eq3
eq4
=
=
=
=
[
[
[
[
' ( 1 / ( 2 ∗BW)−a r e a / ( 2 ∗ Sr ∗ r ) − r ) ˆ2 − a r e a / Sr − r ˆ2 ' ] ;
'BW = ' , num2str ( BW ) ] ;
' a r e a = ' , num2str ( area ) ] ;
' Sr = ' , num2str ( Sr ) ] ;
soln = solve ( eq1 , eq2 , eq3 , eq4 ) ;
%P i c k i n g t h e a p r o p r i a t e s o l u t i o n
% I f t h e r e are
i f i s r e a l ( soln . r )
r = eval ( soln . r ( 2 ) ) ;
else
r = eval ( soln . r ( 1 ) ) ;
end
g = Sr ∗ r ;
%Area o f t h e e n t i r e t r a p e z o i d
Area_r = area +r ∗ g ;
function [ hrewind , trap_rewind , my_hdr ] = calcParams ( BW , RES , Sr , area )
% Parameters f o r r e w i n d i n g g r a d
r1=sqrt ( area / Sr ) ;
g1=sqrt ( area ∗ Sr ) ;
my_hdr . rewinder . r1 = r1 ;
my_hdr . rewinder . g1 = g1 ;
% Parameters f o r t h e f i r s t
r2=sqrt ( 0 . 5 ∗ area / Sr ) ;
g2=sqrt ( 0 . 5 ∗ area ∗ Sr ) ;
h a l f rewinding
my_hdr . halfrewinder . r2 = r2 ;
my_hdr . halfrewinder . g2 = g2 ;
% Parameters f o r t h e T r a p e z o i d
a =1.0;
b = (2 .0∗ r1 −1/BW ) ;
c=(area / Sr ) ;
%i f i m a g i n e r y answer
i f ( bˆ2−4∗a ∗ c ) < 0
error ( ' C a l c u l a t i o n E r r o r : I m a g i n a r y time . I n c r e a s e s p e c t r a l BW o r S p a t i a l ←Resolution ' ) ;
else
r_sol1 =(−1/2∗(b−(bˆ2−4∗a ∗ c ) ˆ ( 1 / 2 ) ) / a ) ;
r_sol2 =(−1/2∗( b+(bˆ2−4∗a ∗ c ) ˆ ( 1 / 2 ) ) / a ) ;
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% Take t h e s h o r t e s t s o l u t i o n
i f r_sol1 < r_sol2
g=Sr ∗ r_sol1 ;
r=r_sol1 ;
else
g=Sr ∗ r_sol2 ;
r=r_sol2 ;
end
Tf= area / g − r ;
Tr= 2∗ r+2∗r1 ;
end
my_hdr . Area_k = Tf ∗ g ;
my_hdr . trapezoid . r = r ;
my_hdr . trapezoid . g = g ;
my_hdr . trapezoid . Tf = Tf ;
my_hdr . SpectBW = 1 / ( Tf+Tr ) ;
my_hdr . Quality = Tf / Tr ;
my_hdr . EffSNR = sqrt ( Tf / ( Tf+Tr ) ) ;
my_hdr . S p a ti a l R e s o l u t i o n= [ num2str ( RES ) , ' cm ' ] ;
% S c a l e g r a d s t o 32766
my_hdr . ampscaling = my_hdr . rewinder . g1 ; % f o r s t u p i d
g = g ∗32766/ abs ( g1 ) ;
g2 = g2 ∗32766/ abs ( g1 ) ;
g1 = g1 ∗32766/ abs ( g1 ) ; %Dangerous b u t oh w e l l
r = r ∗1 E6 ;
r1 = r1 ∗1 E6 ;
r2 = r2 ∗1 E6 ;
Tf =Tf ∗1 E6 ;
%C r e a t e waveform w i t h p r o p e r s c a l i n g
hrewind = [ 0 , 0 ; r2 ,− g2 ; 2∗ r2 , 0 ] ;
trap_rewind = [ 0 , 0 ; r , g ; r+Tf , g ; 2∗ r+Tf , 0 ; . . .
2∗ r+r1+Tf ,− g1 ; 2∗ r+2∗r1+Tf , 0 ] ;
function grad = convtoEvenINT ( y )
%% Grad must c o n s i s t o n l y o u t o f even i n t e g e r s
[ my , ny ]= s i z e ( y ) ;
amp( ny ) = 0 ;
dif ( ny ) = 0 ;
f o r i = 1 : 1 : ny
i f isnan ( y ( i ) )
amp( i ) = 0 ;
dif ( i )= 0 ;
else
i f mod ( y ( i ) , 2 ) == 0
amp( i ) = y ( i ) ;
dif ( i ) = 0 ;
else
i f mod ( c e i l ( y ( i ) ) , 2 ) == 0
amp( i ) = c e i l ( y ( i ) ) ;
dif ( i ) = y ( i ) − c e i l ( y ( i ) ) ;
else
amp( i ) = f l o o r ( y ( i ) ) ;
dif ( i ) = y ( i ) − f l o o r ( y ( i ) ) ;
end
i f mod ( y ( i ) , 2 ) == 1
amp( i ) = y ( i ) + 1 ;
dif ( i ) = 1 ;
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end
end
end
end
grad = int16 (amp) ;
function wave_GradUP = wavetoGradUP ( wave_d )
%% Grad Updated time i s 4 us e n s u r i n g t h e wave i s p r o p e r l y s p a c e d
% I n t e r p o l a t i n g v a l u e s t o t h e 1 us g r i d making s u r e time w i l l be d i v i s i b l e
% i n t o 4 us i n t e r v a l
[ m , n ] = s i z e ( wave_d ) ;
time = wave_d ( m , 1 ) ;
i f mod ( c e i l ( time ) , 4 ) ˜= 0
time = c e i l ( time ) + 4 − mod ( c e i l ( time ) , 4 ) ;
else
time=c e i l ( time ) ;
end
wave_GradUP = interp1 ( wave_d ( : , 1 ) , wave_d ( : , 2 ) , 0 : 4 : time ) ;
x = 0 : 4 : time ;
%f i g u r e , p l o t ( wave d ( : , 1 ) , wave d ( : , 2 ) , ' − or ' , x , wave GradUP , ' . ' ) ;
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Dept. of Medical Physics
Appendix B
function dataout = reconepsi ( data , BW , BWs , BWdes , maxm , freqres , numlobes )
% I n p u t p a r a m e t e r s some i n f o you can g e t from t h e h e a d e r
% This f u n c t i o n w i l l s t r i c t l y u s e manual i n p u t
% L a t e r can implement p r o p e r mechanism f o r automated reco n
%
% Example :
%
%
r e c o n e p s i ( data1 , 3 1 2 5 0 , 4 9 8 . 9 5 6 , 3 0 9 3 5 , 6 2 , 5 5 , 2 5 6 ) ;
%
% I n p u t geometry f o r r e s h a p i n g d a t a from d a t a ( e p s i r e s , avgs , p h r e s )
%
i n t o data ( sres , f r e q r e s , phres )
% Need t o know t h e e p s i waveform i n f o
% BW = 31250 max Sampling Bandwidth
% BWs = S p e c t r a l Bandwidth
%
Can be measured from p h a s e b u t i t s h o u l d be
%
what your e p s i g r a d i e n t s were d e s i g n e d f o r
%
% BWdes = D e s i r e d Bandwidth f o r i n t e r p o l a t i o n
%
i d e a l l y BWs∗maxm
%
% maxm = ( i n t ) How many p o i n t s can you f i t i n t o t h e BWs
%
i . e . round (BW/BWs) or f l o o r (BW,BWs)
% f r e q r e s = ( i n t ) number o f p o i n t s t h a t f i t on t o t h e f l a t p a r t
%
of the trapezoid
% numlobes = maxn = number o f l o b e s , maxn∗BW/BWs < e p s i r e s
%
% Output s t r u c t u r e m by n
% m=maxm
. . . .
...
.
%
.
%
.
%
.
% m=3 . . . .
...
.
% m=2 . . . .
...
.
% m=1. . . .
...
.
%
n=1 n=2 n=3 n = maxn
%
%
%
% T e s t i n g d a t a q u a l i t y echo each p l o t and l o o k a t peak a l l i g n m e n t
% p l o t ( real ( squeeze ( dataout (5 ,: ,1:5 ) ) ) )
% f f t d a t a o u t = f f t s h i f t ( f f t n ( dataout ( : , 1 : 4 8 , : ) ) ) ;
%
c l e a r i pi
i f nargin < 7
error ( ' See ” h e l p r e c o n e p s i ” f o r arguments ' ) ;
end
% I f d a t a i s l a r g e r t h e n 2D ( e p s i r e s x p h r e s ) need t o a v e r a g e
[ epsires , avgs , phres ] = s i z e ( data ) ;
i f ( avgs ˜= 1 ) && ( phres ˜= 1 )
data = avgepsi ( data ) ;
end
[ epsires , phres ] = s i z e ( data ) ;
maxn=numlobes ;
dt=1/BW ;
dtdes = 1/ BWdes ;
% V e c t o r s needed f o r i n t e r p o l a t i o n
x = 0 : dt : ( epsires −1)∗ dt ;
xi = 0 : dtdes : ( epsires −1)∗ dt ;
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p ha se co r in td at a=zeros ( maxm , maxn , phres ) ;
intdata = zeros ( maxm , maxn , phres ) ;
f o r phase = 1 : phres
data_vect = data ( : , phase ) ;
intdata_vect = interp1 ( x , data_vect , xi , ' s p l i n e ' ) ;
% Only need maxn∗maxm p o i n t s t o r e s h a p e
% d a t a t h e r e s t can be t r a s h e d
intdata ( : , : , phase )=reshape ( intdata_vect ( 1 : maxn ∗ maxm ) , maxm , maxn ) ;
% Phase c o r r e c t i o n u s i n g s i n c i n p e r o l a t i o n s h i f t
% Note t h a t d t d e s i s used and n o t d t s i n c e u s i n g
% I n t e r p o l a t e d data
f o r m = 1 : maxm
f o r n = 1 : maxn
p ha se co r in td at a ( m , n , phase ) = intdata ( m , n , phase ) . . .
∗exp ( i ∗2∗ pi ∗ m ∗ n ∗ dtdes ∗ BWs / numlobes ) ;
end
end
end
dataout = shiftdim ( phasecorintdata , 2 ) ;
function avgdata = avgepsi ( data )
[ sres , freqres , phres ] = s i z e ( data ) ;
even_data = squeeze (mean( data ( : , 2 : 2 : freqres , : ) , 2 ) ) ;
odd_data = squeeze (mean( data ( : , 1 : 2 : freqres , : ) , 2 ) ) ;
avgdata = even_data − odd_data ;
function [ data , resi ] = ppc ( DataA , DataB ) ;
% P r i n c i p a l Phase C o r r e c t i o n . Performs p h a s e r o t a t i o n o f
% quadrature data without knowledge o f the phase angle .
%
% I /O: [ data , r e s i ] = ppc ( DataA , DataB ) ;
%
% DataA and DataB a r e t h e two q u a d r a t u r e c h a n n e l s g i v e n
% as column v e c t o r s .
%
%
% Henrik T o f t Pedersen , 14−Jan −2002
%
i f s i z e ( DataA , 1 ) < s i z e ( DataA , 2 )
DataA = DataA ' ;
end
i f s i z e ( DataB , 1 ) < s i z e ( DataB , 2 )
DataB = DataB ' ;
end
[U,S,V]
Int
data
resi
=
=
=
=
svd ( [ DataA , DataB ] , 0 ) ;
U∗S ;
Int ( : , 1 ) ∗ sign (sum( Int ( : , 1 ) ) ) ;
Int ( : , 2 ) ∗ sign (sum( Int ( : , 1 ) ) ) ;
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Appendix C
# 31 P and 1 H Double Tuned Coil
# Solving f o r the Total Impedance of the bellow circuit
# From Rectance Find Resonances and Attempt to get
# parametric equations f o r L1 C1 L2 C2 in order to tune into the desired frequecies
#
# W1 = 2∗ Pi ∗ 1 2 7 . 8 MHz
# W2 = 2∗ Pi ∗ 5 1 . 7 3 MHz
restart ;
# Constants
# Double tuned coil Simulation See above circuit f o r Parameters and placement of ←the components
# R2 is ignored as it does not contribute . . .
L1 :=692 E −9;
C1 : = 1 0 . 4 E −12;
R1 := 0 . 5 ;
L2 :=173 E −9;
C2 : = 1 1 . 8 E −12;
# Impedance
# Z1 = L1C1R1 series impedance
# Z2 = L2 | | C2 impedance
Z1 := R1 + I ∗ w ∗ L1 ∗(1 −1/( w ˆ2∗ L1 ∗ C1 ) ) ;
Z2 := I ∗ ( w ∗ L2 /(1−w ˆ2∗ C2 ∗ L2 ) ) ;
X1 := w ∗ L1 ∗(1 −1/( w ˆ2∗ L1 ∗ C1 ) ) ;
X2 :=( w ∗ L2 /(1−w ˆ2∗ C2 ∗ L2 ) ) ;
Z := R1 + I ∗ ( X1 + X2 ) ;
# Solving f o r Resonances where Reactance = 0
# In our case Z1+Z2 − R1 is the Reactance ;
solve ( X1+X2 =0) ;
eqR1 := subs ( w=2∗Pi ∗v , X1 ) :
eqR2 := subs ( w=2∗Pi ∗v , X2 ) :
eqRsum := subs ( w=2∗Pi ∗v , X1+X2 ) :
plot ( [ eqR1 , eqR2 , eqRsum ] , v=1E7 . . 2 E8 , y=−5E2 . . 5 E2 ) ;
# Solving the Equation f o r the Previous Solution
# The resonances are in Hz and are dead on as expected from the RF Simulation ( see ←Schematic Above )
evalf ( solve ( C2 = ( w ˆ2∗ L1 ∗ C1−1+w ˆ2∗ L2 ∗ C1 ) / ( w ˆ2∗ L2 ∗ ( w ˆ2∗ L1 ∗ C1 −1) ) , w ) / ( 2 ∗ Pi ) ) ;
# Explicit Solution and Numerical Calculations
solve ( C2 = ( w ˆ2∗ L1 ∗ C1−1+w ˆ2∗ L2 ∗ C1 ) / ( w ˆ2∗ L2 ∗ ( w ˆ2∗ L1 ∗ C1 −1) ) , w ) ;
# Explicit Solutions f o r the Resonant Frequencies
# W1 = 2∗ Pi ∗ 1 2 7 . 8 E6 MHz
# W2 = 2∗ Pi ∗ 5 1 . 7 3 E6 MHz
W1 :=1/2∗ sqrt ( 2 ) ∗ sqrt ( C2 ∗ L2 ∗ L1 ∗ C1 ∗ ( L2 ∗ C1+C2 ∗ L2+L1 ∗ C1+sqrt ( L2 ˆ2∗ C1 ˆ2+2∗ L2 ˆ2∗ C1 ∗ C2+2∗←L2 ∗ C1 ˆ2∗ L1+C2 ˆ2∗ L2 ˆ2−2∗ C2 ∗ L2 ∗ L1 ∗ C1+L1 ˆ2∗ C1 ˆ 2 ) ) ) / ( C2 ∗ L2 ∗ L1 ∗ C1 ) ;
W2 :=1/2∗ sqrt ( 2 ) ∗ sqrt ( C2 ∗ L2 ∗ L1 ∗ C1 ∗ ( L2 ∗ C1+C2 ∗ L2+L1 ∗ C1−sqrt ( L2 ˆ2∗ C1 ˆ2+2∗ L2 ˆ2∗ C1 ∗ C2+2∗←L2 ∗ C1 ˆ2∗ L1+C2 ˆ2∗ L2 ˆ2−2∗ C2 ∗ L2 ∗ L1 ∗ C1+L1 ˆ2∗ C1 ˆ 2 ) ) ) / ( C2 ∗ L2 ∗ L1 ∗ C1 ) ;
# L1 Variable is fixed since it is determined by the coil geometry
# L2 Is fixed f o r calculation purposes :
L1 := evalf ( 1 / ( ( 2 ∗ Pi ∗129 E6 ) ˆ 2 ∗ 2 . 2 E −12) ) ;
L2 := L1 / 2 ;
M := solve ( { W1=2∗Pi ∗ 1 2 7 . 8 E6 , W2=2∗Pi ∗ 5 1 . 7 3 E6 } , [ C1 , C2 ] ) ;
M[1];
# Steping through L2 in a loop
# NOTE : after 320 nH Solutions are imaginarry might need the other part of the ←equations ?
f o r n from 1 to 13 do
L2 := 100 E−9+20E −9∗(n −1) :
solve ( { W1=2∗Pi ∗ 1 2 7 . 8 E6 , W2=2∗Pi ∗ 5 1 . 7 3 E6 } , [ C1 , C2 ] ) :
end do ;
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Dept. of Medical Physics
#
# Multiple Breaks
# Now we have n breaks with X2 =Sum ( Xn )
X2 :=( w ∗ L2 /(1−w ˆ2∗ C2 ∗ L2 ) ) ;
L2 :=160 E −9;
C2 : = 1 6 . 9 E −12;
n :=2;
Ln := L2 / n ;
solve ( X2 = n ∗ ( w ∗ Ln /(1−w ˆ2∗ Cn ∗ Ln ) ) ) ;
# The rule seems to be L / n C ∗ n where n is the number of breaks
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