advanced image processing in magnetic resonance imaging.pdf

Signal Processing and Communications

Editorial Board

Maurice G. Ballanger, Conservatoire Nationaldes Arts et Mtiers (CNAM), Paris

Ezio Biglieri, Politecnico di Torino, Italy

Sadaoki Furui, Tokyo Institute of Technology

Yih-Fang Huang, University of Notre Dame

Nikil Jayant, Georgia Institute of Technology

Aggelos K. Katsaggelos, Northwestern University

Mos Kaveh, University of Minnesota

P. K. Raja Rajasekaran, Texas Instruments

John Aasted Sorenson, IT University of Copenhagen

1. Digital Signal Processing for Multimedia Systems, edited by Keshab K. Parhi and Takao Nishitani

2. Multimedia Systems, Standards, and Networks, edited by Atul Puri and Tsuhan Chen

3. Embedded Multiprocessors: Scheduling and Synchronization, SundararajanSriram and Shuvra S. Bhattacharyya

4. Signal Processing for Intelligent Sensor Systems, David C. Swanson5. Compressed Video over Networks, edited by Ming-Ting Sun and Amy R. Reibman6. Modulated Coding for Intersymbol Interference Channels, Xiang-Gen Xia7. Digital Speech Processing, Synthesis, and Recognition: Second Edition,

Revised and Expanded, Sadaoki Furui8. Modern Digital Halftoning, Daniel L. Lau and Gonzalo R. Arce9. Blind Equalization and Identification, Zhi Ding and Ye (Geoffrey) Li10. Video Coding for Wireless Communication Systems, King N. Ngan, Chi W. Yap,

and Keng T. Tan11. Adaptive Digital Filters: Second Edition, Revised and Expanded,

Maurice G. Bellanger12. Design of Digital Video Coding Systems, Jie Chen, Ut-Va Koc, and K. J. Ray Liu13. Programmable Digital Signal Processors: Architecture, Programming,

and Applications, edited by Yu Hen Hu14. Pattern Recognition and Image Preprocessing: Second Edition, Revised

and Expanded, Sing-Tze Bow15. Signal Processing for Magnetic Resonance Imaging and Spectroscopy,

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and Algorithms, Artyom M. Grigoryan, Sos S. Agaian, S.S. Agaian19. High-Resolution and Robust Signal Processing, Yingbo Hua, Alex B. Gershman

and Qi Cheng20. Domain-Specific Processors: Systems, Architectures, Modeling, and Simulation,

Shuvra Bhattacharyya; Ed Deprettere; Jurgen Teich21. Watermarking Systems Engineering: Enabling Digital Assets Security

and Other Applications, Mauro Barni, Franco Bartolini22. Biosignal and Biomedical Image Processing: MATLAB-Based Applications,

John L. Semmlow23. Broadband Last Mile Technologies: Access Technologies for Multimedia

Communications, edited by Nikil Jayant24. Image Processing Technologies: Algorithms, Sensors, and Applications,

edited by Kiyoharu Aizawa, Katsuhiko Sakaue and Yasuhito Suenaga25. Medical Image Processing, Reconstruction and Restoration: Concepts

and Methods, Jiri Jan26. Multi-Sensor Image Fusion and Its Applications, edited by Rick Blum

and Zheng Liu27. Advanced Image Processing in Magnetic Resonance Imaging, edited by

Luigi Landini, Vincenzo Positano and Maria Santarelli

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DK2411_title 7/29/05 12:58 PM Page 1

AdvancedImage

Processing

MagneticResonanceImaging

in

edited by

Luigi LandiniVincenzo Positano

Maria Filomena Santarelli

A CRC title, part of the Taylor & Francis imprint, a member of theTaylor & Francis Group, the academic division of T&F Informa plc.

Boca Raton London New York


Published in 2005 byCRC PressTaylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

2005 by Taylor & Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group

No claim to original U.S. Government worksPrinted in the United States of America on acid-free paper10 9 8 7 6 5 4 3 2 1

International Standard Book Number-10: 0-8247-2542-5 (Hardcover) International Standard Book Number-13: 978-0-8247-2542-6 (Hardcover) Library of Congress Card Number 2005043995

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Advance image processing in magnetic resonance imaging / edited by Luigi Landini, Vicenzo Positno, Maria Santarelli.

p. cm. -- (Signal processing and communications ; 26)Includes bibliographical references and index.ISBN 0-8247-2542-51. Magnetic resonance imaging. 2. Image processing. I. Landini, Luigi. II. Positano, Vicenzo. III.

Santarelli, Maria. IV. Series.

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Preface

Magnetic Resonance (MR) imaging produces images of the human tissues in anoninvasive manner, revealing the structure, metabolism, and function of tissuesand organs. The impact of this image technique in diagnostic radiology is impres-sive, due to its versatility and flexibility in joining high-quality anatomical imageswith functional information.

Signal and image processing play a decisive role in the exploitation of MRimaging features, allowing for the extraction of diagnostic and metabolic informa-tion from images.This book attempts to cover all updated aspects of MR imageprocessing, ranging from new acquisition techniques to state-of-art imaging tech-niques. Because the textbook provides the tools necessary to understand the physicaland chemical principles, and the basic signal and image processing concepts andapplications, it is a valuable reference book for scientists, and an essential sourcefor upper-level undergraduate and graduate students in these disciplines.

The books 18 chapters are divided into five sections. The first section focuseson MR signal and image generation and reconstruction, the basics of MR imaging,advanced reconstruction algorithms, and the parallel MRI field. In the secondsection, the state-of-art techniques for MR images filtering are described. In par-ticular, the second section covers the signal and noise estimation, the inhomoge-neities correction and the more advanced image filtering techniques, taking intoaccount the peculiar features of the noise in MR images. Quantitative analysis isa key issue in MR diagnostic imaging. The third sections topics range from imageregistration to integration of EEG and MEG techniques with MR imaging. Twochapters cover the cardiac image quantitative analysis issue. In the fourth section,MR spectroscopy is described, from both the signal generation and the dataanalysis point of view. Diffusion tensor MR imaging and MR elastography arealso examined. Finally, in the last section, the functional MR image processing isdescribed in detail. Fundamentals and advanced data analysis (as exploratoryapproach), bayesian inference, and nonlinear analysis are also depicted.

Experts from all fields of MRI have cooperated in preparing this book. We wouldlike to thank all authors for their excellent and up-to-date chapters. Some overlap wasunavoidable, due to the multiple connections in MR image processing fields.

The outline of this book was born at the MR Laboratory at the CNR Institute ofClinical Physiology in Pisa, Italy. The MR Laboratory is a multidisciplinary arenafor biomedical engineers, physicists, computer scientists, radiologists, cardiologists,and neuroscientists, and it shares experts from CNR and university departments.For this textbook, we are sincerely grateful to professor Luigi Donato, the chiefof the CNR Institute of Clinical Physiology, and the pioneer of the multidisci-plinary approach to biomedicine.

DK2411_C000.fm Page v Wednesday, August 10, 2005 12:04 PM


Contributors

Antonio Benassi

Institute of Clinical Physiology, CNRPisa, Italy

Alessandra Bertoldo

Department of Information Engineering

University of PadovaPadova, Italy

Sergio Cerutti

Department of Biomedical Engineering

Polytechnic UniversityMilano, Italy

Claudio Cobelli


University of PadovaPadova, Italy

Arjan J. den Dekker

Delft Center for Systems andControl

Delft University of TechnologyDelft, The Netherlands

Francesco Di Salle

Department of Neurological Sciences

University of Naples, ItalyNaples, Italy

Elia Formisano

Department of Cognitive Neurosciences

Faculty of PsychologyUniversity of MaastrichtMaastricht, The Netherlands

Alejandro Frangi

C.N.R. Institute of Clinical Physiology

Department of TechnologyPompev Fabra UniversityBarcelona, Spain

Karl J. Friston

The Wellcome Department of Imaging Neuroscience

University College LondonLondon, U.K.

Rainer Goebel

Department of Cognitive Neurosciences

Faculty of PsychologyUniversity of MaastrichtMaastricht, The Netherlands

Steven Haker

Harvard University School of Medicine

Brigham and Womens HospitalBoston, Massachusetts

Yaroslav O. Halchenko

Psychology DepartmentRutgers UniversityNewark, New Jersey

DK2411_C000.fm Page vii Wednesday, August 10, 2005 12:04 PM


Stephen Jose Hanson

Rutgers UniversityPsychology DepartmentNewark, New Jersey

W. Scott Hoge



Filip Jiru

Section of Experimental MR of the CNS

Department of NeuroradiologyTbingen, Germany

Uwe Klose

Section of Experimental MR of the CNS

Department of NeuroradiologyTbingen, Germany

Walid Elias Kyriakos



Luigi Landini


University of PisaPisa, Italy

Koen Van Leemput

HUS Helsinki Medical Imaging Center

Helsinki, Finland

Boudewijn P.F. Lelieveldt

Laboratory for Clinical and Experimental Image Processing

Department of Radiology, Leiden University Leiden, The Netherlands

Zhi-Pei Liang

Department of Electrical Engineering & Computer Engineering

University of Illinois at UrbanaUrbana, Illinois

Massimo Lombardi

MRI Laboratory Institute of Clinical Physiology, CNRPisa, Italy

Luca T. Mainardi

Department of Biomedical Engineering

Polytechnic UniversityMilano, Italy

Armando Manduca

Mayo Clinic and FoundationRochester, Minnesota

Wiro J. Niessen

Image Sciences InstituteUtrecht UniversityUtrecht, The Netherlands

Lauren ODonnell

Massachusetts Institute of TechnologyHarvard Division of Health, Science

and TechnologyBoston, Massachusetts

Barakk A. Pearlmutter

Hamilton Insitute and Department of Computer Science

Kildare, Ireland

DK2411_C000.fm Page viii Wednesday, August 10, 2005 12:04 PM


William D. Penny

The Wellcome Department of Imaging Neuroscience

University College LondonLondon, U.K.

Vincenzo Positano




Jan Sijbers

Vision Laboratory, Campus Middelheim

University of AntwerpAntwerp, Belgium

Martin Styner

Department of Computer ScienceUniversity of North CarolinaChapel Hill, North Carolina

Nicola Vanello

Department of Electrical Systemsand Automation

University of PisaPisa, Italy

M.A. Viergever

Image Sciences InstituteUtrecht UniversityUtrecht, The Netherlands

Yongmei Michelle Wang

Departments of Statistics, Psychology Bioengineering

University of Illinois at UrbanaChampaign

Champaign, Illinois

Carl-Fredrik Westin

Laboratory of Mathematics in ImagingHarvard Medical SchoolBrigham and Womens HospitalBoston, Massachusetts

Leslie Ying

Department of Electrical Engineering and Computer Science

University of WisonsinMilwaukee, Wisconsin

Alistair A. Young

Department of Anatomy with Radiology

University of AucklandAuckland, New Zealand

Francessca Zanderigo

Department of Information EngineeringUniversity of Padova

Padova, Italy

DK2411_C000.fm Page ix Wednesday, August 10, 2005 12:04 PM


Contents

Part ISignal and Image Generation and Reconstruction

Chapter 1

Basic Physics of MR Signal and Image Generation ......................3

Chapter 2

Advanced Image Reconstruction Methods in MRI ......................39

Chapter 3

Part IISNR Improvement and Inhomogeneities Correction

Chapter 4

Estimation of Signal and Noise Parameters from MR Data ........85

Chapter 5

Retrospective Evaluation and Correction of Intensity Inhomogeneity .............................................................................145

Chapter 6

Noise Filtering Methods in MRI.................................................169

Part IIIImage Processing and Quantitative Analysis

Chapter 7

DK2411_C000.fm Page xi Wednesday, August 10, 2005 12:04 PM



Leslie Ying and Zhi-Pei Liang

Walid Elias Kyriakos and W. Scott Hoge

Parallel MRI: Concepts and Methods...........................................57

Arjan J. den Dekker and J. Sijbers

Martin Styner and Koen Van Leemput

Luigi Landini, Massimo Lombardi and Antonio Benassi

Vincenzo Positano

Image Registration Methods in MRI ..........................................189

Chapter 8

Multimodal Integration: fMRI, MRI, EEG, and MEG ..............223

Chapter 9

A Survey of Three-Dimensional Modeling Techniques for Quantitative Functional Analysis of Cardiac Images.................267

Chapter 10

MRI Measurement of Heart Wall Motion ..................................343

Part IVSpectroscopy, Diffusion, Elasticity: From Modeling to Parametric Image Generation

Chapter 11

Principles of MR Spectroscopy and Chemical Shift Imaging ........................................................................................369

Chapter 12

Data Processing Methods in MRS..............................................411

Chapter 13

Image Processing of Diffusion Tensor MRI Data......................427

Chapter 14

Analysis of Dynamic Magnetic Resonance Elastography Data .......................................................................451

Part VBOLD Contrast MR Imaging and fMRI Signal Analysis

Chapter 15

Functional MRI ...........................................................................481

DK2411_C000.fm Page xii Wednesday, August 10, 2005 12:04 PM


Yaroslav O. Halchenko, Stephen Jos Hanson, and Barak A. Pearlmutter

Alejandro F. Frangi, Wiro J. Niessen, M.A. Viergever, and Bovdewijr P.F. Lelieveldt

Alistair A. Young

Uwe Klose and Filip Jiru

Luca T. Mainardi and Sergio Cerutti

Lauren ODonnell, Steven Haker, and Carl-Fredrik Westin

Armando Manduca

Elia Formisano, Francesco Di Salle, and Rainer Goebel

Fundamentals of Data Analysis Methods in

Chapter 16

Exploratory Data Analysis Methods in Functional MRI ...........505

Chapter 17

Classical and Bayesian Inference in fMRI .................................541

Chapter 18

Modeling and Nonlinear Analysis in fMRI via Statistical Learning................................................................565

Chapter 19

Assessment of Cerebral Blood Flow, Volume, and Mean Transit Time from Bolus-Tracking MRI Images: Theory

DK2411_C000.fm Page xiii Wednesday, August 10, 2005 12:04 PM


Nicola Vanello and Luigi Landini

William D. Penny and Karl J. Friston

Yongmei Michelle Wang

Alessandra Bertoldo, Francesca Zanderigo, and Claudio Cobelli

and Practice .................................................................................587

Part I

Signal and Image Generation and Reconstruction

DK2411_C001.fm Page 1 Thursday, August 4, 2005 11:04 AM


3

1

Basic Physics of MR Signal and Image Generation


CONTENTS

1.1 Introduction..................................................................................................41.2 Nuclear Spin ................................................................................................61.3 Nuclei in a Magnetic Field .........................................................................7

1.3.1 Notes on Larmor Frequency: Changes Due to Dishomogeneities ................................................................9

1.3.2 Bulk Magnetization .......................................................................101.4 RF Excitation for the Resonance Phenomenon Generation .....................111.5 MR Signal Generation and Acquisition....................................................14

1.5.1 Free Induction Decay and the Fourier Transform ........................141.6 MR Signal Characteristics.........................................................................16

1.6.1 Relaxation ......................................................................................161.6.1.1 The Decay of Transverse Magnetization: T2 ................161.6.1.2 The Recovery of Longitudinal Magnetization: T1........161.6.1.3 Pseudo-Relaxation: T2

*

..................................................171.6.2 Proton Density ...............................................................................171.6.3 The Bloch Equations .....................................................................18

1.6.3.1 Rotating Frame of Reference.........................................181.7 Multiple RF Pulses ....................................................................................18

1.7.1 Gradient Echo................................................................................191.7.2 Inversion Recovery ........................................................................191.7.3 Spin Echo ......................................................................................20

1.8 Magnetic Field Gradients ..........................................................................231.9 Spatial Localization of MR Signals..........................................................24

1.9.1 Slice Selection ...............................................................................241.9.2 Frequency Encoding......................................................................261.9.3 Phase Encoding .............................................................................281.9.4 Phase History of Magnetization Vectors during

Phase Encoding .............................................................................281.9.5 Timing Diagram of an Imaging Sequence....................................29



4

Advanced Image Processing in Magnetic Resonance Imaging

1.10 Acquiring MR Signals in the K-Space .....................................................301.10.1 K-Space Trajectories .....................................................................33

1.11 Imaging Methods.......................................................................................33Acronyms ............................................................................................................36References ...........................................................................................................36

1.1 INTRODUCTION

The nuclear magnetic resonance (NMR or MR) phenomenon in bulk matter wasrst demonstrated by Bloch and associates [1] and Purcell and associates [2] in1946. Since then, MR has developed into a sophisticated technique with applicationsin a wide variety of disciplines that now include physics, chemistry, biology, andmedicine. Over the years, MR has proved to be an invaluable tool for molecularstructure determination and investigation of molecular dynamics in solids andliquids. In its latest development, application of MR to studies of living systemshas attracted considerable attention from biochemists and clinicians alike. Thesestudies have progressed along two parallel and perhaps complementary paths. First,MR is used as a spectroscopic method to provide chemical information fromselected regions within an object (magnetic resonance spectroscopy [MRS]). Suchinformation from a localized area in living tissue provides valuable metabolic datathat are directly related to the state of health of the tissue and, in principle, can beused to monitor tissue response to therapy. In the second area of application, MRis used as an imaging tool to provide anatomic and pathologic information.

The rapid progress of MR to diverse elds of study can be attributed to thedevelopment of pulse Fourier transform techniques in the late 1960s [3]. Additionalimpetus was provided by the development of fast Fourier transform algorithms,advances in computer technology, and the advent of high-eld superconductingmagnets. Then, the introduction of new experimental concepts such as two-dimen-sional MR has further broadened its applications [4] to the magnetic resonanceimaging (MRI) technique.

MRI is a tomographic imaging technique that produces images of internal phys-ical and chemical characteristics of an object from externally measured MR signals.Tomography is an important area in the ever-growing eld of imaging science.The Greek term

tomos

(

) means cut, but tomography is concerned withcreating images of the internal (anatomical or functional) organization of an objectwithout physically cutting it open.

Image formation using MR signals is made possible by the spatial informationencoding principles, originally named

zeugmatography

[4,5]. As it will be brieydescribed in this chapter, these principles enable one to uniquely encode spatialinformation into the activated MR signals detected outside an object.

As with any other tomographic imaging device, an MRI scanner outputs amultidimensional data array (or image) representing the spatial distribution ofsome measured physical quantity. But unlike many of them, MRI can generatetwo-dimensional sectional images at any orientation, three-dimensional volumetric




5

images, or even four-dimensional images representing spatialspectral or spatialtemporal (i.e., in cardiac imaging) distributions. In addition, no mechanical adjust-ments to the imaging machinery are involved in generating these images.

Another peculiarity of the MRI technique is that MR signals used for imageformation come directly from the object itself. In this sense, MRI is a form ofemission tomography similar to position emission tomography (PET) and single-photon computed tomography (SPECT). But unlike PET or SPECT, no injectionof radioactive isotopes into the object is needed for signal generation in MRI.There are other forms of tomography in use, including transmission tomographyand diffraction tomography. X-ray computer-assisted tomography (CT) belongsto the rst category, while most acoustic tomography is of the diffraction type.In both cases, an external signal source is used to probe the object being imaged.

MRI operates in the radio frequency (RF) range, as shown in Figure 1.1. There-fore, the imaging process does not involve the use of ionizing radiation and doesnot have the associated potential harmful effects. However, because of the uniqueimaging scheme used, the resulting spatial resolution of MRI is not limited by theprobing (or working) frequency range as in other remote-sensing technologies.

FIGURE 1.1

Electromagnetic wave range for MR and MRI.

MRimaging

1016

Wavelength(m)

Frequency(Hz)

Energy(eV)

1014

1012

1010

108

106

104

102

1

102

104

106

1010

108

106

104

102

1

102

104

106

108

1010

1012

1024

1022

1020

1018

1016

1014

1012

108

1010

106

104

102

Visible lightInfrared

Microwaves

Radiowaves

Ultraviolet

Gamma rays

X-rays



6


Images are extremely rich in information content. The image pixel value isin general dependent on a lot of intrinsic parameters, including the nuclear spindensity (

)

, the spin-lattice relaxation time (T1), the spin-spin relaxation time(T 2), molecular motions (such as diffusion and perfusion), susceptibility effects,and chemical shift differences. The imaging effects of these parameters can besuppressed or enhanced in a specic experiment by another set of operator-selectable parameters, such as repetition time (TR), echo time (TE), and ip angle(

). Therefore, an MR image obtained from the same anatomical site can lookdrastically different with different data acquisition protocols.

In the present chapter, the basic physical principles of MRI are presented.The objective is to allow readers to understand and interpret the MR signal andimage generation in order to introduce them to processing analysis: why and howsignal processing theories and methods, described in the following chapters ofthe book, can be applied on MR signals and images.

For a deeper study of MRI physical principles, extensive literature exists[613], which the reader is encouraged to consult; part of the material of thischapter has been extracted from these texts.

1.2 NUCLEAR SPIN

The basis of NMR lies in a property possessed by certain nuclei, called the

spinangular momentum

(

p

). The spin angular momentum of the nucleus can be con-sidered as an outcome of the rotational or spinning motion of the nucleus aboutits own axis. For this reason, nuclei having spin angular momentum are oftenreferred to as nuclear spins. The spin angular momentum of a nucleus is denedby the nuclear

spin quantum number

I, and is given by the relationship

(1.1)where h

=

h/2

and h is the Plancks constant. The value of spin quantum numberdepends on the structure of the nucleus the number of protons and neutrons

of some selected nuclei is given. Hydrogen (

1

H, with I

=

1/2), the most abundantelement in nature and in the body, is most receptive to NMR experiments. Onthe other hand, the most common isotopes of carbon (

12

C) and oxygen (

16

O) havenuclei with I

=

0 and hence cannot be observed by magnetic resonance experi-ments.

Because the nucleus is a charged particle, spin angular momentum is accom-panied by a

magnetic moment

(

) given by

=

p

(1.2)where

is called the

gyromagnetic ratio

.

Note that both

and

p

are vector quantities having magnitude and direction.The gyromagnetic ratio is characteristic of a particular nucleus (see Table 1.1),and it is proportional to the charge-to-mass ratio of the nucleus.

| | [ ]p h I I 1= + ( )



and can be an integer, half-integer, or zero. In Table 1.1, the spin quantum number


7

1.3 NUCLEI IN A MAGNETIC FIELD

In MR experiments we are concerned with the behavior of nuclei placed in anexternal magnetic eld. According to the

classical model

, the presence of amagnetic moment means that the nucleus behaves like a tiny bar magnet, with anorth and a south pole, and it will therefore interact with a magnetic eld.

In the case of a bar magnet, application of an external magnetic eld wouldcause the magnet simply to align with or against the direction of the eld. However,a nucleus has angular momentum and, consequently, precesses about the directionof the applied eld, just as a spinning top precesses in the Earths gravitationaleld. The interaction between the magnetic moment

and the eld

B

0

tries toalign the two, according to the formula:

L

=

B

0

(1.3)where

L

is the torque or turning force.

magnetic moment vector

about the direction of the external magnetic eld

B

0

in addition to the nuclear spin about its own axis.The torque force causes the nucleus to precess about

B

0

, altering the angularmomentum

p

; in fact,

d

p

/dt

=

L

=

B

0

(1.4)But

p

=

/

, so that

d

/dt

=

B

0

=

0

(1.5)where

0

=

B

0

is the

frequency of precession

, i.e., the rate at which

rotates about

B

0

, and iscalled the

Larmor frequency

; the minus sign describes the rotation direction.As said before, this result can be deduced from the laws of classical physics.

However, in order to obtain a complete description of the behavior of nuclei ina magnetic eld, quantum mechanical theory must be considered.

TABLE 1.1NMR Properties of Some Selected Nuclei

NucleusNuclear

SpinGyromagnetic Ratio (MHz/T)

Natural Abundance (%)

Relative Sensitivity*

1H 1/2 42.58 99.98 113C 1/2 10.71 1.11 0.01619F 1/2 40.05 100 0.87031P 1/2 17.23 100 0.066

23Na 3/2 11.26 100 0.093

* calculated at constant eld for an equal number of nuclei



This precessional motion is schematized in Figure 1.2 as a rotation of the

8


According to the

quantum mechanical model

, the orientation of the spinangular momentum vector, and hence the magnetic moment, of the nucleus withrespect to the direction of

B

0

cannot be arbitrary and it is subjected to certainrestrictions. In fact, the nuclear magnetic moment can only have 2I + 1 orientationsin a magnetic eld, corresponding to 2I

+

1 allowed energy levels, so that theproton (

1

H) nucleus can either align with or against the applied eld. These twoorientations are similar to the two directions in which a bar magnet may orient ina magnetic eld and are consequently called the parallel (spin-up) and the anti-parallel (spin-down) orientations. These orientations correspond to low and high

The difference in energy,

E

, between the two states is proportional to thestrength of the magnetic eld

B

0

, and it is given by

E

=

B

0

(1.6)

FIGURE 1.2

Precession of nuclear magnetic moment about the magnetic eld

B

0

.

Precessionalmotion

B0

Spin motion



energy states respectively, as schematically shown in Figure 1.3.


9

So far, the behavior of a single isolated nucleus has been considered. However,in practice, the mean result due to a large number of similar nuclei is observed.When an ensemble of nuclei is subjected to an external magnetic eld, nucleidistribute themselves in the allowed orientations. At equilibrium, the population ofnuclei in the parallel orientation (lower energy state) exceeds that in the antiparallelorientation (higher energy state) by a small amount, according to Boltzman statis-tics. The excess population in the lower energy state,

N

, is dependent on theenergy difference between the spin states and the absolute temperature, T:

(1.7)

where

N

0

is the total number of nuclei in the sample, and

k

is the Boltzmanconstant. The fractional excess of population (

N

/

N

0

) in the lower energy stateis extremely small; for example, for hydrogen nuclei at body temperature (310 K)in a magnetic eld B

0

=

1 T,

N/N

0

=

3.295

10

6

.

1.3.1 N

OTES

ON

L

ARMOR

F

REQUENCY

: C

HANGES

D

UE

TO

D

ISHOMOGENEITIES

We have seen that the precession frequency of

experiencing a B0 eld is givenby 0 = B0 and, considering the modulus:

0 = B0 (1.8)

This relation, commonly called the Larmor equation, is important because theLarmor frequency is the natural resonance frequency of a spin system.Equation 1.8 shows that the resonance frequency of a spin system is linearlydependent on both the strength of the external magnetic eld B0 and the valueof the gyromagnetic ratio . This relationship describes the physical basis for

1

FIGURE 1.3 Energy level diagram for I = 1/2.

Energy

E

B0

Low energy state

High energy state

N N EkT

0 2



achieving nucleus specicity. In fact, as shown in Table 1.1, the nuclei of H and

10 Advanced Image Processing in Magnetic Resonance Imaging

31P in an object resonate at 42.58 MHz and 11.26 MHz, respectively, when theobject is placed in a magnetic eld B0 = 1 T; this difference in resonance frequencyenables us to selectively image one of them without interacting with the other.

Actually, a specic spin system (i.e., hydrogen protons) may have a range ofresonance frequencies. In this case, each group of nuclear spins that share thesame resonance frequency is called an isochromat.

There are two main reasons for a magnetized spin system to have multipleisochromats: (1) the presence of inhomogeneities in the B0 eld and (2) thechemical shift effect, which is exploited for chemical components studies.

When B0 is not homogeneous, spin with the same value will have differentLarmor frequencies at different spatial locations. Such a condition can be usefullyexploited, as we will see in MR image generation, when the inhomogeneity ofB0 is known. However, if B0 inhomogeneities are not known, they are consideredas bringing a negative effect, that is artifacts.

The chemical shift effect is due to the fact that nuclei in a spin system are partof different molecules in a chemically heterogeneous environment. Because eachnucleus of a molecule is surrounded by orbiting electrons, these orbiting electronsproduce their own weak magnetic elds, which shield the nucleus to varyingdegrees depending on the position of the nucleus in the molecule. As a result, theeffective magnetic eld that a nucleus sees is

(1.9)

where is a shielding constant taking on either positive or negative values. Basedon the Larmor relationship, the resonance frequency for the nucleus is

(1.10)

Equation 1.10 shows that spins in different chemical environments will haverelative shifts in their resonance frequency even when B0 is homogeneous. Thefrequency shift depends on both the strength of the magnetic eld B0 andthe shielding constant . Usually, the value of is very small, on the order ofa few parts per million (ppm), and it depends on the local chemical environmentin which the nucleus is embedded. Knowledge of these chemical shift frequenciesand the corresponding spin densities is of great importance for determining thechemical structures of an object, which is the subject of MR spectroscopy.

1.3.2 BULK MAGNETIZATION

To describe the collective behavior of a spin system, we introduce a macroscopicmagnetization vector M, which is the vector sum of all the microscopic magneticmoments in the object:

(1.11)

( )B B0 0 1=

( ) 0 0 0 1= =

M ii

Ns

=

=

1



Basic Physics of MR Signal and Image Generation 11

where i is the magnetic moment of the i-th nuclear spin, and Ns is the totalnumber of spins.

Although there is a microscopic transverse component for each magneticmoment vector, the transverse component of M is zero at equilibrium becausethe precessing magnetic moments have random phases. The macroscopic effectof an external magnetic eld B0 on an ensemble of nuclei with nonzero spins isthe generation of an observable bulk magnetization vector M pointing along thedirection of B0.

The magnitude of the equilibrium magnetization M is equal to that of the spinexcess predicted by the quantum model. M itself behaves like a large magneticdipole moment, and if perturbed from its equilibrium state, it will precess at 0about B0. By analogy with Equation 1.5, we can write:

dM/dt = 0 M = M B0 (1.12)

1.4 RF EXCITATION FOR THE RESONANCE PHENOMENON GENERATION

It is the precessing bulk magnetization that we detect in an MR experiment. Inorder to detect it, we must somehow perturb the system from its equilibrium state,and get M to precess about B0. This is done by applying a second magnetic eldB1, perpendicular to B0, rotating about B0 at 0 in synchronism with the precessingnuclear magnetic moments.

The B1 eld causes M to tilt away from B0 and to execute a spiral path, as

10/2 is normally between 1MHz and 500 MHz, corresponding to radio waves.The eld is usually turned on for a few microseconds or milliseconds. Also, incontrast to the static magnetic eld B0, the B1 eld is much weaker (i.e., B1 =50 mT while B0 = 1.5 T).

A typical B1 eld takes the following form:

(1.13)

where B1(t) is the envelope function, 0 is the excitation carrier frequency, and is the initial phase angle.

Equation 1.13 describes a circularly polarized RF pulse, perpendicular to thez axis, and hence to the B0 eld. The initial phase angle , if it is a constant, hasno signicant effect on the excitation result so that we assume it is equal to zero.The excitation frequency 0 can be considered as a constant for almost all RFpulses, and it is determined by the resonance conditions. The envelope functionB1(t) is the heart of an RF pulse. It uniquely species the shape and duration ofan RF pulse and thus its excitation property. In fact, many RF pulses are named

B B B B1 1 1 1 0( ) ( ) ( ) ( ), , ( )t t i t t ex y i t= + = +



schematically described in Figure 1.4.The term RF pulse is a synonym of the B eld generation, so called because


based on the characteristics of this function. For example, the envelope function

(1.14)

where is the pulse time width. Another frequently used RF pulse is the followingsinc pulse (see Figure 1.5b):

(1.15)

When B1 is switched off, M continues to precess, describing a cone at someangle to B0 as shown in Figure 1.4. This is called the flip angle (FA); it dependson the strength of the B1 eld and how long it is applied.

We can adjust the value of the FA by changing the duration or the amplitudeof the envelope function B1(t). If we turn off the transmitter RF after M hasprecessed down into the transverse plane, this is called a /2 (or 90) pulse. A/2 pulse is commonly used as an excitation pulse, because it elicits the maxi-mum signal from a sample that is in equilibrium. An RF pulse twice as long, a

FIGURE 1.4 Trajectory of M after an RF pulse.

B0

M

Bl

z

x

y

B t B tB t

1 112 0

0( ) /= =

otherwise

B tB c f t t

11 2 0

0( ) sin [ ( / )]=

otherwise



of the widely used rectangular RF pulse (see Figure 1.5a) is given by:


(or 180) pulse, will place M along the negative z axis. This is often calledan inversion pulse. In general, for a constant B1 amplitude, an pulse causesthe magnetization vector M to precess through the ip angle:

= B1 (1.16)

FIGURE 1.5 Typical RF pulse shapes: (a) rectangular pulse, (b) sinc pulse.

Rectangular RF pulse

t(a)

Sinc RF pulse

t(b)




More generally, if we allow the amplitude of the applied RF to vary withtime, the FA is given by:

(1.17)

1.5 MR SIGNAL GENERATION AND ACQUISITION

Once displaced from the z axis by the RF pulse, the net magnetization M isno longer at equilibrium. We denote this nonequilibrium magnetization vectorby M, and the magnitudes of its components along the x, y, and z axes willbe denoted by Mx, My, and Mz, respectively. The magnitude of the componentof M in the transverse xy plane (i.e., the resultant of Mx and My) will bedenoted as Mxy. The equilibrium magnetization M0 represents the situation inwhich M is aligned along the z axis corresponding to the case of Mz = M0 andMxy = 0.

During the period following the pulse, M experiences a torque due to the B0eld and thus precesses about the B0 eld at the Larmor frequency. The effect ofthe precessing magnetization is similar to that of a rotating bar magnet and henceis equivalent to producing a periodically changing magnetic eld in the transverseplane. If the sample investigated is surrounded by a suitable oriented coil of wire,an alternating voltage will be induced in the coil according to Faradays law ofinduction. The coil used to receive the signal can be the same as or different fromthat used to produce the B1 eld so that precession of M corresponds to reorien-tation of Mxy in the xy plane at a rate equal to the Larmor frequency (i.e., Mxand My are oscillatory with time at a frequency equal to the Larmor frequency)while Mz is unchanged.

It can be shown that the amplitude of the alternating voltage induced in thereceiver coil is proportional to the transverse magnetization component Mxy.Hence, a maximum amplitude voltage signal is obtained following a /2 pulsebecause such a pulse creates a maximum Mxy component equal to M0. In general,for RF pulse of ip angle , the amplitude of the alternating voltage is propor-tional to M0 sin().

As a result of relaxation (such phenomena will be described later), Mxy (i.e.,the amplitudes of oscillation of Mx and My) decays to zero exponentially; thus,the voltage signal observed in practice corresponds to an oscillating signal at theLarmor frequency, with exponentially decaying signal amplitude. This type ofdecaying signal, obtained in the absence of B1, is called a free induction signalor free induction decay (FID).

1.5.1 FREE INDUCTION DECAY AND THE FOURIER TRANSFORM

The FID induced in the receiver coil is extremely weak and has a frequency inthe RF range: very-high-frequency signal information cannot easily be stored in

= B t dt10 ( )




a computer, and direct amplication without introducing distortion is a difculttask. Therefore, low-level amplitude of the MR signal at the Larmor frequencyis converted to a low-frequency signal by subtracting a frequency componentequal to the frequency of a chosen reference signal (RF detection). The latterusually is chosen to be equal to that of B1 (i.e., 0); the resulting low-frequencysignal has a frequency of in the Hz to KHz range. This signal is ampliedagain to the required level. The low-frequency signal (we will call it base-bandFID) contains all the information, previously present in the RF signal, which isrequired to generate the MR spectrum.

In order to store the base-band FID in a computer, the analog signal is sampledat specic times (analog-to-digital conversion) and an array of numbers repre-senting the sampled voltages is stored in the computer memory. The process ofanalog signal sampling must be performed in accordance with the Nyquist sam-pling theorem to ensure that the analog signal is correctly represented in digitalform; the time duration in which the FID is sampled is referred to as the acqui-sition or readout time.

When a single group of equivalent nuclei (i.e., 1H in a water sample) isconsidered, the FID represents a simple decay oscillation at a particular fre-quency, as described before. This frequency can be determined simply by mea-suring the period of the oscillation, tosc, and calculating the value of 1/tosc.However, if the sample being examined contains different chemical shift fre-quencies, the observed FID represents the composite of several individual FIDsignals, with slightly different frequencies. The individual frequency componentsof any FID are most conveniently identied by subjecting the FID to Fouriertransformation. The Fourier relationship between the FID and the MR spectrumis shown in Figure 1.6. All the characteristics of the FID are represented in thespectrum, but in a different format: the frequency of oscillation of the FID isindicated by the horizontal scale of the spectrum, the rate of decay of the FIDis inversely related to the width of the spectral line at its half maximum, andthe height (maximum amplitude) of the spectral line is directly proportional to

FIGURE 1.6 Fourier transform relationship between FID in the time domain and fre-quency domain.

exp (1/T2)

Fouriertransform

Time Frequency

1/T2




the initial amplitude of the FID or, more exactly, it is equal to the area underthe FID envelope.

1.6 MR SIGNAL CHARACTERISTICS

1.6.1 RELAXATION

Following the perturbation by the RF pulse, the spin system returns to theequilibrium population distribution between the energy levels by releasing excessenergy into the surroundings. In the classical vector model, this corresponds tothe return of the magnetization M to the equilibrium position along the z axis.Thus, during the relaxation period, any transverse magnetization component Mxycreated by the RF pulse decays to zero, and, at the same time, the longitudinalmagnetization component Mz returns to the equilibrium value of M0. The decayof Mxy and the recovery of Mz are two distinct processes, and they are referredto as spin-spin and spin-lattice relaxation, respectively.

1.6.1.1 The Decay of Transverse Magnetization: T2

The decay of transverse magnetization following an -degree pulse, due to spin-spin relaxation can be described with the equation:

(1.18)

where the parameter T 2 is called the transverse relaxation time or spin-spinrelaxation time; it represents the time interval required for the transverse mag-netization to decay to 36.7% of its initial value M0.

1.6.1.2 The Recovery of Longitudinal Magnetization: T1

After the application of an -degree RF pulse, the longitudinal magnetization Mzrecovers back to equilibrium at a rate that is linearly proportional to the differencebetween its current value and the equilibrium value; this rate is also characteristicof the sample. It can be derived that:

(1.19)

where the parameter T1 is called the longitudinal relaxation time or spin-lattice relaxation time, and it represents the time interval needed for thelongitudinal magnetization to recover to a value of 63.2% of the equilibriumvalue M0.

M t M e exy i t t T( ) sin ( ) /= + 0 20

M t M ez t T( ) ( ( cos ) )/= 0 11 1




This process of recovery of longitudinal magnetization is very important whencalculating the contrast between tissues with different T1 values and determiningthe imaging method that obtains the greatest signal-to-noise ratio (SNR).

The relaxation times T1 and T 2 are determined by the molecular environmentand thus are dependent on the sample. If the sample contains chemically shiftedresonances, the nuclei in different chemical environments exhibit T1 and T 2values characteristic of the particular environment. Knowledge of T1 and T2values is obtained experimentally by perturbing the equilibrium magnetization inappropriated multiple RF pulses. For pure liquids T1 = T2 and for biologicalsamples T2 < T1. Molecules in a mobile liquid environment have T1 and T 2values in the range of tens of hundreds of milliseconds. For tissues in the body,the relaxation times are in the ranges 250 msec < T1 < 2500 msec and 25 msec< T 2 < 250 msec and, usually, 5T 2 < = T1 < = 10T 2.

1.6.1.3 Pseudo-Relaxation: T2*

In practice, variations in the value of the magnetic eld throughout the samplealso cause dephasing of the magnetic moments, mimicking the decay of transversemagnetization caused by relaxation. It is convenient to dene another relaxationtime, T2*, describing the observed rate of decay of the FID:

(1.20)

where T2 describes the decay in signal due to the magnetic eld inhomogeneity:

(1.21)

where B0 is the extent of variation of the applied magnetic eld strength overthe region occupied by the sample. This causes the nuclei in different regions ofthe sample to experience slightly different magnetic elds and hence to precessat different Larmor frequencies. The FID obtained in such a eld is seen to decayfaster than that determined by T 2 (T 2* < T 2).

1.6.2 PROTON DENSITY

Most of the hydrogen atoms in the tissues are within water molecules; it is theseprotons that we detect in an MR experiment. The term proton density (PD) simplyrefers to the number of protons per unit volume and is effectively proportionalto the density of water in the tissue. Thus, for example, bone has very low protondensity, and liver has high proton density, while blood has a very high protondensity.

It is easy to see that we can identify the proton density with M0, the equilib-rium magnetization.

12

12

12T T T*

= +

12 0T

= B




1.6.3 THE BLOCH EQUATIONS

From the preceding description of the observed behavior of M(t) the followingequation can be postulated for the complete description of this motion:

(1.22)

where B(t) is composed of the static eld and the RF eld i.e., B(t) = B0 + B1(t)and R is the relaxation matrix:

(1.23)

and the vector M0 = [0, 0, M0].This is the set of equations used when constructing models of MRI. For most

applications, these equations are transformed into the rotating frame of reference.

1.6.3.1 Rotating Frame of Reference

Usually, MR experiments involve the application of a sequence of RF pulses.Viewed from xed coordinates (stationary frame), a description of the motion ofthe magnetization results can be complicated and difcult to visualize, especiallywhen two or more RF pulses are applied. Hence, we consider the motion of themagnetization from the point of view of an observer rotating about an axis parallelwith B0, in synchronism with the precessing nuclear magnetic moments: this isthe so-called rotating frame of reference.

A rotating frame is a coordinate system whose transverse plane is rotatingclockwise at an angular frequency . To distinguish it from the conventional sta-tionary frame, we use x, y, and z to denote the three orthogonal axes of this frame,and correspondingly, i, j, and k as their unit directional vectors. Mathematically,this frame is related to the stationary frame by the following transformations:

(1.24)

1.7 MULTIPLE RF PULSES

Up to now, we have described single-RF-pulse effects and subsequent detectionof FID (one-pulse experiments). Now we consider the effects of multiple pulseson the magnetization.

d tdt

t t R tM M B M M( ) ( ) ( ) { ( ) }= 0

RT

TT

=

1 2 0 00 1 2 00 0 1 1

//

/

=

= +

=

i i j

j i j

k

cos t sin tsin t cos t

( ) ( )( ) ( )

kk




Interrogation of the nuclear spin system with multiple pulses provides infor-mation that is not accessible via one-pulse experiment. In fact, multipulse experi-ments, such as multiple gradient-echo, inversion-recovery, and spin-echo sequences,enable experimental determination of proton density, T1, and T 2, respectively.

1.7.1 GRADIENT ECHO

Gradient-echo pulse (GE) includes an -degree RF pulse (usually a low ip angleis dened for fast imaging) followed by a time-varying gradient magnetic eldin order to generate (and then acquire) an echo signal instead of a decaying FID.

concept underlying GE formation is that a gradient eld can dephase and rephasea signal in a controlled fashion so that one or multiple echo signals can be created.After the application of an -degree RF pulse, a negative gradient (for example,along x axis) is switched on; as a result, spins in different x positions will acquiredifferent phases, which can be expressed as:

(1.25)

so that the loss of spin phase coherence becomes progressively worse as timeelapses after the excitation pulse. When the signal decays to zero, a positivegradient of the same strength is applied; the transverse magnetization componentswill gradually rephase, resulting in a regrowth of the signal. The spin phase angleis now given by:

(1.26)

The phase dispersal introduced by the negative gradient is gradually reduced overtime after positive gradient is switched on at t = t. After a time t, the spin phase is zero for any x value, which means that all the spins have rephased, andtherefore an echo signal is formed. Time t is called echo time, TE.

1.7.2 INVERSION RECOVERY

In the inversion-recovery (IR) pulse, the equilibrium magnetization M0 is ini-tially perturbed by a 180 pulse. Following a short time period TI (time ofinversion), a second perturbation is introduced in the form of a 90 pulse. Thiscan be written as a 180-TI-90-FID sequence. In order to examine the effectsof this pulse sequence, assume that the RF pulses are applied along the x axisof the rotating frame and are on-resonance. Prior to the 180 pulse the magne-

to rotate by 180 about the x axis, and at the end of the pulse it will be orientedalong the negative z axis (Figure 1.7b). Because, in this case, the 180 pulse

( , )x t xG d xG tx xt

= = 0

( , )x t xG t xG dx xt

t

= +



tization is at equilibrium (Figure 1.7a). The 180 pulse causes the magnetization

For physical concepts of magnetic eld gradients, see Section 1.8. The key


inverts the magnetization from positive z axis to negative z axis, it is sometimesreferred to as an inversion pulse. During the period TI, magnetization relaxesexponentially to the equilibrium position at a rate determined by T1; the mag-nitude of magnetization along the z axis (Mz) at a time T I after the 180 pulseis given by

(1.27)

Hence, the magnetization present at the end of the period TI is dependenton the ratio TI/T1. The magnetization following relatively short and long TIperiods is shown in Figure 1.7c and Figure 1.7e, respectively. During the periodTI, the magnetization, though not at equilibrium, is completely oriented alongthe z axis (i.e., Mx = My = 0); hence, an MR signal cannot be observed duringthis delay.

Application of a 90 pulse causes the magnetization present at that time torotate by 90 about the x axis. Thus, at the end of the 90 pulse, the magnetizationwill be directed along the negative y axis or the positive y axis, depending onthe value of TI chosen (Figure 1.7d and Figure 1.7f, respectively). In the subse-quent time period, the transverse magnetization generated by the 90 pulse allowsa FID to be observed. The initial amplitude of the FID will be proportional to

z

the z component of magnetization changes as a function of the parameter TI fordifferent T1 values.

1.7.3 SPIN ECHO

The spin-echo (SE) pulse sequence consists of an initial 90 pulse followed bya 180 pulse after a period TE/2 (time of echo), that is a 90-TE/2-180-echosequence. The rotating frame view of an on-resonance magnetization subjected

FIGURE 1.7 Rotating frame view of a magnetization subjected to an IR sequence.

x x

x xy y

y y

y y

x

z

z

z

x

z

z

zM0

(a) (b)

(c) (d)

(e) (f )

180-degreepulse

90-degreepulse

90-degreepulse

Short T

I

Long TI

M Mz TI T= 0 1 2e[ ]( / )1



M , remaining at the end of the delay period TI. In Figure 1.8 it is shown how


rium magnetization M0 by 90 about the x axis, creating transverse magnetizationoriented along the y axis (Figure 1.9a and Figure 1.9b). During the following

ation and the inhomogeneity of the B0 magnetic eld.As previously said, inhomogeneity of the eld causes the nuclei in different

regions of the sample to have different Larmor frequencies; this means that theisochromats are off-resonance from the B1 eld to different extents and hencewill precess in the xy plane at slightly different frequencies. Therefore, individualisochromats will be seen to dephase in the xy plane following the 90 pulse.The position of a few isochromats at a time TE/2 after the pulse is shown inFigure 1.9c; the isochromats are displayed spread out on both sides of the yaxis, because the Larmor frequency of some isochromats will be greater thanthe nominal value ( > 0) whereas that of others will be lower ( < 0).Consequently, isochromats with greater and less than zero will be seen toprecess in opposite directions in the rotating frame. At sufciently long TE/2values, isochromats will be completely dephased in the xy plane and Mxy willbe reduced to zero. In addition to spin-spin relaxation, inhomogeneity of theeld contributes to the gradual reduction of the net magnetization in the xyplane. However, the additional decay of the magnetization due to eld inhomo-geneity can be reversed, applying a 180 pulse. Application of a 180 pulse attime TE/2 following the initial 90 pulse causes all the isochromats to rotate by180 about the x axis; this brings the isochromats to their mirror-image position(Figure 1.9d). Following the 180 pulse, the frequency and direction of preces-sion of isochromats in the xy plane remain same as prior to the 180 pulse asthe Larmor frequency of each isochromat is unchanged. Hence, immediately

FIGURE 1.8 Plot of Mz vs. time in the IR pulse sequence.

1MO

1MO

0Mz

Short T1

t = TITime

Long T1

Medium T1



to this sequence is shown in Figure 1.9. The initial 90 pulse rotates the equilib-

period, = TE/2, this transverse magnetization decays because of spin-spin relax-


after the 180 pulse, the isochromats precessing faster are seen to lag behindthe slower ones. Precession of isochromats for a period TE/2 after the 180 pulseallows the faster isochromats to rephase with the slower ones, and at this instantall isochromats will be refocused along the negative y axis (Figure 1.9e). Furtherprecession of isochromats following refocusing causes them to dephase in thexy plane (see Figure 1.9f). Therefore: following the 180 pulse, the net magne-tization along the y axis (My) increases, until a maximum is reached at timeTE; after reaching the maximum, the magnetization decreases in a similarmanner to the decay following the initial 90 pulse.

The net magnetization My at this time is determined by the decay due tospin-spin relaxation only and is given by:

(1.28)

The maximum amplitude of the echo is proportional to My, given by Equation 1.28.

FIGURE 1.9 Rotating frame view of a magnetization subjected to an SE sequence.

x

y

z

x

y

z

x

y

z

x

y

z

M0

x

y

z

(a) (b)

(c) (d)

(e) (f )

TE/2

TE

x

y

z

90-degreepulse

180-degreepulse

M M ey TE T = 0 ( / )2




Equation 1.28 is applicable when the effect of molecular diffusion is negli-gible. For complete refocusing of isochromats, each nucleus must experience thesame eld during period TE. Movement of nuclei in an inhomogeneous eldbecause of diffusion causes the echo amplitude to be reduced.

1.8 MAGNETIC FIELD GRADIENTS

Magnetic eld gradients allow spatial information to be obtained from analysisof the MR signal. A eld gradient is an additional magnetic eld in the samedirection as B0, whose amplitude varies linearly with position along a chosenaxis. The application of a eld gradient Gx in the x direction, for example, causesthe magnetic eld strength to vary according to:

Bz(x) = B0 + xGx (1.29)

Magnetic eld gradients are produced by combining the magnetic elds fromtwo sources: the main homogeneous B0 eld, plus a smaller magnetic elddirected primarily along the z axis; such a secondary magnetic eld is producedby current-carrying coils (gradient coils). The design of a gradient coil is suchthat the strength of the magnetic eld produced by it varies linearly along a certaindirection. When such a eld is superimposed on the homogeneous eld B0, iteither reinforces or opposes B0 to a different degree, depending on the spatialcoordinate. This results in a eld that is centered on B0.

0the total magnetic eld that varies linearly along the x axis.

It is important to note that the total magnetic eld is always along the B0 axis(for convention, z axis), while the gradient can be along any axis, x, y, or z:

Gx = dBz /dxGy = dBz /dy (1.30)Gz = dBz/dz

The units of magnetic eld gradient are Tesla per meter (Tm1).To understand how magnetic eld gradients are utilized in MRI, consider

a large sample of water placed in such a eld. If the sample is subjected to aGx gradient, 1H nuclei at different x coordinates possess different Larmorfrequencies:

(x) = (B0 + xGx) (1.31)

Hence, in the presence of Gx, planes of constant eld strength also becomeplanes of constant resonance frequency. Equation 1.31 describes that a relation-ship between the position and the resonance frequency can be established byapplication of a magnetic eld gradient. It follows that the 1H spectrum obtained



In Figure 1.10, the gradient eld is added to the uniform B eld, obtaining


from the water sample subjected to a gradient magnetic eld will show a distri-bution of resonance frequencies.

The spectral amplitude at a particular frequency is proportional to the numberof nuclei in the given constant-frequency plane. It can be seen that the MRspectrum of an object placed in linear magnetic eld gradient corresponds to the

1.9 SPATIAL LOCALIZATION OF MR SIGNALS

There are three main methods of spatial discrimination, all of which use eld

are called slice selection, frequency encoding, and phase encoding.

1.9.1 SLICE SELECTION

Slice selection is the method by which the RF excitation, and therefore the signal,is limited to a chosen slice within the sample. It is achieved by applying theexcitation pulse simultaneously with a gradient perpendicular to the desired slice.So, if we want to image a slice in the xy plane, the selection gradient will beapplied orthogonal to this plane, i.e., in the z direction. The effect of the gradient

FIGURE 1.10 Magnetic eld gradient along x axis. Up: uniform magnetic eld B0; down:the gradient eld; right: the total magnetic eld.

y

x

y

x

y

x

+ =

B0

Bz (x) = B0 +x Gx

x Gx



projection of the object onto the gradient direction (Figure 1.11).

gradients, and that are combined in the imaging pulse sequence. The techniques


is to make the resonance frequency a function of position along the z direction:

(z) = Bz(z) = (B0 + zGz) (1.32)

As seen before, the RF pulse is not just an on - off pulse of radio waves, butit is shaped (i.e., amplitude modulated) so that it contains a narrow spread offrequencies close to the fundamental resonance frequency of the magnet 0.

Then, the width of the slice, z, is given by:

z = / Gz (1.33)

within the selective RF pulse.

FIGURE 1.11 Relationship between the spatial extent of the sample of water and the MRspectrum, in the presence of a linear gradient: the MR spectrum corresponds to the projec-tion of the object in the gradient direction.

y

0

0

x

Frequency

Object

MR spectrum

Gradient



as shown in Figure 1.12, where is the bandwidth of frequencies contained


is related to the shape and the duration of the pulse; if we consider aGaussian-shaped pulse, = 2/t, where is the full width at half maximum(FWHM) of the pulses frequency spectrum, and t is the relevant FWHM of thepulse envelope in seconds. From Equation 1.33, the slice can be made thinner bydecreasing the spectral bandwidth of the pulse (i.e., by making the pulse longerin time) or by increasing the strength of the slice selection gradient Gz.

The slice prole is determined by the spectral contents of the selective pulse,and it is approximately given by the Fourier transform of the RF pulse envelope.Thus, a Gaussian-shaped 90 pulse gives a roughly Gaussian slice prole.

The slice selection pulse sequence can be represented by a pulse timing

as a function of time. The selection gradient is followed by a negative gradientpulse in order to bring the spins back into phase across the slice.

After a signal has been activated by a selective or nonselective pulse, spatialinformation can be encoded into the signal during the free precession period. Wehave essentially two ways to encode spatial information: frequency encoding andphase encoding.

1.9.2 FREQUENCY ENCODING

Frequency encoding makes the oscillation frequency of an MR signal linearlydependent on its spatial origin. Let us consider an idealized one-dimensionalobject with spin distribution (x). If the magnetic eld that the object experiences

FIGURE 1.12 Selective excitation of a thin slice of the sample.

Frequencyspectrumof pulse

Slice 0

z Gz

z

t

Shaped 90-degree pulse

0



diagram, as shown in Figure 1.13, showing the RF pulse and selection gradient


after an excitation is the homogeneous B0 eld plus another linear gradient eld(xGx), the Larmor frequency at position x is

(x) = 0 + xGx (1.34)

Correspondingly, the FID signal ds(x,t), generated locally from spins in aninnitesimal interval dx at point x without considering the transverse relaxationeffect, is

(1.35)

where the constant of proportionality c is dependent on the ip angle, the maineld strength, and so on. The signal in Equation 1.35 is said to be frequencyencoded because its oscillation frequency is linearly related to the spatial location;consequently, Gx is called a frequency-encoding gradient. The signal receivedfrom the entire sample in the presence of this gradient is:

(1.36)In the general case, the received frequency-encoded FID signal after demod-

ulation (i.e., after removal of the carrier signal exp(i0t)), is given by

(1.37)

where Gfreq is the frequency-encoding gradient dened by Gfreq = (Gx, Gy, Gz).

FIGURE 1.13 Example of a pulse timing diagram relevant to the slice selection operation.

RF

90-degree

Time

Gz

ds(x,t) (x)dxe B xG0 x= +c i t( )

s t ds x t c x e dx ci B xGx( ) ( , ) ( ) ( )= = =

+

0 (( )x e dx ei xG t i tx

0

s t c e dri G tfreq( ) ( )=

r r




1.9.3 PHASE ENCODING

Considering the one-dimensional case after a RF pulse, if we turn on a gradientGy for a short interval ty and then we turn it off, the local signal under the inuenceof this gradient is:

(1.38)

where (y) is the spin distribution along y. From Equation 1.38, during the interval0 t ty the local signal is frequency encoded; as a result of this frequencyencoding, signals from different y positions accumulate different phase anglesafter a time interval ty. Therefore, the signal collected after ty will bear an initialphase angle

(1.39)

Because (y) is linearly related to the signal location y, the signal is said tobe phase encoded, the gradient Gy is called phase-encoding gradient, and ty isthe phase-encoding interval.

Phase encoding along an arbitrary direction can be also done for a multi-dimensional object by turning on Gx, Gy, and Gz simultaneously during thephase-encoding period Gphas = (Gx, Gy, Gz) for 0 t ty; the initial angle is (r) = rGphasty. Similar to frequency encoding, the received signal is the sumof all the local phase-encoded signals and is given by:

(1.40)

where the carrier signal exp(i0t) is removed after signal demodulation.

1.9.4 PHASE HISTORY OF MAGNETIZATION VECTORS DURING PHASE ENCODING

Let us consider the evolution of the phase angle of magnetization vectors inthe transverse plane as a function of a different phase-encoding gradient ampli-tude Gy = mGy by varying m; we call this a phase-encoding step. Referring to

value of m, that is, a different amplitude of the phase-encoding gradient. Wecan write the sequence of phase shifts added to a magnetization vector at thelocation y0 as m(y0) = y0mGyty. Therefore, the expression for a set of different

ds t,yy e 0 t ty e

i B yG ty

0 y

( ) ( )( )

)=

+

i yG t i B t yy ye t t 0

(y) yG ty y=

s t ds r t c r e dri rG tphas G( ) ( , ) ( )= =

e i t0



the scheme of Figure 1.14, each phase-encoding step corresponds to a different


phase-encoding gradient amplitudes can be written:

(1.41)

A Fourier transform applied to the sequence sm(t) for different m values, canbe used to compute the position of the object in the y direction. If the signal isfrom a collection of point objects in a column with different y offsets, the Fouriertransform of the resulting signal will yield a spectrum that is proportional to aprole of the column.

1.9.5 TIMING DIAGRAM OF AN IMAGING SEQUENCE

Image sequence timing diagrams, also called sequence diagrams, are commonlyused to describe the implementation of a particular MR sequence, and show themagnitude and duration of the three orthogonal magnetic eld gradients and theRF pulses. An example of pulse sequence diagram is shown in Figure 1.14. Inparticular it shows the two-dimensional Fourier transform image formation method,as described in Reference 14, that is a development of the earlier technique ofFourier zeugmatography [5]. It could be considered the basic imaging sequencefrom which all the hundreds of image sequences existing nowadays are derived.

FIGURE 1.14 Schematic representation of a sequence timing diagram.

RF

Gz

Gy

Gx

Intervals I II III

s t c y e dy em i mG t i ty y( ) = ( )

0




Generally, a sequence diagram can be split into three distinct sections, namely,slice selection, phase encoding, and readout, according to the previous descrip-

but, especially in more recently developed sequences, some of them overlap.The rst event to occur in the imaging sequence represented in Figure 1.14

is to turn on the slice selection gradient, together with the RF pulse. As previouslydescribed, the slice selective RF pulse should be a shaped pulse. Once the RFpulse is complete, the slice selection gradient is turned off, and a phase-encodinggradient is turned on. In order to obtain an MR echo signal, a negative readgradient is switched on. Once the phase-encoding gradient has been turned off,a positive frequency-encoding gradient is turned on and an echo signal is recorded.

This sequence of pulses is usually repeated m times, and each time thesequence is repeated, the magnitude of the phase-encoding gradient is changedaccording to Equation 1.41. The time between the repetitions of the sequenceis called the repetition time, TR.

1.10 ACQUIRING MR SIGNALS IN THE K-SPACE

According to Equation 1.36 and Equation 1.41 we can describe the signals result-ing from a two-dimensional Fourier transform sequence as a function of both thephase-encoding step and the time during the readout period. When M frequency-encoded FIDs are obtained, each one experiences a different value of the phase-encoding gradient amplitude; usually both positive and negative amplitudes areapplied:

(1.42)

In order to obtain digital MR signals, data acquired during frequency gradientactivation are sampled. So that, k-space data are sampled data, memorized in amatrix of N M points, if N is the number of samples along reading gradientand M the number of times the phase gradient is activated.

Then, if each FID is sampled and t is the sampling time interval, and weconsider the demodulated signal, from Equation 1.42 we obtain:

(1.43)

The formula in Equation 1.43 is usually referred to as the imaging equation.So that, if M = 256 and the FID sampling points N are 256, then a 256 256data matrix of complex numbers is the result.

s t cm ( ) ( , ) ( )= +

x y e dxdyi xG t ymG tx y G e i t0

s n,m c x,y e dxdyi xG n t ymG tx y G( ) ( ) ( )= +

+

02 1 2

n NM/ m M/



tions. In Figure 1.14 such sections are separated in time (three time intervals)


Considering the following substitution:

(1.44)

the formula in Equation 1.43 can be rewritten as:

(1.45)

This shows that the data matrix s(kx, ky) is a sampling of the Fourier coefcientsof the function (x, y). Therefore, by applying a two-dimensional inverse Fouriertransform to the data s(n, m), the result will be an estimate of the function (x, y).

Several parameters of interest in the k-space can be dened in terms ofparameters described in the pulse sequence. The sample spacing and width ofthe k-space are:

(1.46)

FIGURE 1.15 Sampling parameters of k-space.

k n tGk mt G

x x

y G y

= =

s k ,k c x,y e dxdyx y

i(xk yk )x y( ) ( )= +

k G t

k G y

W N k G T

W M

x x

y y

kx x x x

ky

=

=

= =

=

2

2

2

k G ty y G=2

2,max

Wy

Wx Wkx

Wky

kx

kx

ky

ky

x

y




kx and ky are the frequency interval step sizes in the x and y directions,respectively,

Wkx and Wky are the maximum frequencies that contain the object infor-mation

x

durationy is the phase-encoding gradient step sizeGy,max is the phase gradient maximum amplitude

Creating an MR image requires sampling the two-dimensional k-space withsufcient density (kx, ky) over a specied extent (Wkx, Wky). The Nyquistsampling theorem dictates the sampling spacing necessary to prevent spatial alias-ing of the reconstructed object (parts of the object can alias to different locations).The unaliased region is known as the field of view (FOV). The extent of theacquisition in Fourier space dictates the high-spatial frequency content and, hence,the spatial resolution.

For data acquired on a two-dimensional rectilinear grid in k-space and recon-structed with a two-dimensional Fast Fourier transform (FFT), the spatial reso-lution and FOV relationships are:

(1.47)

Up to now, we have described the MR signal equations for MRI for a particularpulse sequence (two-dimensional Fourier) and a homogeneous sample. In general,the value of the MR parameters, (T1, T 2, and T 2*) vary with position; thisgenerates the contrast between tissues. So, a more general form of Equation 1.45 is:

(1.48)

where (r, p) is a function of position r, and p is a parameters vector p = ((r),T1(r), T 2(r), , TR, TE, , ) that describes the dependence on the tissueparameters (, T1, T 2, etc.) and the scanner parameters (i.e., TR, TE, etc.). Forexample, in the two-dimensional Fourier sequence, if there are no effects frommagnetic eld inhomogeneities (i.e., T 2 = T 2*):

(1.49)k(t), described in Equation 1.48, is the spatial frequency vector as a function oftime; the Fourier coefcients of the image s(k(t)) can be sampled along the pathwaydened by k(t).

FOV FOVx x y ykx ky

k kx W y W

= =

= =

1 11 1

/ // /

s t e d

t

i t( ( )) ( , )

( ) (

( )k r p r

k G

k r=

=

sample

2

))dt

0

( , ) ( ) ( )/ ( ) / ( )r p r r= e eTE T TR T2 11r



where (see also Figure 1.15):

t is the readout sampling time interval and T is the readout gradient time


1.10.1 K-SPACE TRAJECTORIES

By applying Equation 1.42 to a eld gradient sequence, a path in k-space can be

paths in the k-space generated by the three time intervals as shown in Figure 1.14.Figure 1.16 shows the path(s) in k-space corresponding to intervals II and III.

Each different phase encode takes us to a new starting point on the left duringinterval II; during interval III, only the readout gradient is applied, ensuring that ktravels horizontally from left to right at constant speed. Such trajectory is due tothe x-gradient (readout gradient), shape, which has a negative compensation lobebefore the sampling period that allows the k-space to be sampled symmetricallywith the readout gradient; i.e., during the readout period, k goes from kmax to kmax.

The important parts of the paths are where the MR signal is being sampledfor processing into an image. In a straightforward FT imaging procedure, the wholetrack in k-space should be a rectilinear scan, preferably with a square aspect ratio,because this implies equal spatial resolution along both axes.

1.11 IMAGING METHODS

There are numerous variations on the basic MRI sequences described earlier. Otherthan the RF pulse shapes and repetitions (gradient echo, spin echo, inversion recovery,etc.), there are many aspects that distinguish them from one another; for example, dataacquisition and image reconstruction velocity. There exist a number of rapid imagingtechniques that could be grouped under echo-planar imaging (EPI), spiral imaging,and, more recently, parallel imaging. Another important aspect that characterizes

FIGURE 1.16 k-space path for 2-D Fourier transform.

III

II

kx

First line

Last line

ky



traced out. Consider again the sequence diagram of Figure 1.14; let us trace the


groups of imaging sequences is the k-space sampling method: it can be homogeneousor not, it can include the partial k-space lling instead of full k-space lling, etc.

In the present section, only a quick description is given: the most recent andefcient methods and their applications will be described in more detail in the rest ofthe book.

One feature that distinguishes different image acquisition methods is howquickly one can acquire data for an image and how easily the methods can beextended to generate higher spatial resolution. The two-dimensional Fourier trans-form method described in the preceding text is excellent for generating high-quality, high-spatial resolution images, but it is rather slow because only one linein k-space is acquired for each TR. The EPI method is well known as a very fastmethod that generally has limited spatial resolution. The EPI pulse sequence isuniquely characterized by a zigzag-like pathway that goes back and forth rapidlyin the frequency direction (kx) and moves in small steps in the phase direction(ky), as schematically shown in Figure 1.17.

As in the two-dimensional Fourier transform method, we apply the Fouriertransform in both frequency and phase directions obtaining the two-dimensionalspatial map of MRI data, which is an MR image. As it can be seen, a single RF

FIGURE 1.17 (a) Timing diagram of EPI sequence and (b) relevant k-space path.

RF

Gz

Gy

Gx

(a)

(b)

ky

kx

Last line

First line




pulse is all that is needed to produce sufcient data for an image. This is calleda single-shot imaging method, which is very popular in functional MRI (fMRI).There are limits on how fast the gradient elds can be changed and, thus, thereare limits on the spatial resolution that can be achieved with this method.

Another common rapid imaging method is spiral imaging. It exploits the math-ematics of how k-space trajectories are related to gradient waveforms (Equation1.48) and that allows nearly any continuous trajectory to be acquired. In particular,in spiral imaging, gradients are designed to produce a spiral trajectory in k-spaceand samples are placed along that pathway (as schematically shown in Figure 1.18).

FIGURE 1.18 Timing diagram of (a) spiral sequence and (b) relevant k-space path.

RF

Gy

Gx

(a)

ky

kx

(b)




Then, taking the two-dimensional Fourier transform of data in this plane willproduce an image. Similar to EPI, spiral imaging can produce is suitable for usein rapid imaging such as fMRI and cardiac imaging.

Parallel MRI techniques are characterized by multiple RF receiver coils andassociated RF receiver electronics. The proposed methods, such as SMASH [15],SENSE [16], and SPACE RIP [17] offer improved temporal and/or spatial reso-lution, so that they are good candidates for clinical imaging applications thatrequire both high speed and resolution, such as cardiac imaging. Each parallelMRI method uses a unique reconstruction scheme that exploits the independenceof the spatial sensitivity proles of the RF coils. Most recently, they have beensuccessfully combined with other fast acquisition methods, offering furtherimprovements; for example, the UNFOLD method for increasing temporal res-olution was recently applied to parallel coil acquisition [18], and non-CartesianSENSE [19] renders the use of the SENSE reconstruction technique compatibl

advanced image processing in magnetic resonance imaging.pdf

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