Rapid and Continuous Magnetic Resonance

Imaging Using Compressed Sensing

by

Li Feng

A Dissertation Submitted in Partial Fulfillment

of The Requirements for The Degree of

Doctor of Philosophy

Department of Basic Medical Science

Program in Biomedical Imaging

New York University

May, 2015

____________________

Daniel K. Sodickson, MD, PhD

____________________

Ricardo Otazo, PhD

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© Li Feng

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iii

DEDICATION

To my whole family, for their infinite love and support.

iv

ACKNOWLEDGMENTS

I started my MRI journey at NYU School of Medicine in May 2008

when I was a Master student. The past seven years that I spent here gave

me lots of memories and I owe my gratitude to many people who both

helped me in my studies and research and enriched my life in New York

City. The support, generosity and love from those people have made the

past many years a wonderful part of my life and will be deeply embedded in

my memory forever.

The first two people I would like to thank are Dr. Daniel Sodickson

and Dr. Ricardo Otazo. I am very grateful to have advising from both of

them in my PhD study. Dan’s enthusiasm for research and his passion in

both teaching and presentation are very impressive and encouraging to

inspire me throughout my whole PhD study. Dan gave me tremendous

freedom and kept encouraging me to actively think and test my own ideas

for research. He always gave me great feedback to guide my research

towards the direction that could solve practical problems. All of these

enabled me to grow rapidly and helped me develop good research

independence. Moreover, Dan welcomes any questions and critiques from

the students, and of course always gives excellent advices and answers

whenever we need. He is such a wonderful advisor and role model that all

v

the students can rely upon. On the other hand, Ricardo is really a wonderful

mentor who taught me lots of hands-on skills and helped me with all the

details in my research. My PhD study in the past many years would not

have been so productive without the support from Ricardo and it was such

a fabulous experience working with him. Ricardo was always available

whenever I needed his help. I could knock at his office door directly without

an appointment for discussions and he always welcomed me without any

hesitation. I enjoyed all the discussion with him, in which his critical thinking

and depth of knowledge were extremely helpful in my research. Meanwhile,

Ricardo is such an excellent presenter in giving a talk, and I have learnt a

lot from him in how to make good slides and how to tell a good story in a

presentation.

Dr. Daniel Kim deserves lots of my gratitude. Dan was my first

mentor in the MRI world and he is the person who brought me into this field

when I was still a Master student. I am so grateful and feel so lucky that he

accepted me as a summer intern in 2008 to work with him on a cardiac MRI

project. Dan taught me many things that all beginners need to learn,

including how to use the MRI scanners, how to perform a cardiac MRI

exam, how to efficiently debug a program and how to make good figures for

a paper. His patience and step-by-step guidance in my first MRI project

helped me grow up quickly from a “raw” novice and led to my first lead-

vi

author paper published in Magnetic Resonance in Medicine (MRM) and first

oral presentation acceptance at the 2009 Annual Meeting of the

International Society for Magnetic Resonance in Medicine (ISMRM), even

before I started the PhD study. Although he moved to Utah in the summer

of 2011 and it is hard to meet and talk to him now, I will never forget those

days that we worked together in his office and the scanner rooms in the

later evenings.

I would like to also thank Jian Xu from Siemens Healthcare USA for

his infinite support in both research and living. Jian was the first few people

I met when I came to the US in 2007. He was the first person I have known

in the MRI field and was the person who introduced me to NYU School of

Medicine. I enjoyed those countless weekends I spent with him in the MRI

scanner room. He taught me many things about sequences and cardiac

MRI, and gave me tremendous support in cardiac MRI sequences for my

research. I feel very lucky to be good friends with him.

I am grateful for Drs. Leon Axel and Hersh Chandarana, who gave

me clinical advising in my PhD study. They always pointed out practical

clinical needs for me and guided me to find good solutions for them. I also

appreciate their great support for clinical patient studies in both cardiac and

abdominal MRI.

vii

Special thanks go to Drs. Tobias Block and Florian Knoll from NYU,

Robert Grimm from University of Erlangen, and Dr. Jing Liu from UCSF. I

started working with Tobias in 2011 and I was happy that he joined NYU in

the end of 2011 and brought his radial imaging sequence here. My

dissertation could not have been finished without his support on radial

sequences. Meanwhile, I also learnt a lot from Tobias’ critical thinking and

attitude in research. Robert is Tobias’s PhD student and I enjoyed the time

we spent together here at NYU in 2012. His great effort led to successful

application of our compressed sensing approaches for clinical studies.

Florian shared his GPU implementation of 3D non-Cartesian gridding with

me and this was extremely helpful for the 3D radial imaging reconstructions

in my research. I was very lucky that I could be always the first person who

has access to his latest version of 3D non-Cartesian gridding code. Jing is

an expert in both cardiac imaging and radial imaging at UCSF. She gave

me lots of help, suggestions in both my research projects and career

planning.

We started the collaboration with Dr. Matthias Stuber’s research

group at the University of Lausanne in Switzerland in 2014. I am very

grateful to the whole team at Lausanne, including Dr. Matthias Stuber, Dr.

Davide Piccini, Simone Coppo, Gabriele Bonanno and many others, for the

opportunity to have this wonderful collaboration. Davide provided me with

viii

support for the 3D golden-angle spiral phyllotaxis sequence; Simone and

Gabriele were always willing to share so many coronary MRA datasets with

me without any hesitation and gave me great help with coronary artery MR

image post-processing. All of these valuable supports helped me finish the

last part of my dissertation very quickly. I believe our collaboration will

continue to be fruitful and can keep moving forward successfully in the

future.

I want to thank Dr. Ruth Lim for her help in the evaluation of MRI

image quality in many of my projects. Her response was always very fast,

and she gave me very helpful comments from a clinical point of view. I hope

we would have more opportunities in the future to work together on more

projects.

I would like to thank rest of my Committee members, including Dr.

Christopher Collins, Dr. Riccardo Lattanzi and Dr. Reza Nezafat. It is my

great honor to have Dr. Reza Nezafat from Beth Israel Deaconess Medical

Center at Harvard Medical School joins my Committee team as my external

dissertation reviewer.

In additional to the people who gave me direct support in research, I

need to express my gratitude to many lab mates and friends at NYU School

of Medicine, including Ding Xia, Cem Deniz, Leeor Alon, Gene Cho, Gang

Chen, Manushka Vaidya, Gillian Haemer, Alicia Yang, Nicole Wake,

ix

Gregory Lemberskiy Ke Zhang, Hong-Hsi Lee, Harikrishna Rallapalli and

Yuan Wang. These friends enriched my life in the lab and always gave me

great suggestions, help, positive energy and encouragement whenever I

needed them. I would like to specially thank Alicia, Nicole and Gillian, who

gave me great help in proofreading of the dissertation and language editing

in general. I also need to thank many friends in the Chinese Student and

Scholar Association (CSSA) of NYU Medical Center. I will never forget the

times we spent together for karaoke and poker games.

Finally, my greatest gratitude goes to my wife, my parents and all my

other family members. I would not have been able to finish my study without

their infinite support, encouragement and love.

x

ABSTRACT

Magnetic Resonance Imaging (MRI) is a powerful and multifaceted

imaging modality widely used for routine clinical practice. However, the

stringent constraints on MR imaging speed have resulted in comparatively

long examination times, and/or in limited spatial resolution, temporal

resolution and volumetric coverage. Meanwhile, slow image acquisitions

also lead to increased sensitivity to motion, particularly in abdominal and

cardiovascular exams, which require patient- or anatomy-specific scan

planning and reliable motion compensation strategies. The cost of these

complex and cumbersome imaging acquisitions is substantial “dead time”

between successive imaging protocols, as well as the potential discomfort

for patient during prolonged imaging examinations.

Rapid imaging approaches have the potential to shift the balance

from complex and tailored acquisitions to a continuous process that leads to

a simple and efficient imaging paradigm. Compressed sensing is such an

approach that can be applied to accelerate data acquisitions, and that could

further change the imaging paradigm in MRI. In this dissertation, novel

imaging techniques are developed using compressed sensing to enable

rapid and continuous MRI. In particular, golden-angle radial sampling is

combined with compressed sensing and parallel imaging to enable

xi

continuous data acquisitions. Moreover, a new use of sparsity to handle

physiological motion is also proposed for improved rapid and continuous

free-breathing MRI using compressed sensing ideas. The performance of

the proposed techniques is demonstrated for a wide range of clinical

applications in MRI.

The contributions presented in this dissertation enable rapid and

continuously updated image acquisitions, which eliminate “dead time” and

complex anatomy-specific scan planning, and which are also compatible

with flexible reconstructions that can be tailored retrospectively for various

clinical needs.

xii

TABLE OF CONTENTS

DEDICATION ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙iii

ACKNOWLEDGMENT ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙iv

ABSTRACT ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙x

LIST OF FIGURES ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙xix

LIST OF TABLES ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙xxxviii

1. Chapter1 ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙1

1.1. Overview and Motivation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙1

1.2. Thesis Contributions and Outline ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙9

2. Background ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙12

2.1. MR Signal ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙12

2.1.1. NMR Phenomenon ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙12

2.1.2. Signal Excitation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙14

2.1.3. Relaxation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙16

2.2. Signal Localization ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙17

2.2.1. Slice Selection ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙19

2.2.2. Spatial Encoding and k-Space Formalism ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙19

xiii

2.3. MR Image Acquisition ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙21

2.4. Imaging Requirements ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙23

2.4.1. Field of View and Spatial Resolution ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙23

2.4.2. Signal to Noise Ratio ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙26

2.5. MR Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙28

2.5.1. Generalized Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙28

2.5.2. Reconstruction of Non-Cartesian k-Space Data ∙∙∙∙∙∙∙∙∙∙∙∙∙30

2.6. Parallel MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙34

2.6.1. The Need for Speed in MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙34

2.6.2. Spatial Encoding Using Coil Arrays ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙35

2.6.3. Generalized Parallel MRI Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙36

2.6.4. Estimation of Coil Sensitivities ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙38

2.6.5. SNR in Parallel MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙39

2.7. Compressed Sensing MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙41

2.7.1. Introduction to Compressed Sensing ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙41

2.7.2. The Sensing Problem ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙43

2.7.3. Sparsity ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙45

2.7.4. Conditions for Sparse Signal Recovery ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙46

xiv

2.7.5. Sampling and Incoherence ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙47

2.7.6. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙51

2.7.7. Combination of Compressed Sensing and Parallel Imaging

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙54

2.7.8. Low Rank Matrix Completion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙57

2.8. Motion in MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙59

2.8.1. Influence of Motion in MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙59

2.8.2. Free-Breathing MRI Techniques ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙61

3. Accelerated T2 Measurement of the Heart Using k-t

SPARSE-SENSE ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙64

3.1. Prologue ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙64

3.2. Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙65

3.3. Low Rank Property in MR T2 Mapping ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙67

3.4. Imaging Studies ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙70

3.4.1. k-Space Undersampling and Pulse Sequence ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙71

3.4.2. Phantom Validation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙74

3.4.3. T2 Mapping of the Heart ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙74

3.4.4. Improving Sparsity Using Preconditioning RF Pulses ∙∙∙∙∙74

xv

3.5. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙77

3.6. Image Analysis and Statistics ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙77

3.7. Results ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙78

3.8. Discussion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙84

3.9. Conclusion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙86

4. Accelerated Real-Time Cardiac Cine MRI Using k-t

SPARSE-SENSE ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙88

4.1. Prologue ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙88

4.2. Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙89

4.3. Imaging Strategies ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙91

4.3.1. k-Space Undersampling: Incoherence and Self-Calibration

of Coil Sensitivities ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙91

4.3.2. Comparison of Sparsifying Transforms ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙92

4.3.3. Comparison of Acceleration Rates ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙98

4.4. Imaging Studies ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙99

4.4.1. Pulse Sequence ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙99

4.4.2. Cardiac Imaging ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙99

4.5. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙102

xvi

4.6. Image Analysis and Statistics ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙103

4.7. Results ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙106

4.8. Discussion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙110

4.9. Conclusion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙115

5. GRASP: Golden-Angle Radial Sparse Parallel MRI ∙∙∙∙117

5.1. Prologue ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙117

5.2. Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙116

5.3. Golden-Angle Radial Sampling ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙121

5.4. GRASP Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙122

5.5. Reconstruction Implementation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙123

5.6. Imaging Applications ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙126

5.7. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙129

5.8. Image Analysis and Statistics ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙130

5.9. Results ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙133

5.10. Discussion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙139

5.11. Conclusion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙144

6. XD-GRASP: Extra-Dimensional Golden-Angle Radial

Sparse Parallel MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙146

xvii

6.1. Prologue ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙146

6.2. Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙147

6.3. A Simple Example of XD-GRASP ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙152

6.4. Motion Estimation and Data Sorting ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙154

6.5. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙160

6.6. Imaging Applications ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙163

6.7. Results ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙170

6.8. Discussion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙176

6.9. Conclusion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙182

7. Towards Five-Dimensional Cardiac and Respiratory

Motion-Resolved Whole-Heart MRI Using XD-GRASP

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙184

7.1. Prologue ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙184

7.2. Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙186

7.3. 3D Phyllotaxis Golden-Angle Radial Sampling ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙188

7.4. Free-Breathing Whole-Heart MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙190

7.4.1. ECG-Triggered Whole-Heart Coronary MRA ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙190

7.4.2. Free-Breathing Continuous Whole-Heart MRI ∙∙∙∙∙∙∙∙∙∙∙∙∙191

xviii

7.5. Motion Estimation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙192

7.6. Data Sorting ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙194

7.7. Image Reconstruction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙195

7.8. Image Quality Comparison ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙198

7.9. Results ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙199

7.10. Discussion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙203

7.11. Conclusion ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙207

8. Summary and Future Work ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙209

8.1. Chapter Summaries ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙210

8.2. An Outlook for the Future ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙214

9. List of Publications ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙219

9.1. Journal Papers ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙219

9.2. Conference Contributions ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙221

Bibliography ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙230

xix

LIST OF FIGURES

Figure 2.1. (a) Without a strong external magnetic field (B0), the spins are

randomly oriented and the total magnetic moments have a vector sum of

zero. (b) Alignment of spins either parallel or anti-parallel to the direction of

B0 when exposed to an external magnetic field. (c) A net magnetization

vector Mz (also known as M0) is generated as the vector sum of all the spin

angular momenta at the thermal equilibrium state. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙13

Figure 2.2. Excitation of the spins. Following the excitation, the excess z-

population is at least partially converted into a transverse magnetization

component (Mxy), and the ensemble of spins retain their relative alignment,

or phase coherence. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙16

Figure 2.3. Comparison of Cartesian sampling and radial sampling

schemes. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙22

Figure 2.4. In Cartesian, the sampling intervals (∆kx and ∆ky) must be

smaller than the reciprocal the object size in the corresponding spatial

dimensions in order to avoid aliasing. In radial sampling, the maximum

interval between two adjacent radial lines (∆d) has to be small than or equal

to ∆k in order to reconstruct an image without aliasing artifacts .

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙24

xx

Figure 2.5. An example of multicoil brain images with corresponding coil

sensitivity maps with 8 coil elements. Each individual coil element has a

different spatially-varying sensitivity pattern. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙35

Figure 2.6. Sparse representation of a brain image in wavelet transform

domain. By keeping only the largest 10% coefficients and discarding the

rest, the image can still be recovered without loss of important information

but with 10-folder smaller size. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙44

Figure 2.7. A cardiac cine image series has temporal correlation because

dynamic region is limited in only a small region, while the background is

static. An FFT can be employed along the temporal dimension to sparsify

the dataset. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙46

Figure 2.8. Sampling matrix and the corresponding Gram matrix HA A . Ψ

is set as the identity matrix and Φ is the fully sampling Fourier matrix (a)

and partial Fourier matrices with regular (b) and random (c) undersampling

schemes. The off-diagonal entries in (c) are very small, suggesting that

random undersampling is good for compressed sensing∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙49

Figure 2.9. An example of one dimensional variable density undersampling

pattern and the corresponding incoherence, represented by the point

spread function (PSF) of the undersampling pattern. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙51

xxi

Figure. 2.10. Combination of compressed sensing and parallel imaging

enables reduced incoherent artifact level when compared with a single coil

model. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙54

Figure 3.1. Low rank property of T2 mapping. (a): An example of cardiac

T2 mapping image series, in which images at different echo times have

similar anatomical structures but with different T2-weighted contrast. (b):

The Casorati matrix generated from the image series. The Casorati matrix

can be represented by a few dominant singular values and the

corresponding singular vectors. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙68

Figure 3.2. Schematics details of estimating a PCA basis. By concatenating

each time signal vector along column direction, a matrix V is constructed. A

basis set for PCA is then estimated by conducting eigen-decomposition of

the covariance matrix C of V. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙69

Figure 3.3. (a): A simulated monoexponential decay curve. (b): FFT

representation of (a). (c): PCA representation of (a). These plots clearly

show that a monoexponential decay curve is sparser in PCA domain than in

FFT domain. To further validate this finding, a reference cardiac T2

mapping image series is displayed in both (d) FFT and (e) PCA domains.

The results were consistent with the ideal curves shown (a–c). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙71

xxii

Figure 3.4. (a): Six-fold accelerated ky-t undersampling pattern with 16

dynamic frames. (b) Corresponding PSF in the sparse y-PCA space using

PCA as the sparsifying transform. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙72

Figure 3.5. Schematic diagram of the proposed accelerated T2 mapping

pulse sequence with preconditioning RF pulses. ECG triggering was used

to image at mid to late diastole, to image at a cardiac phase where there is

minimal cardiac motion. Three presaturation RF modules and a single fat

suppression module were applied before ME-FSE readout. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙75

Figure 3.6. (a): Representative short-axis scout image displaying positions

and thicknesses of three presaturation RF pulses (displayed as meshed-

strip lines). Resulting images with none (b), fat suppression (c), three

spatial presaturation RF pulses (d), and fat suppression plus three spatial

presaturation RF pulses (e). The combined use of fat suppression and

spatial presaturation RF pulses produced the best suppression of bright

signals unrelated to the heart. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙76

Figure 3.7. Representative T2 mapping images acquired using the

reference and accelerated T2 mapping pulse sequences: (top row)

GRAPPA and (bottom row) k-t FOCUSS. When compared with GRAPPA,

k-t FOCUSS consistently yielded higher spatial resolution in the phase-

encoding direction (1.7 x 1.7 mm2 vs. 1.7 x 4.2 mm2; accelerated vs.

reference, respectively). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙80

xxiii

Figure 3.8. Zoomed cardiac T2 mapping images and the T2 maps

corresponding to Figure 3.7: (top row) GRAPPA and (bottom row) k-t

FOCUSS. When compared with GRAPPA image, k-t FOCUSS image

produced higher spatial resolution in the phase-encoding direction, as

shown by the intensity profiles of the muscle–blood border. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙81

Figure 3.9. Example cardiac T2 mapping image and the corresponding T2

maps with and without preconditioning RF pulses. For the latter case, note

the signal heterogeneity in the k-t FOCUSS reconstruction, particularly in

the lateral wall, as well as the corresponding T2 error. These results are

corroborated with zero-filled FFT reconstruction images which show more

residual aliasing artifacts for the latter case. The results clearly demonstrate

the usefulness of increasing sparsity in cardiac T2 mapping through the use

of preconditioning RF pulses. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙85

Figure 4.1. (a): Eight-fold accelerated ky–t sampling pattern varied along

time. (b): A schematic illustrating how the kx–ky–t sampling pattern is

averaged over time to produce the resulting kx–ky sampling pattern. This

kx–ky pattern represents the sampling used to perform self-calibration of

coil sensitivities. White lines represent acquired samples. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙92

Figure 4.2. Simulation results comparing the fully sampled reference

cardiac cine data to retrospectively eight -fold accelerated k– t

SPARSESENSE results with different sparsi fying transforms with

xxiv

regularization weight 0.01: temporal FFT, temporal PCA, and temporal TV.

(a): In the zoomed view of the heart, temporal TV yielded the lowest RMSE.

(b): In the chest wall, temporal FFT yielded the lowest RMSE. (c) and (d):

Corresponding plots of RMSE for the heart and chest wall regions,

respectively, as a function of regularization weight ranging from 0.005 to

0.05. These results show that temporal TV is superior to the other two

sparsifying transforms for the dynamic region, whereas temporal FFT is

superior to the other two transforms for the static region. Based on these

results, we elected to use temporal TV as the primary sparsifying transform

with regularization weight 0.01 and temporal FFT as the secondary

transform with regularization weight 0.001. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙94

Figure 4.3. Numerical simulation results comparing the (a) fully sampled

data (R = 1) to the retrospectively eight-fold undersampled reconstruction

results using four different sparsifying transforms: (b) temporal FFT, (c)

temporal PCA, (d) temporal TV, and (e) temporal TV + FFT. (First row) end-

systolic SAX image, (second row) spatial-temporal profile from the SAX

image, (third row) end-systolic LAX image, and (fourth row) spatial-temporal

profile from the LAX image. Both temporal FFT and temporal PCA yielded

more temporal blurring artifacts within the wall (arrows) than temporal TV

and temporal TV + FFT. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙96

xxv

Figure 4.4. Numerical simulation results (top row: end-diastolic images,

middle row: end-systolic images, bottom row: spatial-temporal plots through

the blood-myocardium boundary) comparing different R values using

temporal TV with weight 0.01 and temporal FFT with weight 0.001: (first

column) R = 1, (second column) R = 2, (third column) R = 4, (fourth column)

R = 6, (fifth column) R = 8, and (sixth column) R = 10. These results show

good results can be obtained up to R = 8. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙98

Figure 4.5. (a): Coil sensitivities calculated using an (left column) external

reference acquisition (pre-scan) and (right column) self-calibration method.

(b): The resulting k–t SPARSE-SENSE images using externally and self-

calibrated coil sensitivities. Note that two sets of data are very similar,

suggesting that our self-calibration of coil sensitivities was robust. ∙∙∙∙∙∙∙∙∙102

Figure 4.6. Schematic flowchart of the image reconstruction method. (a):

Coil sensitivity maps were self-calibrated by averaging undersampled k-

space data over time and computed using the adaptive array combination

method. (b): Multicoil, zero-filled k-space data, along with the corresponding

coil sensitivity maps, were reconstructed using both temporal TV and

temporal FFT as the sparsifying transforms, where regularization weight of

temporal TV is 10 times larger than that for temporal FFT. 104

Figure 4.7. (Rows 1–2) End-diastolic and (rows 3–4) end-systolic images at

multiple cardiac phases comparing (rows 1 and 3) breath-hold cine MRI and

xxvi

(rows 2 and 4) real-time cine MRI. Both image sets were acquired from a

29-year-old (male) healthy subject. Note that the breath-hold cine MR

images had higher spatial resolution than the real-time cine MR images (1.6

mm2 vs. 2.3 mm2, respectively). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙106

Figure 4.8. Bland–Altman plots illustrating good agreement between

breath-hold cine MRI and real-time cine MRI for the following LV function

measurements: (top, left) EDV (mean difference = 15.2 mL [solid line];

lower and upper 95% limits of agreement = 27.6 and 2.8 mL [dashed lines],

respectively), (top, right) ESV (mean difference = 2.1 mL [solid line]; lower

and upper 95% limits of agreement = 4.7 and 8.9 mL [dashed lines],

respectively), (bottom, left) SV (mean difference = 17.3 mL [solid line]; lower

and upper 95% limits of agreement = 31.3 and 3.3 mL [dashed lines],

respectively), and (bottom, right) EF (mean difference = 5.7% [solid line];

lower and upper 95% limits of agreement = 11.3% and 0.1% [dashed lines],

respectively). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙107

Figure 4.9. Proposed real-time cine MRI protocol with prospective ECG

triggering to capture true end diastole, where images are continuously

acquired through the second R-wave to visually identify true end diastole.

This proposed approach produced global function measurements in

excellent agreement with breath-hold cine MRI with retrospective ECG

gating. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙110

xxvii

Figure 4.10. Representative end-diastolic and end-systolic real-time cine

images: (top row) SAX view of a 26-year-old (female) patient and (bottom

row) LAX view of a 36-year-old (male) patient. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙111

Figure 5.1. (a) Continuous acquisition of radial lines with stack-of-stars

golden-angle scheme in GRASP. (b) Point spread function (PSF) of an

undersampled radial trajectory with 21 golden-angle spokes and 256

sampling points in each readout spoke for a single element coil (top) and for

a sensitivity-weighted combination of 8 RF coil elements (bottom). The

Nyquist sampling requirement is 256*π/2≈402. The standard deviation of

the PSF side lobes was used to quantify the power of the resulting

incoherent artifacts (pseudo-noise) and incoherence was computed using

the main-lobe to pseudo-noise ratio of the PSF. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙120

Figure 5.2. GRASP reconstruction pipeline. (a) Estimation of coil sensitivity

maps in the image domain, where the multicoil reference image (x-y-coil) is

given by the coil-by-coil NUFFT reconstruction of the composite k-space

data set that results from grouping all the acquired spokes. (b)

Reconstruction of the image time-series, where the continuously acquired

data are first re-sorted into undersampled dynamic time series by grouping

a number of consecutive spokes. The GRASP reconstruction algorithm is

then applied to the re-sorted multicoil radial data, using the NUFFT and the

coil sensitivities to produce the unaliased image time-series (x-y-t). ∙∙∙∙∙∙∙123

xxviii

Figure 5.3. Reconstruction of one representative partition from the contrast-

enhanced volumetric liver dataset acquired with golden-angle radial

sampling scheme using NUFFT (a) and GRASP with three different

weighting parameters (b-d) by grouping 21 consecutive spokes in each

temporal frame. Results with λ = M0*0.05 achieved an appropriate

compromise between image quality and temporal fidelity. This value was

therefore chosen for GRASP reconstruction with temporal resolutions of 21

spokes per frame. The weighting parameter was adjusted for different

temporal resolutions according to the level of incoherent aliasing artifacts or

pseudo-noise in the PSF. M0 was the maximal magnitude value of the

NUFFT images that were also used to initialize the GRASP reconstruction.

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙132

Figure 5.4. Comparison of GRASP (top) reconstruction with coil-by-coil

compressed sensing (middle) and iterative SENSE (bottom) reconstructions

in the liver dataset with the same acceleration rate and temporal resolution

of 21 spokes/frame = 3 seconds/volume. GRASP showed superior image

quality compared to both coil-by-coil compressed sensing and iterative

SENSE reconstructions. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙134

Figure 5.5. (a) GRASP reconstruction of free-breathing contrast-enhanced

volumetric abdominal imaging of a 10-year old patient referred for tuberous

sclerosis. Representative images with three different temporal resolutions

xxix

are shown, including (top) 34 spokes/frame = 8 seconds/volume, (middle)

21 spokes/frame = 5 seconds/volume and (bottom) 13 spokes/frame = 3

seconds/volume. The reconstructed image matrix size was 256 x 256 in

each dynamic frame, with in-plane spatial resolution of 1 mm and the

weighting parameters of different temporal resolutions were adjusted

according to the acceleration rate. b) Signal-intensity time courses for the

aorta and portal vein, which do not show significant temporal blurring as

compared with the corresponding NUFFT results. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙136

Figure 5.6. (a) GRASP reconstruction of free-breathing contrast-enhanced

volumetric unilateral breast imaging in an adult patient referred for

fibroadenoma with fibrocystic changes. Temporal resolution is 21

spokes/frame = 3 seconds/volume. The reconstructed image matrix size is

256 x 256 for each dynamic frame, with in-plane spatial resolution of 1.1

mm. b) Signal-intensity time courses for the breast lesion, which is a

fibroadenoma with fibrocystic changes (white arrow), for the heart cavity

(white ROI), and for a blood vessel and breast tissue (white arrows),

showing no significant temporal blurring. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙137

Figure 5.7. (a) GRASP reconstruction of contrast-enhanced volumetric

neck imaging in an adult patient referred for neck mass and squamous cell

cancer. Temporal resolution is 21 spokes/frame = 7 seconds/volume. The

reconstructed image matrix size is 256 x 256 for each dynamic frame, with

xxx

in-plane spatial resolution of 1 mm. b) Signal-intensity time courses

evaluated for the carotid arteries show no significant temporal blurring.

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙138

Figure 6.1. Schematic illustration of the XD-GRASP method: (a)

Continuously acquired radial k-space data are sorted into respiratory states

from expiration (top) to inspiration (bottom), using a respiratory motion

signal extracted directly from the data. Different colors indicate different

motion states. The number of spokes grouped in each motion state is the

same. (b) Approximately uniform coverage of k-space, with distinct

sampling patterns in each respiratory motion state, is achieved using the

golden-angle acquisition scheme. (c) Data sorting removes blurring and

clearly resolves respiratory motion, at the expense of introducing

undersampling artifacts. The purple dashed line shows the distinct

respiratory motion states after data sorting. (d) Sparsity is exploited along

the extra dimension to remove aliasing artifacts due to undersampling.

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙151

Figure 6.2. Data sorting procedure for XD-GRASP in abdominal MRI

without contrast ejection. Respiratory motion was first sorted from end-

expiration to end-inspiration and the corresponding set of spokes were

evenly distributed into multiple respiratory states so that the number of

spokes is the same in each motion state. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙153

xxxi

Figure 6.3. XD-GRASP motion estimation and data sorting for cardiac cine

imaging. (a) 2D golden-angle radial trajectory. Motion signals are estimated

from the central k-space position of each radial line (gray dot). (b-c)

Estimation of cardiac and respiratory motion signals using information from

multiple coils. The signals with the highest peaks in the frequency range of

0.1-0.5Hz and 0.5-2.5Hz are automatically selected for respiratory and

cardiac motion signals, respectively. (d) Conventional GRASP sorting of

cardiac phases, given by grouping consecutive spokes in each frame. (e)

XD-GRASP sorting, in which all the cardiac cycles are concatenated into an

expanded dataset with one cardiac dimension (tC) and an extra respiratory

dimension (tR), so that sparsity along tC and tR can be exploited in the

multidimensional compressed sensing reconstruction. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙155

Figure 6.4. Selection of cardiac and respiratory motion signals from

multiple coils. (a) 2D golden-angle radial trajectory for free-breathing 2D

cardiac cine MRI and (b) estimation of cardiac and respiratory motion

signals using information from multiple coils. The motion signal in the coil-

element with the highest peak in the frequency range of 0.1-0.5Hz was

automatically selected to represent respiratory motion; and the motion

signal in the coil-element with the highest peak in the frequency range of

0.5-2.5Hz was automatically selected to represent cardiac motion. A

xxxii

filtering procedure can be performed on the detected motion signals for

denoising. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙156

Figure 6.5. XD-GRASP motion estimation and data sorting for DCE-MRI

imaging. (a) 3D stack-of-stars radial trajectory with golden-angle rotation,

where all spokes along kz for a given rotation angle are acquired before

rotating the sampling direction to the next angle. (b) A 1D Fourier transform

along the series of k-space central points of each slice is performed to

obtain a projection profile of the entire volume for each angle and all the

projection profiles from all coils are concatenated into a large two-

dimensional matrix, followed by principal component analysis (PCA) along

the z+coil dimension. (c-d) The principal component with the highest peak

in the frequency range of 0.1-0.5Hz is selected to represent respiratory

motion. (e-g) Contrast-enhancement effect is approximately removed by

estimating and subtracting the envelope of the composite signal. (h-i)

Processed respiratory motion signals are shown superimposed on the z-

projection profiles for normal breathing (left) and heavy breathing (right),

demonstrating reliable motion estimation. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙158

Figure 6.6. For DCE-MRI, the respiratory motion sorting procedure

described in Figure 6.2 is performed in each contrast-enhancement phase

separately. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙160

xxxiii

Figure 6.7. Conventional NUFFT reconstruction without respiratory sorting

(motion average) and XD-GRASP reconstruction with 6 respiratory states

for datasets acquired in transverse, coronal and sagittal orientations. XD-

GRASP significantly reduces motion-blurring, as indicated by the white

arrows. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙169

Figure 6.8. XD-GRASP reconstruction results for four representative

respiratory sparsity regularization parameters ( 2 ) in cardiac imaging and

liver DCE-MRI. Utilization of a sparsity constraint along the extra

respiratory-state dimension improved the removal of undersampling

artifacts, when compared with the non-regularized case ( 2 =0). Very low

values of 2 resulted in residual aliasing artifacts, while very high values of

2 introduced blurring. A 2

of 0.01 in cardiac cine imaging and 0.015 in

liver DCE-MRI provided a good tradeoff between residual aliasing artifacts

and temporal fidelity. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙171

Figure 6.9. Comparison of XD-GRASP against the standard breath-hold

approach used in routine clinical studies (i.e., with retrospective ECG-

gating) at end-diastolic and end-systolic cardiac phases in the volunteer

scan. XD-GRASP provided similar performance to the routine clinical

breath-hold method. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙172

xxxiv

Figure 6.10. (a) XD-GRASP provides access to respiratory motion

information for each cardiac phase, where respiratory-related motion of the

interventricular septum, especially at diastolic cardiac phases (top row) can

be seen, indicating left-right ventricular interaction during respiration. Gray

arrows indicate different respiratory motion states. (b) Comparison of XD-

GRASP reconstruction exploiting sparsity along two dynamic dimensions

(right-hand column) with GRASP reconstruction exploiting sparsity along a

single dynamic dimension only (left-hand column), using the same data set

acquired during free breathing. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙174

Figure 6.11. (a) Comparison of XD-GRASP and the standard breath-hold

approach with retrospective ECG-gating for the patients. Conventional

breath-hold scans achieved good image quality in a patient with normal

sinus rhythm, but it produced poor image quality for patients with

arrhythmia. XD-GRASP achieved consistent image quality by separating

the cardiac phases with arrhythmia. (b) In the patient with 2nd degree AV

block, the arrhythmic cardiac cycles were further sorted for a separate XD-

GRASP reconstruction to provide additional physiological information. (c)

Corresponding cardiac motion signals for three patients with varying length

of the cardiac cycle indicated by gray arrows. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙176

Figure 6.12. Comparison of GRASP with XD-GRASP in both aortic and

portal-venous enhancement phases in two representative partitions each

xxxv

from two volunteer datasets. XD-GRASP improved delineation of the liver

and vessels with enhanced vessel-tissue contrast. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙177

Figure 6.13. Comparison of GRASP with XD-GRASP in a total of five

representative partitions from two volunteers and one patient. Volunteer 4

was asked to breathe deeply. XD-GRASP achieved superior overall image

quality, with reduced motion-blurring. The white arrow indicates a

suspected liver tumor, which is better delineated in XD-GRASP. ∙∙∙∙∙∙∙∙∙∙∙∙179

Figure 6.14. Comparison of XD-GRASP reconstructions with different

number of respiratory motion states in abdominal DCE-MRI (end-expiratory

motion state only). 4 and 6 respiratory states achieved better resolved

respiratory motion than 2 states and 1 state. 6 respiratory states resulted in

slightly lower performance than 4 respiratory states. White arrows indicate

motional blurring for a choice of 1 motion state, and residual blurring for a

choice of 2 motion states. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙182

Figure 7.1. Comparison of golden-angle radial sampling schemes that are

based on stack-of-stars pattern (a) and spiral phyllotaxis pattern (b),

respectively. When compared with the stack-of-stars scheme, radial

sampling is also employed along the kz dimension in the 3D phyllotaxis

sampling trajectory, so that each k-space line passes through the center of

k-space and an image can be reconstructed with isotropic spatial resolution.

The 3D radial sampling pattern in (b) can be segmented into multiple

xxxvi

heartbeats for cardiac MRI, with golden-angle rotation along the z-axis

between every two successive data interleaves. An additional spoke

oriented along the superior-inferior (SI) direction (red lines) can be acquired

at the beginning of each data interleave for respiratory motion detection and

self-navigation. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙189

Figure 7.2. (a) Data sorting procedure in XD-GRASP reconstruction for

ECG-triggered whole-heart coronary MRA, in which the 3D golden-angle

radial k-space data are sorted into 4 respiratory motion states spanning

from expiration (top) to inspiration (bottom) (x-y-z-respiratory) using the

respiratory motion signals drived from the acquired data. The sorting

procedure is performed so that the number of spokes grouped in each

motion state is the same. Approximately uniform coverage of k-space with

distinct sampling patterns in each motion state can be achieved, as shown

in (b)&(c)∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙193

Figure 7.3. Five-dimensional data sorting in free running continuous whole-

heart imaging, with one cardiac motion dimension (20 cardiac phases) and

one respiratory motion-state dimension (4 respiratory states). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙195

Figure 7.4. Comparison of XD-GRASP reconstruction (end-expiratory

motion states) with the 1D respiratory motion correction reconstruction in

two representative datasets. XD-GRASP improves the delineation of

xxxvii

coronary arteries and removes the blurring effects by resolving the

respiratory motion. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙200

Figure 7.5. End-expiratory myocardial wall (SAX and 4CH), proximal

coronary arteries, right coronary artery (RCA) and left anterior descending

coronary artery (LAD) in diastolic (top) and systolic (bottom) phases. All the

images are reformatted from a single continuous data acquisition with 5D

XD-GRASP reconstruction. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙201

Figure 7.6. 5D XD-GRASP reconstruction achieved reduced blurring,

improved sharpness and better visualization of myocardium and the RCA

compared with 4D reconstruction with respiratory motion correction (MC) in

one representative volunteer with irregular respiratory pattern. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙202

xxxviii

LIST OF TABLES

Table 3.1. Mean Segmental and Whole Myocardial T2 Measurements

Obtained Using GRAPPA and k-t SPARSE-SENSE Datasets. Not that

these values represent results analyzed by observer 1 and analysis 1.

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙82

Table 3.2. Bland–Altman Statistics of T2 Measurements Obtained Using

GRAPPA and k-t SPARSE-SENSE Datasets. Note that these values

represent results analyzed by observer 1 and analysis 1. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙83

Table 3.3. Intraobserver and Interobserver Agreements for T2 Calculations

Based on Manual Segmentation of LV Contours. Intraobserver difference

was defined as T2 (analysis 1)-T2 (analysis 2), and interobserver difference

was defined as T2 (observer 1)-T2 (observer 2). ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙84

Table 4.1. Mean scores of image quality, temporal fidelity of wall motion

and artifact, produced by Breath-Hold cine MRI and Real-Time cine MRI.

∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙108

Table 4.2. Bland–Altman and CV analyses of four global function

measurements between Real-Time and Breath-Hold cine MRI pulse

sequences. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙109

Table 4.3. ICC analysis of interobserver variability of EDV, ESV, SV, and

EF within each pulse sequence type. ICC scale: 0-0.2 indicates poor

xxxix

agreement, 0.3-0.4 indicates fair agreement, 0.5-0.6 indicates moderate

agreement, 0.7-0.8 indicates strong agreement, and >0.8 indicates almost

perfect agreement. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙109

Table 5.1. Representative imaging parameters of dynamic volumetric MRI

in different applications. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙128

Table 5.2. Image quality assessment scores represent mean ± standard

deviation for each reconstruction category for different applications. ∙∙∙∙∙∙139

Table 7.1. Readers’ scores for comparison of 1D self-navigation motion

correction reconstruction v.s. XD-GRASP reconstruction (end-expiration

only) in visualization/sharpness of RCA, LAD and left main coronary artery.

0-4: non-diastolic to excellent. * Indicates statistical significance. LM: Left

Main Coronary Artery. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙203

Table 7.2. Readers’ scores for comparison of 1D self-navigation motion

correction reconstruction v.s. XD-GRASP reconstruction (end-expiration

only) in diastolic quality of RCA, LAD and left main coronary artery. 0 = not

visible, 1 = visible, and 2 = diagnostic. * Indicates statistical significance.

LM: Left Main Coronary Artery. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙204

Table 7.3. Reader’s scores for comparison of 4D reconstruction with motion

correction v.s. 5D XD-GRASP reconstruction (end-expiration only) in

visualization/sharpness of myocardium, the proximal segment of RCA, LAD

xl

and left main coronary artery. 1-5: non-diastolic to excellent. * Indicates

statistical significance. LM: Left Main Coronary Artery. ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙204

1

Chapter 1

Introduction

1.1. Overview and Motivation

Magnetic Resonance Imaging (MRI) is a multifaceted, non-invasive

and powerful imaging modality, with a broad range of applications in both

clinical diagnosis and basic scientific research. Comparing to other medical

imaging modalities, MRI does not use ionizing radiation and provides

superior soft-tissue characterization with high resolution and flexible image

contrast parameters. Moreover, MRI allows good visualization of anatomical

structure, physiological function, blood flow, and metabolic information,

making it compelling in a variety of clinical applications.

MRI is based on the phenomenon of nuclear magnetic resonance

(NMR) that was discovered in the 1940s (1,2) and has been applied in a

variety of research experiments in chemistry, biology, physics and medicine.

2

In a simple NMR experiment, the signal is generated by applying a resonant

radiofrequency (RF) pulse to excite the spin of the atomic nucleus in an

object that is placed inside a strong static magnetic field (B0). Following the

excitation, the object being studied emits a decaying RF signal that can be

detected in the form of radiofrequency voltage in a receiver coil. In order to

distinguish the received signals from different spatial positions, additional

magnetic field gradients are superimposed on the main magnetic field, so

that the field strength varies linearly with spatial position, allowing the exact

origins of NMR signal emitted from the object to be localized (3). Based

upon the idea of gradient encoding, Fourier imaging was proposed (4), in

which measurements representing the spatial frequency of the object,

termed k-space, can be acquired using a specific trajectory (5,6). The most

common acquisition scheme is Cartesian sampling, where k-space points

are acquired on a uniform rectangular grid and image reconstruction is

performed in a robust and efficient fashion by applying an inverse Fast

Fourier Transform (FFT).

Fourier imaging led to revolutionary progress in MRI and formed the

basis of most variants of MRI techniques that are used today. However, the

major limitation of Fourier imaging is the relatively slow data acquisition

process, in which only one k-space position can be encoded per unit time

and this process has to be sequentially repeated until the entire k-space

3

region for the target spatial resolution is covered. Low imaging speed

increases patient discomfort and imposes strict limits in spatiotemporal

resolution and volumetric coverage. Meanwhile, in order to make the best

use of precious scan time, data acquisitions are often planned in oblique

image planes adjusted to the target anatomy, which results in complex and

cumbersome scan planning and also requires extensive training for scan

operators. Furthermore, because of the wide range of contrasts that MR is

capable of producing, the acquisition planning is often repeated multiple

times for different imaging sequences and protocols, which results in

substantial “dead time” between successive data acquisitions. The exam

workflow is even more complicated and tedious in the imaging of moving

organs, such as liver, kidney or heart. For example, in a typical cardiac MRI

exam, the data acquisition has to be synchronized with the contraction of

the heart and is usually performed during multiple breath-holds in order to

avoid respiratory-motion induced artifacts (7). Since breath-hold capabilities

are subject dependent and can be significantly limited in patients, repeated

data acquisitions are often required in the case of failed breath-holds, or in

the presence of different types of arrhythmias, which further increase

patient discomfort and prolong the examination times, making MRI more

challenging in some applications such as cardiac imaging or abdominal

imaging.

4

Rapid imaging approaches can help shift the balance from complex

tailored acquisitions to a simple and continuous acquisition paradigm in MRI.

Since the introduction of MRI, researchers have devoted tremendous effort

to the acceleration of MR scans, and the speed with which data can be

acquired has already increased dramatically with a combination of

advances in MR hardware and innovations in imaging techniques. For

example, fast switching magnetic field gradients have allowed the intervals

between data collections to be reduced substantially. The invention of fast

imaging strategies, such as Echo-Planar Imaging (EPI) (8), Fast Spin-Echo

(FSE) imaging (9), Fast Low Angle SHot (FLASH) imaging (10), balanced

Steady-State Free Precession (bSSFP) imaging (11), and spiral imaging

sequences (12,13) all significantly increased imaging efficiency. However,

the nature of sequential data acquisition in conventional Fourier imaging still

limits achievable imaging speed.

An alternative approach to increase imaging speed in MRI is to

reduce the quantities of phase-encoding measurements while maintaining

the target resolution. The idea of partial Fourier imaging was proposed in

the 1980s and early 1990s for accelerated MRI exams, in which the

conjugate symmetry of k-space is exploited to reduce scan times by

acquiring approximately half of the k-space data (14-16). Although partial

Fourier imaging is still used in clinical exams today, the maximum

5

acceleration factor that can be achieved is only close to two. Beginning in

the late 1990s, a variety of parallel imaging techniques, such as

Simultaneous Acquisition of Spatial Harmonics (SMASH) (17), Sensitivity

Encoding (SENSE) (18), and Generalized Autocalibrating Partially Parallel

Acquisition (GRAPPA) (19), were proposed to accelerate the data

acquisition in MRI using an array of receive coils with spatially-varying

sensitivities. The knowledge of coil sensitivities, which is usually estimated

using additional reference data, can be employed to perform some portion

of spatial encoding that is normally accomplished via gradients, thus

enabling reconstruction of an image without aliasing from only a subset of

k-space data (20). Temporal parallel imaging techniques, such as TSENSE

(21) or TGRAPPA (22), further eliminate the need to acquire extra

reference data for coil sensitivity calibration in dynamic imaging exams,

estimating the coil sensitivities by combining different temporal frames

acquired with shifted lattice undersampling patterns, under the assumption

that the sensitivity maps are smooth and do not change significantly over

time. However, the acceleration in parallel imaging is fundamentally limited

by noise amplification in the reconstruction (also known as g-factor), which

increases non-linearly with increasing acceleration factor (18,20). The

presence of extensive spatial and temporal correlations in dynamic MRI can

be also exploited to accelerate data acquisition, and these methods are

6

usually combined with parallel imaging for better performance. For example,

k-t acceleration methods, such as k-t BLAST/k-t SENSE (23), k-t GRAPPA

(24) and k-t PCA (25), are based on the fact that the representation of

dynamic images in the combined spatial and temporal Fourier domain (x-f

space) is typically sparse, which reduces the signal overlap in x-f space due

to regular k-t undersampling and thus enables higher accelerations. K-t

techniques represented the first attempt to exploit compressibility or

sparsity to reconstruct undersampled data. However, one potential

drawback of k-t techniques is the need to explicitly compute the signal

distribution in x-f space, which is usually performed by acquiring a low

spatial resolution reference image. Therefore, it can be challenging to

recover some detailed features using k-t techniques, and their use may

introduce residual aliasing artifacts at edges.

The idea of compressed sensing (26,27), proposed in the 2000s,

represents another powerful approach for increasing imaging speed in MRI

by exploiting the compressibility or sparsity of an image (28). Since its

introduction, compressed sensing has already generated great excitement

and enabled significant advances in coding and information theory.

According to the Nyquist theorem, the sampling rate in a conventional

sampling scheme must be at least twice the maximum bandwidth presented

in the signal. Unfortunately, in many applications, Nyquist sampling is time-

7

consuming and data-intensive, posing a challenge for sampling system

design, data storage and transmission. In order to address this logistical

and computational challenge, high-dimensional data are often compressed

after acquisition by transforming to a basis that provides a sparse or

compressible representation for the signal, and discarding insignificant

components. This transform coding framework has been widely used in the

JPEG, JPEG2000 and MPEG image/video compression standards. The

ability to compress images so effectively raises an interesting question, one

which underlies compressed sensing: instead of first sampling a signal at a

high sampling rate and then discarding most of the sampled measurements

in the compression process, why not directly acquire the data in a

compressed form at a lower sampling rate? In other words, can we build the

compression process directly into the acquisition or sensing step, so that

one does not have to perform so many measurements only to discard most

of them afterwards? Candes, Romberg, Tao and Donoho proved the

feasibility of this hypothesis (26,27) and proposed the compressed sensing

framework by which a sparse or compressible signal could be successfully

recovered from undersampled measurements that are far below the Nyquist

limit (26,27).

After rapid development in the past decade, compressed sensing

has already achieved notable impact in a wide range of application areas,

8

including medical imaging, sensor design in high-resolution cameras,

geophysical data analysis, computational biology, radar analysis and many

others. One of the applications that can substantially benefit from

compressed sensing is MRI, in which the imaging speed can be

dramatically improved by reconstructing the sparse representation of an

image from undersampled measurements without loss of important

information (28). Meanwhile, since multicoil data acquisition is widely used

in MRI nowadays, compressed sensing can be combined with parallel

imaging to further increase imaging speed and improve reconstruction

performance exploiting the idea of joint multicoil sparsity (29-31). These two

reconstruction approaches can be synergistically combined, because image

sparsity and coil sensitivity encoding are complementary sources of

information. On one hand, compressed sensing can serve as a regularizer

for the inverse problem in parallel imaging, and can thus prevent heavy

noise amplification due to high accelerations. On the other hand, parallel

imaging can reduce the level of incoherent aliasing artifact in compressed

sensing, by exploiting joint sparsity in sensitivity-weighted combinations of

multicoil images (30).

Remarkable advances in rapid MRI have been achieved over the last

two decades, improving the performance of existing techniques and

enabling new imaging methods that were not feasible before due to limited

9

imaging speed. However, the paradigm of routine clinical imaging still

remains complex, given the rich diversity of acquisition choices and the

adjustments needed to reduce the influence of unwanted effects, such as

respiratory motion, cardiac motion, relaxation effects, and others. Therefore,

it is desirable to shift the day-to-day clinical workflow from time-consuming,

inefficient, and tailored acquisitions to rapid, continuous and comprehensive

acquisitions with user-defined reconstructions adapted retrospectively for

different clinical needs. The combination of compressed sensing and

parallel imaging has the potential to enable such an efficient imaging

paradigm. The overall goal of this dissertation is to develop novel imaging

techniques that support simple and efficient MRI protocols, and begin to

enable a rapid continuous data acquisition paradigm for clinical and

research MRI exams.

1.2. Thesis Contributions and Outline

Chapter 1 (current chapter) gives an introductory overview and

motivation for this dissertation.

Chapter 2 presents a brief overview of fundamental principles of

MRI, parallel MRI and compressed sensing.

Chapter 3 and Chapter 4 present highly-accelerated MR parameter

(T2) mapping and real-time cardiac cine MRI using k-t SPARSE-SENSE,

which is a framework combining compressed sensing and parallel imaging

10

using Cartesian k-space sampling. The purpose and contribution of these

two chapters are to demonstrate the performance of k-t SPARSE-SENSE

for different clinical applications and also compare the performance of

different sparsifying transforms that can be used in the subsequent

chapters.

Chapter 5 presents a highly-accelerated dynamic imaging technique

called Golden-angle RAdial Sparse Parallel MRI (GRASP), which

synergistically combines compressed sensing and parallel imaging

reconstruction with golden-angle radial sampling. Golden-angle radial

sampling provides a continuous data acquisition scheme that is robust to

motion and well-suited for compressed sensing acceleration. GRASP

represents a promising imaging paradigm for clinical workflow, based on

rapid continuous data acquisition with flexible spatiotemporal resolution

tailored retrospectively to different clinical needs. The performance of

GRASP is demonstrated in a wide range of clinical applications, including

dynamic contrast-enhanced imaging of the liver, kidney, breast, neck,

prostate, etc.

Chapter 6 presents a novel framework for free-breathing MRI called

eXtral-Dimensional Golden-angle RAdial Sparse Parallel MRI (XD-GRASP),

which uses the same continuous data acquisition as GRASP, but

reconstructs additional motion dimensions using compressed sensing.

11

Instead of explicitly removing or correcting for motion, XD-GRASP takes a

different approach to handling various types of periodic motion by sorting

and reconstructing the acquired data with multiple resolved motion states.

Besides motion compensation, XD-GRASP also provides access to new

physiological information that could be of potential clinical value.

Chapter 7 presents an extension of XD-GRASP to 3D golden-angle

radial sampling based on the spiral phyllotaxis sampling pattern. 3D radial

sampling not only enables volumetric isotropic spatial coverage, but also

provides increased motion robustness and allows exploitation of

incoherence along all spatial dimensions. The performance of the technique

is first demonstrated for free-breathing ECG-triggered whole-heart coronary

MR angiography (MRA) with improved motion compensation. It is then

applied for continuous five-dimensional cardiac and respiratory motion-

resolved whole-heart MRI that enables simultaneous assessment of

myocardial function in arbitrary planes and visualization of whole-heart

arterial anatomy (including aorta and coronary arteries, etc).

Chapter 8 summarizes the contributions presented in this

dissertation and discusses an outlook for the future.

Chapter 9 is a list of publications.

12

Chapter 2

Background

This chapter presents a brief overview of basic principles of MRI,

parallel imaging and compressed sensing. The discussion is focused on the

aspects that are relevant to the subsequent chapters of the dissertation.

2.1. MRI Signal

2.1.1. NMR Phenomenon

The physical phenomenon behind MRI is nuclear magnetic

resonance (NMR), which was first discovered in the 1940s (1,2). An atomic

nucleus with an odd number of protons possesses an angular momentum J

called spin, which generates a tiny magnetic moment μ . The magnetic

moment is directly proportional to the angular moment as

γμ J (2.1)

13

Figure. 2.1: (a) Without a strong external magnetic field (B0), the spins are randomly oriented and the total magnetic moments have a vector sum of zero. (b) Alignment of spins either parallel or anti-parallel to the direction of B0 when exposed to an external magnetic field. (c) A net magnetization vector Mz (also known as M0) is generated as the vector sum of all the spin angular momenta at the thermal equilibrium state.

Here γ is a constant called the gyromagnetic ratio. The proton in hydrogen

(1H) is particularly interesting because it is abundant in water and other

molecules in the human body. At room temperature and without a strong

external magnetic field (B0), the spins are randomly oriented and their

magnetic moments have a vector sum of zero, as shown in Figure 2.1a.

When the spins are placed in a strong external magnetic field, they

align themselves with B0, as shown in Figure 2.1b. In fact, in the so-called

thermal equilibrium state, proton spins are divided among two populations,

one population (n+) oriented parallel and the other (n-) oriented anti-parallel

to B0. (In general, at thermal equilibrium, spins populate their quantized

energy states according to a Boltzmann distribution; for spin-1/2 species

14

such as 1H, there are two such states corresponding to oppositely oriented

angular momenta.) The n+ spin population is at a relatively lower energy

state and thus has a slightly larger number of spins than the n- spin

population. This results in a net magnetization vector 0M , which is the

vector sum of all the spin angular momenta and is aligned in the direction of

B0, known as the z direction or longitudinal direction, as shown in Figure

2.1c. The magnitude of 0M can be calculated as

2 2

0

04

h B NM

KT

(2.2)

Here h is Planck’s constant, K is the Boltzmann’s constant and T is the

absolute temperature. Equation 2.2 suggests that the net magnetization

that can be measured in MRI is proportional to the magnetic field B0.

When spins are perturbed away from the axis of the applied B0 field,

they precess around the direction of B0 at a frequency that is proportional to

the strength of B0, as given by the Larmor equation

γ 0ω B (2.3)

Now let us consider how such perturbations are accomplished.

2.1.2. Signal Excitation

In order to generate MR signal that can be measured by a detector,

the net magnetization 0M needs to be tipped towards the direction

15

perpendicular to B0, which is known as the x-y plane or transverse plane.

This process is known as “excitation” and is achieved by applying a

radiofrequency (RF) pulse that creates a magnetic field (B1) perpendicular

to B0 and rotating at the Larmor frequency. The RF pulse causes 0M to

move away from the z direction into the transverse plane until the pulse is

switched off. The nutation angle through which 0M moves, or the “flip angle”

of the pulse, is given by

1

0( )

T

B t dtt (2.4)

Here B1 is the strength of the RF magnetic field and T is the duration of the

pulse. Following the excitation, the excess z-population 0M is at least

partially converted into a transverse magnetization component ( xyM ), as

shown in Figure 2.2, and the ensemble of spins retain their relative

alignment, or phase coherence. The precession of xyM generates an

oscillating magnetic field, and the changing magnetic flux associated with

this field can then induce a voltage in a suitably configured receive coil. This

voltage constitutes the MR signal, which can subsequently be demodulated

or manipulated otherwise as desired.

16

Figure. 2.2: Following the excitation, the excess z-population is at least partially converted into a transverse magnetization component (Mxy), and the ensemble of spins retain their relative alignment, or phase coherence.

2.1.3. Relaxation

After the RF pulse is switched off, the precessing spins gradually

lose their coherence and return to the z-directed equilibrium state, in

processes known collectively as relaxation. The loss of spin coherence

results from differences in local field strength and precession frequency, or

else from other interactions between spins and their environment, and it is

characterized by a time constant T2. The return to longitudinal equilibrium is

associated with loss of the energy the spins absorbed from the RF pulse,

and is characterized by a time constant T1.

These two relaxation mechanisms, together with the behavior of the

magnetization vector when exposed to an external magnetic field, can be

described by the Bloch equation:

17

0

2 1

( )γ

i j kMM B

x y zM M M Md

dt T T

(2.5)

Here ( )M x y zM ,M ,M , and i , j , and k are unit vectors along x, y, and z

directions respectively. The cross-product term describes the precession

behavior and the relaxation terms describe the exponential behavior of

transverse dephasing and longitudinal recovery. For a flip angle of 90o, the

solution to Equation 2.5 is given by

1

2 0

-t /Tz 0

-iω t-t /Txy 0

M (t)= M (1- e )

M (t)= M e e (2.6)

2.2. Signal Localization

In a hypothetical uniform-sensitivity receiver coil, the MR signal

following an RF pulse contains contributions from all the transverse

magnetization within the entire volume of interest (VOI). This signal can be

expressed as

( )

( ) ( ) ( ) 02 -i t-t /T

vol volS t = M ,t dr= M e e dr

rr r

(2.7)

Here M is the object to be imaged and r is a vector indicating spatial

position. For the purposes of simplicity, the transverse relaxation term

exp( )2-t /T may be ignored for the time being, and the exp( )0-i t term

can be also removed via demodulation in the process of signal detection.

18

Therefore, the MR signal that is detected immediately after RF excitation

can be simplified as

( )vol

S = M drr (2.8)

Equation 2.8 suggests that the detected signal arises from the entire

sample volume and thus the spatial information in the acquired signal

cannot be differentiated. In order to obtain the spatial positions of the signal,

additional magnetic field gradients must be used to generate spatially

varying longitudinal field strength (3).

In general, the field gradients applied in MRI are time-varying and

can be expressed as a vector:

( ) ( ) ( ) ( )G i j kx y z

t G t G t G t (2.9)

Thus, the total magnetic field with all three gradients turned on is

0( ) ( )B r, G rt B t (2.10)

where r is the spatial position in the sample. Since the magnetic field now

varies as a function of position, spins at different positions have different

precession frequencies, with the differences given by

( ) γr G r (2.11)

Therefore, when gradients are turned on, Equation 2.8 can be expressed as

γ( ) ( ) exp( γ ( ) )

ti t

vol volS = M e dr M i d drG r

0r r G r (2.12)

19

where t is the duration of the applied gradients. The orientation of the

gradient fields is parallel to B0 and thus the Larmor frequency, which is

proportional to the field strength, becomes dependent on signal location.

2.2.1. Slice Selection

An RF pulse is usually designed at the resonance condition, with a

bandwidth centered at the Larmor frequency. Thus, without the presence of

gradients, the RF pulse excites all the spins in the object being imaged

since they are all rotating at the same frequency. In order to excite a slice in

2D MR imaging, a slice-selective gradient ( zG ) needs to be first imposed

along an axis perpendicular to the imaging slice to be excited, so that the

Larmor frequency of the spins varies along that axis. Besides, a specially

tailored RF pulse needs to be applied simultaneously with the gradient, with

its frequency bandwidth matching the range of frequencies contained in the

desired slice. This slice-selective excitation ensures that only spins within a

slice or slab of interest can be excited, while leaving other spins

unperturbed. Accordingly, the relationship between slice thickness and RF

bandwidth or gradient strength can be given by

RF bandwidth

slice thisknesszG

(2.13)

2.2.2. Spatial Encoding and k-Space Formalism

20

Following slice selection, the in-plane spatial information now can be

further encoded with two additional gradients, known as frequency-

encoding and phase-encoding gradients.

In 2D imaging, the received MRI signal in Equation 2.12 can be

described as

0

γ( ) ( , )exp( ( ) )t

x ys t m x y i d dxdy G r (2.14)

Here ( )m x,y represents the spin density in the 2D object to be imaged.

Considering only the x and y gradient fields ( )xG t and ( )yG t , Equation 2.14

becomes

0 0( ) ( , )exp γ ( ) exp γ ( )

t t

x yx y

s t m x y i G d x i G d y dxdy (2.15)

Setting

0

0

γ( ) ( )

2

γ( ) ( )

2

t

x x

t

y y

k t G d

k t G d

(2.16)

Equations 2.15 can be rewritten as

2 [ ( ) ( ) ]

( , ) ( , )x y

x yx y

i k t x k t ys k k m x y e dxdy

(2.17)

Here xk and yk are the spatial-frequency variables in a two-dimensional

space that is known as k-space. Equation 2.17 is usually referred as the

signal equation, which suggests that the MRI signal represents the spatial

21

Fourier transform of the object being imaged. The signal equation can be

extended into three dimensions as

2 [ ( ) ( ) ( ) ]

( , , ) ( )x y

x y zx y z

zi k t x k t y k t zs k k k m x,y,z e dxdydz

(2.18)

Here additional phase-encoding is employed along the partition dimension.

2.3. MR Image Acquisition

It can be seen from Equation 2.15 that the position in k-space at a

given time t depends on the area of the gradient waveform at that time

point. Therefore, theoretically an MR image can be acquired by traversing

k-space using any trajectory designed by altering the gradients in an

appropriate way according to Equation 2.16. A variety of sampling

trajectories (i.e., Cartesian, radial, spiral, and many other variants) have

been proposed since the introduction of MRI.

In Cartesian sampling, k-space data is acquired in a sequential line-

by-line fashion on a Cartesian grid until sufficient data are acquired to form

an image, as shown in Figure 2.3a.

Instead of sampling parallel lines, k-space data can be also acquired

in a radial scheme, as shown in Figure 2.3b, where each radial line is

acquired with a combination of gradients in both x and y directions. This

combination can be altered to enable rotated sampling of k-space lines at

different angles according to

22

Figure. 2.3: Comparison of Cartesian sampling and radial sampling schemes.

0

0

cos( )

sin( )

x

y

G G

G G

(2.19)

Here is the acquisition angle of a given radial line and 0G denotes the

gradient amplitude needed to sample the central k-space line.

It is straightforward to extend two-dimensional acquisitions, including

both Cartesian and radial sampling, into three dimensions by adding an

additional phase-encoding step along the slice dimension. In particular,

radial and Cartesian sampling can be combined in a hybrid three-

dimensional acquisition scheme known as stack-of-stars sampling, where

radial sampling is employed in the kx-ky plane and Cartesian sampling is

23

employed along the kz dimension. Alternatively, radial sampling can be also

employed in all three dimensions, with isotropic volumetric coverage.

The distinct acquisition geometry in radial sampling offers some

unique imaging properties, such as higher inherent incoherence, improved

robustness to motion, self-navigation and readout oversampling in multiple

spatial dimensions. Those features are exploited in Chapters 5-7.

2.4. Imaging Requirements

2.4.1. Field of View and Spatial Resolution

Although the discussion so far has been focused on continuous time

signals, k-space is always sampled discretely in practice, with the sampling

intervals defined as ∆kx and ∆ky in the frequency-encoding and phase-

encoding dimensions in the case of Cartesian sampling. Considering the

one-dimensional case first, the discrete sampling of each k-space line

represents multiplication of the continuous signal by a comb function with

interval width ∆k. This corresponds to a convolution of the 1D object being

imaged with the Fourier transform of the comb-function, which is also a

comb function with reciprocal interval width 1/∆k. Therefore, the discrete

sampling leads to periodic replication of the imaged object at a distance of

1/∆k. According to the Nyquist sampling theorem, a band-limited signal with

given bandwidth B must be sampled at an interval (∆k) no larger than 1/2B

24

Figure. 2.4: In Cartesian, the sampling intervals (∆kx and ∆ky) must be smaller than the reciprocal the object size in the corresponding spatial dimensions in order to avoid alishing. In radial sampling, the maximum interval between two adjacent radial lines (∆d) has to be small than or equal to ∆k in order to reconstruct an image without aliasing artifacts.

in order to reconstruct it without aliasing. Generalized to the two-

dimensional case in MRI, and considering the case of Cartesian sampling,

the following requirements must be satisfied:

1

1

x

x

y

y

kW

kW

(2.20)

Here xW and

yW are the size, in the x and y dimensions, respectively, of the

object being imaged. Since the field of view (FOV), which indicates the

extent of the region to be imaged, can be defined as

25

1

1

x

x

y

y

FOVk

FOVk

(2.21)

it suggests that the FOV must be greater than the size of the object in order

to avoid aliasing.

The relationship in Equation 2.21 can also be applied in non-

Cartesian imaging. In the case of radial sampling, for example, the

maximum interval between two adjacent radial lines, as denoted by ∆d in

Figure 2.4b, must be less than or equal to ∆k, which can be satisfied when

2

sn n

(2.22)

Here sn is the number of radial lines and n is the number of sampling

points in each readout line. Equation 2.22 suggests that radial sampling

requires more measurements than Cartesian sampling in order to

reconstruct an image without aliasing artifacts.

Theoretically, an image can be reconstructed unambiguously from

infinite k-space sampling. However, k-space sampling in practice is always

finite, with the highest spatial frequency sampled represented by kxmax and

kymax in two spatial dimensions. This can be described as a multiplication of

infinite k-space sampling by a rectangular function with width 2×kmax (from –

kmax to +kmax) in each dimension. Therefore, the reconstructed image

26

always represents the convolution of the true object with a point spread

function (PSF), which is the Fourier transform of the rectangular truncation

window (a sinc function). The sinc function limits the ability to resolve

detailed structures and thus the spatial resolution is limited to the effective

width, often measured as the full-width at half maximum (FWHM), of the

PSF. The spatial resolution in MRI, denoted as x and y in two

dimensions, can be calculated as

1

1

x

x x x

y

y y y

FOVx

N k N

FOVy

N k N

(2.23)

Since max2x x xk N k and

max2y y yk N k , we also have

max

max

1

2

1

2

x

y

xk

yk

(2.24)

2.4.2. Signal to Noise Ratio

The intensity values of an MR image represent combinations of the

true signal intensity and the noise. Therefore, another practical

consideration for MRI is signal to noise ratio (SNR). The noise in an image

can result from a number of factors, including noise in the body due to

movement of charged particles and other similar mechanisms, and noise

27

from the measurement electronics. Although SNR is usually expressed as a

function of various parameters including voxel size, number of averages,

number of sampling points, and receiver bandwidth, it can be simplified as

follows (32):

( )( )( ) sSNR x y z T (2.25)

Here sT indicates the total readout duration in a scan. This equation

indicates that the SNR is in general proportional to the voxel size and to the

square root of the total acquisition time. Proportionality to the voxel size

imposes a trade-off between spatial resolution and SNR in MRI. Meanwhile,

factors that change the Ts also tend to change the SNR. For example, the

SNR decreases in the case of undersampling (see parallel imaging in

section 2.6) and higher readout bandwidth (decreased sampling dwelltime

in each readout line and thus shorter TR). On the other hand, SNR can be

increased by acquiring the same image multiple times and averaging.

Meanwhile, we can double the sampling rate in each readout line to

increase FOV in order to avoid aliasing without SNR penalty, because it

neither changes the voxel size nor the sampling time Ts.

2.5. MR Image Reconstruction

2.5.1. Generalized Image Reconstruction

28

As shown in Equation 2.17, MR signal from a two-dimensional plane

is a spatial integration of the spin density against the sinusoidal spatial

modulation generated by encoding gradients. In other words, the MR signal

comprises projections of the spin density against Nx×Ny distinct functions,

in which a total number of Nx×Ny measurements are obtained in the

presence of frequency- and phase-encoding gradients. As discussed in the

previous sections, appropriate discretization is needed in practical MRI

acquisition and thus the integration in Equation 2.17 can be approximated

as

,

( , ) ( )exp( 2 [ ])yx

x y

yxx y

kks k k m x,y i x y

N N (2.26)

Equation 2.26 represents the Fourier transform of the spin density in the

two-dimensional plane with appropriate discretization of the continuous

positions. Therefore, the reconstruction of the image can be formulated as

the inverse discrete Fourier transform (DFT) of the MR measurements

,

1( ) ( , )exp( 2 [ ])

yxx y

yx x ykx ky

kkm x,y s k k i x y

N N N N (2.27)

The spatial encoding part in Equation 2.27 can be written as

( ) exp( 2 )exp( 2 )yx

x y

yx

kkE x, y,k ,k i x i y

N N (2.28)

Equation 2.28 is usually referred as “encoding function”, because it

29

represents the way that spatial encoding is performed. Thus, Equation 2.26

can be rewritten in matrix notation as

s = Em (2.29)

Here s is the measured MR signal vector in k-space (with size Nk×1), m is

the image vector to be reconstructed (with size N𝜌×1) and E is the

encoding matrix that transforms the image vector into the signal vector (with

size Nk×N𝜌). Reconstruction of the image m is given by inverting Equation

2.29

-1m=E s (2.30)

Supposing the measurement matrix E is full rank, the reconstruction

described in Equation 2.30 is discussed in three different situations.

i) When Nk=N𝜌, the system is well-determined and then s can be uniquely

reconstructed from Nk linearly independent measurements.

ii) When Nk>N𝜌, the system is over-determined and, strictly speaking, there

is no solution. In general, a good guess for the solution can be found by

minimizing the mean square error

2

2s

ˆmin E m-s (2.31)

Here 2 is the L2-norm to quantify the energy or power of the difference

between the measurements and the estimated solution. The solution of

Equation 2.31 is given by the Moore-Penrose pseudoinverse

30

H H

m̂ = (E E)E s (2.32)

Here H defines the Hermitian conjugate and m̂ represents the

approximate solution.

iii) When Nk<N𝜌, the system is under-determined. This is usually the case

for undersampling in MRI, and the solution is not unique. In this case,

one classic way of arriving at a solution is to solve the following

optimization problem:

2

2s

s t 2

ˆmin m

ˆ. . E m-s (2.33)

Here is the estimated noise level. Based on the isometric property of

the Fourier transform, minimization of energy in the image domain is

equivalent to minimization of energy in k-space, and thus the solution to

Equation 2.33 is the zero-padded reconstruction, which can lead to

aliasing artifacts in the result. Additional regularizations can be applied

to reconstruct a better solution, which will be discussed more in the

section on compressed sensing.

2.5.2. Reconstruction of Non-Cartesian k-Space Data

As mentioned above, MRI reconstruction can be performed by

applying an inverse DFT on the measured k-space data as shown in

Equation 2.27. For Cartesian sampling, where all the k-space points are

31

sampled on an equidistant grid, the image reconstruction can be efficiently

implemented with an inverse Fast Fourier Transform (FFT) instead of

inverse DFT, which reduces the computational problem from N2 to N∙logN

and thus enables faster reconstruction.

In the case of non-Cartesian sampling, k-space data is not located

on a conventional Cartesian grid, and thus the use of inverse FFT for

reconstruction is not valid anymore. A straightforward approach for non-

Cartesian image reconstruction would be direct evaluation of Equation 2.27.

However, this approach is computationally expensive and is impractical for

clinical use.

A generalized and efficient approach for non-Cartesian image

reconstruction is the use of non-uniform FFT, which is also known as

gridding (33,34). The reason this works is because the image support is

finite and thus each point in k‐space can be estimated by convolution with a

sinc function, which corresponds to the multiplication of the image by a

rectangular function in the image domain. In a standard gridding

reconstruction, the measured k-space points on an arbitrary trajectory (i.e.,

radial or spiral) are first interpolated onto a Cartesian grid, and then

subjected to the inverse FFT to reconstruct the final images. In actual

practice, gridding reconstruction typically requires multiple steps.

32

The sampling density of a non-Cartesian trajectory is typically not

uniform, with denser sampling usually performed in the center of k-space

than in the outer regions. This leads to significant image blurring in the

reconstructed image, because signal from the dense sampling regions

accumulates and thus results in overestimation of the central k-space

regions. Therefore, a density compensation process is usually needed

before the interpolation. This compensation can be achieved by weighting

each k-space point with the inverse density of the corresponding sampling

positions in k-space, which is called a density compensation function

(DCF). The DCF can be estimated by computing a Voronoi diagram of the

trajectory (35), or in the case of radial sampling, the DCF can be simplified

as a Ram-Lak filter (also known as an M filter) because the sampling

density of each k-space point is inversely proportional to its distance to the

k-space center.

The interpolation process in a gridding reconstruction can be

achieved by convolving the measured k-space data with a pre-defined

kernel at the desired k-space positions

( ) ( )ii

i

s k s K k k ˆ (2.34)

Here ik k is the distance between the ith k-space point and the desired

position in Cartesian grid, and K is the kernel used for convolution. The

33

convolution with a kernel in k-space leads to a multiplication of the image

with the Fourier transform of the kernel, thus it would in a sense be ideal to

choose the sinc function because it can preserve the original image

information. However, it is generally impractical to compute the convolution

with a full sinc function, and thus a compact kernel has to be used to

approximate the properties of the sinc function. The Kaiser-Bessel kernel

has been shown to provide good image quality (33) and thus it is commonly

used in gridding reconstruction. Moreover, in order to avoid aliasing effects

due to the interpolation with a finite kernel, the convolution is usually

evaluated on a two-fold oversampled Cartesian grid with half of the

sampling distance. This leads to doubled FOV in the final results, and the

reconstructed image can be cropped back to the original FOV at the end of

the reconstruction.

After the interpolation process, a two-dimensional inverse FFT can

be directly applied to reconstruct the image. However, due to the

convolution with a finite kernel in k-space, the reconstructed image is

modulated by the Fourier transform of the kernel. This undesired

modulation, which is called apodization or roll-off effect, can be

compensated by dividing the image by the Fourier transform of the kernel,

which is known as deapodization or roll-off correction.

2.6. Parallel MRI

34

2.6.1. The Need for Speed in MRI

MR imaging speed is of critical importance in many clinical

applications. However, the imaging speed with which gradient-encoded MR

images can be acquired is fundamentally limited by the sequential nature of

gradient-based MR acquisitions, in which only one k-space line can be

acquired per unit time. In order to accelerate data acquisition, conventional

MRI requires stronger field gradients, faster gradient switching rates, and/or

more frequent RF excitations. Faster gradient switching can lead to

peripheral nerve stimulation, and more frequent RF pulses can result in

increased RF power deposition, with an increased risk of damaging

biological tissues.

Another way of accelerating MR imaging is to acquire reduced

quantities of measurements without compromising image information or

spatiotemporal resolution. Partial Fourier imaging is one of the simplest

approaches for accelerated data acquisition in MRI. In this approach,

approximately half of the k-space lines are acquired and the rest are

estimated by exploiting the conjugate symmetry property of the Fourier

transform for real-valued objects (14-16). However, the maximum

acceleration factor in Partial Fourier imaging is limited to less than two.

Parallel MRI, introduced in earnest in the late 1990s (17-19), is a more

popular and more flexible approach for increasing imaging speed. Parallel

35

Figure. 2.5: An example of multicoil brain images with corresponding coil sensitivity maps with 8 coil elements. Each individual coil element has a different spatially-varying sensitivity pattern.

MRI uses spatial information from an array of RF receive coils to perform

some portion of the spatial encoding that is normally accomplished via field

gradients. Since signals in multiple receive coil elements may be acquired

simultaneously, coil-based encoding enables acquisition of multiple lines of

image data at the same time (20). Over the past two decades, parallel MRI

has evolved rapidly, and it is implemented on most clinical MR scanners in

use today. A generalized formulation of parallel MRI is briefly reviewed in

this section.

2.6.2. Spatial Encoding Using Coil Arrays

In parallel MRI, an array of RF receiver coils is used for data

acquisition. Each individual coil element has different spatially-varying

sensitivities, as shown in Figure 2.5. In the case of two-dimensional

imaging, the k-space data acquired in each coil-element can be described

by adapting Equation 2.17 as follows:

36

2 [ ( ) ( ) ]

( , ) ( , ) ( , )x y

x yl lx y

i k t x k t ys k k c x y m x y e dxdy

(2.35)

Here, cl 1,2,3, ,N is the coil index ( cN is the total number of coil

elements) and ( )lc x, y is the coil sensitivity map for the l th coil element. A

hybrid encoding function can then be formulated as

( ) ( )exp( 2 )exp( 2 )x y x yl lE x,y,k ,k c x, y i k x i k y (2.36)

Equations 2.35 and 2.36 suggest that as long as the combination of

different coil sensitivities can imitate the gradient encoding, some k-space

lines that are traditionally acquired using gradient encoding can be

generated retrospectively using the coil sensitivities. In other words, the use

of multiple receiver coils with different spatial sensitivities allows for

acquisition of multiple k-space lines simultaneously.

2.6.3. Generalized Parallel MRI Reconstruction

The hybrid encoding function formulated in Equation 2.36 does not

form a pure Fourier basis, and thus reconstruction is more involved than an

inverse Fourier transform. Combining Equation 2.35 and 2.36, we have

( , ) ( , ) ( , , , )x yx ylx y

s k k m x y E x y k k dxdy (2.37)

Or, with appropriate discretization

,

( , ) ( , ) ( , , , )x yx yl

x y

s k k m x y E x y k k (2.38)

Equation 2.38 can be rewritten in matrix notation as

37

s =Ε m (2.39)

Here vector s contains the measured multicoil MR signal (with size

NcNk×1), m is the image vector to be reconstructed (with size N𝜌×1), and

E is the hybrid encoding matrix (with size NcNk×N𝜌). It can be seen from

Equation 2.39 that theoretically the encoding matrix E remains invertible as

long as gradient encoding steps are reduced by no more than a maximum

factor of Nc. In general, the reconstruction problem is over-determined, and

thus reconstruction of image m̂may be performed by minimizing the mean

square error

2

2m

ˆmin E m-s (2.40)

As is the case for Equation 2.32, the solution of Equation 2.40 can be found

by computing the Moore-Penrose pseudoinverse

1 1 H H

m̂ = (E E)E s (2.41)

Here, the pseudoinverse has been generalized by including a noise

covariance matrix , which accounts for correlations in noise contributions

from distinct array elements, and which can be computed from

measurements in the absence of RF excitation (i.e. with a flip angle of

zero). The noise covariance matrix is a practical tool to evaluate the noise

distribution in different coil elements in the measurement system. The best

reconstruction result can be obtained when any noise shared by more than

38

one coil is removed from the problem. This step is commonly known as

noise decorrelation or noise whitening.

Practical parallel MRI reconstruction methods are usually

implemented based on particular sampling patterns (i.e., regular

undersampling), or else using iterative reconstructions in the case of non-

Cartesian sampling. Different parallel imaging techniques, such as SMASH,

SENSE, and GRAPPA, have been extensively developed and broadly

implemented for day-to-day clinical exams. The discussion of specific

parallel imaging algorithms can be found in many classic papers and

textbooks (36-39).

2.6.4. Estimation of Coil Sensitivities

In parallel MRI, accurate coil sensitivity information is required in

order to reconstruct the image from undersampled k-space data. Since the

sensitivities can depend not only on the geometry of the coil array, but also

on the object be imaged, estimation of coil sensitivities is required for each

individual scan. Additional reference data, which can be either acquired

separately or self-calibrated, is used for coil sensitive estimation.

In parallel image reconstruction methods like SENSE, coil sensitivity

maps must be calculated explicitly and then used to compute the encoding

matrix so that the aliased image can be unfolded from undersampled

measurements. In parallel image reconstruction methods like GRAPPA (19)

39

or SPIRiT (31), on the other hand, the coil sensitivity maps do not need to

be calculated explicitly and the missing k-space lines are estimated using

autocalibrated measurements acquired as an integral part of the data

measurements. Autocalibration has the advantage that reconstructions are

less sensitive to temporal changes such as motion. In dynamic imaging, the

sampling pattern in different temporal frames can be specifically designed,

so that the coil sensitivities can be estimated from the averaging of all the

dynamic frames. In these methods, additional reference data is not

necessary for coil sensitivities estimation.

The raw sensitivity maps are generally corrupted by noise, and many

approaches have been proposed to produce smooth sensitivity maps, such

as the adaptive array combination (40).

2.6.5. SNR in Parallel MRI

Although parallel imaging has already enabled significant progress in

rapid MRI, it has been found that acceleration in parallel imaging is limited

by noise amplification in the reconstructed images, whose extent is

ultimately determined by RF receiver coil design (41). The classic SNR

equation for parallel imaging was proposed by Pruessmann (18) as

0SNR

SNRg R

(2.42)

40

It can be seen from Equation 2.42 that the SNR in parallel imaging is

determined by three terms, including:

i) The baseline SNR0 of the system or the acquisition protocol, which is

related to many factors such as echo time (TE), repetition time (TR), and

other sequence parameters.

ii) The undersampling factor R , which is related to the parallel imaging

acceleration. This is consistent with the SNR Equation described in

section 2.4.2, which indicates that the SNR drops whenever the

sampling time Ts is decreased.

iii) The geometry-factor, or simply g-factor, g , which is related to the coil

design or coil geometry. The g-factor results from the nonunitarity of the

combined coil-sensitivity- and gradient-based encoding functions, and

the corresponding ill-conditioning of the encoding matrix inverse.

Intuitively speaking, high degrees of overlap in coil sensitivities results in

amplification of noise out of proportion to signal, especially in the case of

excessive accelerations.

It has been demonstrated that there is a fundamental upper limit to

acceleration for a given level of acceptable noise amplification (41,42). This

limit arises from the smoothness of physically realizable electromagnetic

fields, as described by Maxwell’s equations. Generally speaking,

accelerations beyond a factor of 3 or 4 for 2D imaging or a factor of 6 to 10

41

for 3D imaging are starved for SNR, even when arrays with very large

numbers of elements are used.

2.7. Compressed Sensing MRI

Parallel imaging has led to revolutionary progress in the field of rapid

MRI in the past two decades. However, as discussed in the previous

section, the maximum acceleration that can be achieved in parallel imaging

is limited by the number and the design of coils, and ultimately by

fundamental electrodynamic principles. Compressed sensing (26,27) is

another powerful approach that can be applied to accelerate data

acquisitions in MRI (28), and has attracted enormous attention since its

introduction. Compressed sensing can be combined with parallel imaging in

MRI to further increase imaging speed by exploiting joint sparsity in multicoil

images (29-31). In this section, the basics of compressed sensing, its

application for MRI, and its combination with parallel imaging, are briefly

reviewed.

2.7.1. Introduction to Compressed Sensing

Conventional schemes for sampling a signal must satisfy the

requirements of the Nyquist theorem: namely, that the sampling rate must

be at least twice the maximum bandwidth presented in the signal.

Unfortunately, in many applications, Nyquist sampling is time-consuming

42

and data-intensive, posing a challenge for sampling system design, data

storage, and transmission. In order to address this logistical and

computational challenge, high-dimensional data are usually compressed

after acquisition by transforming to a basis that provides a sparse or

compressible representation for the signal and discarding insignificant

components. This transform coding framework has been widely used in the

JPEG, JPEG2000, and MPEG standards.

The ability to compress images so effectively raises an interesting

question, one which underlies compressed sensing: instead of first

sampling a signal at a high sampling rate and then discarding most of the

sampled measurements in the compression process, why not directly

acquire the data in a compressed form at a lower sampling rate? In other

words, can we build the compression process directly into the acquisition or

sensing step, so that one does not have to perform so many measurements

and discard most of them afterwards? Candes, Romberg, Tao, and Donoho

proved the feasibility of this hypothesis and proposed a framework by which

a sparse or compressible signal could be successfully recovered from

undersampled measurements that are far below the Nyquist limit (26,27).

After rapid development in the past decade, compressed sensing

has already achieved notable impact in a wide range of application areas,

such as sensor design in high-resolution cameras and medical imaging.

43

One of the applications that can substantially benefit from compressed

sensing is MRI, in which the imaging speed can be dramatically improved if

three requirements can be satisfied, including: i) the image is sparse or has

a sparse representation in some transform domain; ii) incoherent sampling,

seen in the structural preservation of image content and the structural

disintegration of artifacts in the sparse domain; iii) a non-linear

reconstruction to recover the image by removing the incoherent artifact. The

discussions in the following sections will focus on these aspects.

2.7.2. The Sensing Problem

The sampling of a signal vector mwith size N×1 can be interpreted

as a projection of the signal onto the sampling waveforms i

,i is i 1,2,3,...,M m (2.43)

Here s is a measurement vector with size M×1. In matrix notation, we have

s =Φm (2.44)

Here the waveforms form the sampling matrix Φ with size of M×N.

In the case of undersampling, the number of measurements is

smaller than the dimensionality of the signal (M<N) and therefore the

system is under-determined, assuming that the sampling matrix Φ has full

rank. As shown in section 2.5, the solution to Equation 2.44 is not unique.

However, if the signal m is known to be sparse and the sampling matrix Φ

44

Figure. 2.6: Sparse representation of a brain image in wavelet transform domain. By keeping only the largest 10% coefficients and discarding the rest, the image can still be recovered without loss of important information but with 10-folder smaller size.

satisfies certain specific requirements, it is possible to recover the signal

from an incomplete set of measurements. For example, supposing a signal

is K-sparse with K<M, which means that there are at most K nonzero

components in the signal vector, and the locations of the nonzero

components are also known. In this case, the measurement matrix Φ can

be reduced to KΦ , with size M×K, by keeping only the K columns

corresponding to nonzero locations. The signal m can be similarly reduced

to Km with only the nonzero coefficients. Therefore, the system becomes

K Ks =Φ m (2.45)

In this situation, only K measurements are sufficient to recover the signal.

2.7.3. Sparsity

From the previous subsection, we know that the first requirement in

compressed sensing is sparsity of the signal. A signal is said to be sparse if

45

most of its coefficients are zero or close to zero, and the relatively few large

coefficients can capture most of the information. In other words, sparsity

implies that one can discard the small coefficients without significant

perceptual loss. The signal can be sparse itself, or it can be compressible,

which means that the signal has a sparse representation in some

appropriate transform domains. Mathematically speaking, a compressible

signal m can be expanded in a transform basis (such as a wavelet basis)

Ψ as follows:

m=Ψx (2.46)

Here, x is the sparse representation of signal m in the basis defined by Ψ .

In Figure 2.6 we can see how a dataset that is not sparse in the

image domain, a brain MR image, may have a sparse representation in a

different domain, the wavelet transform domain. By keeping only the largest

10% of coefficients and discarding the rest in the transform domain, the

image can still be recovered without losing important information, but with

10-folder smaller size. Dynamic images have much higher sparsity than

static images because of the the presence of significant temporal

correlations associated with periodic motion or gradual evolution.

Therefore, a dynamic image series can have sparse representation with an

appropriate transform applied along the dynamic dimension. Figure 2.7

46

Figure. 2.7: A cardiac cine image series has temporal correlation because dynamic region is limited in only a small region, while the background is static. An FFT can be employed along the temporal dimension to sparsify the dataset.

shows an example of sparse representation of a cardiac cine image series,

in which a temporal FFT is employed as the sparsifying transform.

2.7.4. Conditions for Sparse Signal Recovery

In order to reconstruct the sparse signal from undersampled

measurements, the measurement matrix must be designed properly so that

it can preserve the information of the sparse signal. Equations 2.44 and

2.46 can be combined as

s =ΦΨx Ax (2.47)

47

Here the measurement matrix Φ and the sparse representation matrix Ψ

are combined as a single matrixA . We can define the isometry constant

K of A as the smallest number such that

2 2 2

2 2 2(1 ) (1 )K K x Ax x (2.48)

holds for all K-sparse vectors x (26). A is said to obey the restricted

isometry property with a constant K if Equation 2.48 is satisfied. Good

performance of compressed sensing can be ensured when K is close to

zero. Equation 2.48 implies that the matrix A can preserve at least part the

energy of a K-sparse vector. In other words, it suggests that any K-sparse

vectors cannot be in the null space of matrix A and all subsets of K

columns taken from A must be linearly independent or even nearly

orthogonal. Note that in the case of undersampling, all of the columns of

matrix A cannot be exactly orthogonal, because A is a wide matrix, with

more columns than rows. As proposed by Candes (26), a sufficient

condition to ensure stable recovery of a K-sparse signal is that A satisfies

2 2 2

2 22 2 2(1 ) (1 )K K x Ax x (2.49)

with an isometry constant 2 2 1K for any 2K-sparse vectors.

2.7.5. Sampling and Incoherence

48

Although restricted isometry property guarantees sparse signal

recovery, it is computationally expensive to verify in practice. Therefore, it is

preferable to exploit some properties of A that can be easily evaluated. The

mutual coherence of a matrix, which is defined as the largest absolute inner

product between any two columns, is one such property, and can be

formulated as follows:

2 2

,μ( ) max

i j

i ji j

A A

A AA (2.50)

The mutual coherence can be estimated by calculating the Gram matrix

HA A . When A is very close to the identity matrix, the off diagonal entries

in matrix HA A are small, suggesting high incoherence and therefore

ensuring good performance in compressed sensing. One can relate the

mutual coherence to the restricted isometry property because μ must be

small when Equation 2.48 is satisfied.

The restricted isometry property also implies that the measurement

basis Φ must be incoherent with sparse representation basisΨ . In other

words, a signal having a sparse representation in a given transform domain

Ψ must spread out in the sampling domain Φ , thus ensuring sparse signal

recovery from only a small subset of measurements. Supposing Φ and Ψ

49

Figure. 2.8: Sampling matrix and the corresponding Gram matrix HA A . Ψ is set as

the identity matrix and Φ is the fully sampling Fourier matrix (a) and partial Fourier matrices with regular (b) and random (c) undersampling schemes. The off-diagonal entries in (c) are very small, suggesting that random undersampling is good for compressed sensing.

are both orthonormal bases, incoherence can be measured by calculating

the coherence between two bases as

1 ,

( , ) max ,jkk j N

N

Φ Ψ (2.51)

Herek

and j

are normalized to 1, and [ ]1, N . Equation 2.47

calculates the largest correlation between any two elements of Φ and Ψ .

Small coherence indicates that Φ and Ψ do not contain correlated

elements, and thus are paired well for compressed sensing.

Since MRI k-space data are acquired in the spatial frequency

domain, the measurement matrix is the partial Fourier matrix in the case of

50

undersampling. Therefore, the mutual coherence can be used as a metric

to evaluate different undersampling schemes. Figure 2.8 shows the Gram

matrix HA A , where Ψ is set as the identity matrix and Φ is the full Fourier

matrix (a) and partial Fourier matrices that were taken from (a) in a regular

(b) and random (c) fashion. It can be seen from Figure 2.8 that regular

undersampling leads to aliasing artifacts, while random undersampling

produces high incoherence (small off-diagonal coefficients) that is preferred

in compressed sensing. In fact, random undersampling is broadly employed

for Cartesian sampling in compressed sensing MRI, and a more common

way of measuring incoherence is to evaluate the point spread function

(PSF) of the undersampling mask, as proposed by Lustig (28). Moreover,

some practical restrictions have to be considered when designing an

undersampling pattern. For example, since most of the energy is

concentrated in the low-frequency region of k-space, taking more

measurements from the k-space center is preferred. Because

undersampling along the frequency-encoding dimension does not save

acquisition time in Cartesian MRI, undersampling is only performed along

the phase-encoding dimensions. Therefore, the ability to exploit

incoherence in conventional Cartesian imaging is limited to the phase-

encoding dimensions only. Figure 2.9 shows an example of a 1-dimensional

variable density undersampling pattern and the corresponding PSF along

51

Figure. 2.9: An example of one dimensional variable density undersampling pattern and the corresponding incoherence, represented by the point spread function (PSF) of the undersampling pattern.

the phase-encoding dimension. More freedom can be achieved by

extending the sampling scheme into three-dimensional or non-Cartesian

sampling, in which undersampling can be performed along more than one

dimension. Moreover, dynamic MRI is particularly interesting for

compressed sensing, because different undersampling patterns can be

applied in different dynamic frames, providing the possibility of exploiting

incoherence along the temporal dimension.

2.7.6. Image Reconstruction

The compressed sensing theory states that as long as the signal is

sparse and the requirement for the restricted isometry property or

incoherence is satisfied, the sparse solution is unique and guaranteed (26).

52

The measurement of sparsity in a signal vector can be performed by

calculating the zero-norm (L0-norm), which is defined as:

0

0x i

i

x (2.52)

The L0-norm simply counts the number of nonzero coefficients in a vector.

Therefore, the reconstruction of the sparse signal x in compressed sensing

can be formulated by minimizing the L0-norm problem

0arg min

. .s txx

s =ΦΨx (2.53)

or

0arg min

. .s txx

s = xA (2.54)

Because the measurements are usually contaminated by noise, Equation

2.54 is modified as follows:

0

2

2

arg min

. .s t

xx

s - Ax (2.55)

Here is the estimated noise level and the L2-norm is applied to quantify

the power or energy in the difference between the measurements and the

estimated solution. A more general formulation for solving the problem is to

find the signal m that has a minimized L0-norm in the sparse domain

53

0

2

2

arg min

. .s t

H

fΨ m

s -Φm (2.56)

Here H is defined as the Hermitian conjugate. While the L0-norm can

exactly determine the sparsest solution, minimizing the L0-norm is an NP-

hard problem, which is computationally intensive. Therefore, different

approaches that can be applied to solve the compressed sensing problem

more efficiently have attracted lots of attention. Currently the main type of

sparse recovery algorithms is related to the basis pursuit (BP), in which a

L1-norm minimization is used to replace the L0-norm minimization as

follows:

1

2

2

arg min

. .s t

H

fΨ m

s -Φm (2.57)

Here the L1-norm is defined as the summation of the absolute values of all

the coefficients in a signal vector

1

x i

i

x (2.58)

The L1-norm minimization problem in Equation 2.58 is a convex problem

and thus guarantees a global minimum over all the possible solutions. It has

been demonstrated that as long as certain conditions can be satisfied (e.g.,

the restricted isometry property), the solution of minimizing the L1-norm can

be equivalent to the solution of minimizing the L0-norm (26). The

54

Figure. 2.10: Combination of compressed sensing and parallel imaging enables reduced incoherent artifact level when compared with a single coil model.

constrained minimization problem in Equation 2.57 can be alternatively

formulated as an unconstrained problem using Lagrange multipliers

(Equation 2.59), so that it can be efficiently solved using algorithms like

gradient descent or conjugate gradient methods.

2

2 1arg min H

ms -Φm Ψ m (2.59)

The regularization parameter in Equation 2.52 controls the trade-off

between the data consistency (L2-norm) and the promotion of sparsity (L1-

norm).

2.7.7. Combination of Compressed Sensing and Parallel Imaging

Compressed sensing can be combined with parallel imaging to

further increase imaging speed by exploiting the idea of joint multicoil

sparsity. These two approaches can be synergistically combined as image

55

sparsity and coil sensitivity encoding are complementary sources of

information. On one hand, compressed sensing can serve a regularizer for

the inverse problem in parallel imaging and prevent noise amplification. On

the other hand, the additional spatial encoding capabilities of multiple

receiver coils in parallel imaging enable exploitation of joint sparsity in

multicoil images, and thus higher acceleration rates. In this section, a

reconstruction framework called k-t SPARSE-SENSE (30), which combines

compressed sensing and parallel imaging using a SENSE-type

reconstruction scheme, is briefly reviewed.

Instead of performing compressed sensing reconstructions for each

individual coil separately, a joint compressed sensing reconstruction that

enforces joint sparsity in the multicoil model is applied in k-t SPARSE-

SENSE. With this approach, the additional spatial encoding capabilities of

multiple receive coils can be exploited to reduce the incoherent aliasing

artifacts in the multicoil combination, thus enabling higher acceleration

rates. Figure 2.10 shows the reduction of incoherent artifact when

comparing a joint 8-coil array model to a single coil model.

The reconstruction of k-t SPARSE-SENSE can be formulated as

2

2 1arg min H

m-ΦCm s Ψ m (2.60)

56

Here Φ is the same measurement matrix as before and 21, ,... nC C CC

are the coil sensitivities in n coil elements. When compared with Equation

2.59, the L1-norm term in 2.60 enforces joint multicoil sparsity, since m

represents the combined multicoil image and is comprised of contributions

from all coils. The coil sensitivity maps can be self-calibrated or estimated

from reference data acquired separately. Reconstruction algorithms that

solve the compressed sensing problem in Equation 2.59 can be applied for

the joint compressed sensing problem with the incorporation of coil

sensitivities.

In addition to the combination with parallel imaging in a SENSE-type

framework, compressed sensing can be also combined with

GRASP/SPIRiT-type parallel imaging methods, such as L1-SPIRiT

proposed in (31). The performances of different combinations would

achieve similar performance though the coil sensitivities are used in

different ways, e.g., autocalibrated or explicitly calibrated. The differences

between SENSE and GRAPPA, however, will affect the corresponding

combination. For example, because GRAPPA or SPIRiT use a kernel with a

pre-defined kernel size to estimate the missing k-space points, standard

variable density random undersampling patterns may fail to achieve good

performance because there are larger gaps in the outer part of k-space

than the central part, and thus the estimation of high-frequency k-space

57

points can be challenging. Therefore, different random undersampling

strategies (e.g., poisson-disc undersampling) are required. On the other

hand, variable density random undersampling would achieve good

performance in the combination of compressed sensing with a SENSE-type

framework because SENSE effectively uses all of the k-space point to

estimate the missing k-space points (20).

2.7.8. Low Rank Matrix Completion

The undersampled dataset in MRI can be described in terms of a

matrix with missing entries, and correspondingly, the reconstruction of an

image from undersampled data can be formulated as a procedure to fill the

missing k-space lines. If a matrix is in low-rank condition, it can be

represented by only a few dominant singular values, and their

corresponding singular vectors, without loss of important information. In

other words, an image can have a sparse representation in the singular

value decomposition (SVD) domain if it has low rank. Therefore, the low

rank property of a matrix can be applied to enforce sparsity and exploit the

correlation in MRI images (43). Since dynamic image series usually have

much higher correlation with much lower rank along the temporal

dimension, the low rank constraint is often applied in the reconstruction of

undersampled dynamic MRI.

58

As shown in Equation 2.46, the measurement of an image can be

described as

s =Φm (2.61)

Here m is the image to be reconstructed and s are the measurements. In

the case of undersampling, the system is underdetermined and the solution

is not unique. However, if the image to be reconstructed is known to have

low rank, and if the requirement for incoherent sampling, as described in

section 2.7.5, is satisfied, a natural way to complete the missing entries in a

matrix is to search for the solution that has the lowest rank:

2

2

min rank( )

. .s t

m

s -Φm (2.62)

Unfortunately, similar to minimization of the L0-norm, rank-minimization is

also an NP-hard problem and there is no efficient way to solve this problem.

Usually the nuclear norm is used to replace the rank function in Equation in

2.57 as follows:

*

2

2

min

. .s t

m

s -Φm (2.63)

Here * is the nuclear norm of a matrix, given by the sum of its singular

values. In dynamic MRI, the nuclear norm is enforced on the corresponding

Casorati matrix of the image series, in which each image frame is

concatenated into an individual column in the Casorati matrix.

59

The major difference between standard compressed sensing

reconstruction with L1-norm minimization (Equation 2.63) and low rank

completion with nuclear norm minimization (Equations 2.57) is that SVD is

performed as the sparsifying transform to exploit the sparsity of images,

and therefore, it does not require an explicit sparsifying transform. Image

series in specific applications, such as MR parameter mapping, have very

low rank, and thus reconstruction with nuclear norm minimization can

achieve superior performance to standard compressed sensing

reconstructions that use a temporal sparsifying transform such as temporal

FFT, as will be seen in Chapter 3. The exploitation of the low rank property

can be further combined with a sparsity constraint to enable reconstruction

with improved performance (44).

2.8. Motion in MRI

2.8.1. Influence of Motion in MRI

As discussed in the previous sections, MRI data acquisition is

comparatively slow, which leads to sensitivity to subject motion (45).

Loosely speaking, motion in MRI can be divided into three major categories:

rigid body motion (or bulk motion), blood flow, and elastic motion (i.e.,

cardiac motion, respiratory motion, and gastrointestinal peristalsis) (46).

According to the shift property of the Fourier transform, any motion of the

object being imaged generates a phase modulation in k-space, which

60

results in ghosting artifacts in conventional Cartesian imaging. Ghosting

artifacts appear as replicated copies of the moving object exclusively along

the phase-encoding direction. Since the ghosting artifacts are generally

overlaid with the true image representation, they also lead to blurring in the

image.

Although the sensitivity to motion in MRI can sometimes be used to

produce useful information (e.g., velocity mapping of blood flow), they

usually degrade image quality and can result in diagnostic

misinterpretations. Since the introduction of MRI, tremendous efforts have

been made to mitigate or correct for motion artifacts. There are various

strategies to handle different types of motion in MRI. For example, the

subjects being imaged can be immobilized. Specific strategies in sequence

design (e.g., gradient moment nulling) can be used to compensate artifacts

induced by blood flow. Cardiac motion can be eliminated by synchronizing

and gating the data acquisition with the cardiac contraction of the subjects,

so that the data acquisitions are only performed in specific quiescent period

(e.g., mid-diastole). Among various types of motion, respiratory motion is

one of the most frequent sources of artifacts. Although it can be avoided

with suspension of breathing during scans, breath-hold capabilities are

subject-dependent and can be significantly limited in some patients.

Meanwhile, typical breath-hold durations (10-15 seconds) also limit spatial

61

resolution and volumetric coverage. Therefore, free-breathing MRI is

desirable and attractive. Different techniques that can be used for free-

breathing MRI exams are surveyed in the following subsection.

2.8.2. Free-Breathing MRI Techniques

The most popular free-breathing approach is to use either navigator

signals (47) or respiratory bellows (48) to monitor respiratory motion, so that

data are only acquired at a specific respiratory state (e.g., end-expiration).

This approach is widely used in cardiovascular MR exams. However, this

gated data acquisition significantly reduces imaging efficiency and further

prolongs the total examination times. Real-time MRI is another approach

that can be used for free-breathing cardiac cine imaging (49), but the

acquisitions usually comprise only a single slice with limited spatial and

temporal resolution. Therefore, advanced image reconstruction techniques,

such as temporal parallel imaging or dynamic compressed sensing imaging,

are needed. Non-Cartesian k-space sampling schemes, such as radial or

spiral sampling, are substantially less sensitive to motion, and thus enable

free-breathing imaging, sometimes at the expense of increased scan times

(50-52). For example, radial imaging eliminates k-space gaps due to

motion-related phase shifts, by repeated sampling of the k-space center.

However, substantial motion is still a challenge for non-Cartesian imaging

and can result in blurring and undersampling artifacts, e.g. streaks in radial

62

imaging. Non-Cartesian imaging also offers the potential benefit of

retrospective self-gating, owing to the continuous passage of the radial lines

through the center of k-space, and can therefore eliminate the need to use

navigator signals or other external gating devices (53,54). However, the

self-gating approaches are still inefficient, since typically only data within a

predefined motion state (e.g., close to expiration) is used for reconstruction.

Another type of approach has also been proposed to correct for

respiratory motion by integrating an image registration framework into the

reconstructions. For example, a rigid-body motion registration framework

can be applied for respiratory motion correction in coronary MR

angiography (MRA) or cardiac perfusion imaging (55). More complex

deformable registration techniques that account for non-rigid body motion

can be also employed for reconstruction of cardiac cine images, cardiac

perfusion images, and abdominal DCE-MRI images (56-58). Moreover,

some approaches are even able to learn the motion fields to guide image

reconstruction, which, in addition to performing motion correction, can also

provide access to specific motion information (59). These motion correction

schemes, however, require the use of specific motion models, which may

be insufficient to account for the complex movement of the organs during

respiration, especially for patients with pronounced respiration or irregular

respiratory patterns. A novel free-breathing technique is proposed in this

63

dissertation to perform motion sorting and the reconstruction of additional

respiratory motion dimension in place of conventional motion correction.

The technique is called XD-GRASP (eXtra-Dimensional Golden-angle

RAdial Sparse Parallel imaging) and will be described in Chapters 6 and 7,

with demonstration of applications to both abdominal and cardiovascular

imaging.

64

Chapter 3

Accelerated T2 Measurement of the Heart Using k-t

SPARSE-SENSE

3.1. Prologue

As briefly reviewed in Chapter 2, compressed sensing can be

combined with parallel imaging (e.g. with approaches like k-t SPARSE-

SENSE) to further increase imaging speed and improve reconstruction

performance by exploiting joint sparsity in multicoil images. In this chapter,

we investigate the feasibility of applying a tailored k-t SPARSE-SENSE

reconstruction framework for accelerated parameter mapping by exploiting

the low rank property of the dynamic image series. Specifically, we focus on

rapid T2 measurement in the heart, which is particularly interesting but

challenging due to the influence of both cardiac motion and respiratory

motion.

65

The contents presented in this chapter were published in the journal

Magnetic Resonance in Medicine (MRM 2011 Jun; 65(6):1661-9) (60).

3.2. Introduction

T2-weighted MRI is a valuable method for detecting T2 changes that

are induced by a variety of diseases in the liver (61-64), heart (65-68), and

other organs. The most widely exploited contrast mechanism in T2-weighted

MRI is the alteration of water content by diseases, with edematous tissue

exhibiting increased T2-weighted signal and excessive iron accumulation

resulting in decreased T2-weighted signal. However, the resulting clinical

interpretation in T2-weighted images is often hindered by surface coil effects,

which can yield non-uniform MR signals unrelated to pathology.

Quantitative tissue characterization with T2 measurement or T2

mapping can overcome the limitations associated with T2-weighted MRI and

thus can improve the accuracy of assessing the severity of diseases.

Multiple single spin-echo pulse sequence with different echo times is

currently considered the reference technique for T2 quantification. However,

this technique is substantially inefficient for clinical application because

sufficient time is needed for spin relaxation after acquisition of each k-space

measurement. An alternative approach for T2 quantification is multiecho fast

spin echo (ME-FSE) pulse sequence (9), which accelerates the data

acquisition with the echo train length. While the ME-FSE sequence

66

considerably reduces the total acquisition time comparing to multiple single

spin-echo acquisitions, the imaging efficiency is still relatively low,

especially for abdominal or cardiac imaging that are affected by cardiac

motion and respiratory motion. Moreover, in the presence of radio

frequency field (B1) inhomogeneity, the flip angle variation can produce

indirect echoes and thus result in systemically error in T2 quantification. The

use of phase-cycling in Carr-Purcell-Meiboom-Gill (CPMG) sequence

(69,70) has been used to correct the flip angle variation for even echoes

and thus only even echoes can be kept to form the images with improved

accuracy in T2 quantification at the expense of reduced imaging efficiency.

In order to overcome the limitations associated with the ME-FSE

sequence, the sequence has been modified previously with “reverse centric”

k-space reordering, in which two successive echoes are used to form an

image with the odd and even echoes acquired at the outer and inner halves

of k-space, respectively (71). Therefore, this new ME-FSE sequence can

characteristically behave like an even-echo CPMG sequence with two-fold

acceleration. The accuracy of this sequence has been validated in vivo

against the even-echo only CPMG sequence, as described in (71).

Combining the two-fold acceleration in this new ME-FSE sequence with

conventional parallel imaging enables data acquisition within single breath-

hold for abdominal and cardiac imaging with spatial resolution on the order

67

of 2.7 x 3.8 mm2 (71). However, this spatial resolution may still not be

adequate, particularly in cardiac imaging, which is sensitive to partial

volume effects, especially in patients with thinned myocardial wall. Thus,

faster imaging speed is desirable for further accelerating T2 measurements.

T2 mapping is a good candidate for compressed sensing, because

the transverse magnetization (e.g., signal) as a function of time is

approximately mono-exponential and thus can be sparsely represented with

an appropriate transform basis along the echo dimension. In this work, we

propose to accelerate the ME-FSE pulse sequence described above for

cardiac T2 mapping with increased spatial resolution, using a tailored k-t

SPARSE-SENSE reconstruction framework adapted from (30). The results

were validated against that obtained from a reference approach using the

ME-FSE pulse sequence with parallel imaging only.

3.3. Low Rank Property in MR T2 Mapping

In MR T2 mapping, a dataset consists of a series of images that are

acquired at the same position but at different echo times and the signal

decay in each image pixel across the dynamic series is a function of T2

value to be estimated. Thus, the images along the echo dimension usually

have similar anatomical structures but with different T2-weighted contrast,

resulting substantial redundancy along the dynamic dimension. The data

redundancy in a T2 mapping dataset can be expressed as the low rank

68

Figure. 3.1: Low rank property of MRPM. (a): An example of cardiac T2 mapping image series, in which images at different echo times have similar anatomical structures but with different T2-weighted contrast. (b): The Casorati matrix generated from the image series. The Casorati matrix can be represented by a few dominant singular values and the corresponding singular vectors.

property of the corresponding Casorati matrix (Figure 3.1), in which each

image frame is concatenated into an individual column. Due to the low rank

property, the Casorati matrix can be represented by a few dominant

singular values and their corresponding singular vectors, suggesting that a

T2 mapping dataset is sparse and accordingly, the low rank constraint can

be used to reconstruct an accelerated T2 mapping dataset. This is known

as low rank matrix completion as reviewed in Chapter 2. The low rank

constraint can be enforced by performing a principal component analysis

(PCA) along the echo dimension as the sparsifying transform. Figure 3.2

shows the schematic details of performing a PCA on a reference cardiac T2

69

Figure. 3.2: Schematics details of estimating a PCA basis. By concatenating each time signal vector along column direction, a matrix V is constructed. A basis set for PCA is then estimated by conducting eigen-decomposition of the covariance matrix C of V.

mapping dataset. By concatenating each time signal vector along column

direction, a matrix V is constructed. Then, by conducting eigen-

decomposition of the covariance matrix C of V, a basis set for PCA can be

estimated.

In order to further demonstrate the low rank property of a T2 mapping

dataset, numerical simulation was performed to compare two different

sparsifying transforms, fast Fourier transform (FFT) and PCA along echo

dimension for a representative T2 mapping dataset. Using the

corresponding sequence parameters used in this study (see Pulse

Sequence section) and assuming T2 = 50 msec, an ideal monoexponential

curve representing the transverse magnetization was generated and plotted

70

in Figure 3.3a. The same monoexponential time curve was replicated for

each point on the image plane (x-y) to emulate the spatial resolution studied

in vivo, where x and y are two spatial dimension in an image. The temporal

FFT and PCA representations of this curve are shown in Figure 3.3b&c.

These plots clearly show that a monoexponential decay curve is sparser in

temporal PCA domain than in temporal FFT domain. To further validate this

finding, both temporal FFT and PCA representations of a reference cardiac

T2 mapping dataset are shown in Figure 3.3d&e. These preliminary results

prove that temporal PCA enables sparser representation than temporal FFT

for T2 mapping datasets, and confirms the rationale behind the exploitation

of low rank property in T2 mapping studies with compressed sensing.

3.4. Imaging Studies

The aim of the work was to achieve relatively high spatial resolution

(less than 2 x 2 mm2) for cardiac T2 mapping within clinically acceptable

breath-hold duration (less than 20 sec) using the ME-FSE sequence

developed in (71) and k-t SPARSE-SENSE reconstruction. An acceleration

rate (R) of 6, was needed to acquire a 192 x 192 x 16 (echoes) data matrix

within single breath-hold duration of less than 20 heart beats. For a field of

view of 320 x 320 mm2 and an inter-echo spacing of 5 ms, the acquisition

matrix corresponds to a scan time of 18 heart beats (one heart beat to

acquire the coil sensitivity data; one heart beat to acquire dummy scans to

71

Figure. 3.3: (a): A simulated monoexponential decay curve. (b): FFT representation of (a). (c): PCA representation of (a). These plots clearly show that a monoexponential decay curve is sparser in PCA domain than in FFT domain. To further validate this finding, a reference cardiac T2 mapping image series is displayed in both (d) FFT and (e) PCA domains. The results were consistent with the ideal curves shown (a–c).

approach steady-state of magnetization; and 16 heart beats to acquire the

image data) and an echo-train duration of ~160 ms.

3.4.1. k-Space Undersampling and Pulse Sequence

A different pseudo-random undersampling pattern with higher

density at the center of k-space for each time point was proposed for

applications of compressed sensing to dynamic imaging, in order to

maximize incoherence and reduce the resulting aliasing artifacts by

distributing them along two dimensions (72). We generated a similar 6-fold

accelerated variable-density random undersampling pattern along ky for

72

Figure. 3.4: (a): Six-fold accelerated ky-t undersampling pattern with 16 dynamic frames. (b) Corresponding PSF in the sparse y-PCA space using PCA as the sparsifying transform.

each t, where ky is the spatial frequency in the phase-encoding direction

and t is the echo dimension (Figure 3.4a). Figure 3.4b shows the resulting

point-spread-function (PSF) after applying fast Fourier transform (FFT)

along ky and PCA along t, where ky is the phase-encoding direction. The

ratio of the peak and standard deviation of PSF, which has been proposed

as a measure of incoherence (28), was 31.7.

For the purposes of this work, the ME-FSE pulse sequence with

parallel imaging alone is defined as the reference T2 mapping pulse

sequence, and the ME-FSE pulse sequence with k-t SPARSE-SENSE is

defined as the accelerated T2 mapping sequence. The details of the

reference T2 mapping pulse sequence can be found in (71).

Both the reference and accelerated T2 mapping pulse sequences

were implemented on a whole-body 3T MRI scanner (MAGNETOM

73

TimTrio, Siemens AG, Healthcare Sector, Erlangen, Germany) equipped

with a gradient system capable of achieving a maximum gradient strength

of 45 mT/m and a slew rate of 200 T/m/s. The RF excitation was performed

using the body coil, and a 32-element cardiac coil array (Invivo, Orlando,

FL) was employed for signal reception. Relevant imaging parameters for

both pulse sequences were: field of view = 320 x 320 mm2, slice thickness

= 10 mm, inter-echo spacing = 5 ms, number of images = 16, echo train

length per shot = 32, echo train duration = 163 ms, receiver bandwidth =

531 Hz/pixel, and double-inversion black-blood preparation pulses. As

mentioned previously, the sequence was employed with “reverse centric” k-

space reordering, in which two successive echoes form an image with the

odd and even echoes acquired at the outer and inner halves of k-space,

respectively.

The accelerated T2 mapping pulse sequence used R = 6, acquisition

matrix = 192 x 192 x 16, breath-hold duration = 18 heartbeats (one

heartbeat to acquire the coil sensitivity maps, one heartbeat to acquire

dummy scans to approach steady-state of magnetization, and 16

heartbeats to acquire the image data). The reference T2 mapping pulse

sequence used GRAPPA with R = 1.8, acquisition matrix = 192 x 76, and

breath-hold duration = 21 heat beats (one heart beat to acquire dummy

scans to approach steady-state of magnetization and 20 heart beats to

74

acquire the image data). Note that in order to maintain single breath-hold

duration on the order of 20s, the spatial resolution in the phase-encoding

direction in the reference T2 mapping pulse sequence was set at only 40%

of the resolution achieved in the accelerated T2 mapping pulse sequence.

3.4.2. Phantom Validation

For in vitro validation, we imaged a phantom consisting of five tubes

containing different concentrations of manganese chloride (MnCl2) in

distilled water: 0.135, 0.270, 0.405, 0.540, and 0.675 mM. MnCl2 was

chosen because it has T1/T2 on the order of 10, and these concentrations

were chosen to emulate clinically relevant T2 values in the myocardium.

3.4.3. T2 Mapping of the Heart

Twelve adult volunteers (seven men and five women; mean age =

26.1 ± 1.8 years) were imaged in a middle ventricular short-axis plane.

Electrocardiogram triggering was used to acquire image at middle to late

diastole, where cardiac motion is minimal. Human imaging was performed

in accordance with protocols approved by the New York University School

of Medicine Institutional Review Board and was found to comply with the

HIPAA. All subjects provided written informed consent before the imaging.

3.4.4. Improving Sparsity Using Preconditioning RF Pulses

75

Figure. 3.5: Schematic diagram of the proposed accelerated T2 mapping pulse sequence with preconditioning RF pulses. ECG triggering was used to image at mid to late diastole, to image at a cardiac phase where there is minimal cardiac motion. Three presaturation RF modules and a single fat suppression module were applied before ME-FSE readout.

As discussed in Chapter 2, the performance of compressed sensing

is determined by image sparsity. We hypothesize that suppression of bright

signals (e.g., fat signal) unrelated to the heart increases the sparsity of T2

mapping datasets and thus improve the performance of reconstruction.

Given that the proposed ME-FSE echo train is acquired at middle to late

diastole (~163 ms) per heartbeat, the pulse sequence permits the use of

multiple preconditioning RF pulses before ME-FSE readout (Figure 3.5),

without exceeding clinically acceptable specific absorption rate limit. We

explored the use of fat suppression and spatial pre-saturation RF pulses to

suppress bright signals unrelated to the heart (e.g., chest wall and back).

Note that pulse duration of a spatial pre-saturation module is ~4.8 ms and

that pulse duration of a fat suppression module is ~12.2 ms. Figure 3.6

shows a short-axis scout image to illustrate how preconditioning RF pulses

were utilized to increase the sparsity for cardiac T2 mapping imaging, as

well as the resulting four cases of results with GRAPPA (see Pulse

76

Figure. 3.6: (a): Representative short-axis scout image displaying positions and thicknesses of three presaturation RF pulses (displayed as meshed-strip lines). Resulting images with none (b), fat suppression (c), three spatial presaturation RF pulses (d), and fat suppression plus three spatial presaturation RF pulses (e). The combined use of fat suppression and spatial presaturation RF pulses produced the best suppression of bright signals unrelated to the heart.

Sequence section for the relevant imaging parameters) using different

preconditioning RF pulses: (i) none, (ii) fat suppression only, (iii) three

spatial pre-saturation pulses only, and (iv) fat suppression plus three

spatial pre-saturation RF pulses. The slice thickness of the pre-saturation

bands was graphically adjusted between 50 and 100 mm. Among the four

cases of preliminary results, the combined use of fat suppression and

spatial pre-saturation RF pulses produced the best suppression of bright

signals unrelated to the heart (Figure 3.6). Given the lack of consequential

penalty associated with the combined use of fat saturation and three spatial

pre-saturation pulses, we acquired accelerated T2 mapping images with fat

saturation plus the preconditioning RF pulses. To validate the usefulness of

increasing sparsity with preconditioning RF pulses, we compared both the

77

image quality and T2 accuracy of accelerated data with and without

preconditioning RF pulses.

3.5. Image Reconstruction

The GRAPPA image reconstruction was performed online, using

commercially available reconstruction algorithm. The k-t SPARSE-SENSE

reconstruction was performed offline using customized software developed

in a graphics processing unit (GPU) platform with Compute Unified Device

Architecture programming. The compressed sensing optimization problem

was solved using focal underdetermined system solver (FOCUSS) (73,74).

Temporal PCA was used as the sparsifying transform in k-t SPARSE-

SENSE reconstruction and the PCA basis was calculated and updated in

each iteration of the reconstruction. Using a NVIDIA Tesla GPU with 4 GB

global memory, the total reconstruction time of each dynamic image series

using k-t SPARSE-SENSE was ~1 min.

3.6. Image Analysis and Statistics

3.6.1. Image Analysis

Image analysis was performed using software developed in MATLAB

(MATLAB Statistics ToolboxTM). For each MnCl2 phantom bottle, a mask

covering the whole bottle was manually generated and the corresponding

pixel-by-pixel T2 map was calculated. For in vivo data, both reference and

78

accelerated T2 mapping datasets of all subjects were pooled and

randomized for independent blinded analysis. The left ventricle (LV)

endocardial and epicardial contours were manually drawn to mask the

whole LV. The corresponding pixel-by-pixel T2 map was calculated by

nonlinear least square fitting for three parameters of the monoexponential

signal relaxation equation:

2/2 2 1/2

0( ) ( ) ;

t T

ideal idealS t S S S e

[3.1]

where S(t) is the signal amplitude at time t, Sideal is the ideal signal, and the

three unknown parameters are: the initial signal amplitude (S0), the mean

background noise (σ), and T2. This approximate estimation procedure is

similar to the method of McGibney and Smith (75).

3.6.2. Statistical Analysis

For each bottle of the phantom, the mean transverse relaxation rate

(R2 = 1/T2) was calculated. The five mean R2 measurements were plotted

as a function of MnCl2 concentration, and the transverse relaxivity of MnCl2

was calculated by performing linear regression analysis.

According to the 16-segment model recommended by American

Heart Association, the middle ventricular short-axis view of the myocardium

was divided into six segments including: anterior, anteroseptal, inferoseptal,

inferior, inferolateral, and anterolateral. Statistical analysis was performed

79

using SPSS version 18 (SPSS, Chicago, IL). The mean T2 and segmental

T2 measurements were compared using the paired-sample t-test (two-

tailed). A P < 0.05 was considered to be statistically significant. Reported

values represent mean ± standard deviation. Bland–Altman analysis was

performed to compare T2 measurements obtained using the accelerated T2

mapping pulse sequence and the reference T2 mapping pulse sequence.

To assess the influence of manual segmentation of LV contours on

T2 calculation, we assessed the intra-observer and inter-observer variability

of T2 calculation using the Bland–Altman analysis. For intra-observer

variability assessment, one blinded observer repeated the image analysis

(i.e., contour segmentation and T2 calculation) with 2 weeks of separation

from the first analysis, and Bland–Altman analysis was performed on the

resulting two sets of T2 measurements (analysis 1 vs. analysis 2). Intra-

observer difference was defined as T2 (analysis 1) – T2 (analysis 2). For

inter-observer variability assessment, the second blinded observer

independently analyzed the datasets, and Bland–Altman analysis was

performed on the resulting two sets of T2 measurements (observer 1,

analysis 1 vs. observer 2). Inter-observer difference was defined as T2

(observer 1, analysis 1) – T2 (observer 2).

3.7. Results

80

Figure. 3.7: Representative T2 mapping images acquired using the reference and accelerated T2 mapping pulse sequences: (top row) GRAPPA and (bottom row) k-t SPARSE-SENSE. When compared with GRAPPA, k-t FOCUSS consistently yielded higher spatial resolution in the phase-encoding direction (1.7 x 1.7 mm

2 vs. 1.7 x 4.2

mm2; accelerated vs. reference, respectively).

The two sets of in vitro T2 maps calculated from the reference and

accelerated ME-FSE data sets were in good agreement. The reference and

accelerated T2 maps yielded transverse relaxivities of 88.6 ± 1.2 and 88.3 ±

1.4 s–1/ mM, respectively.

Figure 3.7 shows representative cardiac T2 mapping images

acquired using the reference and accelerated T2 mapping pulse sequences.

The accelerated T2 mapping pulse sequence consistently yielded higher

spatial resolution in the phase-encoding direction (1.7 x 1.7 mm2 vs. 1.7 x

4.2 mm2; accelerated vs. reference, respectively). In all subjects, k-t

SPARSE-SENSE yielded good image quality. Figure 3.8 shows the

corresponding zoomed images and the resulting T2 maps. Mean T2 values

81

Figure. 3.8: Zoomed cardiac T2 mapping images and the T2 maps corresponding to Figure 3.7: (top row) GRAPPA and (bottom row) k-t FOCUSS. When compared with GRAPPA image, k-t FOCUSS image produced higher spatial resolution in the phase-encoding direction, as shown by the intensity profiles of the muscle–blood border.

of this subject calculated from GRAPPA and k-t SPARSE-SENSE images

were 51.7 ± 3.6 and 51.6 ± 4.0, respectively. When compared with the

GRAPPA image, k-t SPARSE-SENSE image yielded higher spatial

resolution, as highlighted by the intensity profiles at the edge of the muscle–

blood border.

For the pool of 12 subjects, mean myocardial T2 values measured by

observer 1 (analysis 1) using the GRAPPA and k-t SPARSE-SENSE data

were 49.9 ± 1.5 and 50.0 ± 1.5, respectively, and they were not significantly

different (P > 0.05). The corresponding mean segmental T2 values are

summarized in Table 3.1, and mean T2 values for every segment were not

significantly different (P > 0.05). These findings are consistent with

82

Myocardium GRAPPA (msec) k-t SPARSE-SENSE (msec)

Anterior 49.1±3.2 48.7±2.6

Anteroseptal 52.0±3.0 51.5±2.8

Inferoseptal 51.4±2.6 51.6±2.9

Inferior 50.3±1.8 49.9±1.9

Inferolateral 50.2±1.5 49.6±2.7

Anterolateral 47.1±1.9 49.1±2.6

Whole Myocardium 49.9±1.5 50.0±1.5

Table. 3.1: Mean Segmental and Whole Myocardial T2 Measurements Obtained Using GRAPPA and k-t SPARSE-SENSE Datasets. Not that these values represent results analyzed by observer 1 and analysis 1.

previously reported T2 measurements of healthy subjects at 3T. These

mean T2 values between GRAPPA and k-t SPARSE-SENSE were not

significantly different (P > 0.05).

According to the Bland–Altman analysis, T2 measurements obtained

from k-t SPARSE-SENSE and GRAPPA images were in excellent

agreement (mean difference = 0.04 ms; upper/lower 95% limits of

agreement were 2.26/-2.19 ms), suggesting that the corresponding T2

measurements are quantitatively equivalent. The corresponding Bland–

Altman statistics for the segmental T2 measurements are summarized in

Table 3.2.

The intra-observer agreements for T2 calculations using the same

set of GRAPPA and k-t SPARSE-SENSE data were -0.18 and 0.04, the

upper 95% limits of agreements were 0.99 and 0.91, and the lower 95%

limits of agreements were -1.34 and -0.83, respectively. The inter-observer

83

Myocardium Difference (msec) Upper 95%

Limit (msec) Lower 95%

Limit (msec)

Anterior -0.40 5.43 -6.23

Anteroseptal -0.52 5.69 -6.72

Inferoseptal 0.21 4.01 -3.60

Inferior -0.36 3.29 -4.00

Inferolateral -0.59 5.31 -6.49

Anterolateral 1.99 8.14 -4.17

Whole Myocardium 0.04 2.26 -2.19

Table. 3.2: Bland–Altman Statistics of T2 Measurements Obtained Using GRAPPA and k-t SPARSE-SENSE Datasets. Note that these values represent results analyzed by observer 1 and analysis 1.

agreements for T2 calculations using the same set of GRAPPA and k-t

SPARSE-SENSE data were 0.36 and 0.35, the upper 95% limits of

agreements were 1.56 and 1.37, and the lower 95% limits of agreements

were -0.83 and -0.68, respectively. These Bland–Altman statistics (Table

3.3) suggest that T2 calculations from a given set of data are highly

repeatable and reproducible.

Figure 3.9 shows examples of cardiac T2 mapping images and the

corresponding T2 map with and without preconditioning RF pulses. For the

latter case, signal heterogeneity was found in the k-t SPARSE-SENSE

reconstruction, particularly in the lateral wall, and the corresponding T2 error

was produced. In this subject, the mean T2 measurements within the

segmented myocardium were 50.0 ± 4.0 ms and 60.8 ± 12.9 ms for with

and without preconditioning RF pulses, respectively. These results are

corroborated with zero-filled FFT reconstruction images which show more

84

Agreement Type Difference (msec) Upper 95%

Limit (msec) Lower 95%

Limit (msec)

Intra (GRAPPA) -0.18 0.99 -1.34

Intra (k-t SPARSE-SENSE) 0.04 0.91 -0.83

Inter (GRAPPA) 0.36 1.56 -83

Inter (k-t SPARSE-SENSE) 0.35 1.37 -0.68

Table. 3.3: Intraobserver and Interobserver Agreements for T2 Calculations Based on Manual Segmentation of LV Contours. Intraobserver difference was defined as T2 (analysis 1)-T2 (analysis 2), and interobserver difference was defined as T2 (observer 1)-T2 (observer 2).

aliasing artifacts for the latter case. These findings clearly demonstrate that

the use of preconditioning RF pulses for k-t SPARSE-SENSE

reconstruction increases sparsity in T2 mapping datasets and thus improve

the accuracy of T2 measurement.

3.8. Discussion

Our results demonstrate the feasibility of performing a 6-fold

accelerated breath-hold cardiac T2 mapping acquisition using k-t SPARSE-

SENSE. When compared with the reference T2 mapping pulse sequence

with GRAPPA, the accelerated T2 mapping pulse sequence with k-t

SPARSE-SENSE produced in vivo results of comparable accuracy. In all

the subjects, k-t SPARSE-SENSE reconstruction consistently produced

good image quality. The intra-observer and inter-observer agreements for

T2 calculations from a given set of images were excellent.

This study demonstrates that the proposed method is a promising

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Figure. 3.9: Example cardiac T2 mapping image and the corresponding T2 maps with and without preconditioning RF pulses. For the latter case, note the signal heterogeneity in the k-t FOCUSS reconstruction, particularly in the lateral wall, as well as the corresponding T2 error. These results are corroborated with zero-filled FFT reconstruction images which show more residual aliasing artifacts for the latter case. The results clearly demonstrate the usefulness of increasing sparsity in cardiac T2 mapping through the use of preconditioning RF pulses.

investigational tool for myocardial T2 measurement with relatively high

spatial resolution (1.7 x 1.7 mm2). The reconstruction approaches can be

also applied in other application other than cardiac imaging. Preliminary

results for accelerated parameter mapping in musculoskeletal MRI using

the proposed approach have been reported in (76).

Our initial studies also have limitations that warrant discussion. First,

commercially available fat suppression pulse did not completely suppress

the undesirable fat signals. The use of improved fat suppression pulse,

such as chemically selective inversion recovery, should further suppress

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the fat signal and, subsequently, increase sparsity of cardiac T2 mapping

datasets. Second, we used three spatial pre-saturation pulses before ME-

FSE readout to suppress bright signals outside the heart. More work is

needed to explore the optimal use of spatial pre-saturation pulses (e.g.,

quantity, width, location, and orientation, etc) for maximal suppression of

undesirable background signal. Third, our accelerated T2 mapping pulse

sequence acquired an echo train for ~160 ms during middle to late diastole,

and our data analysis was performed assuming no cardiac motion.

However, even with a perfectly still breath hold, gradual ventricular

relaxation occurs during 160 ms of data acquisition. Advanced image

registration methods could be further used to eliminate this potential source

of error in data fitting for T2 calculation. Fourth, our study was carried out in

a small number of healthy subjects at 3T, without edema or iron overload.

Further studies in a larger number of patients with the whole spectrum of T2

encountered in clinical practice are necessary to fully evaluate the clinical

utility of the accelerated T2 mapping pulse sequence. Fifth, the breath-hold

duration of 18 heartbeats could be still too long for some patients with

impaired breath-hold capacity. In such patients, it may be necessary to

sacrifice some spatial resolution to reduce the breath-hold duration.

3.9. Conclusion

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In conclusion, the proposed accelerated breath-hold T2 mapping

approach can be used to perform rapid T2 mapping of the heart with

relatively high spatial resolution. The proposed technique is a promising

investigational method for quantitative assessment of myocardial edema

and iron overload, and can be also applied in other organs such as liver, hip

or knee.

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Chapter 4

Accelerated Real-Time Cardiac Cine MRI Using k-t

SPARSE-SENSE

4.1. Prologue

In the previous chapter, we have demonstrated that k-t SPARSE-

SENSE can be applied for accelerated MR T2 mapping. In this chapter, we

extend the applications of k-t SPARSE-SENSE to cardiac cine imaging,

which is a very important tool for evaluating myocardial function. We

demonstrate that k-t SPARSE-SENSE can be used to achieve 8-fold

accelerated real-time cardiac cine imaging, which allows cardiac exams

during free-breathing.

The contents presented in this chapter were published in the journal

Magnetic Resonance in Medicine (MRM 2013 Jul; 70(1): 64-74) (77).

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4.2. Introduction

Non-invasive assessment of myocardial function plays an important

role in the management of various cardiac diseases (78-85). Breath-hold

cine MRI with balanced steady-state free precession (bSSFP) readout,

which is now considered the gold standard for imaging myocardial function,

offers good spatial resolution, high blood-to-myocardium contrast, and

exquisite image quality (86). However, in patients with impaired breath-hold

capacity and/or arrhythmias, breath-hold cine MRI may yield non-diagnostic

image quality. In these patients, real-time cine MRI may be more clinically

useful than breath-hold cine MRI (87-97).

Commercially available real-time cine MRI methods using dynamic

parallel imaging, such as TSENSE (21) or TGRAPPA (22), typically yield

relatively low spatio-temporal resolution, due to their poor image acquisition

speed. For example, our clinical real-time cine MRI protocol using

TGRAPPA with acceleration factor (R) = 3 produces 2.5 x 2.5 mm2 spatial

resolution and 114 ms temporal resolution. For patients with tachycardia

(heart rate > 100 bpm) or imaging during stress testing, higher temporal

resolution is desirable.

Many investigational image processing methods, in conjunction with

parallel imaging, have been reported to further accelerate real-time cine

MRI. For example, one method utilizes a Karhunen-Loeve transform filter to

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improve the performance of TSENSE or TGRAPPA (98). Deformable

registration has been incorporated into the image reconstruction framework

to perform motion compensation (99). Besides, real-time cine MRI can be

also performed with retrospective reconstruction to improve spatio-temporal

resolution, in which multiple cardiac cycles are synchronized and averaged

with motion correction techniques (88,92). However, such retrospective

reconstruction method may not perform well in patients with arrhythmias,

because of its need for averaging results over multiple heart beats.

More advanced investigational image acquisition methods have also

been reported to further accelerate real-time cine MRI. The presence of

spatial and temporal correlations in cardiac cine datasets can be exploited

to further accelerate data acquisition, due to the sparsity in the spatial-

temporal Fourier domain. One such method is the partially separable

function model (100), which has been shown to produce high spatio-

temporal resolution for imaging rat hearts (101). Another method is radial k-

t SENSE (102), which has been shown to yield 8-fold accelerated real-time

cine MR images with 2.3 x 2.3 mm2 spatial resolution and 40 ms temporal

resolution.

An alternative method to further accelerate real-time cine MRI is

compressed sensing (26-28), which has been applied to accelerate breath-

hold cardiac cine MRI (73,74,103). These pioneering works have been

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important developments but have not been applied for real-time cardiac

cine MRI yet. Real-time cardiac cine MRI is a good candidate for

compressed sensing, because the background is static (due to steady state

of magnetization) and the dynamic region (e.g., heart) is relatively small.

This condition produces a high degree of spatio-temporal correlation, which

could be exploited with an appropriate transform, such as temporal fast

Fourier transform (FFT), temporal principal component analysis (PCA) or

temporal finite differences (also known as temporal total-variation or

temporal TV).

Based on our clinical experiences, we are aiming to achieve spatial

resolution on the order of 2.5 x 2.5 mm2 and temporal resolution on the

order of 40 ms, to produce high-quality real-time cine MR images that could

be applied clinically for wall motion assessment and measurement of global

LV function. In this chapter, k-t SPARSE-SENSE is used to highly

accelerate real-time cine MRI with 8-fold acceleration. The resulting image

quality and global function measurements is compared and validated

against the reference approaches.

4.3. Imaging Strategies

4.3.1. k-Space Undersampling: Incoherence and Self-Calibration of

Coil Sensitivities

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Figure. 4.1: (a): Eight-fold accelerated ky–t sampling pattern varied along time. (b): A schematic illustrating how the kx–ky–t sampling pattern is averaged over time to produce the resulting kx–ky sampling pattern. This kx–ky pattern represents the sampling used to perform self-calibration of coil sensitivities. White lines represent acquired samples.

The first major component in compressed sensing is incoherent

aliasing artifact due to k-space undersampling. Similar to the studies in

Chapter 3, a random k-space undersampling pattern with higher density at

the center of k-space was applied to achieve a high degree of incoherence

in time-series data, with different undersampling pattern at each time frame,

in order to distribute the resulting aliasing artifacts along both ky, the spatial

frequency in the phase-encoding dimension, and t, the temporal dimension.

An 8-fold accelerated ky-t undersampling pattern, as shown in Figure 4.1a,

was applied in this study. Note that temporal average of the sampling

pattern, kx-ky, represents the sampling used to perform self-calibration of

coil sensitivities, as shown in Figure 4.1b.

4.3.2. Comparison of Sparsifying Transforms

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The second major component in compessed sensing is sparse

representation of the image with a known transform basis. We performed

retrospective undersampling experiments on fully sampled, breath-held cine

MR datasets (one short-axis (SAX) and one long-axis (LAX)) in order to

determine the optimal sparsifying transform and the corresponding

regularization parameter. The relevant imaging parameters (e.g., spatial

resolution, temporal resolution, receiver bandwidth, and flip angle) were

similar to the proposed real-time cine MRI protocol (see Pulse Sequence

section below). The k-t SPARSE-SENSE reconstruction was performed

using the steps described in the Image Reconstruction section below. This

subsection describes the methods and results of the preliminary

retrospective simulation experiment which was needed for the prospective

acceleration strategies.

4.3.2.1. Primary Sparsifying Transform: Dynamic Region

k-t SPARSE-SENSE reconstruction was first performed on

retrospectively 8-fold undersampled datasets using three different

sparsifying transforms, including temporal FFT, temporal PCA and temporal

TV (see Figure 4.1a for the sampling pattern). The regularization parameter

was determined empirically by comparing between k-t SPARSE-SENSE

images and fully-sampled cine MR images. Our prior cardiovascular MRI

applications of k-t SPARSE-SENSE reported regularization parameter

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Figure. 4.2: Simulation results comparing the fully sampled reference cardiac cine data to retrospectively eight-fold accelerated k–t SPARSESENSE results with different sparsifying transforms with regularization weight 0.01: temporal FFT, temporal PCA, and temporal TV. (a): In the zoomed view of the heart, temporal TV yielded the lowest RMSE. (b): In the chest wall, temporal FFT yielded the lowest RMSE. (c) and (d): Corresponding plots of RMSE for the heart and chest wall regions, respectively, as a function of regularization weight ranging from 0.005 to 0.05. These results show that temporal TV is superior to the other two sparsifying transforms for the dynamic region, whereas temporal FFT is superior to the other two transforms for the static region. Based on these results, we elected to use temporal TV as the primary sparsifying transform with regularization weight 0.01 and temporal FFT as the secondary transform with regularization weight 0.001.

values ranging from 0.01 to 0.05. Thus, we repeated the experiment with

regularization weight ranging from 0.005 – 0.05 (0.005 increment) and

calculated the root-mean-square-error (RMSE) between k-t SPARSE-

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SENSE and fully-sampled results. Note that, for each sparsifying transform

per regularization weight, RMSE was calculated for a cropped region

containing mainly the heart. Figure 4.2c shows the plot of RMSE as a

function of regularization parameter for the heart region. Compared with

temporal FFT and temporal PCA, temporal TV yielded lower RMSE, and

regularization weight = 0.01 yielded the minimal RMSE for temporal TV

(see Figure 4.2a). We also performed visual inspection to confirm high

temporal fidelity of myocardial wall motion at weight 0.01. Given these

results, we elected to use temporal TV as our sparsifying transform with

regularization weight 0.01.

4.3.2.2. Secondary Sparsifying Transform: Static Region

We also performed the same analysis on the static region (e.g.,

chest wall). Figure 4.2d shows the plot of RMSE as a function of

regularization parameter. Compared with temporal PCA and temporal TV,

temporal FFT yielded lower RMSE, suggesting that it is superior for

suppression of residual aliasing artifacts arising from the static region.

Therefore, to further suppress residual incoherent aliasing artifacts arising

from the background, we elected to add temporal FFT as a secondary

orthogonal sparsifying transform. The resultant reconstruction algorithm is a

combination of temporal TV and temporal FFT (temporal TV+FFT)), where

the regularization weight for temporal FFT was empirically chosen to be ten

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Figure. 4.3: Numerical simulation results comparing the (a) fully sampled data (R = 1) to the retrospectively eight-fold undersampled reconstruction results using four different sparsifying transforms: (b) temporal FFT, (c) temporal PCA, (d) temporal TV, and (e) temporal TV + FFT. (First row) end-systolic SAX image, (second row) spatial-temporal profile from the SAX image, (third row) end-systolic LAX image, and (fourth row) spatial-temporal profile from the LAX image. Both temporal FFT and temporal PCA yielded more temporal blurring artifacts within the wall (arrows) than temporal TV and temporal TV + FFT.

times lower than that for temporal TV - small enough to not introduce

temporal blurring artifacts but large enough to help suppress residual

aliasing artifacts arising from the static regions. Given that the regularization

weight for temporal FFT is small (0.001), we did not perform a systematic

search for the optimum value. Note that both temporal TV and temporal

FFT terms were solved simultaneously during image reconstruction.

4.3.2.3. Preliminary Evaluation

97

For completeness, we also performed simulation experiments

comparing the performances of temporal FFT, temporal PCA, temporal TV,

and temporal TV + FFT, using both fully-sampled SAX and LAX datasets as

references. Both temporal FFT and temporal PCA, as shown in Figure

4.3b&c (x-y plane), produced temporal blurring artifact in the myocardial

wall (white arrows). In contrast, both temporal TV and temporal TV + FFT

(Figure 4.3d&e) did not produce the specific temporal blurring artifact. The

signal intensity profiles, through the blood-myocardium boundary, along y-t

were also evaluated for all datasets along the white dotted lines drawn on

Figure 4.3a. These spatial-temporal profiles also show more temporal

blurring artifacts for the temporal FFT and temporal PCA than temporal TV

and temporal TV+FFT. To further evaluate the temporal fidelity, the

reconstructed SAX (4 cases) and LAX (4 cases) datasets were randomized

for blind evaluation by 4 readers: two cardiologists, one pediatric

cardiologist, and one radiologist. Each reader independently ranked the

temporal fidelity of myocardial wall motion (1-4: highest-lowest). The

results, averaged over 4 readers, showed that both temporal TV+FFT (1.9)

and temporal TV (2.0) produced better temporal fidelity of myocardial wall

motion than temporal FFT (2.9) and temporal PCA (3.0). Based on these

preliminary results, we elected to use temporal TV + FFT as the sparsifying

transform for accelerated real-time cine MR data reconstruction.

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Figure. 4.4: Numerical simulation results (top row: end-diastolic images, middle row: end-systolic images, bottom row: spatial-temporal plots through the blood-myocardium boundary) comparing different R values using temporal TV with weight 0.01 and temporal FFT with weight 0.001: (first column) R = 1, (second column) R = 2, (third column) R = 4, (fourth column) R = 6, (fifth column) R = 8, and (sixth column) R = 10. These results suggest that good results can be obtained up to R = 8.

4.3.3. Comparison of Acceleration Rates

To determine the maximal acceleration with acceptable image

quality, we performed repeated undersampling simulation with different

acceleration rates ranging from 2-10, where temporal TV with regularization

parameter 0.01 and temporal FFT with regularization parameter 0.001 were

used for k-t SPARSE-SENSE reconstruction. Figure 4.4 shows the end-

diastolic and end-systolic images for the different accelerations, as well as

their corresponding spatial-temporal plots through the blood-myocardium

boundary. These results show that good results can be obtained up to R =

8. Based on this preliminary experiment, we elected to use R =8 for

prospective accelerated acquisitions.

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4.4. Imaging Studies

4.4.1. Pulse Sequence

Our proposed 8-fold accelerated, real-time cine MRI sequence was

implemented on a whole-body 3T MRI scanner (MAGNETOM TimTrio,

Siemens AG, Healthcare Sector, Erlangen, Germany). A 6-element body

matrix coil array and a spine coil array (with only 6 elements on) were

employed for signal reception.

The relevant imaging parameters for real-time cine were: field of

view = 300 x 300 mm2, acquisition matrix size = 128 x 128, spatial

resolution = 2.3 x 2.3 mm2, slice thickness = 8 mm, flip angle = 40o,

repetition time / echo time = 2.7/1.37 ms, receiver bandwidth = 1184

Hz/pixel, and temporal resolution = 43.2 ms. The total scan time was 2

heart beats per slice, with 1 heart beat to achieve steady-state of

magnetization, and another heart beat to acquire the cine data.

The relevant imaging parameters for breath-hold cine MRI were: field

of view = 300 x 300 mm2, acquisition image matrix = 192 x 192, spatial

resolution = 1.6 x 1.6 mm2, slice thickness = 8 mm, flip angle = 40o,

retrospective electrocardiogram (ECG) gating with 25 reconstructed cardiac

phases, receiver bandwidth = 1300 Hz/pixel, and 1.6-fold acceleration with

GRAPPA.

4.4.2. Cardiac Imaging

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Twelve healthy human volunteers with no known cardiac disease (11

males and 1 female; mean age = 26.2 ± 2.7 years) and one male patient

(age = 45 years) with history of heart transplantation were imaged using

both real-time cine and breath-hold cine MRI sequences, to perform pre-

clinical testing and confirmation of comparability of global function

measurements with breath-hold cine MRI. Fifteen consecutive clinical

patients (7 males and 8 females; mean age = 49 ± 21 years) were recruited

for image quality assessment of real-cine MRI in a clinical setting. Human

imaging was performed in accordance with protocols approved by the New

York University School of Medicine Institutional Review Board and was

found to comply with the HIPAA. All subjects provided written informed

consent before the imaging.

4.4.2.1. Experiment I: Image Quality and Global Function

Measurement Comparison (real-time vs. breath-hold cine MRI)

In the first experiment, twelve healthy human volunteers were

imaged using both pulse sequences in a stack of 12 short-axis planes

covering the entire LV, in order to compare their resulting image quality

(only three views; base, mid, and apex) and global function measurements

including end diastolic volume (EDV), end systolic volume (ESV), stroke

volume (SV), and ejection fraction (EF). Real-time cine imaging of the entire

LV was performed during free breathing with prospective ECG triggering,

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and breath-hold cine imaging was performed with 6 separate breath-holds

(2 slices per breath-hold) with retrospective ECG gating.

4.4.2.2. Experiment II: Comparison of LV Function by Real-time Cine

MRI with prospective ECG triggering vs. Breath-hold Cine MRI with

Retrospective ECG gating

In the second experiment, one male patient with history of heart

transplantation was imaged to compare global function measurements

between two different end-diastolic frames for real-time cine MRI with

prospective ECG triggering. We note that in prospective ECG triggering, it

may be difficult to capture true end diastole, which may lead to

underestimation of EDV, SV and EF when compared with retrospective

ECG gated acquisitions. In this patient, we imaged a stack of 14 short-axis

planes covering the entire LV, for 2 heart beats, in order to capture true end

diastole between the first and second heart beats with prospectively ECG

triggered real-time cine MRI. We calculated two sets of global function

measurements from this data set: i) first frame defined as end-diastole and

ii) visually identified end-diastolic frame, acquired between the first and

second heart beats. For reference, another stack of 14 short-axis slices

were acquired using breath-hold cine MRI with retrospective ECG gating.

4.4.2.3. Experiment III: Clinical application of real-time Cine MRI

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Figure. 4.5: (a): Coil sensitivities calculated using an (left column) external reference acquisition (pre-scan) and (right column) self-calibration method. (b): The resulting k–t SPARSE-SENSE images using externally and self-calibrated coil sensitivities. Note that two sets of data are very similar, suggesting that our self-calibration of coil sensitivities was robust.

Fifteen patients (7 males and 8 females; mean age = 49 ± 21 years)

were recruited to evaluate the performance (e.g., image quality, wall

motion, artifacts) of our proposed real-time cine MRI pulse sequence. For

patient recruitment, the only inclusion criterion was normal sinus rhythm,

and our patient population included different cardiac disease conditions.

Given that our study was not aimed at a particular clinical indication, only

one cardiac view was acquired per patient (12 patients had SAX, 3 patients

had LAX).

4.5. Image Reconstruction

K-t SPARSE-SENSE reconstruction was performed off-line in

MATLAB (R2011b software; Mathworks, Natick, MA). The coil sensitivity

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maps were self-calibrated by averaging undersampled k-space data over

time (see Figure 4.1b, Figure 4.6a) and computed using the adaptive array-

combination technique (40). This subsection describes the results of the

preliminary experiment to demonstrate that the self-calibration of coil

sensitivities is robust. We acquired a data set with external coil calibration

data, and compared the results using external and self-calibrated coil

sensitivities. Figure 4.5 shows a comparison between self-calibrated and

externally acquired (as a pre-scan) coil sensitivities, as well as their

resulting k-t SPARSE-SENSE images. Note that the two sets of data are

similar. The benefits of self-calibrated coil sensitivities are that they are

intrinsically registered with the undersampled data and do not require

additional time for acquisition.

For the k-t SPARSE-SENSE reconstruction, the optimization

problem (Figure 4.6b) was solved iteratively using a non-linear conjugate

gradient algorithm originally proposed in (28). The overall flowchart of the

image reconstruction is illustrated in Figure 4.6. Reconstruction time per

slice was about 4.6 min in a server equipped with an Intel Xeon CPU at

2.27 GHz with 24 GB RAM.

4.6. Image Analysis and Statistics

4.6.1. Image Analysis

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Figure. 4.6: Schematic flowchart of the image reconstruction method. (a): Coil sensitivity maps were self-calibrated by averaging undersampled k-space data over time and computed using the adaptive array combination method. (b): Multicoil, zero-filled k-space data, along with the corresponding coil sensitivity maps, were reconstructed using both temporal TV and temporal FFT as the sparsifying transforms, where regularization weight of temporal TV is 10 times larger than that for temporal FFT.

For each subject, three slices (apex, mid, base) were selected from

both real-time and breath-hold cine sets for image quality assessment. A

total of 72 (36 real-time cine; 36 breath-hold cine) datasets were pooled and

randomized for blinded qualitative evaluation by four readers described

previously. Readers independently scored the image quality (1=non-

diagnostic, 2=poor, 3=adequate, 4=good, 5=excellent), temporal fidelity of

myocardial wall motion (1=non-diagnostic, 2=poor, 3=adequate, 4=good,

5=excellent), and artifact level (1=none, 2=mild, 3=moderate, 4=severe,

5=non-diagnostic).

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For global function assessment, 24 (12 real-time cine; 12 breath-hold

cine) short-axis stacks of cine datasets were pooled and randomized for

blinded quantitative evaluation. The same 4 readers independently

calculated the EDV, ESV, SV, and EF for each data set.

For comparison of prospective ECG triggered real-time cine MRI and

retrospective ECG gated breath-hold cine MRI, one cardiologist analyzed a

short-axis stack of real-time cine MRI data with prospective ECG triggering

using two different end diastolic frames (1 and 16, where frame 16 was

visually defined as the true end diastole) and compared their global function

measurements with those obtained using a stack of breath-hold cine MR

data with retrospective ECG gating.

For evaluation of real-time cine MRI in patients, 15 cine datasets

were pooled and randomized for blinded evaluation. The same four readers

independently scored the image quality, temporal fidelity of myocardial wall

motion and artifact for each data set.

4.6.2. Statistical Analysis

For image quality comparison, the reported scores, which were

averaged over four readers, represent mean ± standard deviation.

Statistical analysis was performed using Excel (Microsoft Corporation,

Redmond, WA). Wilcoxon signed-rank sum test was used to compare the

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Figure. 4.7: (Rows 1–2) End-diastolic and (rows 3–4) end-systolic images at multiple cardiac phases comparing (rows 1 and 3) breath-hold cine MRI and (rows 2 and 4) real-time cine MRI. Both image sets were acquired from a 29-year-old (male) healthy subject. Note that the breath-hold cine MR images had higher spatial resolution than the real-time cine MR images (1.6 mm

2 vs. 2.3 mm

2, respectively).

mean scores between two groups, where P < 0.05 was considered to be

statistically significant.

For global function comparison, Bland-Altman and coefficient of

variation (CV) analysis were performed. In addition, inter-observer

variability within each pulse sequence was also assessed by inter-class

correlation (ICC).

4.7. Results

4.7.1. Experiment I

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Figure. 4.8: Bland–Altman plots illustrating good agreement between breath-hold cine MRI and real-time cine MRI for the following LV function measurements: (top, left) EDV (mean difference = 15.2 mL [solid line]; lower and upper 95% limits of agreement = 27.6 and 2.8 mL [dashed lines], respectively), (top, right) ESV (mean difference = 2.1 mL [solid line]; lower and upper 95% limits of agreement = 4.7 and 8.9 mL [dashed lines], respectively), (bottom, left) SV (mean difference = 17.3 mL [solid line]; lower and upper 95% limits of agreement = 31.3 and 3.3 mL [dashed lines], respectively), and (bottom, right) EF (mean difference = 5.7% [solid line]; lower and upper 95% limits of agreement = 11.3% and 0.1% [dashed lines], respectively).

Figure 4.7 shows representative sets of real-time cine MR images

and breath-hold cine MR images, in five short axis slices from one healthy

subject. Both breath-hold and real-time images produced good diagnostic

overall image quality. Table 4.1 shows the mean scores of image quality,

temporal fidelity of wall motion and artifact level for breath-hold cine and

real-time cine results (n=36). Compared with breath-hold cine, real-time

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Sequence Image Quality Temporal Fidelity

of Wall Motion Artifact

Breath-Hold 4.6±0.5 4.7±0.4 1.4±0.6

Real-Time 3.5±0.5 4.5±0.4 2.3±0.5

Table. 4.1: Mean scores of image quality, temporal fidelity of wall motion and artifact, produced by Breath-Hold cine MRI and Real-Time cine MRI.

cine yielded significantly (P < 0.05) worse scores for all four categories.

However, for real-time cine MRI, the image quality and temporal fidelity of

wall motion scores were above 3.0 (adequate) and artifacts and noise

scores were below 3.0 (moderate), suggesting that acceptable diagnostic

image quality can be achieved. According to the Bland-Altman (Figure 4.8)

and CV analysis (Table 4.2), all four global function measurements,

averaged over 4 readers, were in good agreement, with CV less than 10%.

The inter-class correlation (Table 4.3) shows that the inter-observer

variability in calculating global function measurements ranged from

moderate to strong for real-time cine MRI and moderate to near perfect for

breath-hold cine MRI.

4.7.2. Experiment II

Figure 4.9 shows two potential candidates for an end-diastolic frame

from the same series of mid-ventricular short-axis images acquired with

prospective ECG triggering. In this example, the first candidate is cardiac

frame 1, and the second candidate is cardiac frame 16 (denoted as frame N

in the Figure). Note that cardiac frame 16 shows larger LV cavity than

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Measurement Mean Mean

Difference Upper

95% Limit Low 95%

Limit CV(%)

EDV 143 mL -15.2 mL -2.8 mL -27.6 mL 6.22

ESV 55.7 mL 2.1 mL 8.9 mL -4.7 mL 4.42

SV 87.3 mL -17.3 mL -3.3 mL -31.3 mL 8.79

EF 0.61 -5.7% -0.1% -11.3% 4.68

Table. 4.2: Bland–Altman and CV analyses of four global function measurements between Real-Time and Breath-Hold cine MRI pulse sequences.

Measurement ICC (BH) ICC (RT)

EDV 0.88 0.78

ESV 0.76 0.77

SV 0.76 0.64

EF 0.64 0.64

Table. 4.3: ICC analysis of interobserver variability of EDV, ESV, SV, and EF within each pulse sequence type. ICC scale: 0-0.2 indicates poor agreement, 0.3-0.4 indicates fair agreement, 0.5-0.6 indicates moderate agreement, 0.7-0.8 indicates strong agreement, and >0.8 indicates almost perfect agreement.

cardiac frame 1. In the male patient with heart transplantation, defining

cardiac frame 1 as end diastole produced EDV = 86ml, ESV = 44ml, SV =

42ml, and EF = 49%. Defining cardiac frame 16 as end diastole produced

EDV = 94ml, ESV = 44ml, SV = 49ml, and EF = 53%. The reference breath-

held cine MRI with retrospective ECG gating yielded EDV = 99ml, ESV =

44ml, SV = 55ml, and EF = 56%. This comparison shows that the proposed

prospective ECG triggered approach of acquiring real-time cine MR data

through 2 heart beats would ensure capture of true end diastole.

4.7.3. Experiment III

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Figure. 4.9: Proposed real-time cine MRI protocol with prospective ECG triggering to capture true end diastole, where images are continuously acquired through the second R-wave to visually identify true end diastole. This proposed approach produced global function measurements in excellent agreement with breath-hold cine MRI with retrospective ECG gating.

Figure 4.10 shows representative real-time cine MR images in two

different patients in different cardiac imaging planes. Real-time cine MRI

with k-t SPARSE-SENSE consistently yielded high-quality images in all

patients. The mean scores of image quality, temporal fidelity and artifact

were 3.7 ± 0.6, 4.3 ± 0.7 and 1.7 ± 0.7, which are similar to the

corresponding scores from volunteer data. Again, the image quality and

temporal fidelity of wall motion scores were above 3.0 (adequate) and the

artifact scores was below 3.0 (moderate).

4.8. Discussion

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Figure. 4.10: Representative end-diastolic and end-systolic real-time cine images: (top row) SAX view of a 26-year-old (female) patient and (bottom row) LAX view of a 36-year-old (male) patient.

This work demonstrates the feasibility of performing 8-fold

accelerated real-time cine MRI using k-t SPARSE-SENSE, by exploiting a

high degree of spatiotemporal correlation in cardiac cine MRI data. Our 8-

fold accelerated real-time cine MRI protocol can achieve adequate spatial

resolution of 2.3 x 2.3 mm2 and relatively high temporal resolution of 43.2

ms for global cardiac function assessment, with diagnostically acceptable

image quality and high temporal fidelity. This work also demonstrates an

approach to capture end diastole with prospective ECG triggering, by

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continuously performing real-time cine MRI through the second heart beat

and visually identifying the end-diastolic frame.

While real-time cine MRI yielded significantly worse image quality,

temporal fidelity and artifact scores compared with breath-hold cine MRI,

the image quality and temporal fidelity scores were above 3.0 (adequate),

and artifact score was below 3.0 (moderate). This trend was also true in 15

patients examined. Meanwhile, the temporal fidelity scores were above 4.0

(good) in both healthy subjects and patients. Temporal fidelity of myocardial

wall motion is particularly important to assess regional wall motion

abnormalities in the context of coronary artery diseases.

Previously reported k-t acceleration methods, such as k-t GRAPPA

(24), k-t SENSE (23), and PEAK-GRAPPA (104), also exploit

spatiotemporal correlations in the time series in combination with coil

sensitivities. However, sparsity and coil sensitivity encoding are exploited in

a different way than in k-t SPARSE-SENSE. These k-t acceleration

methods take advantage of sparsity to reduce signal overlap in the sparse

domain due to regularly undersampled data and perform a linear

reconstruction to reconstruct the sparse representation of the images using

prior information on this sparse representation and coil sensitivity

information. These linear algorithms are computationally less demanding.

Acceleration is achieved at the expense of signal-to-noise ratio and residual

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coherent aliasing artifacts, and additional training data are usually needed

to learn the sparse representation. In contrast, a non-linear reconstruction is

used in k-t SPARSE-SENSE to recover the sparse signal coefficients

contaminated by incoherent aliasing artifacts produced by an irregular

(pseudo-random) undersampling pattern. This non-linear reconstruction is

computationally more demanding. Acceleration is achieved at the expense

of residual incoherent aliasing artifacts and loss of low signal coefficients in

the sparse domain, and it does not require training data.

The proposed 8-fold accelerated real-time cardiac cine method is a

promising investigational tool for rapid imaging of myocardial function,

particularly for patients with impaired breath-hold capacity, arrhythmias

and/or tachycardia. While this new pulse sequence may be clinically useful,

we describe several issues that warrant discussion below.

First, a combination of temporal TV and temporal FFT was utilized as

sparsifying transforms in this work, where the regularization weight for

temporal TV was empirically set to be ten times larger than that for the

temporal FFT. We note that the use of temporal FFT as a secondary

sparsifying transform with a low regularization weight reduces residual

incoherent aliasing artifacts. Temporal FFT and temporal PCA are good

sparsifying transforms to reduce aliasing artifacts because artifacts are

spread incoherently over the entire y-f space and y-PCA space,

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respectively. They also exploit the correlation and redundancy over the

whole dynamic series. However, some signal coefficients with small signal

intensity are also suppressed along with incoherent aliasing artifacts, and

this leads to temporal blurring artifacts, as shown in Figure 4.3b&c.

Temporal TV, on the other hand, exploits temporal correlation by taking the

gradient of two adjacent temporal frames, producing less temporal blurring

than temporal FFT and temporal PCA, as shown in Figure 4.3d&e.

Second, we observed that regularization parameters in k-t SPARSE-

SENSE are crucial. For example, larger parameters produce less residual

artifacts at the expense of temporal blurring, and vice versa. In the current

implementation, we did not apply any rigorous mathematical criteria to

systematically select the regularization parameter. Instead, they were

determined empirically based on numerical analysis. This empirical

approach is based on our prior compressed sensing work in cardiovascular

MRI applications, where a fully sampled dataset was retrospectively

undersampled and reconstructed to “fine tune” the regularization parameter.

Third, compared with breath-hold cine MRI using retrospective ECG

gating, our accelerated real-time cine MRI with prospective ECG triggering

with cardiac frame 1 defined as end diastole underestimated EDV, SV, and

EF, because of the finite time needed to detect the ECG trigger and acquire

an image. We have proposed an alternative approach to overcome this

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limitation in the future studies, by acquiring real-time cine data with

prospective ECG triggering for 2 heart beats and visually identifying a frame

that best represents end diastole (Figure 4.9).

Fourth, our study was carried out in a small number of healthy

volunteers and patients with cardiac disease and no arrhythmias. Further

studies in a larger cohort of patients with a variety of heart disease

encountered in clinical practice are necessary to fully evaluate the clinical

utility of the proposed accelerated real-time cine MRI. Sixth, we performed

real-time cine MRI during free breathing. As such, our LV function

measurements may be contaminated with respiratory motion. We note that

this is an issue for all other real-time cine MRI methods performed during

free breathing as well. One possible solution to minimize this problem is to

perform imaging with breath-hold. Lastly, the current implementation of k-t

SPARSE-SENSE performs the reconstruction of each slice series serially.

Parallel computing could be used to reduce the reconstruction time in

further works.

4.9. Conclusion

In conclusion, 8-fold real-time cine MRI with k-t SPARSE-SENSE

can be used to achieve adequate spatial resolution (2.3 x 2.3 mm2) and

relatively high temporal resolution (43 ms), with good image quality and

relatively accurate global function measurements. This 8-fold accelerated

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real-time cine MRI method may be useful for patients with reduced breath-

hold capacity, arrhythmia, and/or tachycardia for qualitative assessment of

wall motion and quantitative assessment of LV function.

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Chapter 5

Golden-Angle Radial Sparse Parallel MRI: Combination of

Compressed Sensing, Parallel Imaging, and Golden-Angle

Radial Sampling for Fast and Flexible Dynamic Volumetric

MRI

5.1. Prologue

In the previous two chapters, we have demonstrated the

performance of k-t SPARSE-SENSE for accelerated cardiac T2 mapping

and real-time cardiac cine MRI with Cartesian sampling. The required

incoherence was achieved using variable density random undersampling.

K-t SPARSE-SENSE has also been successfully implemented for several

additional applications (105,106). However, the performance of compressed

sensing using Cartesian sampling is fundamentally limited, since the

frequency-encoding dimension is usually fully sampled and thus

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incoherence can only be exploited along the phase-encoding dimension.

Cartesian sampling, moreover, is sensitive to respiratory motion and the

results can be corrupted by ghosting artifacts or blurring. In this chapter, we

extend the reconstruction framework of k-t SPARSE-SENSE into radial

sampling and develop a new reconstruction approach called Golden-angle

RAdial Sparse Parallel MRI (GRASP), which combines compressed

sensing and parallel imaging with golden-angle radial sampling. We will

demonstrate that GRASP allows a simple and flexible means of performing

rapid dynamic MRI for a variety of clinical applications.

The contents presented in this chapter were published in the journal

Magnetic Resonance in Medicine (MRM 2014 Sep;72(3):707-17) (107), In

addition, Applications of GRASP were published in 1 paper in the Journal

Investigative Radiology (108) and 1 paper in the journal of Magnetic

Resonance Imaging (JMRI) (109) .

5.2. Introduction

Dynamic MRI requires rapid data acquisition to provide an

appropriate combination of spatial resolution, temporal resolution, and

volumetric coverage for clinical studies. For example, rapid imaging speed

is needed for dynamic contrast-enhanced (DCE) examinations, in which

fast signal-intensity changes must be monitored during the passage of the

contrast agent (110,111). As discussed in Chapter 1, a variety of fast MRI

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techniques have been developed over the last two decades to accelerate

the data acquisition. These methods include various parallel imaging

techniques, k-t acceleration techniques, and many others.

Compressed sensing is becoming as a powerful approach to

accelerate data acquisition in dynamic MRI. Compressed sensing methods

exploit spatial and temporal correlations by employing irregular

undersampling schemes to create incoherent aliasing artifacts and using a

non-linear reconstruction to enforce sparsity in a suitable transform domain.

As seen in previous two chapters, incoherent aliasing artifacts are often

created using Cartesian k-space trajectories with random undersampling

patterns. However, because undersampling is only performed along the

phase-encoding dimension, the incoherence achievable in this way is

relatively low, which limits the performance of compressed sensing. Radial

k-space trajectories are an interesting alternative due to the inherent

presence of incoherent aliasing in multiple dimensions, even for regular

(non-random) undersampling. Moreover, radial sampling is known to have

less sensitive to motion, which improves capturing dynamic information

(50,112). When acquiring radial data according to the golden-angle ordering

scheme (113), where the angle of the radial lines is increased continuously

by 111.25°, a rather uniform coverage of k-space with high temporal

incoherence is obtained for any arbitrary number of consecutive lines. This

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Figure. 5.1: (a) Continuous acquisition of radial lines with stack-of-stars golden-angle scheme in GRASP. (b) Point spread function (PSF) of an undersampled radial trajectory with 21 golden-angle spokes and 256 sampling points in each readout spoke for a single element coil (top) and for a sensitivity-weighted combination of 8 RF coil elements (bottom). The Nyquist sampling requirement is 256*π/2≈402. The standard deviation of the PSF side lobes was used to quantify the power of the resulting incoherent artifacts (pseudo-noise) and incoherence was computed using the main-lobe to pseudo-noise ratio of the PSF.

enables dynamic imaging studies using continuous data acquisition and

retrospective reconstruction of image series with arbitrary temporal

resolution by grouping different numbers of consecutive radial lines into

temporal frames.

In this work, the framework of k-t SPARSE-SENSE is extended to

golden-angle radial acquisitions and demonstrated for various clinical

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dynamic imaging applications, including free-breathing liver DCE MRI,

pediatric body MRI, breast and neck imaging. The performance of the

proposed approach, entitled Golden-angle RAdial Sparse Parallel MRI

(GRASP), is compared to reconstructions with coil-by-coil compressed

sensing and parallel imaging alone.

5.3. Golden-Angle Radial Sampling

Continuous 3D data acquisition was implemented using a stack-of-

stars k-space trajectory, where Cartesian sampling is employed along the

partition dimension (kz) and golden-angle radial sampling is employed in the

kx-ky plane, as summarized in Figure 5.1a. The golden-angle acquisition

scheme (113), which has previously been applied for accelerated dynamic

imaging, ensures approximately uniform coverage of k-space for any

arbitrary number of consecutive spokes, in particular if the number belongs

to the Fibonacci series (defined as F(k+2) = F(k) + F(k+1), where k ≥ 0, and

F(0) = 0 and F(1) = 1, e.g. 1,2,3,5,8,13,21,34,…). Figure 5.1b shows the

point spread function (PSF) for a golden-angle radial acquisition with 21

spokes using a single element receiver coil (top) and a sensitivity-weighted

combination of 8 RF coil elements (bottom). The PSF for the single coil is

calculated by performing gridding on a simulated k-space matrix with ones

along an undersampled radial trajectory with 21 golden-angle spokes and

256 sampling points along each spoke, followed by an inverse nonuniform

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fast Fourier transform (NUFFT) operation. The Nyquist sampling

requirement for this case is 256*π/2≈402, corresponding to a simulated

acceleration rate of 19.1. As mentioned in previously chapters, the PSF is

indicative of the degree of incoherence associated with particular

undersampling prior to the compressed-sensing reconstruction. The PSF of

the 8-coil acquisition with identical acceleration was computed using the

multicoil SENSE model, which performs a sensitivity-weighted combination

of individual PSFs using simulated sensitivity maps. The resulting

incoherence, which was computed as the ratio of the main-lobe to the

standard deviation of the side-lobes in the PSF, was 83.1 for the single-coil

case and 106.9 for the 8-coil case. As shown in the Figure 5.1b, the use of

the multicoil SENSE model reduces the side-lobes, which correspond to

aliasing artifacts due to undersampling. The higher encoding capabilities

provided by the coil array therefore are expected to improve the

performance of compressed sensing.

5.4. GRASP Reconstruction

Figure 5.2 shows the GRASP reconstruction pipeline. Since the kz

dimension is uniformly sampled, a fast Fourier transform (FFT) is applied

along this dimension to enable slice-by-slice reconstructions, which reduces

the computational burden and enables straightforward parallelization of the

reconstruction. Coil sensitivity maps are computed with the adaptive array-

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Figure. 5.2: GRASP reconstruction pipeline. (a) Estimation of coil sensitivity maps in the image domain, where the multicoil reference image (x-y-coil) is given by the coil-by-coil NUFFT reconstruction of the composite k-space data set that results from grouping all the acquired spokes. (b) Reconstruction of the image time-series, where the continuously acquired data are first re-sorted into undersampled dynamic time series by grouping a number of consecutive spokes. The GRASP reconstruction algorithm is then applied to the re-sorted multicoil radial data, using the NUFFT and the coil sensitivities to produce the unaliased image time-series (x-y-t).

combination technique (40) using coil-reference data from the temporal

average of all acquired spokes, which is usually fully sampled as shown in

Figure 5.2a. Afterwards, the continuously acquired radial spokes are re-

sorted by grouping a Fibonacci number (e.g., 34, 21, or 13) of consecutive

spokes to form each temporal frame with the desired temporal resolution.

The GRASP reconstruction is formulated as follows:

2

2 1argmin

x

x F C x y S x [5.1]

where x is the image series to be reconstructed in x-y-t space, S is the

sparsifying transform (temporal finite differences, also known as temporal

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total-variation or temporal TV in this work) imposed on the L1 norm,

1

n

y

y

y

are the acquired multicoil radial k-space data with n coils, F is the

NUFFT operator defined on the radial acquisition pattern,

1

n

c

c

c

are the

coil sensitivity maps in x-y space, and is the regularization weight that

controls the tradeoff between parallel imaging data consistency and

sparsity. A ramp filter in the kx-ky plane was applied to each spoke to

compensate for variable density sampling.

5.5. Reconstruction Implementation

5.5.1. Implementation of the Reconstruction Algorithm

The GRASP reconstruction was initially implemented in customized

software developed in MATLAB (Mathworks, MA), using a tailored version

of the non-linear conjugate gradient algorithm originally proposed in (28).

In order to achieve reconstruction times that allow for more practical

evaluation in clinical settings, the reconstruction was also implemented as a

stand-alone application using the C++ language. Several algorithmic

optimizations were incorporated to achieve high reconstruction speed. First,

a channel-compression procedure was applied to reduce the amount of k-

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space data, which combined the receiver channels into eigenmodes based

on a principal component analysis and discarded higher-order modes such

that 95% of the signal power was preserved (114). Second, the

reconstruction was parallelized across slices using the OpenMP framework

(115), yielding an almost linear reduction of the reconstruction time with the

number of processor cores. The NUFFT was implemented via convolution

with a Kaiser-Bessel kernel. Interpolation coefficients were pre-calculated

and shared across threads. Corner rounding was applied to allow for

differentiation of the TV L1 norm.

5.5.2. Selection of Reconstruction Parameters

To determine the optimal weighting parameter , the performance of

several values was first evaluated on one representative dataset for one

temporal resolution and then adjusted for other temporal resolutions

according to the difference in aliasing artifacts (pseudo-noise). First,

GRASP reconstructions were performed using different weights ranging

from 0.01* M0 to 0.1* M0 (step size 0.01), where M0 was the maximal

magnitude value of the NUFFT images that are also used to initialize the

GRASP reconstruction, for the case of 21 spokes per temporal frame. An

adequate value for was selected by an experienced radiologist, who

identified the appropriate value corresponding to the adequate balance

between preservation of fine detail and residual noise or pseudo-noise

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level, and also evaluated the signal intensity of regions of interest (ROI)

along time. The parameter for different temporal resolutions was then

obtained with 21 21/ *

tA A , where t

A is the pseudo-noise at the target

temporal resolution and 21A is the pseudo-noise at 21 spokes per frame.

The pseudo-noise was computed as described before. In this way, higher

temporal resolutions (or equivalently, use of fewer spokes for each temporal

frame) will be regularized more strongly, proportionally to the higher level of

pseudo-noise. This parameter estimation procedure needs to be performed

only once for a certain target temporal resolution and application, and the

value can then be used for different temporal resolutions and applications.

5.6. Imaging Applications

GRASP dynamic imaging was clinically implemented and evaluated

for a variety of representative imaging applications, as described in the

subsequent subsections. Human imaging was performed in accordance

with protocols approved by the New York University School of Medicine

Institutional Review Board and was found to comply with the HIPAA. All

subjects provided written informed consent before the imaging.

5.6.1. Dynamic Contrast-Enhanced Liver Imaging

DCE liver MRI was performed in six healthy volunteers (age

34.5±5.2 years) and seven patients (age 51±8.4 years) in axial orientation

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during free breathing using whole-body 3T or 1.5T scanners (MAGNETOM

Verio / Avanto, Siemens AG, Healthcare Sector, Erlangen, Germany) with a

combination of body-matrix and spine coil elements with 12 channels in

total. Data acquisition was initiated simultaneously with intravenous

injection of 10 ml of gadopentate dimeglumine (Gd-DTPA) (Magnevist,

Bayer Healthcare, Leverkusen) followed by a 20-ml saline flush, both

injected at a rate of 2 ml/second. A radial stack-of-stars 3D Fast Low Angle

SHot (FLASH) pulse sequence with golden-angle ordering was employed

for the data acquisitions. Two-fold readout oversampling was applied to

avoid spurious aliasing along the spokes. All partitions corresponding to

one radial angle were acquired sequentially before moving to the next

angle. The ordering scheme along kz was switched between linear (from

kz=-kxmax/2 to kz=+kmax/2) and centric out (starting at kz=0) depending

on the number of slices, as done in most of the modern 3D gradient echo

(GRE) sequences. Frequency-selective fat suppression was used and 60

initial calibration lines were acquired to correct system-dependent gradient-

delay errors as described in (116). Relevant imaging parameters are listed

in Table 5.1.

5.6.2. Dynamic Contrast-Enhanced Pediatric Body Imaging

Abdominal DCE MRI was performed in five pediatric patients (age

4.8±4.1 years) in axial orientation on a 1.5T scanner (MAGNETOM Avanto,

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DCE Liver DCE Pediatrics DCE Breast DCE Neck

#Sampling in Each Readout (2x)

512~768 512 512 512

#Partitions 29~40 48 35 69

#Spokes in Each Partition

600~800 800 2280 800

Slice Thickness (mm)

3 3 2 2

FOV (mm2) 370x370 250x250 270x270 256x256

TR/TE (ms) 3.83/1.71 4.24/2.07 3.6/1.47 4.57/2.06

Flip Angle (Degree) 12 12 12 12

Acquisition Time (s) 90 193 331 283

Table. 5.1: Representative imaging parameters of dynamic volumetric MRI in different applications.

Siemens AG, Healthcare Sector, Erlangen, Germany) using a body/spine

coil array with 12 elements. Acquisitions were performed during free

breathing because the patients were sedated during the exam. The imaging

and contrast-injection protocols were comparable to the liver example

described above. Relevant parameters are listed in Table 5.1.

5.6.3. Dynamic Contrast-Enhanced Breast Imaging

Free-breathing unilateral breast DCE MRI was performed in six

patients (age 55.3±6.7 years) in sagittal orientation prior to MRI-guided

biopsy using the radial 3D FLASH protocol on a 3T scanner (MAGNETOM

TimTrio, Siemens AG, Healthcare Sector, Erlangen, Germany), equipped

with a 7-element breast-coil array (InVivo Corporation, Gainesville, FL). A

single dose of Gd-DTPA with concentration of 0.1 mM/kg body weight was

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injected at 3 ml/second into an antecubital vein. Relevant imaging

parameters are listed in Table 5.1.

5.6.4. Dynamic Contrast-Enhanced Neck Imaging

DCE MRI of the neck was performed in ten patients (age 66.2±19.9

years) in axial orientation using the radial 3D FLASH protocol on a 1.5T

scanner (MAGNETOM Avanto, Siemens AG, Healthcare Sector, Erlangen,

Germany), using a head/spine coil with 15 elements. The contrast-injection

protocol was identical to the liver example. Relevant imaging parameters

are listed in Table 5.1.

5.7. Image Reconstruction

Iterative SENSE, coil-by-coil compressed sensing, and GRASP

reconstructions were performed on all the datasets using 21 spokes for

each temporal frame. The reconstructed in-plane matrix size was 256x256

or 384x384, depending on the number of readout samples. The achieved

temporal resolution was about 3 seconds/volume for the liver application, 5

seconds/volume for the pediatric application, 3 seconds/volume for breast

imaging and 7 seconds/volume for neck imaging. Compared to the Nyquist

sampling rate, the reconstructions correspond to an acceleration rate of

19.1 or 28.7.

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The iterative SENSE reconstruction was performed using the

GRASP implementation with a regularization weight of =0. In coil-by-coil

compressed sensing reconstruction, image reconstruction was performed

for each coil element separately, followed by sum of square combination.

The regularization parameter was selected only once, as described for

GRASP.

In order to demonstrate the flexibility of GRASP, reconstructions

were also performed with different temporal resolutions for one of the

pediatric datasets (13 and 34 spokes, corresponding to 3 and 8

seconds/volume).

Image reconstruction was performed using the C++ implementation

on a Linux server equipped with four Intel Xeon E5520 quad core CPUs at

2.27 GHz and 96 GB of RAM. The reconstruction time ranged between 30 -

45 minutes for a complete 3D data set, depending on the size of datasets.

5.8. Image Analysis and Statistics

In order to evaluate the image quality and temporal fidelity achieved

with GRASP, one representative partition was selected from each

reconstructed dataset for image quality assessment. Images were

compared between GRASP vs. iterative SENSE, GRASP vs. coil-by-coil

compressed sensing, and temporal fidelity assessment was compared

between GRASP vs. NUFFT.

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5.8.1. Image Quality Assessment

A total of 39 liver datasets (13 iterative SENSE, 13 coil-by-coil

compressed sensing and 13 GRASP), 15 pediatrics datasets (5 iterative

SENSE, 5 coil-by-coil compressed sensing and 5 GRASP), 18 breast

datasets (6 iterative SENSE, 6 coil-by-coil compressed sensing and 6

GRASP) and 30 neck datasets (10 iterative SENSE, 10 coil-by-coil

compressed sensing and 10 GRASP) were pooled and randomized for

blinded qualitative evaluation by 3 radiologists with expertise on abdominal

imaging, breast imaging and neuroimaging respectively. The score levels

for all the image quality assessments were: 1 = non-diagnostic, 2 = poor, 3

= adequate, 4 = good and 5 = excellent.

The reported scores in each reconstruction category from all five

applications were pooled together to represent mean ± standard deviation.

Wilcoxon signed-rank sum test was chosen to compare the scores between

GRASP vs. iterative SENSE and GRASP vs. coil-by-coil compressed

sensing (n=34), using Excel (Microsoft, Redmond, WA), where P < 0.05

was considered to be statistically significant different.

5.8.2. Temporal Fidelity Assessment

For each of the GRASP datasets, a ROI was manually drawn to

evaluate the signal-intensity time courses. The upslope was computed

using a linear fit of the curve points selected between 10% and 90% of the

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Figure. 5.3: Reconstruction of one representative partition from the contrast-enhanced volumetric liver dataset acquired with golden-angle radial sampling scheme using NUFFT (a) and iGRASP with three different weighting parameters (b-d) by grouping 21 consecutive spokes in each temporal frame. Results with λ = M0*0.05 achieved an appropriate compromise between image quality and temporal fidelity. This value was therefore chosen for iGRASP reconstruction with temporal resolutions of 21 spokes per frame. The weighting parameter was adjusted for different temporal resolutions according to the level of incoherent aliasing artifacts or pseudo-noise in the PSF. M0 was the maximal magnitude value of the NUFFT images that were also used to initialize the iGRASP reconstruction.

relative peak enhancement, which usually corresponded to the first pass of

contrast agent. The corresponding NUFFT data set was evaluated using

the same ROI, and the upslope was calculated using the same length of

curve points as reference. The analysis was performed on all GRASP

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datasets (n=34) and the corresponding NUFFT results. The upslope of

NUFFT and GRASP reconstructions were pooled separately and the

relative accuracy was evaluated by performing linear correlation and

Intraclass correlation (ICC) in Excel (Microsoft, Redmond, WA).

5.9. Results

5.9.1. Selection of Reconstruction Parameters

Figure 5.3 shows the results from the NUFFT reconstruction of one

DCE liver data set (a) and the GRASP reconstructions with three

representative values of the weighting parameter (b-d). It should be

noted that although the dynamic curves from the NUFFT reconstruction are

contaminated by streaking artifacts, they still preserve good contrast-time

evolution due to the fact that intensities were averaged over a relatively

large ROI. Therefore, it can be used as a first rough measure to assess

temporal fidelity. The results suggest that = M0*0.05 yields a good

balance between image quality and temporal fidelity (Figure 5.3c). Higher

weighting (Figure 5.3d, = M0*0.09) produces lower residual artifact and

slightly better image quality but also stronger temporal blurring, and vice

versa for a lower weight (Figure 5.3b, = M0*0.01). Based upon these

results, = M0*0.05 was selected by the radiologist for GRASP

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Figure. 5.4: Comparison of GRASP (top) reconstruction with coil-by-coil compressed sensing (middle) and iterative SENSE (bottom) reconstructions in the liver dataset with the same acceleration rate and temporal resolution of 21 spokes/frame = 3 seconds/volume. GRASP showed superior image quality compared to both coil-by-coil compressed sensing and iterative SENSE reconstructions.

reconstructions with 21 spokes. As shown in the following sections, this

weight led to similarly good results in other applications.

5.9.2. GRASP vs. Coil-by-Coil Compressed Sensing and Iterative

SENSE

Figure 5.4 shows the comparison of GRASP with coil-by-coil

compressed sensing and iterative SENSE reconstructions for a liver dataset

with in-plane matrix size of 384x384. GRASP showed better image quality

than coil-by-coil compressed sensing reconstruction, largely as a result of

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the incorporation of coil sensitivities in the reconstruction and the reduction

of aliasing artifacts provided by the parallel-imaging component (Figure

5.1b). The reduction of aliasing artifacts enabled recovery of more signal

coefficients, particularly those corresponding to high resolution features,

which generally have lower values. GRASP also outperformed iterative

SENSE and showed significantly lower residual aliasing artifacts due to the

temporal TV constraint, which exploits additional temporal correlation and

sparsity.

5.9.3. Dynamic Pediatric Body Imaging

Figure 5.5a shows one representative partition of DCE-MRI from a

10-year old patient. The reconstructed images clearly show distinct aorta,

portal vein, and liver contrast enhancement over time. Note that the same

data set was used to reconstruct dynamic images with different temporal

resolutions by grouping 34 (top), 21 (middle), and 13 (bottom) spokes.

Figure 5.5b evaluates the corresponding signal intensity changes over time

for the aorta (AO) and portal vein (PV). For comparison, the signal intensity-

time curves of the NUFFT reconstruction are included as reference.

5.9.4. Dynamic Breast Imaging

Figure 5.6a shows unilateral breast DCE-MRI of a patient referred for

fibroadenoma with fibrocystic changes. The images reconstructed with

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Figure. 5.5: (a) GRASP reconstruction of free-breathing contrast-enhanced volumetric abdominal imaging of a 10-year old patient referred for tuberous sclerosis. Representative images with three different temporal resolutions are shown, including (top) 34 spokes/frame = 8 seconds/volume, (middle) 21 spokes/frame = 5 seconds/volume and (bottom) 13 spokes/frame = 3 seconds/volume. The reconstructed image matrix size was 256 x 256 in each dynamic frame, with in-plane spatial resolution of 1 mm and the weighting parameters of different temporal resolutions were adjusted according to the acceleration rate. b) Signal-intensity time courses for the aorta and portal vein, which do not show significant temporal blurring as compared with the corresponding NUFFT results.

GRASP show appropriate contrast enhancement over time in the normal

breast tissue and in a suspicious breast lesion indicated by the white arrow.

GRASP also provided good image quality and depiction of relevant

morphological features, such as fibroglandular tissue, skin layer, and the

suspicious lesion. Figure 5.6b shows the corresponding signal intensity

changes over time of the breast lesion, heart cavity, vessel and breast

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Figure. 5.6: (a) GRASP reconstruction of free-breathing contrast-enhanced volumetric unilateral breast imaging in an adult patient referred for fibroadenoma with fibrocystic changes. Temporal resolution is 21 spokes/frame = 3 seconds/volume. The reconstructed image matrix size is 256 x 256 for each dynamic frame, with in-plane spatial resolution of 1.1 mm. b) Signal-intensity time courses for the breast lesion, which is a fibroadenoma with fibrocystic changes (white arrow), for the heart cavity (white ROI), and for a blood vessel and breast tissue (white arrows), showing no significant temporal blurring.

tissue (white arrows and ROI). The GRASP reconstruction did not introduce

significant notable temporal blurring.

5.9.5. Dynamic Neck Imaging

Figure 5.7 shows representative images of two partitions from a

patient referred for neck mass and squamous cell cancer, together with the

corresponding signal-intensity changes for the carotid artery (white arrows).

The reconstruction shows good image quality in different phases and

similar contrast enhancement to the NUFFT curves.

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Figure. 5.7: (a) GRASP reconstruction of contrast-enhanced volumetric neck imaging in an adult patient referred for neck mass and squamous cell cancer. Temporal resolution is 21 spokes/frame = 7 seconds/volume. The reconstructed image matrix size is 256 x 256 for each dynamic frame, with in-plane spatial resolution of 1 mm. b) Signal-intensity time courses evaluated for the carotid arteries show no significant temporal blurring.

5.9.6. Image Quality Comparison

Table 5.2 summarizes the mean scores and standard deviations for

different reconstruction strategies in each application. GRASP yielded

significantly better scores (P < 0.05) when compared with both iterative

SENSE and coil-by-coil compressed sensing reconstructions. The score of

GRASP was above 3.0 in all applications, suggesting that good image

quality can be achieved with the proposed acceleration rate and temporal

resolution.

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DCE Liver DCE Pediatrics DCE Breast DCE Neck

GRASP 3.38±0.65 4.20±0.84 4.67±0.52 3.80±0.79

Coil-by-Coil CS 1.62±0.77 1.80±0.45 2.33±1.03 2.10±0.74

Iterative SENSE 1.38±0.65 1.40±0.55 2.17±1.17 1.00±0.00

Table. 5.2: Image quality assessment scores represent mean ± standard deviation for each reconstruction category for different applications.

5.9.7. Temporal Fidelity Comparison

For the upslope calculated from the data pairs (n=34, GRASP vs.

NUFFT), the linear regression coefficient was 0.99 and ICC was 0.99,

indicating strong agreement between the upslope obtained from GRASP

and NUFFT. This result suggests that GRASP does not introduce

significant temporal blurring.

5.10. Discussion

This chapter introduces a robust imaging approach for rapid dynamic

volumetric MRI named GRASP, which is applicable for a broad spectrum of

clinical applications. Even though individual components of the method

have been proposed before, the synergistic combination of compressed

sensing, parallel imaging, and golden-angle radial sampling results in a

technique that is particularly well-suited to obtain high spatial resolution,

high temporal resolution, and large volumetric coverage at the same time.

GRASP achieved significantly better performance than either parallel

imaging or compressed sensing alone and demonstrated high value for

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clinical studies that require robustness to patient motion and simultaneous

high spatial and temporal resolution. GRASP can be also used in other

applications such as cardiac cine imaging.

The motion robustness can be mainly attributed to the use of radial

k-space sampling. Radial sampling is well-known for being less susceptible

than Cartesian sampling due to (i) lower sensitivity to motion-induced phase

shifts and (ii) signal averaging at the center of k-space. Moreover, it is well-

suited for compressed sensing because radial undersampling creates

incoherent low-intensity streaking artifacts. The golden-angle ordering

scheme additionally introduces temporal incoherence of the k-space

acquisition.

In radial sampling, the image contrast corresponds to the average

over the acquisition window because all lines cover k-space center. In this

regard, radial sampling introduces a certain amount of temporal blurring,

which manifests as slightly lower vessel-tissue contrast compared to

Cartesian acquisitions that use a narrow time window for the acquisition of

the k-space center. However, as opposed to other radial approaches that

use a broad temporal view-sharing filter to extract different temporal phases

without streaking artifacts, GRASP enforces data fidelity only within a

relatively small temporal window (e.g., 21 spokes), which enables to

preserve high temporal sharpness.

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GRASP reconstruction removes streaking artifacts in the

undersampled time-series of images at the expense of suppressing small

coefficients in the temporal TV domain, which can compromise temporal

fidelity for high acceleration factors because rapidly oscillating intensity

changes may be dampened in this case while the temporal onset of sharp

intensity changes remains unaffected due to the use of the L1 norm.

However, unlike reconstruction approaches that employ TV constrains in

the spatial domain, GRASP does not lead to spatial image blurring or

synthetic appearance. In cases where there is motion between temporal

frames, temporal blurring artifacts might under certain circumstances

appear as spatial blurring artifacts, but these artifacts originate in the

temporal dimension. This penalty, which is common to all compressed

sensing methods, is due to the fact that MR images are compressible rather

than truly sparse, presenting a few high coefficients and many small

coefficients. If the small coefficients fall below the pseudo-noise level

created by the undersampling artifacts, they may not be robustly

recoverable. For the particular case of temporal TV, abrupt temporal

variations usually result in high coefficients that are well recovered by the

reconstruction. However, moderate or smooth signal variations might result

in low-value coefficients below the pseudo-noise level, which could be

suppressed by the reconstruction. Although minor compromises in the

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temporal fidelity may result, it is unclear whether these effects are clinically

relevant. Future studies are planned to assess the impact on the diagnostic

performance of dynamic imaging, although the preliminary results obtained

so far indicate that the technique does not introduce clinically significant

temporal distortions. From a clinical perspective, it is presumably of higher

relevance that GRASP enables dynamic abdominal imaging in patients who

have difficulty in suspending their respiration, including severely sick,

pediatric, or sedated patients, and thus it is infeasible to perform dynamic

imaging with adequate diagnostic quality using established conventional

techniques.

GRASP provides a simple and flexible way of performing dynamic

MRI studies in these patients and can help to improve clinical workflow by

enabling data acquisition without the need for synchronization with breath-

hold commands or for selection of a predefined rigid temporal resolution.

While a typical clinical use case does not require reconstruction and

evaluation of image series at multiple temporal resolutions, which would

increase the workload of radiologists if used indiscriminately, the flexibility

of reconstructing different temporal resolutions without the need to re-

acquire data can be another advantage for specific clinical questions or in

the event of a suggestive finding. Formal studies are currently in progress

using a prototypic workflow integration to investigate the clinical potential of

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multi-resolution reconstructions and to determine the range of effectively

achievable temporal resolutions.

The current implementation of GRASP has some limitations that will

be addressed in future work. First, a stack-of-stars k-space sampling

pattern is employed to enable parallelized slice-by-slice reconstructions.

This reduces the computational burden of GRASP reconstructions, but

prevents employing compressed sensing along the slice dimension. The

use of full 3D golden-angle radial sampling along with a volumetric

reconstruction are expected to further increase the performance, at the

expense of higher computational demand. Second, although temporal TV

has been used before for different dynamic MRI reconstructions (77,117-

119) and was shown to be better in some specific applications, it may be

not optimal to use it as the only sparsifying transform for all cases and

applications. Other advanced temporal sparsifying transforms, such as

dictionary learning, might be also useful to increase temporal fidelity for

high undersampling factors. Third, the current work did not use rigorous

mathematical criteria to select the weighting parameter , which controls

the tradeoff between removal of streaking aliasing artifacts and temporal

fidelity. The empirical rule to make proportional to the pseudo-noise level

in the PSF produced reasonable performance for different undersampling

factors. The same was also used in different applications for a given

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temporal resolution, which suggests that the reconstruction can be

automated without intervention. However, evaluation on a larger set of

patients comparing with standard clinical techniques is required to test the

robustness of this new approach. Finally, because it is impossible in

practice to acquire a fully-sampled volumetric DCE dataset with the target

spatial and temporal resolution, the current study employed NUFFT

reconstructions as temporal reference. While NUFFT reconstructions

provide time curves without artificial temporal blurring effects, they can be

affected by strong streaking artifacts at high accelerations that limit their

value for assessing the ground-truth signal evolution. A comprehensive

analysis of the temporal fidelity achieved with GRASP using numerical

simulations and dynamic phantom scans is currently in progress.

5.11. Conclusion

The combination of compressed sensing, parallel imaging, and

golden-angle radial sampling employed in GRASP enables rapid dynamic

volumetric MRI studies with high spatial resolution, temporal resolution, and

motion robustness. Because of the continuous data acquisition and the

flexibility to reconstruct images retrospectively at different temporal

resolutions, dynamic imaging with GRASP can be integrated easily into

clinical workflow. GRASP can be used for a wide range of clinical

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applications and demonstrated particular value for examinations of patients

that are unable to suspend respiration.

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Chapter 6

XD-GRASP: Golden-Angle Radial MRI with Reconstruction

of an Extra Motion-State Dimensions Using Compressed

Sensing

6.1. Prologue

The GRASP technique presented in Chapter 5 has been

successfully demonstrated in a variety of clinical applications and it is

currently being tested in a wide range of routine clinical studies. However,

despite its robustness to respiratory motion, initial clinical studies suggest

that GRASP still suffers from residual respiratory blurring, which may

reduce the vessel-tissue contrast and thus hinder the clinical interpretation

of the images. In this chapter, we develop a novel framework for free-

breathing MRI called XD-GRASP, which exploits the self-navigation

properties of radial imaging and reconstructs an extra respiratory-state

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dimension using compressed sensing. The proposed method represents a

novel way to handle respiratory motion using compressed sensing ideas.

Instead of seeking to remove motion, an additional respiratory-state

dimension is reconstructed, which improves image quality and also offers

new physiological information that could be of potential clinical value.

The contents presented in this chapter were published in the journal

Magnetic Resonance in Medicine (MRM 2015 Mar 25. doi:

10.1002/mrm.25665. [Epub ahead of print]) (120).

6.2. Introduction

Respiratory motion remains a major challenge in MRI, particularly for

abdominal and cardiovascular imaging. Due to the limited encoding speed

of conventional MRI, k-space lines may be acquired in different respiratory

motion states during free breathing, resulting in ghosting artifacts and

image blurring (45,121). As reviewed in Chapter 2, the simplest approach to

avoid respiratory motion effects is to suspend respiration during data

acquisition (7) – an approach that is currently widely used in routine clinical

MRI exams. However, breath-hold capabilities are subject-dependent and

can be significantly limited in some patients. In addition, typical breath-hold

durations (10-15 seconds) also limit achievable spatial resolution and

volumetric coverage. An alternative approach is to use either navigator

signals (47) or respiratory bellows (48) to monitor respiratory motion and

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acquire data only at a specific respiratory state (e.g., end-expiration).

However, such gated data acquisition significantly reduces imaging

efficiency and further prolongs the total examination times. Real-time MRI

can be used for free-breathing cardiac cine imaging (77,101,122,123), but

the acquisitions usually comprise only a single slice with limited spatial and

temporal resolution. Non-Cartesian k-space sampling schemes, such as

radial or spiral, are substantially less sensitive to respiratory motion and

enable free-breathing imaging at the expense of increased scan times (50-

52). For example, radial imaging eliminates k-space gaps due to motion-

related phase shifts, by repeated sampling of the k-space center. However,

substantial motion is still a challenge for non-Cartesian imaging and can

result in blurring and aliasing artifacts, e.g., streaks for radial trajectories

(124,125). Non-Cartesian acquisitions also offer the potential benefit of

retrospective self-gating, owing to the continuous passage of the radial lines

through the center of k-space, and thus can eliminate the need to use

navigator signals or external devices (53,54). For example, Liu et al. have

proposed an image reconstruction approach for free-breathing cardiac cine

MRI, in which the cardiac and respiratory motion signals retrospectively

obtained from the data are used for self-gating and view-sharing

reconstruction with less motion blurring (53). However, these approaches

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are still time-inefficient, since typically only data within a predefined motion

state (e.g., close to expiration) are used for the image reconstruction.

Compressed sensing with temporal sparsifying transforms has

enabled high accelerations in dynamic MRI studies (30,72,126). However,

respiratory motion generally leads to inter-frame misalignments, which

reduce temporal sparsity and result in temporal blurring. Several

approaches have been proposed to integrate an image registration

framework into the reconstruction problem, to correct for respiratory motion.

For example, rigid-body motion registration techniques have been applied

to compressed sensing cardiac perfusion imaging (127) and more complex

deformable registration techniques that account for non-rigid body motion

were employed in compressed sensing reconstruction of cardiac cine (58),

cardiac perfusion (56), and abdominal DCE-MRI examinations (57). A more

advanced method, which learns the motion fields from the data itself to

guide image reconstruction, was recently introduced, which, in addition to

performing motion compensation, can also provide access to specific

motion information (59).

With the goal of combining the motion-robustness of radial imaging

and the acceleration capabilities of compressed sensing, the Golden-angle

RAdial Sparse Parallel (GRASP) technique (107) has been proposed in

Chapter 5 for highly-accelerated motion-robust DCE-MRI. Successful

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applications of GRASP for free-breathing imaging have been demonstrated

in various organs affected by respiratory motion, such as liver (108),

prostate (109) and small bowel (128). However, our clinical evaluation

suggests that GRASP still suffers from some degree of respiratory motion

blurring, especially in sick or elderly patients, who tend to be less

cooperative in following a shallow breathing pattern during data acquisition.

The resulting motion-blurring effects reduce vessel-tissue contrast and may

prevent the detection of small lesions.

In this work, we propose a novel image reconstruction framework,

called eXtra-Dimensional GRASP (XD-GRASP), which combines GRASP

with the self-navigation property of radial imaging and uses motion

detection schemes adapted from previous work (129,130). Instead of

removing or correcting the motion in question, XD-GRASP reconstructs

extra motion dimensions, where continuously acquired k-space data are

sorted into multiple sets of undersampled datasets with distinct motion

states, using motion signals extracted directly from the data (131,132). This

approach may also be generalized to account for multiple sources of motion

or dynamic signal change simultaneously, such as cardiac motion and

contrast enhancement in addition to respiratory motion, by sorting the data

into multiple additional motion-state dimensions. A compressed sensing

algorithm is employed to reconstruct the motion-sorted datasets by

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Figure. 6.1: Schematic illustration of the XD-GRASP method: (a) Continuously acquired radial k-space data are sorted into respiratory states from expiration (top) to inspiration (bottom), using a respiratory motion signal extracted directly from the data. Different colors indicate different motion states. The number of spokes grouped in each motion state is the same. (b) Approximately uniform coverage of k-space, with distinct sampling patterns in each respiratory motion state, is achieved using the golden-angle acquisition scheme. (c) Data sorting removes blurring and clearly resolves respiratory motion, at the expense of introducing undersampling artifacts. The purple dashed line shows the distinct respiratory motion states after data sorting. (d) Sparsity is exploited along the extra dimension to remove aliasing artifacts due to undersampling.

exploiting sparsity along the corresponding motion-state dimensions. From

a clinical perspective, the extra dimensions may also provide new

physiological information, since images of different kinds of motion states

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may be disentangled during reconstruction. The performance of XD-

GRASP is demonstrated by comparing against reconstructions without

motion sorting in representative free breathing imaging applications,

including: i) 3D abdominal imaging with respiratory motion only; ii) 2D

cardiac cine imaging with cardiac and respiratory motion, and iii) 3D liver

DCE-MRI with respiratory motion and time-dependent contrast-

enhancement.

6.3. A Simple Example of XD-GRASP

Successful implementation of XD-GRASP has two principal

requirements: (a) reliable physiological (e.g., respiratory and/or cardiac)

motion signals and (b) preservation of approximately uniform k-space

coverage in each motion state after data sorting. Golden-angle radial

sampling (113), which uses ~111.25o angular increment between

consecutive spokes, is employed for data acquisition, since the repeated

sampling of the k-space center enables extraction of motion-state signals,

and it allows the possibility of arbitrary data sorting with approximately

uniform k-space coverage in each motion state while maintaining sufficient

incoherence in the sampling pattern along the new motion-state dimension

for robust compressed sensing reconstruction.

Figure 6.1 illustrates the basic concept of XD-GRASP, in which the

continuously acquired radial k-space data are sorted into a specific number

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Figure. 6.2: Data sorting procedure for XD-GRASP in abdominal MRI without contrast ejection. Respiratory motion was first sorted from end-expiration to end-inspiration and the corresponding set of spokes were evenly distributed into multiple respiratory states so that the number of spokes is the same in each motion state.

of respiratory states spanning from expiration (top) to inspiration (bottom)

using a respiratory motion signal derived from the acquired data (Figure

6.1a). The sorting procedure is performed so that the number of spokes

grouped in each motion state is the same (as shown in Figure 6.2).

Approximately uniform coverage of k-space with distinct sampling patterns

in each motion state is achieved by using the golden-angle acquisition

scheme (Figure 6.1b). Data sorting removes blurring and clearly resolves

respiratory motion (indicated by the purple dashed line), at the expense of

generating undersampling artifacts (Figure 6.1c). A compressed sensing

reconstruction that exploits sparsity along the new respiratory-state

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dimension can be employed to remove undersampling artifacts (Figure

6.1d).

6.4. Motion Estimation and Data Sorting

Estimation of motion signals and data sorting were performed in a

slightly different way for each target application (e.g., cardiac vs. abdominal

imaging) and k-space trajectory (2D golden-angle radial vs. 3D stack-of-

stars golden-angle radial).

6.4.1. Motion Estimation and Data Sorting in 2D Cardiac Cine MRI

For cardiac cine imaging, the center of k-space (DC component) in

each spoke (Figure 6.3a), which reflects the change in average signal level

due to changes of the volume of lung and heart in the excited slab, was

used to extract information about physiological motion over time (54).

Information from multiple coils was used to obtain separate signals

representing respiratory or cardiac motion, as shown in Figure 6.4.

Conceptually, the motion signal from the coil nearest to the heart provides

predominantly cardiac motion information, and the motion signal from the

coil nearest to the diaphragm provides predominantly respiratory motion

information. Since these motions are known to have different frequency

contents, the motion signal in the coil-element with the highest peak in the

frequency range of 0.1-0.5Hz was automatically selected to represent

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Figure. 6.3: XD-GRASP motion estimation and data sorting for cardiac cine imaging. (a) 2D golden-angle radial trajectory. Motion signals are estimated from the central k-space position of each radial line (gray dot). (b-c) Estimation of cardiac and respiratory motion signals using information from multiple coils. The signals with the highest peaks in the frequency range of 0.1-0.5Hz and 0.5-2.5Hz are automatically selected for respiratory and cardiac motion signals, respectively. (d) Conventional iGRASP sorting of cardiac phases, given by grouping consecutive spokes in each frame. (e) XD-GRASP sorting, in which all the cardiac cycles are concatenated into an expanded dataset with one cardiac dimension (tC) and an extra respiratory dimension (tR), so that sparsity along tC and tR can be exploited in the multidimensional compressed sensing reconstruction.

respiratory motion; and the motion signal in the coil-element with the

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Figure. 6.4: Selection of cardiac and respiratory motion signals from multiple coils. (a) 2D golden-angle radial trajectory for free-breathing 2D cardiac cine MRI and (b) estimation of cardiac and respiratory motion signals using information from multiple coils. The motion signal in the coil-element with the highest peak in the frequency range of 0.1-0.5Hz was automatically selected to represent respiratory motion; and the motion signal in the coil-element with the highest peak in the frequency range of 0.5-2.5Hz was automatically selected to represent cardiac motion. A filtering procedure can be performed on the detected motion signals for denoising.

highest peak in the frequency range of 0.5-2.5Hz was automatically

selected to represent cardiac motion. A filtering procedure can be

performed on the detected motion signals for denoising (129). Figure 6.3b

shows an example of detected motion signals and Figure 6.3c shows the

corresponding frequency information. End-systolic motion-states were

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identified as the valleys in the cardiac motion signal and thus any abnormal

cardiac cycles, in case of arrhythmias, can be identified according to the

difference between cycle lengths for rejection or a separate reconstruction.

Given the selected motion signals, the continuously acquired 2D cardiac

dataset can be sorted into an expanded dataset containing two dynamic

dimensions, representing predominantly cardiac and respiratory motions,

respectively. Specifically, the continuously acquired golden-angle radial

dataset were first sorted into a dynamic cardiac series by grouping a

number of consecutive spokes (e.g., 15 spokes) as one cardiac phase

(Figure 6.3d). All the cardiac cycles, identified using the cardiac motion

signal, were then sorted into an expanded dataset to generate an extra

respiratory state dimension tR (Figure 6.3e), so that sparsity along both

cardiac and respiratory dimensions can be exploited in the compressed

sensing reconstruction.

6.4.2. Motion Estimation and Data Sorting in 3D Abdominal MRI

The 3D stack-of-stars sampling scheme (Figure 6.5a), in which

golden angle radial sampling is employed in the kx-ky plane and Cartesian

sampling is employed along the kz dimension, acquires all spokes along kz

for a given rotation angle and then repeats the procedure for the next

rotation angle, i.e., an inner loop is defined along kz and an outer loop along

the rotation angle. A straightforward approach for motion detection would

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Figure. 6.5: XD-GRASP motion estimation and data sorting for DCE-MRI imaging. (a) 3D stack-of-stars radial trajectory with golden-angle rotation, where all spokes along kz for a given rotation angle are acquired before rotating the sampling direction to the next angle. (b) A 1D Fourier transform along the series of k-space central points of each slice is performed to obtain a projection profile of the entire volume for each angle and all the projection profiles from all coils are concatenated into a large two-dimensional matrix, followed by principal component analysis (PCA) along the z+coil dimension. (c-d) The principal component with the highest peak in the frequency range of 0.1-0.5Hz is selected to represent respiratory motion. (e-g) Contrast-enhancement effect is approximately removed by estimating and subtracting the envelope of the composite signal. (h-i) Processed respiratory motion signals are shown superimposed on the z-projection profiles for normal breathing (left) and heavy breathing (right), demonstrating reliable motion estimation.

be to use the DC component of central spokes along the kz dimension (133)

and perform the same procedure as was just described for 2D imaging.

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However, prior study has shown that motion detection is more robust using

the projections along the slice dimension for 3D stack-of stars imaging

(134). In this work, an adapted version of the projection approach was

employed for respiratory motion detection in 3D liver imaging. Specifically, a

projection profile of the entire volume was computed for each acquisition

angle by taking a 1D partition-direction Fourier transform of the series of

kx=ky=0 central points (gray lines in Figure 6.5a). Respiratory motion

detection was performed by first concatenating the projection profiles from

all coils into a large two-dimensional matrix, followed by principal

component analysis (PCA) along the concatenated z+coil dimension

(Figure 6.5b). As proposed in (130), PCA can be interpreted as a procedure

to determine the most common signal variation mode among all coils, and

the principal component with the highest peak in the frequency range of

0.1-0.5Hz was selected to represent respiratory motion (Figure 6.5c&d),

since respiratory motion is known to occur within this frequency range. For

DCE-MRI, contrast-enhancement has to be separated from respiratory

motion. In this work, the envelope of the detected motion signal was

estimated using a spline data fitting procedure and then subtracted to

generate the respiratory motion signal (Figure 6.5e-g). Figure 3h&i show

two representative examples of respiratory motion in both normal breathing

(left) and deep breathing (right) detected using the proposed approach,

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Figure. 6.6: For DCE-MRI, the respiratory motion sorting procedure described in Figure 6.2 is performed in each contrast-enhancement phase separately

where motion signals were superimposed on the slice projection profiles.

Given the respiratory motion signal, the continuously acquired golden-angle

radial dataset was first divided into successive contrast-enhancement

phases (dynamic dimension tcontrast) and each phase was then further

sorted into multiple respiratory states (dynamic dimension tR), in which the

number of spokes is the same in each motion state, as shown in Figure 6.6.

6.5. Image Reconstruction

XD-GRASP reconstruction extends the GRASP pipeline by enforcing

a different sparsity constraint along each dynamic dimension. Specifically

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for 2D free-breathing cardiac imaging, reconstruction was performed by

solving the following optimization problem:

2

1 1 2 22 1 1argmin

d

d F C d m S d S R d (6.1)

Here F is the non-uniform fast Fourier transform (NUFFT) operator (135)

defined for the radial sampling pattern, represents the n-elements

coil sensitivity maps with dimensionality of x-y-coil, where x and y represent

two spatial dimensions. d is the 2D dynamic image-series with one cardiac

motion dimension and one respiratory-state dimension (x-y-tC-tR), and

1

2

n

m

mm

m

are the corresponding multicoil radial k-space data sorted

according to the new dimensions (x-y-tC-tR-coil). 1S is the sparsifying

transform applied in the cardiac motion dimension with regularization

parameter 1 and 2

S is the sparsifying transform applied along the extra

respiratory-state dimension with regularization parameter 2 . R is a

reordering operator along the tR dimension that sorts all the respiratory

phases at a given cardiac position from expiratory state to inspiratory state.

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This sorting procedure will ensure a smooth transition between adjacent

motion states, which improves the performance of total-variation

minimization along the dynamic dimensions as suggested in (136).

For 3D liver imaging, reconstruction was performed by solving the

following optimization problem:

2

1 1 2 22 1 1argmin

d

d F C d m S d S d (6.2)

Here F is the same as before,

1

2

n

C

CC

C

represents the n-elements coil

sensitivity maps with dimensionality of x-y-z-coil, where z is the partition

dimension. d is the 3D dynamic image- series with one contrast-

enhancement dimension and one respiratory-state dimension (x-y-z-tContrast-

tR), and

1

2

n

m

mm

m

are the corresponding multicoil radial k-space data

sorted according to the new dimensions (x-y-z-tContrast-tR-coil). 1S is the

sparsifying transform applied in the contrast-enhancement dimension with

regularization parameter 1 and 2

S is the sparsifying transform applied

along the extra respiratory-state dimension with regularization parameter

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2 . For liver imaging without contrast injection, 1

is just set as zero. Since

each contrast enhancement phase is already sorted from end-expiratory

state to end-inspiratory state, which promotes a smooth transition between

respiratory motion states, the reordering operator R applied in Equation 6.1

is not needed for Equation 6.2.

In this work, temporal finite differences (also known as total-variation

minimization) along the dynamic dimensions was selected for both 1S and

2S based on the experiences in previous chapters, but with different

weights 1 and 2

tailored to reflect the different degrees of sparsity along

each dynamic dimension. For example, stronger regularization was applied

along the sparser dynamic dimension and vice versa.

6.6. Imaging Applications

The performance of XD-GRASP was tested in free-breathing 3D

abdominal imaging, 2D cardiac cine imaging and 3D liver DCE-MRI, on

both healthy volunteers and patients. Human imaging was performed in

accordance with protocols approved by the New York University School of

Medicine Institutional Review Board and was found to comply with the

HIPAA. All subjects provided written informed consent before the imaging.

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6.6.1. Free-Breathing 3D Abdominal Imaging: Respiratory Motion

Only

3D abdominal imaging (without contrast injection) was performed on

one healthy volunteer (female, age=32) on a whole-body 3T scanner

(MAGNETOM TimTrio, Siemens AG, Healthcare Sector, Erlangen,

Germany) equipped with the standard 12-element body matrix coil array. A

3D stack-of-stars golden-angle radial FLASH pulse sequence with

frequency-selective fat suppression was employed and three scans were

performed in transverse, coronal and sagittal orientations, in order to test

the robustness to motion in different imaging planes. Relevant imaging

parameters included: TR/TE = 3.52/1.41 ms, FOV = 300 x 300 x 140 mm3,

number of points in each spoke = 192, number of partitions = 28 and spatial

resolution = 1.5 x 1.5 x 5 mm3. 510 spokes were acquired for each partition,

with a total scan time of ~57 seconds.

Six respiratory motion-states (84 spokes in each state) were

generated by sorting the continuously acquired data as described in Figure

6.1 and Figure 6.2. XD-GRASP was performed with one sparsifying

transform along the respiratory-state dimension ( 2S in Equation 6.2). The

results were compared with NUFFT reconstruction of the whole dataset

without motion sorting.

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6.6.2. Free-Breathing 2D Cardiac Cine Imaging: Cardiac and

Respiratory Motions

2D cardiac cine data were acquired in one healthy volunteer (female,

age=32), one patient with normal sinus rhythm (male, age=46), one patient

with premature ventricular contractions (PVCs) (female, age=33) and

another patient with second-degree atrioventricular block (female, age=49).

Imaging was performed during normal free breathing on a whole-body 1.5T

scanner (MAGNETOM Avanto, Siemens AG, Healthcare Sector, Erlangen,

Germany) without any external triggering or gating, using a 2D radial

bSSFP pulse sequence with golden-angle acquisition scheme. Three short-

axis slices (SAX) in apical, middle, and basal ventricular positions, and one

slice in a four-chamber plane (4CH) were acquired in the volunteer scans.

Relevant imaging parameters included: TR/TE = 2.8/1.4 ms, FOV = 320 x

320 mm2, number of points in each spoke = 160, spatial resolution = 2 x 2

mm2, slice thickness = 8 mm, and the total acquisition time for each slice

was ~20 seconds. For comparison purposes, cardiac cine images with

similar imaging orientations and parameters were also acquired using the

routine clinical approach with breath-hold, Cartesian k-space sampling and

retrospective ECG gating. In the patient scans, one middle ventricular SAX

slice was acquired in each subject with the following imaging parameters:

TR/TE = 2.8/1.4 ms, FOV = 256 x 256 mm2, number of points in each

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spoke = 128, spatial resolution = 2 x 2 mm2, slice thickness = 8 mm, and

the total acquisition time for each slice was ~15-20 seconds. For

comparison purposes, cardiac cine images with similar imaging orientations

but with relatively higher spatial resolution (~1.6 x 1.6 mm2) were also

acquired using the routine clinical approach with breath-hold, Cartesian k-

space sampling and retrospective ECG gating.

Cardiac cycles with arrhythmias were first detected and separated

using the cardiac motion signals in the patient datasets. Every 15

consecutive spokes were grouped to generate one dynamic phase,

achieving a temporal resolution of ~ 45 ms, as showing in Figure 6.3d. XD-

GRASP reconstruction was performed on an expanded time-series of

undersampled datasets (Figure 6.3e) with ~18-26 cardiac phases and ~10-

16 respiratory phases, depending on the heart rate of the subjects. The

cardiac cycles with arrhythmias in the patient with PVCs were rejected

because there were only 2 cardiac cycles with arrhythmias in the entire

acquisition. Thus the gain of performing XD-GRASP in arrhythmia cycles is

small because of the limited number of respiratory phases and limited

sparsity along the respiratory dimension. In the patient with second-degree

atrioventricular block, there were more arrhythmia cardiac cycles and thus a

separate XD-GRASP reconstruction was performed on the cardiac cycles

with arrhythmias. For comparison, GRASP reconstruction (without

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respiratory sorting) was also performed on a time-series of undersampled

cardiac phases, where each cardiac phase was formed by grouping 15

consecutive spokes in the volunteer datasets. 1.5-fold zero-filling was

performed in all the results for visualization purposes, and a 5th-order

temporal medial filter was performed along the cardiac dimension after

image reconstructions, in order to further reduce the residual streaking

artifact (49).

6.6.3. Free Breathing 3D Liver DCE-MRI: Contrast Enhancement and

Respiratory Motion

3D liver DCE-MRI was performed on four volunteers (males, age =

32.5±1.3) as well as one patient (male, age=69) with a suspected liver

tumor on a whole-body 3T scanner (MAGNETOM Verio, Siemens AG,

Healthcare Sector, Erlangen, Germany) equipped with the standard 12-

element body matrix coil. Three volunteers and the patient were asked to

breathe normally and one volunteer was asked to breath heavily during the

scans. The 3D stack-of-stars pulse sequence was employed to acquire data

in transversal orientation and intravenous injection of 10 ml of

gadopentetate dimeglumine (Gd-DTPA) (Magnevist; Bayer Healthcare,

Leverkusen) was initialized simultaneously with the onset of data

acquisition, followed by a 20-ml saline flush, both injected at a rate of 2 ml /

second. Relevant imaging parameters for the volunteer scan included:

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TR/TE ≈ 3.52/1.41 ms, FOV = 360 x 360 x 240 mm3, number of points in

each spoke = 256, spatial resolution = 1.4 x 1.4 x 3 mm3, number of

partitions = 80, with 60% slice resolution reduction and 6/8 partial Fourier

applied along the slice dimension. 600 spokes were continuously acquired

in each partition, for a total scan time of ~95 seconds. Imaging parameters

for the patient scans were similar, except that the number of points in each

spoke was 320, resulting in a spatial resolution of 1.1 x 1.1 x 3 mm3.

For comparison purposes, GRASP reconstruction (without

respiratory sorting) was first performed on a time-series of undersampled

contrast-enhancement phases, where each phase was formed by grouping

84 consecutive spokes (temporal resolution of ~13 seconds). XD-GRASP

reconstruction was then performed on a multidimensional undersampled

dataset, in which the 84 spokes in each contrast-enhancement phase were

further sorted into 4 respiratory states spanning from end-expiration to end-

inspiration. The sorting procedure was performed such that the number of

spokes in each motion state was the same, as shown in Figure 6.6.

6.6.4. Image Reconstruction Implementation

A tailored version of non-linear conjugate gradient optimization,

originally proposed in, was used to solve the optimization problem in both

Equations 6.1&6.2. Coil sensitivity maps were computed from a fully-

sampled reference given by NUFFT reconstruction of the whole dataset,

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Figure. 6.7: Conventional NUFFT reconstruction without respiratory sorting (motion average) and XD-GRASP reconstruction with 6 respiratory states for datasets acquired in transverse, coronal and sagittal orientations. XD-GRASP significantly reduces motion-blurring, as indicated by the white arrows.

using the adaptive array combination method. Regularization parameters

1 and 2

were empirically selected by two experienced cardiac and body

radiologists. Specifically, the best value of 1 was selected first ( 2

was set

as zero) by testing different values and comparing image quality as well as

temporal fidelity, as previously performed in previous chapters. In the next

step, different values of 2 were then compared in combination with the 1

value selected in the first step and the radiologist selected the best value of

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2 . According to our prior experience with compressed sensing MRI,

regularization parameters selected in this fashion can be used reliably in

the reconstruction of similar datasets. Image reconstruction was performed

in MATLAB (Mathworks, Natick, MA), using a workstation with a 16-core

Intel Xeon CPU and 96 GB RAM. XD-GRASP reconstruction time was ~5

minutes/slice for 3D abdominal imaging, ~40-60 minutes for 2D cardiac cine

imaging, and ~15 minutes/slice for 3D liver DCE-MRI.

6.7. Results

6.7.1. Free Breathing 3D Abdominal Imaging

Figure 6.7 compares the conventional NUFFT reconstruction of the

full dataset without respiratory sorting (corresponding to the motion

average) to XD-GRASP reconstruction with 6 respiratory motion states. XD-

GRASP improves the depiction of vessels and removes the blurring effects

at the edges of the liver (white arrows).

6.7.2. Comparison of Different Regularization Parameters for the

Extra Respiratory Dimension

Figure 6.8 shows XD-GRASP reconstruction results for four

representative respiratory sparsity regularization parameters ( 2 ) in cardiac

imaging and liver DCE-MRI. Utilization of a sparsity constraint along the

extra respiratory-state dimension improved the removal of undersampling

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Figure. 6.8: XD-GRASP reconstruction results for four representative respiratory

sparsity regularization parameters ( 2 ) in cardiac imaging and liver DCE-MRI.

Utilization of a sparsity constraint along the extra respiratory-state dimension improved the removal of undersampling artifacts, when compared with the non-regularized case (

2 =0). Very low values of 2

resulted in residual aliasing artifacts, while very high

values of 2 introduced blurring. A 2

of 0.01 in cardiac cine imaging and 0.015 in

liver DCE-MRI provided a good tradeoff between residual aliasing artifacts and temporal fidelity.

artifacts, when compared to the non-regularized reconstruction ( 2 =0).

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Figure. 6.9: Comparison of XD-GRASP against the standard breath-hold approach used in routine clinical studies (i.e., with retrospective ECG-gating) at end-diastolic and end-systolic cardiac phases in the volunteer scan. XD-GRASP provided similar performance to the routine clinical breath-hold method.

Very low values of 2 resulted in residual aliasing artifacts, while very high

values of 2 introduced blurring. A 2

on the order of 0.01 in cardiac cine

imaging and 0.015 in liver DCE-MRI provided a good tradeoff between

residual aliasing artifacts and temporal fidelity.

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6.7.3. Free Breathing 2D Cardiac Cine Imaging

Figure 6.9 compares XD-GRASP with the standard breath-hold

approach using retrospective ECG-gating at end-diastolic and end-systolic

cardiac phases in the volunteer scans. Free-breathing XD-GRASP

achieved similar image quality to the conventional breath-hold approach but

also enabled the evaluation of the effects of respiratory motion at each

cardiac phase, which can be potentially valuable for examination of

conditions such as constrictive pericardial heart disease (137). Figure 6.10a

shows respiratory-related motion of the interventricular septum, especially

near end- diastolic cardiac phases, which indicates left-right ventricular

interaction during respiration.

Figure 6.10b show the comparison of XD-GRASP reconstruction

exploiting sparsity along two dynamic dimensions (right-hand column) with

GRASP reconstruction exploiting sparsity along a single dynamic dimension

only (left-hand column), using the same data set acquired during free

breathing. XD-GRASP reconstruction achieved superior image quality,

particularly in the removal of aliasing artifacts due to the separation of

cardiac and respiratory motion into different dimensions, which enables

exploitation of extra sparsity along the respiratory dimension.

Figure 6.11a compares XD-GRASP and the standard breath-hold

approach with retrospective ECG-gating for the patients. Although the

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Figure. 6.10: (a) XD-GRASP provides access to respiratory motion information for each cardiac phase, where respiratory-related motion of the interventricular septum, especially at diastolic cardiac phases (top row) can be seen, indicating left-right ventricular interaction during respiration. Gray arrows indicate different respiratory motion states. (b) Comparison of XD-GRASP reconstruction exploiting sparsity along two dynamic dimensions (right-hand column) with GRASP reconstruction exploiting sparsity along a single dynamic dimension only (left-hand column), using the same data set acquired during free breathing.

conventional breath- hold approach produced good image quality in the

patient with normal sinus rhythm, it produced poor image quality for patients

with arrhythmia, due to the failure to properly synchronize cardiac cycles

with different length in the reconstruction. XD-GRASP, on the other hand,

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achieved consistent image quality by enabling separation of cardiac cycles

with arrhythmia. In addition, the data from cardiac cycles with arrhythmia

can be used for a separate XD-GRASP reconstruction to provide additional

physiological information, as shown in the case of 2nd degree AV block

(Figure 6.11b). Figure 6.11c shows the corresponding cardiac motion

signals for three patients, where the cardiac cycle length is varying in the

patients with arrhythmia, indicated by gray arrows. The availability of these

signals offers the possibility to identify and separately reconstruct images of

the cardiac cycles affected by arrhythmia.

6.7.4. Free-Breathing 3D Liver DCE-MRI

Figure 6.12 shows the aortic and portal venous contrast-

enhancement phases in four representative partitions, for both GRASP and

XD-GRASP reconstruction of the first two volunteer datasets. The reduction

of respiratory motion blurring in XD-GRASP improved the delineation of

vessels and vessel-tissue contrast compared to GRASP.

The first four rows of Figure 6.13 show the portal vein

enhancement phase for two representative partitions each from volunteers

3 (normal breathing) and 4 (heavy breathing). GRASP suffered from

significant intra-frame respiratory motion blurring, especially in the dataset

acquired during heavy breathing. XD-GRASP improved the delineation of

vessels and borders in the liver, improved vessel-tissue contrast and

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Figure. 6.11: (a) Comparison of XD-GRASP and the standard breath-hold approach with retrospective ECG-gating for the patients. Conventional breath-hold scans achieved good image quality in a patient with normal sinus rhythm, but it produced poor image quality for patients with arrhythmia. XD-GRASP achieved consistent image quality by separating the cardiac phases with arrhythmia. (b) In the patient with 2

nd

degree AV block, the arrhythmic cardiac cycles were further sorted for a separate XD-GRASP reconstruction to provide additional physiological information. (c) Corresponding cardiac motion signals for three patients with varying length of the cardiac cycle indicated by gray arrows.

enhanced the depiction of the kidney and bowel. The bottom row of Figure

6.13 shows the same comparison for one representative partition from the

patient dataset. The white arrow indicates a suspected liver lesion that can

be seen in GRASP but is better delineated in XD-GRASP.

6.8. Discussion

XD-GRASP provides a novel way to handle respiratory and other

types of motion in free-breathing MRI. Instead of removing motion, e.g.,

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Figure. 6.12: Comparison of GRASP with XD-GRASP in both aortic and portal-venous enhancement phases in two representative partitions each from two volunteer datasets. XD-GRASP improved delineation of the liver and vessels with enhanced vessel-tissue contrast.

using self-gating, extra motion-state dimensions are reconstructed and a

compressed sensing approach is used to exploit compressibility in these

dynamic dimensions. XD-GRASP does not require the use of specific

motion models, and therefore it is immune to interpolation errors and offers

notable advantages as compared with previously proposed registration-

based compressed sensing reconstruction approaches (58,88), which co-

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register images in different respiratory states to correct motion. In addition,

XD-GRASP also enables access to motion information that was not

available before, and thus it could potentially be used to study interesting

clinical problems, such as evaluation of the interaction of the left and right

ventricles during respiration, e.g., for the diagnosis of conditions such as

constrictive pericardial heart disease (137), and evaluation of the

respiration-dependent flow patterns in “Fontan physiology”. Furthermore,

there have been recent concerns about dyspnea caused by certain

hepatobiliary contrast agents, and the use of XD-GRASP for evaluation of

the impact of contrast injection on respiratory motion in abdominal DCE-

MRI is currently underway. Moreover, the ability to reconstruct images in

both inspiratory and expiratory phases from the same acquisition may also

be helpful, for example, in discriminating persistent stenosis of the celiac

artery from physiologic celiac artery narrowing during expiration.

The XD-GRASP approach is not limited to golden-angle radial

sampling, and it can be extended to other trajectories, as long as reliable

physiological motion information can be obtained (e.g., using external ECG

or respiration monitor devices) and arbitrary data sorting can be performed

with approximately uniform k-space coverage in each motion state. For

example, novel trajectories based on 3D Cartesian sampling with butterfly

navigators have been recently introduced for continuous data acquisition,

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Figure. 6.13: Comparison of GRASP with XD-GRASP in a total of five representative partitions from two volunteers and one patient. Volunteer 4 was asked to breathe deeply. XD-GRASP achieved superior overall image quality, with reduced motion-blurring. The white arrow indicates a suspected liver tumor, which is better delineated in XD-GRASP.

following a golden-angle spiral pattern in the ky-kz plane (138). These

trajectories could be well-suited for XD-GRASP reconstruction (139).

In addition to cardiac cine imaging, other applications, such as

coronary MR angiography, can also benefit from simultaneous cardiac and

respiratory motion sorting. In these applications, data acquisition is usually

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performed in a quiescent cardiac phase (e.g., mid-diastole) when cardiac

motion is minimal, and a navigator echo is employed to monitor the

movement of the diaphragm in order to minimize respiratory motion effects.

XD-GRASP can be used to reconstruct datasets acquired continuously

covering the entire cardiac cycles and cardiac phases with best delineation

of a particular artery can be retrospectively selected for visualization. The

application of XD-GRASP for coronary MRA will be exploited in next

chapter.

The temporal resolution demonstrated in this work for abdominal

DCE-MRI (11-12 seconds) may not be adequate for perfusion analysis,

which usually requires 2-3 second temporal resolution. Higher temporal

resolutions for DCE-MRI are restricted due to the fact that the contrast

enhancement is a non-periodic process, which limits the number of spokes

that can be combined for each respiratory state. One way to achieve higher

effective temporal resolutions would be to use a soft-gating approach,

which weights k-space data according to the respiratory motion signal, as

proposed in (140).

For abdominal DCE-MRI, the number of respiratory states was

selected such that they can adequately resolve respiratory motion without

introducing residual aliasing artifacts (As the number of motion states

increases, the number of radial spokes available for each state decreases,

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resulting in increased undersampling). A small number of respiratory states

(e.g., 2) facilitate the removal of aliasing artifacts at the expense of limited

depiction of respiratory motion. A large number of respiratory states (e.g., 8

or 10), on the other hand, potentially enhances the visualization of

respiratory motion and reduces blurring at the expense of introducing

residual aliasing artifacts. As shown in Figure 6.14, we found empirically

that 4 respiratory motion states represent a good compromise between

removal of aliasing artifacts and motion-related blurring. Using more

respiratory states (e.g., 6) led to lower image quality due to increased

undersampling ratio and intrinsic limits in the performance of compressed

sensing reconstruction.

The reconstruction of the additional motion-state dimensions

increases the computational burden, particularly because one forward and

one backward NUFFT operation must be performed separately for each

motion state in each iteration. This issue can be addressed using parallel

computing, following the parallelization concept of the clinical

implementation of GRASP.

The stack-of-stars acquisition scheme employed in this work is still

sensitive to respiratory motion along the kz dimension (Cartesian sampling)

and limits the spatial resolution and slice coverage along z direction (i.e.,

only a limited number of partitions can be acquired, in order to maintain

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Figure. 6.14: Comparison of XD-GRASP reconstructions with different number of respiratory motion states in abdominal DCE-MRI (end-expiratory motion state only). 4 and 6 respiratory states achieved better resolved respiratory motion than 2 states and 1 state. 6 respiratory states resulted in slightly lower performance than 4 respiratory states. White arrows indicate motional blurring for a choice of 1 motion state, and residual blurring for a choice of 2 motion states.

adequate temporal resolution). A true 3D golden-angle radial trajectory,

such as the spiral phyllotaxis sampling approach (141), can be helpful to

overcome these limitations which will be exploited in the next chapter.

6.9. Conclusion

XD-GRASP demonstrates a new use of sparsity for motion

compensation and offers a new way to handle respiratory or other types of

motion in free-breathing MRI. Instead of removing or correcting motion,

extra motion-state dimensions are reconstructed using a compressed

sensing approach that exploits sparsity along the new dimensions. XD-

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GRASP reduces motion-induced blurring and allows separation of

respiratory motion from cardiac motion in cardiac cine MRI and from

contrast enhancement in DCE-MRI. Moreover, the reconstruction of

additional motion dimensions offers additional complementary information,

which can be of potential value for specific clinical applications.

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Chapter 7

Towards Five-Dimensional Cardiac and Respiratory

Motion-Resolved Whole-Heart MRI Using XD-GRASP

7.1. Prologue

The stack-of-stars 3D radial sampling employed in Chapters 5 and 6

is relatively easy to implement and reconstruct. However, this sampling

scheme has several challenges. First of all, although radial sampling in the

kx-ky plane leads to reduced sensitivity to motion, the acquisition is still

sensitive to respiratory motion along the kz dimension, in which Cartesian

sampling is employed. In addition, stack-of-stars sampling also limits spatial

resolution in the partition dimension and the corresponding slice coverage.

In order to maintain adequate temporal resolution, only a limited number of

partitions can be acquired and interpolation is usually performed afterwards

to retrospectively increase the number of partitions. Accordingly, the

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application of stack-of-star sampling to free-breathing 3D cardiac cine

imaging is challenging because the acquisition may fail to capture the

cardiac contraction when a large number of slices are needed. Moreover, in

the context of compressed sensing, the incoherence along the partition

dimension is also limited in stack-of-stars sampling and thus the overall

acceleration is limited. Although random undersampling could also be

implemented along the kz dimension, it would not be as efficient as in the

radial plane. The solution to address these challenges is to extend the

compressed sensing reconstruction into true 3D radial “kushball” sampling.

In this chapter, the XD-GRASP technique presented in Chapter 6 is

extended into a 3D golden-angle radial sampling scheme that is based on

the spiral phyllotaxis pattern. 3D radial sampling not only offers

reconstructions with volumetric isotropic resolution, but also allows

acceleration and exploitation of incoherence along all the spatial

dimensions. The proposed imaging framework is first applied for

electrocardiogram (ECG)-triggered respiratory motion-resolved whole-heart

coronary MR angiography (MRA), and then applied to demonstrate a

continuous five-dimensional whole-heart imaging framework with high

spatial and temporal resolutions, which allows simultaneous assessment of

myocardial function and visualization of cardiac and respiratory motion-

resolved whole-heart great vessels and coronary artery anatomy.

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Some of the results presented in this chapter have been published in

an abstract at the 2015 Annual Meeting of the International Society for

Magnetic Resonance in Medicine (ISMRM 2015, page 27) (142).

7.2. Introduction

3D whole-heart MRI allows for a multifaceted assessment of the

cardiovascular system and is attractive due to the high SNR, large spatial

coverage, and simplified data acquisition. However, 3D imaging usually

requires prolonged data acquisition, and thus the sensitivity to motion,

particularly respiratory and cardiac motions during the scans, remains one

of the major challenges in this field. Conventional 3D cardiac MRI

acquisitions can be performed with ECG triggering, so that the effects of

cardiac motion can be minimized by acquiring data only within a short time

window (e.g., ~50-100 ms in mid-diastole or early-systole) in each cardiac

cycle. The influence of respiratory motion can also be minimized by

employing a navigator-gating (47), which tracks the movement of the right

diaphragm during the scan, so that only datasets at a consistent end-

expiratory phase are required. Although this image acquisition scheme has

been widely used in MR angiography (MRA) exams, it has several

challenges, including low scan efficiency, sensitivity to respiratory drifts, as

well as a relatively complicated measurement setup. In order to increase

the scan efficiency and thus imaging speed, various approaches have been

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proposed to enable 3D free-breathing cardiac MRI by retrospectively

coregistering images at different respiratory states, so that nearly 100%

imaging efficiency can be achieved (55,143). However, the nature that data

acquisition is performed during a limited window in each cardiac cycle still

results in “deadtime” in the scan, and images can be only obtained at

specific cardiac phases. In order to make the best use of the scan time, a

number of 4D cardiac MRI methods have been proposed (144-146), either

employing navigator-gating/self-gating, or specific registration-based

respiratory motion correction, so that the anatomical and functional

information of the heart can be obtained simultaneously. Two self-navigated

4D cardiac MRI methods have recently been proposed for simultaneous

visualization of cardiac function and cardiac motion-resolved coronary

arteries with isotropic higher spatial resolution, in which respiratory motion

was corrected using either displacement-based or affine transform-based

registrations (130,147). Despite improved scan efficiency compared to the

navigator-gated acquisition, motion correction usually requires the use of

various registration algorithms that are based on specific motion models,

which may be insufficient to account for the complex 3D movement of the

heart during respiration, especially for patients with pronounced respiration

or irregular respiratory patterns.

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Compressed sensing has become a powerful approach for fast

cardiac imaging, and in addition to enabling increased acquisition speed, it

has been shown that sparsity can be also used to resolve respiratory

motion by reconstructing an extra motion-state dimension (Chapter 6). A

corresponding technique called XD-GRASP was developed in Chapter 6 to

combine the self-navigation properties of radial sampling and the

acceleration capability of compressed sensing (120). XD-GRASP offers a

different way to handle various types of motion without performing motion

correction, while nearly 100% scan efficiency can be maintained. The

purpose of this work is to extend the XD-GRASP framework into 3D golden-

angle radial sampling and test the new imaging framework in 3D whole-

heart MRI. Specifically, XD-GRASP is first applied for ECG-triggered

respiratory motion-resolved whole-heart coronary MRA, and then applied to

demonstrate a continuous five-dimensional (x-y-z-cardiac-respiration)

whole-heart imaging framework with both high spatial and temporal

resolutions, which allows simultaneous evaluation of myocardial function

and cardiac and respiratory motion-resolved whole-heart arterial anatomy.

7.3. 3D Phyllotaxis Golden-Angle Radial Sampling

Figure 7.1 shows a comparison of golden-angle radial sampling

schemes that are based on stack-of-stars pattern (Figure 7.1a) and spiral

phyllotaxis pattern (141) (Figure 7.1b), respectively. When compared with

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Figure. 7.1: Comparison of golden-angle radial sampling schemes that are based on stack-of-stars pattern (a) and spiral phyllotaxis pattern (b), respectively. When compared with the stack-of-stars scheme, radial sampling is also employed along the kz dimension in the 3D phyllotaxis sampling trajectory, so that each k-space line passes through the center of k-space and an image can be reconstructed with isotropic spatial resolution. The 3D radial sampling pattern in (b) can be segmented into multiple heartbeats for cardiac MRI, with golden-angle rotation along the z-axis between every two successive data interleaves. An additional spoke oriented along the superior-inferior (SI) direction (red lines) can be acquired at the beginning of each data interleave for respiratory motion detection and self-navigation.

the stack-of-stars scheme, radial sampling is also employed along the kz

dimension in the 3D phyllotaxis sampling trajectory, so that each k-space

line passes through the center of k-space and an image can be

reconstructed with isotropic spatial resolution. As described in (141), this 3D

radial sampling pattern can be segmented into multiple heartbeats for

cardiac MRI, with golden-angle rotation along the z-axis between every two

successive data interleaves. Besides, an additional spoke oriented along

the superior-inferior (SI) direction (red lines in Figure 7.1b) can be acquired

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at the beginning of each data interleave for respiratory motion detection and

self-navigation. Similar to stack-of-stars golden-angle radial sampling, the

phyllotaxis 3D radial acquisition scheme also allows for sorting of all the

interleaves into different respiratory states according to their corresponding

respiratory phases, in which approximately uniform and distinct k-space

coverage can be achieved in each motion state.

7.4. Free-Breathing Whole-Heart MRI

The XD-GRASP framework with 3D golden-angle radial sampling

was implemented and tested for two whole-heart cardiac MRI studies, as

described in the following subsections. All the data were acquired at the

University of Lausanne. Human imaging was performed in accordance with

protocols approved by the University of Lausanne Institutional Review

Board and was found to comply with the HIPAA. All subjects provided

written informed consent before the imaging.

7.4.1. ECG-Triggered Free-Breathing Whole-Heart Coronary MRA

Free-breathing whole-heart coronary MRA was performed on 11

healthy volunteers on a 1.5T clinical MRI scanner (MAGNETOM Aera,

Siemens AG, Healthcare Sector, Erlangen, Germany) equipped with a total

number of 30 receiver coil elements (18 chest and 12 spine). Imaging was

performed with ECG-triggering and the acquisition window, which was set

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around ~80-100ms, was placed in mid-diastole and adapted to the heart

rate of individual subject. The trigger delay time was set by visual inspection

of the most quiescent diastolic phase on a mid-ventricular short axis cine

image series acquired in free breathing with 3 averages, prior to the whole-

heart coronary MRA scan. k-Space data were acquired without any external

gating using a prototype 3D radial b-SSFP sequence with golden-angle

rotation scheme based on the spiral phyllotaxis pattern. The data

acquisition was employed with non-slice-selective pulses, T2 magnetization

preparation and fat-saturation. Relevant imaging parameters included:

TR/TE = 3.1/1.56ms, FOV = 220 x 220 x 220 mm3, matrix size = 192 x 192

x 192, voxel size = 1.15 x 1.15 x 1.15 mm3, RF excitation angle=90°, and

receiver bandwidth=898 Hz/Pixel. A total number of 12320 radial readouts

were acquired over 385 heartbeats in each subject, including 385 golden-

angle interleaves and 32 spokes in each of them. As described in the

previous section, each interleave started with a spoke oriented along the SI

direction for self-navigation

7.4.2. Free-Breathing Continuous Whole-Heart MRI

Continuous whole-heart MRI was performed on 9 healthy volunteers

during free-breathing without ECG triggering and any external gating on the

same 1.5T clinical scanner (MAGNETOM Aera, Siemens AG, Healthcare

Sector, Erlangen, Germany). k-Space data were continuously acquired

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using the same prototype 3D radial b-SSFP sequence with golden-angle

rotation scheme. Imaging parameters included: TR/TE = 3.1/1.56 ms, FOV

= 220 x 220 x 220 mm3, matrix size = 192 x 192 x 192, voxel size = 1.15 x

1.15 x 1.15 mm3, and flip angle=90o. A total number of 126478 spokes were

acquired in each subject in 14 minutes and 17 seconds, including 5749

golden-angle interleaves and 22 spokes in each of them. Each interleave

started with a spoke oriented along the SI direction for self-navigation and

was preceded by CHESS fat saturation. More information on this

continuous whole-heart cardiac imaging sequence can be found in (147).

7.5. Motion Estimation

For ECG-triggered free-breathing whole-heart coronary MRA,

respiratory motion signal was detected using a technique recently proposed

by Bonanno (148). Specifically, the k-space center amplitude (KCA) in each

coil element was first obtained by averaging the absolute value of three

central points of the self-navigation readouts acquired at the beginning of

each data segment. Independent component analysis (ICA) (149) was then

performed on the KCA signals from all the coils to identify different

components. The one that can best represent respiration was selected and

used as the respiratory motion signal for subsequent data binning.

For continuous whole-heart imaging, the self-navigation spokes

acquired at the beginning of each interleave (red lines in Figure 7.1b) were

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Figure. 7.2: (a) Data sorting procedure in XD-GRASP reconstruction for ECG-triggered whole-heart coronary MRA, in which the 3D golden-angle radial k-space data are sorted into 4 respiratory motion states spanning from expiration (top) to inspiration (bottom) (x-y-z-respiratory) using the respiratory motion signals drived from the acquired data. The sorting procedure is performed so that the number of spokes grouped in each motion state is the same. Approximately uniform coverage of k-space with distinct sampling patterns in each motion state can be achieved, as shown in (b)&(c).

used to extract respiratory motion signals of the acquired datasets. Similar

to the approach employed in Chapter 6 for abdominal imaging, a 1D

partition-direction FFT was performed on the self-navigation spokes and

principal component analysis was performed along the concatenated z+coil

dimension to determine the most common signal variation mode among all

the coil elements. The principal component with the highest peak in the

frequency range of 0.1-0.5Hz was selected to represent the respiratory

motion signal. Cardiac motion signal was obtained retrospectively from the

ECG trace.

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7.6. Data Sorting

7.6.1. ECG-Triggered Free-Breathing Whole-Heart Coronary MRA

Figure 7.2a illustrates the data sorting procedure in XD-GRASP

reconstruction for ECG-triggered whole-heart coronary MRA, in which the

3D golden-angle radial k-space data are sorted into 4 respiratory motion

states spanning from expiration (top) to inspiration (bottom) (x-y-z-

respiratory) using the respiratory motion signals, so that respiratory motion

can be resolved and additional sparsity can be exploited along the new

dynamic dimension. Similar to the procedure described in Chapter 6, the

sorting procedure is performed such that the number of spokes grouped in

each motion state is the same. Due to the golden-angle acquisition scheme,

approximately uniform coverage of k-space with distinct sampling patterns

in each motion state can be achieved, as shown in Figure 7.2b. It has been

demonstrated in Chapter 6 that four respiratory states achieved a good

balance between the depiction of respiratory motion and residual streaking

artifact in abdominal imaging. Similarly, four respiratory states were also

found to provide a good balance in this study.

7.6.2. Free-Breathing Continuous Whole-Heart MRI

The continuously acquired whole-heart cardiac datasets were first

sorted into 20 cardiac phases (temporal resolution of ~40-50ms depending

on the heart rate) without any view sharing using the cardiac motion signal

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Figure. 7.3: Five-dimensional data sorting in free running continuous whole-heart imaging, with one cardiac motion dimension (20 cardiac phases) and one respiratory motion-state dimension (4 respiratory states).

obtained from the ECG trace, and each cardiac phase was further sorted

into 4 respiratory motion states spanning from end-expiration to end-

inspiration using the estimated respiratory motion signal, thus generating a

5D image set (x-y-z-cardiac-respiratory), as shown in Figure 7.3.

7.7. Image Reconstruction

For ECG-triggered whole-heart coronary MRA, XD-GRASP

reconstruction was performed by employing a temporal sparsity constraint

along the new respiratory dimension by solving the following optimization

problem:

2

2 1min

dd F C d m S d (7.1)

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Here, F is the non-uniform fast Fourier transform (NUFFT) operator (135)

defined for the 3D golden angle radial sampling pattern, C represents the

multiple-elements coil sensitivity maps with dimensionality of x-y-z-coil,

where x, y and z represent three spatial dimensions. d is the 4D dynamic

image-series with the sorted respiratory-state dimension

(size=192x192x192x4), and m is the corresponding multicoil radial k-space

data sorted according to the new dimensions. S is the sparsifying transform

applied in the respiratory motion dimension with regularization parameter .

For continuous whole-heart MRI, XD-GRASP reconstruction was

performed by solving:

2

1 1 2 22 1 1min

dd F C d m S d S d (7.2)

where F is the NUFFT operator as before, C represents the multiple-

elements coil sensitivity maps with dimensionality of x-y-z-coil, d is the 5D

image set to be reconstructed (size=192x192x192x20x4), and m is the

corresponding multicoil radial k-space data. S1 and S2 are the sparsifying

transforms applied along the cardiac and respiratory motion dimensions,

respectively, with regularization parameters 1 and 2 .

All the reconstructions were performed offline in MATLAB

(Mathworks, Natick, MA, USA) using a server equipped with two 16-core

Opteron CPUs, 256 GB RAM, and two NVIDIA graphics processing unit

(GPU) cards with 6 GB memory in each of them. In order to speed up the

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reconstruction, the NUFFT operation was implemented using parallel

computing on GPUs, and was called in the main reconstruction program

implemented in MATLAB (150). Same as in Chapter 6, temporal finite

differences, also known as total-variation minimization, was used in the

reconstructions as the sparsifying transform. The non-linear conjugate

gradient optimization employed in the previous chapters was used to solve

the optimization problems. Coil sensitivity maps were computed from

reference multicoil images given by NUFFT reconstruction of the whole

dataset, using the adaptive array combination method (40). The averaged

XD-GRASP reconstruction time was ~20-30 minutes for each ECG-

triggered whole-heart coronary MRA dataset (size=192x192x192x4) and

was ~8 hours for each continuously acquired whole-heart dataset

(size=192x192x192x20x4).

For comparison purpose, all the ECG-triggered coronary MRA

datasets were also reconstructed using a 1D self-navigation motion

correction algorithm previously described in (143). Meanwhile, 4D

reconstruction with respiratory motion correction (MC), as described in

(147), was also performed on all the continuously acquired cardiac

datasets, in which 20 cardiac phases were generated without any view

sharing (size=192x192x192x20) and were reconstructed using compressed

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sensing that exploits sparsity along the cardiac dimension only using a

total-variation constraint.

7.8. Image Quality Comparison

7.8.1. ECG-Triggered Free-Breathing Whole-Heart Coronary MRA

For ECG-triggered whole-heart coronary MRA, images at the end-

expiratory motion states were selected in XD-GRASP results for image

quality comparison. All the images, including XD-GRASP and 1D motion

correction reconstruction, were randomized for blinded evaluation. Two

coronary MRA experts scored the visualization/sharpness of the left main

coronary artery, the proximal segment of left circumflex coronary artery

(LCX), and the proximal, middle, distal segments of both right coronary

artery (RCA) and left anterior descending coronary artery (LAD) on a 5-

point scale classifying the definition of the vessel borders as follows: 0 = not

visible, 1 = markedly blurred, 2 = moderately blurred, 3 = mildly blurred and

4 = sharply defined. The reported scores represent mean ± standard

deviation, and a paired student's t-test was used for statistical analysis,

where P < 0.05 suggested statistical significance.

Additionally, the diagnostic quality and visibility of left main coronary

artery, the proximal segment of LCX, and each segment of RCA and LAD

were scored for all the coronary datasets on a 3-point scale (diagnostic

grading) by an experienced cardiovascular MR radiologist, who was

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completely blinded to the type of reconstruction used. The scale used was:

0 = nonvisible, 1 = visible but non diagnostic and 2 = visible and diagnostic.

7.8.2. Free-Breathing Continuous Whole-Heart MRI

For continuously acquired whole-heart datasets, one end-systolic

frame and one mid-diastolic frame were manually selected from each

subject (in both the 4D reconstruction and the end-expiratory state of the

5D reconstruction) and all the images were randomized for blinded

evaluation. An experienced cardiovascular MR radiologist scored the

visualization/sharpness of myocardium, the LCX, and the proximal segment

of RCA and LAD on a 1-5 (non-diagnostic to excellent) scale. The reported

scores represent mean ± standard deviation, and a paired student's t-test

was used for statistical analysis, where P < 0.05 suggested statistical

significance.

7.9. Results

7.9.1. ECG-Triggered Free-Breathing Whole-Heart Coronary MRA

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Figure. 7.4: Comparison of XD-GRASP reconstruction (end-expiratory motion states) with the 1D respiratory motion correction reconstruction in two representative datasets. XD-GRASP improves the delineation of coronary arteries and removes the blurring effects by resolving the respiratory motion.

Figure 7.4 compares the XD-GRASP reconstruction (end-expiratory

motion states) with the 1D respiratory motion correction reconstruction in

two representative ECG-triggered whole-heart coronary datasets. XD-

GRASP removes the blurring effects by resolving the respiratory motion,

and thus improves the delineation of coronary arteries, myocardium and the

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Figure. 7.5: End-expiratory myocardial wall (SAX and 4CH), proximal coronary arteries, right coronary artery (RCA) and left anterior descending coronary artery (LAD) in diastolic (top) and systolic (bottom) phases. All the images are reformatted from a single continuous data acquisition with 5D XD-GRASP reconstruction.

papillary muscles. The readers’ scores for qualitative comparison of

different coronary arteries are summarizes in Table 7.1.

The comparison of diagnostic quality and visibility of different

coronary arteries are summarized in Table 7.2. The diagnostic scores for

XD-GRASP reconstructions were higher than the 1D respiratory motion

correction reconstructions in left main coronary artery, and different

segments of both LAD and RCA.

7.9.2. Free-Breathing Continuous Whole-Heart MRI

Figure 7.5 shows the ventricular chambers and the coronary arteries

at end-expiration in one representative continuously acquired whole-heart

dataset, in both systolic (top) and diastolic (bottom) phases, derived from

the 5D XD-GRASP reconstruction. Coronary arteries are reformatted using

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Figure. 7.6: 5D XD-GRASP reconstruction achieved reduced blurring, improved sharpness and better visualization of myocardium and the RCA compared with 4D reconstruction with respiratory motion correction (MC) in one representative volunteer with irregular respiratory pattern.

“Soap-Bubble” software (Philips Healthcare, Netherlands). Good delineation

of both myocardial wall and coronary arteries was obtained without using

any explicit respiratory motion correction algorithm.

Figure 7.6 shows the results from another volunteer who had an

irregular respiratory pattern during data acquisition. In this subject, 5D XD-

GRASP reconstructions exhibited reduced motional blurring and achieved

improved visualization of both myocardial wall and RCA (red arrows) when

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1D Self-

Navigation XD-

GRASP(Exp)

LM 1.8±0.8 2.0±0.5

LDA (Prox) 1.5±0.8 1.9±0.5

LDA (Mid) 1.0±0.8 1.5±0.4*

LDA (Dist) 0.8±0.6 1.1±0.6

LCX (Prox) 1.1±0.7 1.0±0.5

RCA (Prox) 1.9±0.7 2.2±0.4*

RCA (Mid) 1.7±0.6 2.0±0.4*

RCA (Dist) 1.2±0.8 1.5±0.4

Table. 7.1: Readers’ scores for comparison of 1D self-navigation motion correction reconstruction v.s. XD-GRASP reconstruction (end-expiration only) in visualization/sharpness of RCA, LAD and left main coronary artery. 0-4: non-diastolic to

excellent. * Indicates statistical significance. LM: Left Main Coronary Artery.

compared to the 4D reconstruction with 1D respiratory motion correction.

The readers’ scores for qualitative comparison of different coronary arteries

are summarizes in Table 7.3.

7.10. Discussion

This chapter extends the XD-GRASP reconstruction presented in

Chapter 6 into 3D phyllotaixs golden-angle radial sampling, which offers

volumetric data acquisitions with isotropic spatial resolution up to ~1.1mm3,

enabling acceleration and exploitation of incoherence along all the spatial

dimensions in compressed sensing reconstruction.

ECG-triggered acquisition scheme is still widely used in coronary

MRA and many other artery exams. Thus the new XD-GRASP framework

was first tested in ECG-triggered free-breathing whole-heart coronary MRA,

in which XD-GRASP combines the advantages of both conventional

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Myocardium RCA (Prox) LAD (Prox) LM

4D 3.06±0.94 2.83±1.34 2.00±0.69 2.61±1.20

5D (Exp) 3.78±0.73* 3.44±1.20* 2.28±0.57* 3.22±1.17*

Table. 7.3: Reader’s scores for comparison of 4D reconstruction with motion correction v.s. 5D XD-GRASP reconstruction (end-expiration only) in visualization/sharpness of myocardium, the proximal segment of RCA, LAD and left main coronary artery. 1-5: non-diastolic to excellent. * Indicates statistical significance. LM: Left Main Coronary Artery

1D Self-

Navigation XD-

GRASP(Exp)

LM 1.8±0.4 2.0±0.0*

LDA (Prox) 1.6±0.5 2.0±0.0*

LDA (Mid) 1.3±0.6 1.4±0.5

LDA (Dist) 0.9±0.5 1.3±0.5

LCX (Prox) 1.4±0.7 1.4±0.7

RCA (Prox) 1.8±0.4 2.0±0.0

RCA (Mid) 1.3±0.5 1.7±0.5

RCA (Dist) 1.4±0.7 1.7±0.5

Table. 7.2: Readers’ scores for comparison of 1D self-navigation motion correction reconstruction v.s. XD-GRASP reconstruction (end-expiration only) in diastolic quality of RCA, LAD and left main coronary artery. 0 = not visible, 1 = visible, and 2 =

diagnostic. * Indicates statistical significance. LM: Left Main Coronary Artery.

navigator-gated acquisitions and self-navigation techniques. Coronary MRA

datasets are sorted into multiple respiratory motion states using the

respiratory motion signal derived from the acquired data. Instead of

performing registration-based motion correction, sparsity is exploited along

the new respiratory dimension. Therefore, it enables nearly 100% scan

efficiency and potentially avoids the interpolation errors that are typically

associated with the registration-based motion correction.

Compressed sensing has been previously applied for whole-heart

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coronary MRA in many works (151-154). However, most of those

approaches were proposed to reduce the total acquisition time by exploiting

the spatial correlation. XD-GRASP works based on the fact that the

dynamic dimension usually has much higher correlation than the spatial

dimension, and thus the performance of compressed sensing

reconstruction can be substantially improved by exploiting temporal sparsity

instead of spatial sparsity only. Therefore, XD-GRASP represents a new

way of handling respiratory motion in coronary MRA.

Although good results have been achieved with the ECG-triggered

whole-heart coronary MRA, the data acquisition is only performed during

mid-diastole and thus the scan efficiency is not optimal. The reconstruction

performance can be largely improved if the acquisition is extended to cover

the entire cardiac cycles. Therefore, the XD-GRASP framework was also

applied for continuous whole-heart cardiac MRI, in which both ECG

triggering and external gating are not required during the scans and data

are acquired throughout the entire cardiac cycles. The continuously

acquired cardiac datasets are sorted into five dimensions, containing one

cardiac dimension and one respiratory dimension to resolve both cardiac

and respiratory motions. Compressed sensing is then employed to exploit

temporal sparsity along both dynamic dimensions. The proposed 5D XD-

GRASP reconstruction enables both high isotropic spatial resolution and

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high temporal resolution for simultaneous assessment of myocardial

function in arbitrary orientations and visualization of coronary arteries at a

particular cardiac phase and respiratory motion-state.

In 5D whole-heart MRI, fat saturation is necessary to enable

visualization of arteries in small size, such as coronary arteries. Thus, the

steady state in the bSSFP sequence could be interrupted by the fat-

saturation modules. An alternative approach to solve this problem is to

switch the excitation pulses water-excitation pulses, which would maintain

the true steady state during the scans. For example, the image acquisition

proposed in (155) use FLASH sequence with water-excitation RF pulse at

3T scanners following slow contrast agent injection. However, FLASH

imaging may still suffers from reduced SNR when compared to bSSFP

imaging and the requirement for contract agent injection also exclude the

use of the techniques on patients with impaired kidney function.

Although ECG-triggering is not needed in the continuous 5D whole-

heart MRI, the ECG trace is still used to obtain cardiac motion signals.

Alternatively, cardiac motion signal can be also extracted directly from the

centers of k-space data or using the PCA approach as described in (130).

The possibility of extracting both cardiac and respiratory motion signals

simultaneously will be exploited in future works.

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Additional works can be performed in the future to improve the

whole-heart MRI framework described in this chapter. First of all, the data

acquisition in 5D continuous whole-heart MRI is still very long (up to ~14

minutes). Additional experiments are needed in order to determine the

maximum acceleration so that the data acquisition can be shortened

accordingly. Besides, the image reconstruction time, especially in the 5D

image reconstruction, is very extensive and is not yet for clinical use. This

issue can be addressed by implementing the whole reconstruction steps in

GPUs or a workstation with clusters in the future. Moreover, the proposed

imaging framework can be also applied in other types of free-breathing

cardiac scans, such as 4D flowing imaging and late gadolinium

enhancement. Furthermore, these types of cardiac scans may be

potentially combined as a synergistic imaging framework that allows a push

button comprehensive examination of the cardiovascular system. Finally,

only qualitative comparison was performed in this study. Quantitative

comparison, in terms of the vessel sharpness, vessel length, could further

validate and strengthen the improvement and benefit of XD-GRASP

reconstruction compared to the existing reference approaches.

7.11. Conclusion

The extension of XD-GRASP framework into 3D golden-angle radial

sampling is a promising approach for whole-heart MRI. Challenges that are

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associated with conventional navigator-gated acquisitions and self-

navigation approaches can be addressed and the advantages of both

techniques can be combined in XD-GRASP. The 5D whole-heart imaging

framework using XD-GRASP enables high isotropic spatial resolution and

high temporal resolution for assessment of myocardial function in arbitrary

orientations, and visualization of coronary arteries at particular cardiac

phases and respiratory motion-states.

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Chapter 8

Summary and Future Work

The main contributions of this dissertation are the development of

novel MRI techniques that enable rapid and continuous data acquisition

during free breathing. Particularly, two techniques named GRASP and XD-

GRASP have been developed. GRASP represents an efficient and flexible

dynamic imaging framework, which enables continuous data acquisition

and user-defined reconstruction of temporal frames. GRASP offers notable

advantages for dynamic imaging, since the time dimension is defined

retrospectively and the same dataset can be reconstructed in different ways

according to the target application. The tools that enable this flexibility are

golden-angle radial sampling, which allows for quasi-arbitrary data sorting;

and the combination of compressed sensing and parallel imaging, which is

used to reconstruct the sorted undersampled dynamic image series. XD-

GRASP moves a step forward and proposes a new use of sparsity to

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handle different types of motion in MRI. Instead of preventing or correcting

for motion, extra motion dimensions are explicitly reconstructed in XD-

GRASP using compressed sensing ideas. In this chapter, contributions of

this dissertation are summarized first and an outlook for the future is

discussed afterwards.

8.1. Chapter Summaries

k-t SPARSE-SENSE (30) is a reconstruction framework previously

developed in our group to combine compressed sensing and parallel

imaging using a SENSE-type formalism for accelerated Cartesian imaging.

Instead of performing compressed sensing reconstruction in each individual

coil separately, k-t SPARSE-SENSE proposes to exploit joint sparsity in the

multicoil images by taking advantage of the additional sensitivity encoding

capabilities provided by coil arrays, so that the imaging speed and the

reconstruction performance can be significantly improved. The dissertation

started with two applications of k-t SPARSE-SENSE for accelerated

dynamic imaging in Chapter 3 (MR parameter mapping) and Chapter 4

(real-time cardiac cine MRI). The performance of several temporal

sparsifying transforms, including temporal fast Fourier transform (FFT),

temporal principal component analysis (PCA) and temporal total-variation

(TV), were compared and tailored for compressed sensing reconstruction in

different clinical applications.

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Although k-t SPARSE-SENSE has been applied to various clinical

applications with promising results, the performance of compressed sensing

is fundamentally limited in Cartesian imaging, since acceleration can only

be achieved along the phase-encoding dimension, limiting the use of

sparsity and incoherence. The ability to perform compressed sensing in

higher-dimensional datasets, e.g., in 3D datasets or in dynamic datasets

with both spatial and temporal dimensions, would improve the

reconstruction performance, since high-dimensional datasets are more

compressible and incoherent aliasing artifacts can be distributed over a

larger space with lower values. In addition, Cartesian imaging is sensitive to

motion, which usually results in ghosting artifacts in the reconstructed

image. Non-Cartesian imaging can effectively accelerate all spatial

dimensions and thus sparsity can be fully exploited. Furthermore, non-

Cartesian imaging (e.g., radial imaging) has reduced sensitivity to motion

due to the averaging effects associated with repeated sampling of the k-

space center, and thus ghosting artifacts can be avoided. In Chapter 5, the

reconstruction framework of k-t SPARSE-SENSE was applied to golden-

angle radial sampling and a new imaging technique, named Golden-angle

RAdial Sparse Parallel MRI (GRASP), was proposed. GRASP represents

an efficient and simplified imaging paradigm for clinical workflow, which

enables continuous data acquisitions during free breathing and image

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reconstructions with retrospectively selected temporal information.

Meanwhile, the GRASP workflow can also maximize the amount of

information per unit time in a wide range of clinical applications and

facilitate the use of compressed sensing for clinical studies.

As presented in Chapter 5, GRASP allows rapid and continuous

free-breathing abdominal imaging, which is mainly attributed to the

combination of compressed sensing and parallel imaging, and also the

motion robustness of radial sampling. However, it only handles moderate

motion and it is not the ultimate solution for free-breathing imaging. Our

initial clinical evaluation also suggests that in certain patients, the image

quality is still suboptimal due to substantial respiration. The image blurring

introduced in these cases results in loss of vessel-tissue contrast and may

prevent the detection of suspected lesions with small size. Therefore, a

novel image reconstruction framework named eXtra Dimensional GRASP

(XD-GRASP) was developed, as described in Chapter 6. In XD-GRASP,

the continuously acquired data are first sorted into multiple motion states

(e.g., respiratory states) using a motion signal extracted directly from the

data, and a GRASP-type reconstruction is then applied to exploit sparsity in

the new dimension and remove undersampling artifacts. XD-GRASP

represents a novel use of sparsity to handle physiological motion, where

instead of removing or correcting for motion, extra respiratory-state

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dimensions are reconstructed, which improves image quality and offers

access to new physiological information that was previously inaccessible

and could be of potential clinical value.

The GRASP and XD-GRASP techniques proposed in Chapters 5

and 6 employed stack-of-stars radial sampling for volumetric imaging. This

sampling scheme is relatively simple to implement and reconstruct, but the

fact that the kz dimension is sampled on a Cartesian grid introduces some

limitations. First, the incoherence along the slice dimension is limited and

thus the overall acceleration is reduced. Although random undersampling

could be implemented along the kz dimension as well, it would not be as

efficient as in the radial plane. Second, the number of slices that can be

acquired is limited and interpolation is usually required, which leads to

compromises in spatial resolution along the slice dimension. Third,

application of stack-of-stars sampling to 3D cardiac cine imaging is limited

because it may fail to capture the cardiac contraction when a large number

of slices are acquired. In order to increase the spatial coverage and extend

compressed sensing to a true 3D isotropic spatial coverage, a 3D golden-

angle radial sampling based on the spiral phyllotaxis pattern is employed for

XD-GRASP in Chapter 7. The proposed framework was first employed for

ECG-triggered whole-heart coronary MRA with improved respiratory motion

compensation and was then applied to continuous five-dimensional whole-

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heart imaging with high spatial and temporal resolutions, which allows

simultaneous assessment of cardiac function in arbitrary planes and

visualization of cardiac and respiratory motion-resolved whole-heart arterial

anatomy.

8.2. An Outlook for the Future

Despite remarkable progress in imaging speed that has been

achieved over the past decades, the day-to-day practice of MRI is still

fundamentally limited due to the comparatively slow, complex, and

parameter-oriented imaging process compared with other imaging

modalities such as CT. This complexity results in part from the rich diversity

of acquisition approaches, reconstruction algorithms, and tissue contrast

mechanisms enabled by the magnetic resonance phenomenon. Therefore,

it is desirable to have an imaging paradigm that makes MRI simple and

information-rich, and shifts from time-consuming inefficient tailored

acquisitions to a push-button process that allows rapid and continuously

updated acquisitions with comprehensive information. This shift has the

capacity to change the paradigm of everyday clinical imaging, enabling

faster imaging with less operator-dependent planning, and to enhance

information content for research imaging as well.

While the paradigm of rapid, continuous and comprehensive imaging

described above has not been entirely achieved in this dissertation, the

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work presented here can significantly improve the way that MRI exams are

performed every day. There are several directions that can be exploited in

the future to strengthen this work. For example, the GRASP or XD-GRASP

pipeline may be combined with MR Fingerprinting (MRF) (156), which

extracts various physical parameters such as T1, T2 and proton density from

a single MR scan. Currently the implementation of MRF is based on a pixel-

by-pixel pattern recognition using a dictionary database of signal evolution

time courses. However, the resulting images may be contaminated with

artifacts in cases of extreme undersampling, and thus the reliability of the

matching process can be reduced. GRASP can be applied to remove some

of the undersampling artifacts with a temporal sparsifying transform and

thus could potentially improve the pattern-matching performance. In

addition, most of the existing applications of MRF have been focused on

static organs such as the brain, because any physiological motion – and

most particularly through-plane motion – could introduce bias into the

observed spin dynamics. Therefore, it is challenging to extend MRF into

abdominal or chest wall exams. XD-GRASP may be a useful approach to

resolve cardiac, respiratory, and abdominal motion, and to provide reliable

signal evolution mapping in the presence of physiological motion. This

combination could have the potential to achieve rapid and continuous data

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acquisitions with comprehensive information content for different clinical

needs.

The five-dimensional whole-heart imaging framework proposed in

Chapter 7 enables simultaneous evaluation of myocardial function and

whole-heart artery anatomy. The framework can be directly extended into

other scan types in cardiac MRI, such as perfusion imaging, late

gadolinium enhancement imaging, and multidimensional flow imaging.

Several different scan types may also be combined in a synergistic imaging

framework that allows a push-button comprehensive examination of the

cardiovascular system.

The advent of commercial MR-PET scanners has garnered

tremendous attention among researchers from both the MRI and the PET

community in the past few years. Although current state-of-the-art MR-PET

scanners are capable of performing simultaneous acquisition of both MRI

and PET data, the image reconstructions are usually performed separately

and the results are combined at the final stage for visualization. Since MRI

and PET images are acquired in the same organs, they share the same

anatomy and there are correlations between the images that can be

exploited to improve the reconstruction performance and image quality.

Preliminary results have already been reported for a new iterative joint

reconstruction framework based on the combination of compressed sensing

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and parallel imaging that exploits the anatomical correlations between MR

and PET images using a joint sparsity constraint (157). However, most of

the current experiments are still performed in static organs, such as brain,

due to the easy alignment of images from two modalities. XD-GRASP could

be useful for joint MR-PET reconstruction in moving organs, such as liver,

kidney or heart. First, motion detected in the MRI data can be used to sort

both MRI and PET data into extra motion dimensions, so that physiological

motion, such as respiratory motion, can be resolved (158). Afterwards, the

sorted MRI and PET data can be effectively aligned, so that the correlation

between MRI and PET images in different motion states can be exploited

simultaneously for improved joint reconstruction.

One of the major limitations of compressed sensing reconstruction

for clinical studies is the long computation time for the non-linear iterative

reconstruction process. Moreover, large data size further imposes

challenges on computation time and requires adequate hardware and

software implementation. The GRASP and XD-GRASP reconstruction can

be implemented in graphical processor units (GPUs), which are expected to

provide good performance for computation of highly-parallel transforms

such as FFT and wavelets. However, the performance of GPUs may

decrease for the processing of large datasets due to limited GPU memory.

Clusters of multicore computers could offer better performance for

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management of large datasets. Therefore, an appropriate combination of

both platforms is expected to provide faster reconstructions. In additional to

parallel implementation of the reconstruction algorithms, coil compression

methods and numerical optimization algorithms with faster convergence

could also be exploited to further increase the computation speed.

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Chapter 9

List of Publications

9.1. Journal Papers

1. Feng L, Axel L, Chandarana H, Blick KT, Sodickson DK, Otazo R. “XD-GRASP: Golden-Angle Radial MRI with Reconstruction of Extra Motion-State imensions Using Compressed Sensing” Magn Reson Med. 2015 Mar 25. doi: 10.1002/mrm.25665. [Epub ahead of print]

2. Feng L, Grimm R, Block KT, Chandarana H, Kim S, Xu J, Axel L, Sodickson DK, Otazo R. “Golden-Angle Radial Sparse Parallel MRI: Combination of Compressed Sensing, Parallel Imaging, and Golden-Angle Radial Sampling for Fast and Flexible Dynamic Volumetric MRI” Magn Reson Med . 2014 Sep;72(3):707-17

3. Feng L, Srichai MB, Lim RP, Harrison A, King W, Adluru G, Dibella E,

Sodickson DK, Otazo R, Kim D. “Highly-Accelerated Real-Time Cardiac Cine MRI Using k-t SPARSE-SENSE” Magn Reson Med. 2013 Jul; 70(1): 64-74

4. Feng L, Otazo R, Jung H, Jensen JH, Ye JC, Sodickson DK, Kim D.

“Accelerated Cardiac T2 Mapping Using Breath-Hold Multiecho Fast Spin-Echo Pulse Sequence with k-t FOCUSS” Magn Reson Med. 2011 Jun; 65(6):1661-9

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5. Feng L, Donnino R, Babb J, Axel L, Kim D. “Numerical and in vivo validation of fast cine displacement-encoded with stimulated echoes (DENSE) MRI for quantification of regional cardiac function” Magn Reson Med. 2009 Sep;62(3):682-90

6. Chandarana H, Feng L, Ream J, Wang Annie, Babb JS, Block KT,

Sodickson DK, Otazo R. “Respiratory Motion-Resolved Compressed Sensing Reconstruction of Free-Breathing Radial Acquisition for Dynamic Liver MRI” Accepted for publication in Investigative Radiology. May, 2015

7. Chandarana H, Feng L, Block KT, Rosenkrantz AB, Lim RP, Chu D,

Sodickson DK, Otazo R. “Free-Breathing Dynamic Contrast-Enhanced MRI of the Liver with Radial Golden-Angle Sampling Scheme and Advanced Compressed-Sensing Reconstruction” Investigative Radiology. 2013 Jan;48(1):10-6

8. Parasoglou P, Feng L, Xia D, Otazo R and Regatte RR. “Rapid 3D-

Imaging of Phosphocreatine Recovery Kinetics in the Human Lower Leg Muscles with Compressed Sensing” Magn Reson Med. 2012 Dec;68(6):1738-46

9. Rosenkrantz AB, Geppert C, Grimm R, Block TB, Glielmi C, Feng L,

Otazo R, Ream JM, Romolo MM, Taneja SS, Sodickson DK, Chandarana H. “Dynamic Contrast-Enhanced MRI of the Prostate with High Spatiotemporal Resolution using Compressed Sensing, Parallel Imaging, and Continuous Golden-Angle Radial Sampling: Preliminary Experience” Journal of Magnetic Resonance Imaging. 2015 May;41(5):1365-73

10. Kim D, Dyvorne HA, Otazo R, Feng L, Sodickson DK, Lee VS.

“Acelerated phase-contrast cine MRI using k-t SPARSE-SENSE” Magn Reson Med. 2012 Apr;67(4):1054-64

11. Kim D, Jensen JH, Wu EX, Feng L, Au WY, Cheung JS, Ha SY, Sheth

SS, Brittenham GM. “Rapid monitoring of iron-chelating therapy in thalassemia major by a new cardiovascular MR measure: the reduced transverse relaxation rate” NMR Biomed. 2011 Aug;24(7):771-7

12. Wu EX, Kim D, Tosti CL, Tang H, Jensen JH, Cheung JS, Feng L, Au WY, Ha SY, Sheth SS, Brown TR, Brittenham GM. “Magnetic resonance assessment of iron overload by separate measurement of

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tissue ferritin and hemosiderin iron” Ann N Y Acad Sci. 2010 Aug;1202:115-22

9.2. Conference Contributions (First Author)

2015 1. Li Feng, Daniel K Sodickson, Ricardo Otazo “Rapid Free-

Breathing Dynamic Contrast-Enhanced MRI Using Motion-Resolved Compressed Sensing” (IEEE ISBI 2015, Brooklyn, NY, USA)

2. Li Feng, Simone Coppo, Davide Piccini, Ruth P Lim, Matthias Stuber, Daniel K Sodickson, and Ricardo Otazo “Five-Dimensional Cardiac and Respiratory Motion-Resolved Whole-Heart MRI” (ISMRM 2015, Toronto, Canada)

3. Li Feng, Hersh Chandarana, Davide Piccini, Justin Ream, Daniel K Sodickson, and Ricardo Otazo “Rapid Free-Breathing Dynamic Contrast-Enhanced MRI Using Motion-Resolved Compressed Sensing” (ISMRM 2015, Toronto, Canada)

2014 4. Li Feng, Leon Axel, Jian Xu, Daniel K Sodickson, Ricardo Otazo

“Rapid Real-Time Cardiac MRI Exploiting Synchronized Cardio-Respiratory Sparsity” (ISMRM 2014, Milan, Italy)

5. Li Feng, Leon Axel, Darragh Halpenny, Larry Latson, Jian Xu,

Daniel K Sodickson, Ricardo Otazo “Evaluating both “Normal” and “Ectopic” Cardiac Cycles in Patients with Arrhythmias Using Free-Breathing Compressed Sensing MRI with Physiological Motion Synchronization” (ISMRM 2014, Milan, Italy)

6. Li Feng, Daniel K Sodickson, Ricardo Otazo “A Robust and Automatic Cardiac and Respiratory Motion Detection Framework for Self-Navigated Radial MRI” (ISMRM 2014, Milan, Italy)

7. Li Feng, Leon Axel, Larry A Latson, Jian Xu, Daniel K Sodickson

and Ricardo Otazo “Compressed sensing with synchronized cardio-respiratory sparsity for free-breathing cine MRI: initial

222

comparative study on patients with arrhythmias” (SCMR 2014, New Orleans, LA, USA)

8. Li Feng, Leon Axel, Jian Xu, Daniel K Sodickson, Ricardo Otazo “Synchronized cardiac and respiratory sparsity for rapid free-breathing cardiac cine MRI” (SCMR 2014, New Orleans, LA, USA)

2013

9. Li Feng, Leon Axel, Jian Xu, Jing Liu, Daniel K Sodickson, Ricardo Otazo “Rapid Free-Breathing 4D Time-Resolved Non-Contrast Aorta MRA without Cardiac Triggering and External Gating” (2013 International Magnetic Resonance Angiography Workshop, New York, NY, USA)

10. Li Feng, Jing Liu, Kai Tobias Block, Jian Xu, Leon Axel, Daniel K Sodickson, and Ricardo Otazo “Compressed Sensing Reconstruction with an Additional Respiratory-Phase Dimension for Free-Breathing Imaging” (ISMRM 2013, Salt Lake City, USA)

2012

11. Li Feng, Hersh Chandarana, Jian Xu, Kai Tobias Block, Daniel

Sodickson, and Ricardo Otazo “K-t Radial SPARSE-SENSE: Combination of Compressed Sensing and Parallel Imaging with Golden Angle Radial Sampling for Highly Accelerated Volumetric Dynamic MRI” (ISMRM 2012, Melbourne, Australia)

12. Li Feng, Jian Xu, Leon Axel, Daniel Sodickson, and Ricardo Otazo “High Spatial and Temporal Resolution 2D Real Time and 3D Whole-Heart Cardiac Cine MRI Using Compressed Sensing and Parallel Imaging with Golden Angle Radial Trajectory” (ISMRM 2012, Melbourne, Australia)

13. Li Feng, Monvadi Barbara Srichai-Parsia, Ruth P Lim, Alexis Harrison, Wilson King, Ganesh Adluru, Edward Dibella, Daniel Sodickson, Ricardo Otazo, and Daniel Kim “Quantitative Assessment of Highly Accelerated Real Time Cardiac Cine MRI Using Compressed Sensing and Parallel Imaging” (ISMRM 2012, Melbourne, Australia)

2011

223

14. Li Feng, Jian Xu, Kim Dan, Axel Leon, Sodickson K Deniel, Otazo,

Ricardo “Combination of Compressed Sensing, Parallel Imaging and Partial Fourier for Highly-Accelerated 3D First-Pass Cardiac Perfusion MRI” (ISMRM, 2011, Montreal, Canada)

15. Li Feng, Ricardo Otazo, Monvadi B Srichai, Ruth P Lim, Daniel K. Sodickson, Daniel Kim “Highly-Accelerated Real-Time Cine MRI using Compressed Sensing and Parallel Imaging with Cardiac Motion Constrained Reconstruction” (ISMRM, 2011, Montreal, Canada)

16. Li Feng, Ricardo Otazo, Monvadi B Srichai, Ruth P Lim, Daniel K Sodickson, Daniel Kim “Highly-Accelerated Real-Time Cine MRI using Compressed Sensing and Parallel Imaging” (SCMR, 2011, Nice, France)

2010 17. Li Feng, Ricardo Otazo, Monvadi B. Srichai, Ruth P. Lim, Ding

Xia, Daniel K. Sodickson, Daniel Kim “Highly-Accelerated Real-Time Cine MRI Using Compressed Sensing and Parallel Imaging”, (ISMRM, 2010, Stockholm, Sweden)

18. Li Feng, Ricardo Otazo, Jens Jensen, Daniel K. Sodickson, Daniel Kim “Accelerated Breath-Hold Multi Echo FSE Pulse Sequence Using Compressed Sensing and Parallel Imaging for T2 Measurement in the Heart” (ISMRM, 2010, Stockholm, Sweden)

2009 19. Li Feng, Daniel Kim “Theoretical Validation of Fast Cine DENSE

MRI for Quantification of Regional Cardiac Function” (ISMRM, 2009, Honolulu, USA)

20. Li Feng, Robert M. Donnino, James Babb, Leon Axel, Daniel Kim

“In Vivo Validation of Fast Cine DENSE MRI for the Quantification of Regional Cardiac Function” (ISMRM, 2009, Honolulu, USA)

21. Li Feng, Donnino RM, Axel L, Kim D “Quantitative assessment of

intramyocardial function using Cine DENSE MRI: a validation study” (SCMR, 2009, Orlando, Florida, USA)

224

9.3. Conference Contributions (Co-Author)

2015

22. Simone Coppo, Li Feng, Davide Piccini, Jerome Chaptinel, Gabriele Bonanno, Gabriella Vincenti, Juerg Schwitter, Ricardo Otazo, Daniel Sodickson and Matthias Stuber “Improved free-running self-navigated 4D whole-heart MRI through combination of compressed sensing and parallel imaging” (ISMRM 2015, Toronto, Canada)

23. Hersh Chandarana, Li Feng, Justin Ream, Annie Wang, James Babb, Kai T. Block, Mary Bruno, Daniel K. Sodickson, Ricardo Otazo “Respiratory motion-resolved compressed sensing reconstruction of free-breathing radial acquisition for improved dynamic liver MRI with an hepatobiliary contrast agent” (ISMRM 2015, Toronto, Canada)

24. Ding Xia, Li Feng, Tiejun Zhao, and Ravinder R. Regatte “Highly-

Accelerated 3D T1rho Mapping of the Knee Using k-t SPARSE-SENSE” (ISMRM 2015, Toronto, Canada)

25. Daniel K Sodickson, Li Feng, Florian Knoll, Martijn Cloos, Noam Ben-

Eliezer, Leon Axel, Hersh Chandarana, Tobias Block, Ricardo Otazo “The rapid imaging renaissance: sparser samples, denser dimensions, and glimmerings of a grand unified tomography” (SPIE Medical Imaging 2015, Orlando FL, USA)

2014

26. Parisa Amiri Eliasi, Li Feng, Ricardo Otazo, Sundeep Rangan

“Fast Magnetic Resonance Parametric Imaging via Structured Low-Rank Matrix Reconstruction” (2014 48th Asilomar Conference on Signals, Systems and Computers)

27. Jian Xu, Li Feng, Davide Piccini, Ricardo Otazo, Gabriele Bonanno, Florian Knoll, Edward K. Wong, Daniel K Sodickson “Feasibility of Free-Breathing Whole Heart Coronary MRA in Less Than 3 Minutes Using Combination of Compressed Sensing,

225

Parallel Imaging and A 3D Radial Phyllotaxis Trajectory” (ISMRM 2014, Milan, Italy)

28. Jian Xu, Li Feng, Ricardo Otazo, Ruth P Lim, Davide Piccini, Gabriele Bonanno, Yi Wang, Edward K. Wong, Daniel K Sodickson “Free-Breathing 3D Isotropic Whole Chest Non-Contrast MRA Using a Combination of Compressed Sensing and Parallel Imaging with Phyllotaxis Radial Trajectories” (ISMRM 2014, Milan, Italy)

29. Alicia W Yang, Li Feng, Daniel K Sodickson, Ricardo Otazo “Fast

and Simple Patch-Based Sparse Reconstruction Exploiting Local Image Correlations” (ISMRM 2014, Milan, Italy)

30. Jing Liu, Li Feng, David Saloner “Highly Accelerated Free-breathing 4D Cardiac Imaging with CIRCUS Acquisition” (ISMRM 2014, Milan, Italy)

31. Noam Ben-Eliezer, Li Feng, Kai Tobias Block, Daniel K Sodickson, Ricardo Otazo “Accelerated in vivo mapping of T2 relaxation from radially undersampled datasets using compressed sensing and model-based reconstruction” (ISMRM 2014, Milan, Italy)

32. Thomas Koesters, Li Feng, Kai Tobias Block, Michael Fieseler, Klaus P Schafers, Daniel K Sodickson, Frenado Boada “Simultaneous Acquisition of MR and PET data for Motion-Free PET Reconstruction” (ISMRM 2014, Milan, Italy)

33. Jing Liu, Henrik Haraldsson, Li Feng, David Saloner “Free-Breathing Whole Heart CINE Imaging with Inversion Recovery Prepared SSFP Sequence: Feasibility for Myocardium Viability Assessment” (ISMRM 2014, Milan, Italy)

34. Florian Knoll, Thomas Koesters, Ricardo Otazo, Tobias Block, Li Feng, Kathleen Vunckx, David Faul, Johan Nuyts, Fernado Boada, Daniel Sodickson “Simultaneous MR-PET Reconstruction using Multi Sensor Compressed Sensing and Joint Sparsity” (ISMRM 2014, Milan, Italy)

226

35. Elwin Bassett, Ricardo Otazo, Li Feng, Ganesh Adluru, Edward Dibella, Daniel Kim “Highly Accelerated Cine DENSE MRI with k-t SPARSE SENSE” (ISMRM 2014, Milan, Italy)

36. Nathaniel E. Margolis, Linda Moy, Akshat Pujara, Alana Amarosa, Eric E. Sigmund, Christian Geppert, Christopher Glielmi, Melanie Freed, Li Feng, and Ricardo Otazo, Amy N. Melsaether, Sungheon Kim “Initial experience: combination of MR pharmacokinetic modeling and FDG uptake using simultaneous dynamic contrast enhanced MRI and PET imaging” (ISMRM 2014, Milan, Italy)

37. Rosenkrantz AB, Geppert C, Grimm R, Block TK, Glielmi C, Li Feng, Otazo R, Ream JM, Romolo MM, Taneja SS, Sodickson DK, Chandarana H “Combined Compressed Sensing, Parallel Imaging, and Golden-Angle Radial Sampling for High Spatiotemporal Dynamic Contrast-Enhanced MRI of the Prostate” (ISMRM 2014, Milan, Italy)

38. Espagnet, Camilla Rossi; Bangiyev, Lev; Block, Kai Tobias; Grimm, Robert; Feng, Li; Ruggiero, Vito; Babb, James; Davis, Adam; Sodickson, Daniel K.; Fatterpekar, Girish “High Resolution DCE MRI of the Pituitary Gland Using Radial K Space Aquisition with Compressed Sensing Reconstruction” (ISMRM 2014, Milan, Italy)

39. Ream, Justin; Doshi, Ankur M.; Block, Kai Tobias; Kim, Sungheon; Otazo, Ricardo; Feng, Li; Chandarana, Hersh “High Spatiotemporal Dynamic Contrast-Enhanced MRI of the Small Bowel in Active Crohn's Terminal Ileitis Using Compressed Sensing, Parallel Imaging, and Golden-Angle Radial Sampling” (ISMRM 2014, Milan, Italy)

40. Florian Knoll, Thomas Koesters, Ricardo Otazo, Tobias Block, Li

Feng, Kathleen Vunckx, David Faul, Johan Nuyts, Fernando Boada, Daniel K Sodickson “Joint reconstruction of simultaneously acquired MR-PET data with multi sensor compressed sensing based on a joint sparsity constraint” (EJNMMI Physics, July 2014, 1:A26)

41. Thomas Koesters, Florian Buether, Li Feng, Klaus Schafers, David Faul, Fernando Boada “Comparison of PET and MR based

227

data driven gating methods for simultaneous PET/MRI” (Society of Nuclear Medicine Annual Meeting 2014)

2013

42. Prodromos Parasoglou, Li Feng, Ding Xia, Ricardo Otazo, and

Ravinder R Regatte “Three Dimensional Mapping of Oxidative Capacity in Human Lower Leg Muscles Compressed Sensing 31P-MRI” (ISMRM 2013, Salt Lake City, Utah)

43. Hersh Chandarana, Kai Tobias Block, Henry Rusinek, Matthew B Greenberg, Li Feng, Daniel K Sodickson, and Ricardo Otazo “Free-Breathing Dynamic Contrast Enhanced Compressed-Sensing Imaging for Reliable Estimation of Liver Perfusion” (ISMRM 2013, Salt Lake City, Utah)

44. Riccardo Lattanzi, Alicia W Yang, Li Feng, Michael Recht, Daniel K Sodickson, and Ricardo Otazo “Feasibility of accelerating 3 T hip imaging using compressed sensing” (ISMRM 2013, Salt Lake City, Utah)

45. Robert Grimm, Li Feng, Christoph Forman, Jana Hutter, Berthold Kiefer, Joachim Hornegger, and Tobias Block “Automatic Bolus Analysis for DCE-MRI Using Radial Golden-Angle Stack-of-stars GRE Imaging” (ISMRM 2013, Salt Lake City, Utah)

46. Kai Tobias Block, Robert Grimm, Li Feng, Ricardo Otazo, Hersh Chandarana, Mary Bruno, Christian Geppert, and Daniel K. Sodickson “Bringing Compressed Sensing to Clinical Reality: Prototypic Setup for Evaluation in Routine Applications” (ISMRM 2013, Salt Lake City, Utah)

47. Kai Tobias Block, Robert Grimm, Li Feng, Ricardo Otazo, Hersh Chandarana, Daniel Sodickson, Ricardo Otazo “Prototypic Setup for Evaluation of a Compressed-Sensing Technique in Clinical Patient Studies” (ISMRM Workshop on Data Sampling and Image Reconstruction, Sedona, Arizona, 2013)

2012 48. Jian Xu, Li Feng, Ricardo Otazo, Alicia Yang, Kai Tobias Block,

Barbara Srichai, Ruth Lim, Kelly Anne Mcgorty, Joseph Reaume,

228

Leon Axel, Yao Wang, and Daniel Sodickson “Feasibility of 5-Minute Comprehensive Cardiac MR Examination Using Highly Accelerated Parallel Imaging and Compressed Sensing” (ISMRM 2012, Melbourne, Australia)

49. Jian Xu, Li Feng, Ricardo Otazo, Barbara Srichai, Ruth Lim, Bhat Himanshu, Kelly Anne Mcgorty, Joseph Reaume, and Daniel Sodickson “Feasibility of Dynamic 4D Whole Heart Viability Imaging Within a Single Breath-Hold Using Highly Accelerated Parallel Imaging and Compressed Sensing” (ISMRM 2012, Melbourne, Australia)

50. Hersh Chandarana, Li Feng, Tobias Kai Block, Andrew B Rosenkrantz, Ruth P Lim, Dewey Chu, Daniel K Sodickson, and Ricardo Otazo “Free-breathing dynamic contrast-enhanced MRI of the liver with radial golden-angle sampling scheme and advanced compressed-sensing reconstruction” (ISMRM 2012, Melbourne, Australia)

51. Hersh Chandarana, Li Feng, Tobias Kai Block, Joseph P Stepancic, Daniel K Sodickson, and Ricardo Otazo “Contrast-enhanced free-breathing perfusion weighted MR imaging of the whole-liver with high spatial and temporal resolution” (ISMRM 2012, Melbourne, Australia)

52. Sungheon Kim, Li Feng, Linda Moy, Melanie Moccaldi, Kai T. Block, Daniel K. Sodickson, and Ricardo Otazo “Highly-Accelerated Golden-Angle Radial Acquisition with Joint Compressed Sensing and Parallel Imaging Reconstruction for Breast DCE-MRI” (ISMRM 2012, Melbourne, Australia)

53. Alicia Yang, Li Feng, Jian Xu, Ivan Selesnick, Daniel K Sodickson, and Ricardo Otazo “Improved Compressed Sensing Reconstruction with Overcomplete Wavelet Transforms” (ISMRM 2012, Melbourne, Australia)

54. Daniel Kim, Alexis Harrison, Wilson King, Li Feng, Elwin Bassett, Christopher J McGann, Nassir F Marrouche, and Ricardo Otazo “Highly-accelerated, single breath-hold 3D Cine b-SSFP MRI with a combination of compressed sensing and parallel imaging” (ISMRM 2012, Melbourne, Australia)

229

55. Ricardo Otazo, Li Feng, Hersh Chandarana, Tobias Block, Leon Axel, Daniel Sodickson. “Combination of Compressed Sensing and Parallel Imaging for Highly-Accelerated Dynamic MRI” (IEEE ISBI 2012, Barcelona, Spain)

56. Kai Tobias Block, Martin Uecker, Shuo Zhang, Li Feng, Hersh Chandarana, Ricardo Otazo. “Iterative Reconstruction Techniques for Faster Scan Speed in Magnetic Resonance Imaging” (TOPIM 2012, Les Houches)

2010

57. Kim D, Wu EX, Jensen J, Au WY, Feng L, Cheung JS, Ha SY,

Sheth SS, Brittenham GM “A breath-hold R2 mapping pulse sequence detects a decrease in myocardial ferritin iron after one-week of iron chelation” (SCMR, 2010, Phoenix, Arizona, USA)

58. Otazo R, Feng L, Lim R, Duan Q, Wiggins G, Sodickson DK, Kim D “Accelerated 3D carotid MRI using compressed sensing and parallel imaging” (SCMR, 2010, Phoenix, Arizona, USA)

2009

59. Jian Xu, Ricardo Otazo, Sven Zuehlsdorff, Daniel Kim, Xiaoming

Bi, Qi Duan, Sonia Nielles-Vallespin, Monvadi Barbara Srichai, Thoralf Niendorf, Renate Jerecic, Bernd Stoeckel, Yao Wang, Li Feng, Kellyanne Mcgorty, Daniel K. Sodickson “Feasibility of Five-Minute Comprehensive Cardiac MR Examination Using Highly Accelerated Parallel Imaging with a 32-Element Coil Array” (ISMRM, 2009, Honolulu, USA)

230

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