Novel pulse methods for multidimensional NMR imaging and spectroscopy


Material Information

Novel pulse methods for multidimensional NMR imaging and spectroscopy
Physical Description:
xi, 189 leaves : ill. ; 28 cm.
Cockman, Michael D., 1961-
Publication Date:


Subjects / Keywords:
Nuclear magnetic resonance   ( lcsh )
Magnetic resonance imaging   ( lcsh )
Nuclear magnetic resonance spectroscopy   ( lcsh )
Magnetic Resonance Spectroscopy   ( mesh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1988.
Includes bibliographical references.
Statement of Responsibility:
by Michael D. Cockman.
General Note:
General Note:

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001126894
oclc - 20082509
notis - AFM4057
sobekcm - AA00004796_00001
System ID:

Full Text







To my wife and best friend, Lisa


A; "*i


A number of people have contributed to my scientific growth during the past five

years. I would like to thank my chairman, Dr. Wallace S. Brey, Jr., for allowing me to

obtain my degree through the Chemistry Department while working with the Department

of Radiology at Shands Hospital. I also thank him for encouraging me to speak about

topics in NMR at informal and formal meetings. Those experiences were invaluable. I

also thank my co-advisor, Dr. Tom Mareci, who guided me virtually step-by-step through

my years as a member of his research group, letting me stand on my own when he thought

I would learn more that way. For providing generous financial support during my time at

the University of Florida, I thank Dr. Kate Scott. Finally, I thank these three and Drs. W.

Weltner and D. E. Richardson for the time which they have devoted to this dissertation.

Other members of the research group have helped me over the years. 1 especially

value one-on-one discussions with Dr. Richard Briggs and Dr. Ralph Brooker. Richard

often helped me define my personal goals, and Ralph, who rarely accepted a premise until

proven to his satisfaction, clarified much of the science of NMR for me. 1 also thank Pro-

fessor E. Raymond Andrew and Dr. Jeff Fitzsimmons for their lectures and advice. Visit-

ing professors have shown me applications of NMR other than those on which I focused

my research effort. These included Dr. Reszo Gaspar, Dr. Eugene Sczescniak, and Dr.

Attilio Rigamonti, who helped to inspire Chapter 2. The postdocs who have passed

through the group have combined science with fun and taught me a little about both dur-

ing their time at the University of Florida. For this I thank Gareth Barker, Sune

D0nstrup, and Dikoma Shungu. I also acknowledge the students, Bill Brey, Randy Duens-

ing, Willie Kuan, Lori Lewis, Jintong Mao, Laura Pavesi, Dan Plant, and Bill Sattin, all of

whom have affected my way of thinking about things scientific and otherwise.

Without technical support, a grad student's life becomes quite a bit more difficult.

For their contributions at the University of Florida, I thank Barbara Beck, Don Sanford.

and Ray Thomas. I also thank Dave Dalrymple, Chris Sotak, and Subramaniam Sukumar

of Nicolet Instruments and General Electric NMR for their enormous help in understand-

ing and writing the software which drove our NMR instrumentation. Much of this

dissertation could not have been done without them. I also thank Katherine Nash and

Teresa Lyles, the wonderful secretaries of the Magnetic Resonance Imaging Department at

Shands, and Mike Ingeno, Jack Dionis, Jim Kassebaum, Tim Vinson, and Dr. Ray Ber-

geron of the Health Center at the University of Florida.

Finally, I thank my parents for their steadfast support during those periods of doubt

which every graduate student feels from time to time, and my wife's parents, who have

taken such good care of me while I have lived in Florida. I also thank my wife, Lisa.

whose presence has made my life much happier during the production of this dissertation.



ACKNOWLEDGMENTS ....................................................................................................iii

LIST O F TA BLES ............................................................................................................... vii

LIST O F FIG U RES ........................................................................................................... viii

A BSTR A CT ........................................................................................................................... x


1 NMR THEORY AND MULTIDIMENSIONAL NMR .............................................. I

1.0 Introduction.................................................................................................. 1
1.1 Formalism for the Description of NMR Experiments ................................2
1.1.1 The Density Operator .........................................................................2
1.1.2 Equation of Motion of the Density Operator..................................3
1.1.3 The Density Operator at Thermal Equilibrium ..............................5
1.1.4 Basis Operators .............................................................................. 8
1.1.5 Hamiltonian Operators Describing
Spin System Perturbations .................................... ........... 1 The Effect of a Static Magnetic Field.................................13 The Effect of a Linear Field Gradient ...............................17 Application of Radiofrequency (RF) Pulses....................... 18 Application of Phase-Shifted RF Pulses.............................19
1.1.6 Observable Magnetization...................... ........................................21
1.2 The Two-Dimensional NMR Experiment................................... ........... 21
1.2.1 Phase and Amplitude Modulation..............................................22
1.2.2 General Description of the 2D Experiment ....................................24
1.3 Multidimensional NMR...........................................................................27

OF DYNAMIC POLYMER-SOLVENT SYSTEMS............................. ..30

2.0 Introduction................................................................................................30
2.1 Analysis of a Spectral Imaging Method..................................................32
2.2 Experim mental .............................................................................................. 38
2.3 R esults......................................................................................................... 43
2.4 The Susceptibility Model.........................................................................65
2.5 Conclusion ..................................................................................................69

3 CONVOLUTION SPECTRAL IMAGING .............................................................75

3.0 Introduction................................................................................................ 75
3.1 The Convolution Spectral Imaging Method ........................................ ...75

V |r
*' "" "''^ ,

3.2 Experimental .............................................................................................. 86
3.3 Practical Aspects of the Method .................................................................87
3.3.1 The Effect of Sample Geometry ................................................92
3.3.2 The Interaction of TE and and the Spatial Resolution................. 04
3.3.3 Extension to Three Spatial Dimensions .........................................105
3.3.4 Signal-to-Noise Considerations...... ............................................126
3.4 Convolution Spectral Imaging at High Field................................................128
3.5 Conclusion ................................................................................................140

WITH RED NOESY SPECTROSCOPY ................................................142

4.0 Introduction .................................................................................................... 142
4.1 The NOESY Pulse Sequence......................................................................45
4.2 The RED NOESY Pulse Sequence ...........................................................155
4.3 Problems Unique to the RED NOESY Sequence.......................................155
4.4 Experim ental .......................................................................................... ...1. 59
4.5 R esults....................................................................................................... 164
4.6 Discussion .................................................................................................164

R EFER EN C ES......................................................................................................................... 184

BIOGRAPHICAL SKETCH ............................................................................................. 189

vi 1



TABLE page

1-1 Effects of the Single-Element Operators...........................................................10

1-2 Solutions to the Equation of Motion of the
Density Operator Expressed as Cartesian Space Rotations ..................14

1-3 Effect of a Product Operator Hamiltonian on Terms of
a Cartesian Product Operator Basis Set............................................15

1-4 Transformations of Cartesian Operators by Phase-Shifted
R F Pulses ..............................................................................................20

2-1 2D Spectral Imaging: Relative Pulse and Receiver Phases ...............................40

2-2 Calculated Volume Susceptibilities ................................................... ...........67

2-3 Calculated Susceptibility Shifts at 2 Tesla....................................................68

3-1 Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases.....................................................88

3-2 Three-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases.....................................................89

4-1 Relative Pulse and Receiver Phases for RED NOESY.....................................160

4-2 Temperature Dependence of Relaxation Rates, Exchange Rates,
and Free Energies of Activation for DMF, DMA, and DMP.............169

4-3 Activation Parameters of DMF Found Using RED NOESY Data ..................70

4-4 Activation Parameters of DMF: Literature Values.........................................171

4-5 Activation Parameters of DMA: Literature Values ........................................172

4-6 Activation Parameters of DMP: Literature Values .........................................173

4-7 Relaxation Rates of DMF Methyl Protons........................................................ 182

.' *'" 'J



2-1 The pulse sequence corresponding to the evolution period of
a spectral imaging method .....................................................................33

2-2 A pulse sequence for two-dimensional spectral imaging ..................................36

2-3 Sample orientation for studies of PMMA solvation...........................................42

2-4 Spectral images used to observe the quality of the
static field hom ogeneity ................................... .....................................45

2-5 The 'H spectrum of a piece of PMMA partially
dissolved in chloroform at 2 T..............................................................48

2-6 Spectral images of PMMA in deuterated chloroform
after 61 minutes of solvation ................................................................49

2-7 Spectral images of PMMA in deuterated chloroform
after 81 minutes of solvation................................................................52

2-8 Spectral images of PMMA in deuterated chloroform
at late stages of solvation............................... ..........................................54

2-9 Spectral images of PMMA in perdeuterated acetone
after 180 minutes of solvation ..............................................................58

2-10 Spectral images of PMMA in perdeuterated acetone
after 200 minutes of solvation ..............................................................61

2-11 Spectral images of PMMA in perdeuterated acetone
at late stages of solvation............................... ..........................................63

2-12 PM M A solvation ............................................................................................... 71

3-1 The convolution of spectral and spatial information........................................80

3-2 Pulse sequences for convolution spectral imaging .............................................83

3-3 The effect of sample geometry...........................................................................93

3-4 The interaction of TE and the spatial resolution .............................................106

3-5 Three-dimensional convolution spectral imaging........................................... 112

3-6 Convolution spectral imaging at high field ................................................130

viii .

4-1 The NOESY pulse sequence ............................................................................ 146

4-2 Peak intensity behavior as a function of
the mixing time and exchange rate ................................................152

4-3 The RED NOESY pulse sequence .....................................................................156

4-4 A plot of In(k') versus 1000/T for DMF ....................................................165

4-5 A plot of In(k') versus 1000/T (corrected) for DMF.......................................166

4-6 A plot of In(k'/T) versus 1000/T for DMF ................................................167

4-7 A plot of ln(k'/T) versus 1000/T (corrected) for DMF...................................168

4-8 Four NOESY spectra of DMF obtained using the RED NOESY sequence .....174

4-9 Behavior of peak intensities of NOESY spectra of
D M F at 347 K .........................................................................................178


Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy




December, 1988

Chairman: Dr. Wallace S. Brey, Jr.
Major Department: Chemistry

This dissertation examines new methods and applications of existing methods in

multidimensional nuclear magnetic resonance. Chapter 1 contains the outline of a simpli-

fied theory for the description of pulse NMR experiments. The theory is a mathematical

formalism based in quantum theory and describes the effects of the applications of a static

magnetic field, radiofrequency pulses, and linear field gradients to simple spin-1/2 sys-

tems. In the context of the formalism, the basic principles of multidimensional NMR

experiments are described. The material of Chapter 1 is the basis for all subsequent

chapters. Chapter 2 outlines the application of a two-dimensional method to the study of

slices of solid polymethylmethacrylate dissolving in liquid deuterated chloroform and per-

deuterated acetone. The method allowed the correlation of NMR spectra with their spatial

positions. Time-dependent frequency shifts of the spectral frequencies of the polymer

were observed in the vicinity of the polymer-solvent interfaces. The direction of the fre-

quency shifts were found to depend on the orientation of the slice relative to the static

magnetic field. The magnitudes of the shifts depended on the solvent. The effects could

be explained in part by a simple magnetic susceptibility model. Chapter 3 describes new

pulse sequences for two- and three-dimensional spectral imaging. The methods produce

convolutions of spatial and spectral information, which saves experimental time. It is

shown that the methods are best suited for small samples with slowly relaxing nuclei,

immersed in a strong static field. The methods appear to be ideal for NMR microimaging.

Finally, Chapter 4 describes a new multidimensional pulse sequence for the quantification

of exchange rates called RED NOESY. The sequence is applied to three N,N-dimethyl

amides to determine the exchange rates of the N,N-methyl groups. For one of these

molecules, N,N-dimethylformamide, the Arrhenius energy of activation and frequency

factor and the enthalpy and entropy of activation were found by obtaining RED NOESY

data at several temperatures. The values found were 20 kcal mol-1, 28, 19 kcal mol-1, and

-5 cal mol-1 K-', respectively.


1.0 Introduction

Since the original concept of two-dimensional NMR spectroscopy was proposed

(Jee71), the basic principles of 2D NMR have been used to develop a large variety of mul-

tipulse experiments which in turn have found almost limitless applications.

Two-dimensional spectroscopy has been used extensively to unravel networks of coupled

nuclear spins and to estimate distances between nuclei in molecules with molecular

weights up to several thousand daltons. The field has developed such that implementation

of certain 2D NMR techniques is becoming almost routine for organic and inorganic

chemists. Similarly, two-dimensional imaging techniques are now highly automated and

routinely used clinically. Clearly, multidimensional NMR is now a domain shared by

physicists, chemists, and physicians. Yet communication between the three groups has

been impeded in part by their different concerns and professional languages. However,

the multidimensional NMR experiments performed by these groups are united by theory.

One of the functions of this chapter is to describe a theory which can be used to analyze

many NMR experiments.

The chapter first outlines the origin of a formalism for the evaluation of pulsed

NMR experiments. The formalism is then used to describe the behavior of spin systems

when immersed in a static magnetic field and exposed to radiofrequency (RF) magnetic

field pulses and linear magnetic field gradients. In the context of the formalism, the con-

cept of modulation, crucial to the understanding of multidimensional NMR, is explored in

the subsequent section. Next, a general discussion of two-dimensional methods is

presented. The final section of the chapter is devoted to the extension of the ideas of 2D

NMR to three and four dimensions and the problems associated with these experiments.

The material of this chapter forms the basis for the understanding of the multidimensional

experiments described in Chapters 2, 3, and 4.

1.1 Formalism for the Description of NMR Experiments

This dissertation is primarily concerned with the development and application of

pulse sequences to very simple spin-1/2 systems. Therefore, it is usually sufficient to

analyze an experiment applied to a single spin or to two weakly coupled spins. In this

section, a simplified formalism is developed from basic quantum theory to describe the

behavior of spin systems immersed in a static magnetic field and exposed to pulse

sequences made up of time delays, radiofrequency pulses, and linear field gradients. The

classical description of an NMR experiment which arises from the Bloch equations will

not be used because this approach ignores many of the effects created by NMR pulse

sequences. The quantum theory of NMR has been described by Howarth, et al. (How86),

Levitt (Lev88), Mareci (Mar80), Slichter (Sli80), and Sorensen, et al. (Sor83). The discus-

sion of the following sections draws heavily from these references.

1.1.1 The Density Operator

The state of a spin system can be described by a wavefunction V' which in turn can

be represented as a linear combination of orthonormal functions un (the eigenstates)

weighted by coefficients c,,

S'E Cnun* [1-1]

For a system of N spin-1/2 nuclei, there are 2N eigenstates. In the Dirac notation, the

expectation value of an observable, Q, is described by

= < 1QI> [l-2a]

= [l-2b]
m n


= E CmCn [l-2c]
m n

The products of the coefficients, cmCn, may be arranged to form a matrix which is called

the density matrix. If 0 is time-dependent, then either the eigenstates or the coefficients

can carry the time dependence. In the "Schridinger representation", the eigenstates are

allowed to be time-independent and the time-dependence is carried by the coefficients.

Thus, the time dependence of an expectation value is carried by the density matrix. The

density operator, p, is defined by

= Cncm. [1-3]

With this definition, the expectation value for an observable can be written in terms of

the density operator. By Eqs. [1-2] and [1-3] and the orthonormality of the eigenstates un,

'E E [l-4a]
m n

= Tr [p Q], [l-4b]

where Tr is an abbreviation for the trace of the matrix. This important equation estab-

lishes the relationship of the density operator to the operator, Q, which describes an

observable phenomenon. If the trace of the equation is nonzero, the phenomenon is

observed. Mathematical forms of Q and p relevant to NMR are shown in section 1.1.6.

1.1.2 Equation of Motion of the Density Operator

An alternative way of describing the state of a spin system at a particular time is to

use the density operator itself, rather than the density matrix. Thus, perturbations of the

spin system which occur during an NMR pulse sequence appear as changes in the density

operator. Each of the perturbations which will be discussed in this chapter can be

described by a time-independent Hamiltonian operator, H, whose mathematical form

depends on the nature of the perturbation. The behavior of a spin system under the

effect of a perturbation is described by the equation of motion of the density operator.

This is an expression for the rate of change of the density operator with time and may be
derived from the time-dependent Schr6dinger equation using the method described by
Slichter (Sli80). The time-dependent Schridinger equation is

i(-^ =HV>, [1-51

where h = h/2r and h is Planck's constant. Substituting Eq. [1-1] into Eq. [1-5] produces

ixE -un E cnHun [1-6]
n n

since only the can's are time-dependent. Multiplying both sides of the equation from the
left by uk leads to

iE -tUkun E CnUkHun. [1-7]
in n

Integrating and using the orthonormality of the basis functions produces

i = n cEn. [1-8]

The equation of motion of the density operator may be derived by taking the deriva-
tive of Eq. [1-3] with respect to time and substituting Eq. [1-8] into the result:

cm Cn 1-9a]
t c cn- a + cm

= cJ[(-i/)E Cm] + [(-i/N)E Cn]cm [l-9b]
m n

(i/N)[E CnC E CnCm] [ 1-9c]
m n

= (i/l)[E E ] [1-9d]
m n

(i/)[ ]. [ -9e]

Thus the time derivative, or the equation of motion, of the density operator is


= (i/NI) [p,H] [1-10

When H is time-independent, a solution of the equation of motion is

p(t) e(-i/V)Ht p(0) e(i/i)n [1-11]

which may be verified by taking the derivative with respect to time. This equation is of

prime importance because it describes the effect of applying a Hamiltonian operator to

the density operator. In terms of the NMR experiment, the equation describes the effect

of some perturbation, represented by the Hamiltonian, to the state of the spin system,

represented by the density operator. Analyzing the effect of an NMR pulse sequence on a

spin system requires knowledge of the mathematical forms of several operators. One of

these is the density operator which describes a spin system at thermal equilibrium. This is

the normal starting point for a density operator analysis since it describes a spin system at

rest in a static magnetic field. The other required operators are the Hamiltonian operators

describing the various perturbations which can occur. The perturbations commonly

found in NMR pulse sequences include the applications of a static magnetic field,

radiofrequency pulses and linear field gradients to a spin system.

1.1.3 The Density Operator at Thermal Equilibrium

To establish the starting point for a density operator analysis, the form of the density

operator which describes the spin system at thermal equilibrium, p(0), is needed. At ther-

mal equilibrium, 8p/8t = 0. This implies that p(0) has a form determined by the Hamil-

tonian operator describing the interaction of the static magnetic field with the unper-

turbed spin system. For a single spin, this Hamiltonian is

) 1H = --Bols= wl, [1-12]

where the Larmor relationship,

w --Bo, [1-13]

has been used; I is the nuclear gyromagnetic ratio, B0 is the strength of the applied static

field, Is is the operator for the z component of angular momentum, and w is the preces-

sional frequency of the nuclear spin. For a nucleus of spin 1 in eigenstate un,

HI|um> = Em un> $Isumn> = Nmlumn> [1-14]

where m is one of 21 + 1 values in the range I, I-1, -I and Em is the energy of the

mh eigenstate.

The populations of the eigenstates are given by the diagonal terms of the density

matrix, cmcm. At thermal equilibrium these are described by the Boltzmann distribution

factors, pm:

Pm = cmcm = [1-15]

For a set of n states of energies, En, a Boltzmann factor expresses the probability that the

mh state is occupied:

e- Em/kT
Pm -En/kT [1-16]

where k is the Boltzmann constant and T is the temperature of the spin system. If

kT >> En, Em, then the exponential terms can be approximated and the Boltzmann distri-

bution factor becomes

Pm E I-(En/kT) [- 1-17]

Because there are 21 + 1 possible values of En, the sum in the denominator equals 21 + 1.

Collecting equations produces

= (21+1)-(1 -(Em/kT)). [1-18]

Finally, because HIum> = Emlum>, it follows that

p = (21+1)-(1-(H/kT)) = (21+1)- (21+1)-'() wl,/kT) [1-19)

This is the form of the density operator at thermal equilibrium. The constant term,

(21+1)-', cannot be made observable and so it may be dropped. Making the definition,

= (21+1)-(Ow/kT) [1-20]

the reduced density operator at thermal equilibrium may be written

a = PIS [1-21]

Because the density operator, p, and the reduced density operator, a, are related by con-

stant terms, the equation of motion of the reduced density operator may be written


a = (i/) [,H] [1-22]

A solution of this equation for a time-independent Hamiltonian is

o(t) = e(-i/V)Ht o(0) e(i/)Ht [1-23]

and the expectation value of an observable is

= Tr [a Q] [1-24]

Equation [1-21] is the usual starting point for a pulse sequence analysis. For a sys-

tem of N spins, the reduced density operator at thermal equilibrium is

o = 111. + 2,2a + ... + P.NNN [1-25]

where the different spins are labeled by the subscripts. The spins may be treated

independently, applying a perturbation Hamiltonian to each separately. When all the

spins have the same gyromagnetic ratio, all of the 6's are nearly equivalent and the spin

system is referred to as homonuclear. This dissertation deals solely with such systems and

so the P terms will not be written explicitly for the pulse sequence analyses which follow.

The derivation of the expression for a at thermal equilibrium was made using the

assumption that kT >> Em = IIom. To test the validity of the assumption, let T = 298 K,

w = 2r(300 x 106) rad sec-1 and m = 1/2. These are typical values for a proton processing

in a magnetic field of 7.1 tesla at room temperature. A simple calculation with appropri-


ate values of the Boltzmann and Planck constants shows that kT is approximately 40000

times the value of hwm. Thus the "high temperature" assumption is valid for this system.

The nuclei studied for this dissertation were 19F at 2 tesla (w 2r(80.5 x 106) rad sec"') at

room temperature, 1H at 2 tesla (w = 2w(85.5 x 106) rad sec"1) at room temperature, and IH

at 7.1 tesla (w = 2r(300 x 106) rad sec-1) at temperatures ranging from 293 to 363 K. For

all of these cases, the high temperature assumption holds and Eq. [1-21] is a valid starting

point for pulse sequence analysis.

1.1.4 Basis Operators

The density operator can be written as a linear combination of time-independent

basis operators, B,, weighted by time-dependent coefficients, b,(t):

o(t) = E b.(t)B [1-26]

This set of basis operators can also be used to describe the Hamiltonians which describe

possible perturbations to a spin system. The most popular set of basis operators has been

proposed by Sorensen, et al. (Sor83). These are the Cartesian "product operators" pro-

duced by the multiplication of the single-spin, Cartesian angular momentum operators, I,,

ly, and I,, and the unity operator, E. The basis set for N spin-1/2 nuclei consists of 4N

product operators. For a system of two spin-1/2 nuclei the basis consists of:

The unity operator:


One-spin operators:

1lx Ily 11 ', l 12x 2y 12
Two-spin operators:

211yxx 2I1xi2y 2I1x1i 2

21,yI2x 2Iiyl2y 2ly212 ,

211S2.x 21112y 211s12s *

The subscripts 1 and 2 are used to distinguish the two spins. For much of the pulse
sequence analysis of this dissertation, the density operator describing the state of a spin
system will be written in the Cartesian product operator basis. This basis is particularly
suitable for describing the effects of the applications of a static magnetic field, radiofre-
quency pulses, and linear field gradients.
The Cartesian operator basis is not well-suited for describing observable terms of the
density operator, and so it is necessary to convert to another basis set consisting of pro-
ducts of the single-element operators, Ia, IB, I*, and I'. These operators are directly
related to the energy levels of the spin system. It has been shown that a nucleus of
spin-1/2 immersed in a static magnetic field can be in one of two eigenstates, um, with
energies Em = (l/2))w. In this dissertation, the state of lower energy is called 0 and that
of higher energy, a. The application of a single-element operator to an eigenstate can
produce a change in the spin state. The effects of these operators are shown in Table I -1.
Table 1-1 shows that the only operators which cause a change in the spin state and lead to
observable signals are the I+ and I- operators. Thus, the single-element basis is most use-
ful for expressing the observable terms of the density operator (section 1.1.6).
The Cartesian operators are directly related to the single-element operators by the

2Ek .(la+ If), [l-27a]

Ikx +Ik+ ), [1-27b]

Iky -(I+ Ik) ,[1-27c]

Ik = (I- If), [I-27d]

where k indicates a particular spin. The Cartesian product operator basis may be rewrit-
ten as a single-element product operator basis by the use of these relationships.


Table 1-1

Effects of the Single-Element Operators

Initial Spin State

I+ I- Ia IV

Final Spin States

a 0 p a 0

P a 0 0 1

In summary, two basis sets of operators may be used to describe the density operator.
The effect of an NMR pulse sequence on a spin system is described most conveniently in
the Cartesian product operator basis. To determine which terms of the density operator
are observable, however, the single-element product operator basis is more convenient.
The pulse sequence analyses of this dissertation use whichever basis set is most convenient
for emphasizing a particular aspect of the sequence.

1.1.5 Hamiltonian Operators Describing Spin System Perturbations
The form of the density operator which describes a spin system at thermal equili-
brium has been shown, as have two different basis sets of operators which can be used to
describe the density operator in general. Still needed for an NMR pulse sequence analysis
are the Hamiltonian operators which describe the various perturbations which can be
applied to a spin system. These will be shown later in this section, but first an alternative
way of expressing the solution to the equation of motion of the reduced density operator
is described.
The effect of a perturbation on a spin system is described by the solution to the
equation of motion of the reduced density operator. The form previously shown in
Eq. [1-23] is not very convenient for visualizing the behavior of the spin system under the

effect of a perturbation. However, the product operator basis is made up of Cartesian
angular momentum operators and so it is possible to describe the effect of a perturbation
on the density operator in Cartesian terms by showing that the solution to the equation of
motion of the density operator is equivalent to a rotation in a three-dimensional Cartesian
space. The following analysis follows the method described by Slichter (Sli80). Recall
that for a time-independent Hamiltonian a solution to the equation of motion of the
reduced density operator is o(t) = e(-i/)Ht o(0) e(i/)H t. For this analysis, let the initial
state of the density operator be a(0) = Ix and the Hamiltonian be such that )-'Ht = OI,. As

will be seen in section, this describes a spin, not at thermal equilibrium, whose


state is changing under the effect of chemical shift precession. Define

f() = e-il, Ix eIx .

By the relationship between operators A and B,

A eiB = eB A when [A,B] = 0 ,

and the commutators of the angular momentum operators, the first derivative of f(O) is

f'() = e-i9ls IY ei91.,

and the second derivative of f(O) is

f"() = e-i' x'I e'i1I

Thus the relationship of Eq. [1-28] is a solution of the second order differential equation

f"(0) f(O) 0o.

Another solution to this equation is

f(O) = a cos(O) + b sin(4),


which may be verified by substitution. Thus


The coefficient, a, may be found by finding the solution of Eq. [1-33] when 0 equals 0
and using Eq. [1-28]. Finding the solution of the first derivative of Eq. [1-33] when 4
equals 0 and using Eq. [1-30] gives the coefficient, b. The results are that

a = I, [1-35a]

b= l [1-35b]

Finally, the following is obtained:

e-i41 Ix e i, = Ixcos(O) + lysin() .








e-i#IL Ix ei#L a cos(O) + b sin(O) .

Equation [1-36] shows that the quantum mechanical solution to the equation of

motion of the density operator has an analog in a Cartesian space described by axes

defined by the angular momentum operators. The angle of rotation, 0, originates from the

applied Hamiltonian. Similar expressions may be found for Hamiltonians containing Ix or

ly terms. These are outlined in Table 1-2. To find the effect of a particular Hamiltonian

on one of the Cartesian operators, the function 0 and the operator form of the Hamil-

tonian must be determined. A Hamiltonian consisting of a one-spin operator affects the

terms of a product of Cartesian operators separately. For example, let the initial state of

the density operator be the product operator llxl2s and the Hamiltonian be such that

t'1Ht I1s. Then the solution to the equation of motion of the reduced density operator


e -i# 11xs2 ei0 ffi ( 1xcos(o) + Ilysin(O))12,. [1-37]

Only the terms of spin 1 have been affected.

It is also possible for the Hamiltonian to contain products of Cartesian operators.

The only such Hamiltonian relevant to this dissertation is such that /'lHt = 021112, and so

alternative expressions for the reduced density operator solution e'i#211s2s o(t) ei'2 11l12

must be derived for the various product operators which make up o(t). Again this can be

done using the method described by Slichter under the assumption that operators belong-

ing to different spins commute (Sli80). The results are shown in Table 1-3. The Effect of a Static Magnetic Field

By using the expressions of the previous section, it is possible to examine the effects

of a static magnetic field on a weakly coupled spin system. This perturbation occurs after

a spin system is immersed in a static magnetic field and during an NMR pulse sequence

when all radiofrequency pulses and field gradients are removed. The Hamiltonian for the

interaction of the field with a system of N spins is


Table 1-2

Solutions to the Equation of Motion of the Reduced Density Operator
Expressed as Cartesian Space Rotations

o(t) e-i#lx o(t) e'i#x e-i'y o(t) eiol' e-iIs oa(t) ei1,

(1/2)E (1/2)E (1/2)E (1/2)E

Ix Ix Ixcos() Isin(o) Ixcos(o) + lysin(o)

ly lycos() + Isin(o) ly lycos(O) Ilsin(o)

Is I.cos(o) lysin(o) Icos() + Ixsin(o) Is

Table 1-3
Effect of a Product Operator Hamiltonian on
Terms of a Cartesian Product Operator Basis Set

a(t) e-i2 l21112 o(t) e#2 11'sl2

(1/2)E (1/2)E

Ix 1xcos(O) + 211yl2,sin(O)
Ily Ilycos(O) 2I1xI2ssin(o)

lI,, I,.

Ix I2xcos(4) + 211s12ysin(0)

12y I2ycos(O) 2I11I2xsin(o)
12 12s

211x12x 211x12x
211xI2y 211xl2y

211x,12 2I1xI2.cos(() + Ilysin(O)
211yl2x 211ylg
21yl2y 211yl2y

21yI2, 2Ily,,cos() 11xsin(0)

211,2x 211,I2xcos(M) + I2ySin(O)

211,,12 211,2ycos(4) l2xsin(0)
211,12. 211,12,

i=N j=(N 1) k=N
'H E ,wili + E E 2arJjkIjsIks (j 1i=l j=1 k=1

where wi is the angular precession frequency of the ih spin, Jjk is the coupling constant

between spins j and k, and the spins are labeled by the subscripts. The Hamiltonian may

be broken into two parts, which can be applied independently. The chemical shift pre-

cession term is

X'-H = E wil=. .[1-39]

The spin-spin coupling term is

j=(N- ) k=N
N-IH = E E 2rJjkIjIk. (j j=1 k=1

which is rewritten in terms of the Cartesian product operator basis by moving the factor

of two (Sor83):

j=(N 1) k=N
I-H E E rJjk(2IjIks) (j j=1 k=1

The evolution of the angular momentum operators under the chemical shift Hamil-

tonian is described by the fourth column of Table 1-2 where i = wit :

lx licos(wit) + Iiysin(wit) [1-41a]

Iy liycos(wit) lixsin(wit) [ 1-41 b]

Ii Ii,. [1-41c]

Equation [1-41] shows that in the Cartesian frame transverse components of angular

momentum rotate through an angle gi wit under the effects of chemical shift precession.

The longitudinal component is not affected.

From the expressions of Table 1-3 where jk -= rJjkt the evolutions of the one-spin

operators under the coupling Hamiltonian are given by

1jx Ijxcos(lJjkt) + 21jylksin(rJjkt) [1-42a]

ix Jx Jr

Ijy Ijycos(rJjkt) 2IjxIksin(lJjkt) [1-42b]
lj. -. l. [1-42c]

Two-spin operators also evolve under the Hamiltonian which describes spin-spin cou-
pling. Examples are:

21jIks -- 2Ijxlkcos(wJjkt) + Ijysin(iJjkt) [1-43a]
21jylks -. 2IjyIkcos(rJjkt) Ijxsin(rJjkt) [1-43b] The Effect of a Linear Field Gradient
The application of a linear field gradient has effects similar to those of chemical shift
precession but the expression for the angle Oi is different. The Hamiltonian has the form

)I-H -= r, Yl [1-44]

where r x, y, or z and 7, is a vector describing the spatial position of the ih spin along
the r axis. The field gradient is defined by the partial derivative and may be written

=8B [1-45]

for the component of the gradient along the r axis. By this Hamiltonian, i = -yGrrit ; thus
the angular momentum operators evolve as follows:

lix -. Iixcos('Grrt) + Isin(yGrrit) [I-46a]
Iy -. iycos(-7Grrit) lisin(-yGrrt) [1-46b]

lis lis. [1-46c]

The degree of rotation of the transverse components depends upon the strength of the
applied gradient, Gr, and the position of the nuclear spin along the r axis. Two-spin
operators also evolve under the effect of a linear field gradient. Examples are:

2IjxIks -, 2Ijxlk.cos(Grrjt) + 21jylkssin(-yGrjt), [1-47a]


21jylks 21jyIk.cos(yGrrjt) 21xlk.sin(-yGrrjt) [1-47b] Application of Radiofrequency (RF) Pulses
NMR experiments are performed using an alternating magnetic field applied in the

plane transverse to the static field to create a torque on the magnetic moments. The Ham-

iltonian for the total applied magnetic field (static plus alternating) is written

i-H -yBo,I -yBl[Icos(not) + lysin(Not)] [1-48]

where B1 is the strength of the applied alternating field, 0o is its angular frequency and t

is the duration of application. This Hamiltonian is time-dependent, but on transformation

to a frame rotating with angular frequency flo, the time-dependence of the alternating

field vanishes and the Hamiltonian becomes

-H = (fl- wi)I, + x, [1-49]

where the Larmor relationship and the definition Of -B1 have been used (Sli80). The

Hamiltonian can be simplified considerably under two conditions. At resonance, (o = wi

and the off-resonance I, term vanishes. Also, the term becomes negligible when the

applied alternating field is strong enough that 0f >> (flo-wi) (a "hard" pulse). For these

cases, the Hamiltonian becomes

IH -= nlx [1-50]

In terms of Table 1-2, =f fit. However, this product, called the "tip angle", is more com-

monly labeled 0. In this notation, the application of an RF pulse to each of the Cartesian

operators produces:

Ix Ix, [ I-51a]

ly lycos() + Isin(O) [1-51b]

I, l,cos(O) lysin(0) [I-51c]

.:1- ,

 Application of Phase-Shifted RF Pulses

If the alternating field is applied at an angle f with respect to the x axis in the rotat-

ing frame, the Hamiltonian is rotated in the transverse plane by this angle. The applied
field is said to be "phase-shifted" relative to the x axis. In one-dimensional NMR experi-

ments, phase-shifting of the RF pulses of an NMR pulse sequence is done commonly in

conjunction with signal averaging to isolate certain terms of the density operator. This

process is called phase-cycling and has been used for all of the experiments described in
this dissertation. Phase cycles can vary from one pulse sequence to another and so a

description of the effect of a particular cycle will accompany the description of each of

the NMR pulse sequences used.

The density operator under the effect of a phase-shifted RF pulse may be written

o(t) = e- isi e- inx eiIs oa(0) e- ifI, eienx e i1. [1-52]

By using the method outlined in section 1.1.5, the following relationships can be derived:

Ix Ix (cos2() + sin2()cos(0)) + ly cos(f)sin(X)(l-cos(0)) Is sin(e)sin(0) [1-53a]

ly lx cos(C)sin(()( -cos(0)) + ly (cos2(f)cos(0) + sin2(f)) + 1I cos(C)sin(0) [1-53b]

I, Ix sin(f)sin(0) ly cos(f)sin(e) + I, cos(e) [1-53c]

Most older NMR spectrometers are capable of executing RF phase shifts in 90 degree

increments only, that is f can equal 0, 90, 180, or 270 degrees. The work in this disserta-

tion has been done within this limitation. Table 1-4 summarizes the effects of RF pulses

with these shifts on the three angular momentum operators. The terms of this table are

valid only when the phase shifts are perfect increments of 90 degrees and when

off-resonance effects are negligible. These assumptions are made throughout this work.

Table 1-4

Transformations of Cartesian Operators by Phase-Shifted RF Pulses

C (degrees)

0 90 180 270

Initial Operator Final Operators

Ix Ix IxcO-IsO Ix IxcO+I,sO

ly IycO+ISs ly lycO-l,sO ly

1, lcO-lyse 1,cO+lxse lIcO+lyse IScO-lIse

Notation: c = cos and s = sin.

1.1.6 Observable Magnetization

The forms of the operators necessary to analyze an NMR pulse sequence have been

shown. However, the form of an observable term of the density operator has not yet been

derived. In section 1.1.3, it was shown that in terms of the reduced density operator the

expectation value of an observable equals Tr [a Q]. In most modern NMR experiments,

the signal is detected in quadrature. The operator corresponding to the detectable signal

may be written as the sum of the two orthogonal transverse Cartesian angular momentum


I' = (Ix + ly) [1-54]

Thus the expectation value for the detected signal is

= Tr [o(1x + ily)], [1-55]

or transforming to the single-element operator basis,

Tr [al] [1-56]

This implies that o(t) must consist of IY operators to give a nonzero trace and thus observ-

able magnetization. In terns of the two-spin Cartesian product operator basis, only the

terms Il, Ily, 12x, and 12y are directly observable. However, the terms 211x'2,, 211y2z,,

211s'2x, and 21,12,y can evolve under the effects of the chemical shift and coupling Hamil-

tonians to produce observable terms. The unity operator is never observable. Each of the

remaining seven operators of the basis can be made observable only in an indirect way by

subjecting it to an RF pulse Hamiltonian.

1.2 The Two-Dimensional NMR Experiment

The theory necessary for the analysis of most NMR experiments has been intro-

duced. This section makes use of the theory to describe the basic principles of

two-dimensional NMR methods. First, the required concepts of phase and amplitude

modulation are introduced using the density operator formalism. The results of the

application of the Fourier transform to phase- and amplitude-modulated signals are
shown. Next, a general description of the 2D experiment is presented, and a mathematical
expression is derived in terms of the density operator. Finally, a short discussion of some
of the restrictions of 2D NMR follows.

1.2.1 Phase and Amplitude Modulation
The concepts of phase and amplitude modulation are necessary for the understanding
of any NMR experiment and in particular are the keys to understanding two-dimensional
experiments. Both types of modulation can be created by a pulse sequence and appear in
a density operator expression. From the expressions for the evolution of the density
operator introduced in section 1.1, precession due to chemical shifts, spin-spin coupling,
and applied gradients corresponds to the multiplication of product operators by cosine
and sine terms. For example, applying an RF pulse of tip angle 0 i/2 and phase C = 0 to
a single spin initially at thermal equilibrium produces a transverse term which evolves
under chemical shift precession for a time, t, to produce

o(t) = Iy cos(wt) + Ix sin(wt) [1-57]

The amplitude of each of the transverse components oscillates during t according to the
angular frequency, w, of the spin. Thus the components are "amplitude-modulated". If
the components are combined using the single-element operator basis, then

o(t) = (I+e- i I- ei) [1-58]

The phases of the I+ and I' terms oscillate according to w and the terms are said to be
"phase-modulated". This example shows that signal phase modulation may be produced

by properly combining amplitude-modulated signals.
The type of modulation is important because the Fourier transforms of the two types
of signal are very different. By using the notation of Keeler and Neuhaus (Kee85), the
Fourier transform (FT) of a damped phase-modulated signal may be written as


FT [ei' e-/T2] = A' + iD- [1-59]

where A and D are absorption and dispersion Lorentzian functions, respectively. These
functions have the forms:

A-(w) (+( 2 ) [l-60a]
(l+((w lo)2T,2))

(W' O)Ta
D-(w) ( )T22 [l-60b]
(l+((w _o)T22))

where T2 is the damping constant, w is the angular frequency of the spin, and fl is the
angular frequency of the rotating frame. By these functions, the sign of w relative to flo is
unambiguous. For a damped amplitude modulation term,

FT [c(wt)e-/T2] FT [ (ei" + e-i"*)e-tT2] [1-61a]

S[A+ + iD++ A- + iD-], [i-61b]

where A' and D- have been defined and

A+(w) T2 [1 -62a]
(I+((w + O)2T22))
(w + lo)T,2
D+(w) ( + f)T2 [1-62b]
(I+((w + no)2T22))

These results show that amplitude modulation does not allow discrimination of the sign of
a signal. The Fourier transform is a mixture of the two types of absorption and dispersion
functions and so the sign of w relative to the fl0 is ambiguous. This has important conse-
quences for many 2D NMR methods where the density operator describing the detected
signal contains amplitude-modulated terms; these include 2D spectroscopy methods such
as COSY (Jee71) and NOESY (Jee79), and 2D imaging methods such as the "rotating
frame" method of Hoult (Hou79). Because the sign of a signal is desirable information,
methods have been developed to convert the amplitude-modulated signals into


phase-modulated ones. This idea will be used for the experiments of Chapter 4.

1.2.2 General Description of the 2D Experiment

The goal of any 2D NMR experiment is the correlation of two processes which occur

during the experiment. This requires that the signal be doubly-modulated as a function

of the two processes. Such a signal can be produced by a pulse sequence made up of at

least three periods, commonly called preparation, evolution, and detection (Aue76). In

accord with the accepted notation, the evolution and detection periods will be labeled tj

and t2, respectively. Some 2D pulse sequences contain an additional period, often called

mixing, which is sandwiched between the evolution and detection periods (Bax82). The

mixing period will be labeled rm. Each of these periods can be made up of a number of

spin perturbations, including time delays, RF pulses, and linear field gradients.

Each of the four periods of a 2D pulse sequence has a unique function. Occurring at

the outset of the pulse sequence, the preparation period perturbs the spin system from

thermal equilibrium. Assuming the use of the Cartesian product operators, the events of

the preparation period produce transverse terms in the density operator expression which

describes the effect of the pulse sequence. After the preparation period, the terms of the

density operator can change according to the effects of the perturbations of the evolution

period. The terms of the evolution period do not have to be observable. It is the function

of the perturbations of the optional mixing period to convert such terms, if desired, into

observable terms. These are detected when the spectrometer receiver is switched on. The

terms of the density operator then evolve under the effects of the perturbations of the

detection period.

For any two-dimensional NMR method, the density operator expression which

describes the detected signal of a 2D pulse sequence contains modulation terms which

reflect the effects of the perturbations which preceded detection. This can be shown by

using the density operator formalism to derive a general mathematical description of a 2D

NMR experiment. The density operator just before detection begins, o(tl), is a function

of the perturbations of the evolution period, t1. By the equation of motion of the density

operator, a(t1) evolves during the detection period, t2, under the effect of some Hamil-

tonian to produce

o(t1,t2) e(-i/9)Ht2o(t) e(i/g)Ht [1-63]

The detected signal is found by using the trace relation

= Tr[o(tl,t2)I [1-64]

This equation is simply the 2D case of the expression derived earlier for any a, Eq. [1-56].

Substituting for o(tl,t2),

<1+> Tr[e(-i/g)Ht2(t) e(i/)Ht21+] [1-65]

Because the trace is invariant to cyclic permutation of the operators, the expression for


Tr[i(tl)e(i/g)Ht2I+e(-i/$)Ht2] [1-66]

This relationship shows that the initial phases and amplitudes of the terms which evolve

during t2 are determined by the events preceding detection. Pulse sequences can be

designed to control what arrives at the spectrometer receiver, and the detected signal will

have a "memory" of the perturbations which occurred before the receiver was switched


Although o(t,t2) contains information about the events which preceded detection,

the application of a Fourier transform with respect to the detection period would produce

a frequency spectrum which describes the behavior of the spin system only during detec-

tion. This occurs because the modulation functions corresponding to the events preceding

detection are constants for each application of a pulse sequence. To map out the behavior

of the spin system under the effects of the perturbations which precede detection, the

pulse sequence must be repeated with a change in one or more of the perturbations of the

evolution period, while holding the perturbations of the detection period constant. Build-

ing up a matrix of such signals is equivalent to sampling the signal during a changing t1

interval. Application of a Fourier transform with respect to the change made during the

evolution period produces a frequency spectrum of the behavior of the spin system during

that period. This is the principle behind every multidimensional NMR experiment. Since

the terms of the density operator describing the detected signal of a 2D experiment can be

written as a function of the evolution and detection periods, t, and t2, the double Fourier

transform of the signal matrix is a function of two variables, F1 and F2, which have units

of frequency.

Because the Fourier transform applied with respect to the evolution period produces

a frequency spectrum corresponding to only the perturbation which has changed during

the interval, the effects of different spin system perturbations can be completely

separated during evolution. The key to these experiments and others is that although

many perturbations may act simultaneously during evolution, it may be possible to

arrange the pulse sequence such that only one perturbation is changed when the pulse

sequence is repeated while other perturbations remain constant. Thus, the effect of the

changing perturbation can be mapped out. For example, it is possible to produce a fre-

quency spectrum of only the spin-spin coupling information (Aue76). Another possibil-

ity, found in many imaging experiments, is the production of a frequency spectrum

related to the spatial positions of the nuclei (Ede80).

The effects of different perturbations acting during the detection period cannot be

separated, unlike those of the evolution period. Thus, the Fourier transform with respect

to the detection period is a convolution of the various frequency spectra corresponding to

the effects which occur during that period. It may be possible to adjust experimental

parameters such that the effect of a desired perturbation dominates the effects of

unwanted ones. The application of broadband decoupling is a good example of this;

enough decoupling power must be applied to remove the coupling information. Other

examples appear in some NMR imaging experiments where a field gradient term must
dominate the static field term. This can be ensured in many cases by using a strong gra-
dient; just how strong the gradient must be is discussed in Chapter 3. It is possible to take
advantage of convolutions for certain applications, also shown in Chapter 3.

1.3 Multidimensional NMR
The basic 2D NMR method may be extended to include more dimensions by
appending more evolution periods to a 2D pulse sequence. In principle, an N-dimensional
space could be described by a density operator of the form o(tl,t12, tl(N-l),t) where

N 1 processes occur during N 1 evolution periods and are correlated with the events of
the detection period. Data processing requires an N-dimensional (ND) Fourier transform.
At present, the largest value of N described in the literature is four (Hal85).
A number of limitations have impeded the implementation of ND NMR methods.
One of the most troublesome is the amount of spectrometer time required to acquire the
data matrix. Typically, several seconds are required to acquire a single time-domain sig-
nal and allow the spin system to return to equilibrium. If signal averaging is required,
several minutes to several hours may be needed to collect a 2D data set. This time
increases dramatically as the number of dimensions increases, possibly encompassing
several days if good resolution is required in each dimension. The data provided by a
multidimensional data set must be worth the time required to obtain it. This limitation is
not surmountable by technological improvements.
Currently available technology has also imposed some limitations on the implementa-
tion of ND NMR methods. Normally, multidimensional data sets are stored in digital
form on magnetic media. As the number of dimensions increases, so do the media storage
requirements. For example, a typical 2D data set might consist of 256 time-domain sig-
nals, each of which is digitized into 1K, 16-bit words. Thus this matrix requires one-half
megabyte for storage. The data required to describe a third dimension would then require

^ -,

several megabytes. Already this approaches the current limits of modern NMR spectrom-

eters. Thus even if the time is available to acquire large data sets, there is restricted space

to store them. Speedy processing and the display of multidimensional data are also prob-

lems. A solution which is gaining acceptance is processing and display using large

off-line computers with dedicated mathematics hardware. These difficulties are identical

to those encountered in the the early years of 2D NMR, when data handling was done

using small-memory minicomputers.

Because of the difficulties, examples of ND NMR methods are rare but do exist for

both imaging and spectroscopy. Multidimensional NMR methods were first proposed for

imaging applications, possibly because these were concerned with the correlation of spatial

information from the three dimensions of Cartesian space, which is easily visualized. The

first ND NMR experiment was proposed by Kumar and coworkers (Kum75). This was a

method for obtaining a three-dimensional spatial image, but technological limits at the

time precluded its implementation. True three-dimensional results were obtained later by

Maudsley, et al. (Mau83). However, instead of correlating information from the three

spatial dimensions, the method of Maudsley, et al. was used to correlate NMR spectra with

their two-dimensional spatial spin distributions to produce some of the earliest "chemical

shift images". In contrast, the development of ND NMR spectroscopic methods has been

pursued only recently, possibly because these are not concerned with the physical space

defined by imaging methods. Some pseudo-3D methods have been implemented (Bod81,

Bod82, Bol82). These are actually 3D methods in which the effects of two different Ham-

iltonians acting during two different evolution periods are multiplied by stepping two

time intervals in concert. Thus three-dimensional data is compressed into two dimen-

sions. True 3D spectroscopy was first implemented by Plant, et al. by combining the 2D

COSY and J-resolved spectroscopy pulse sequences into one (Pla86). A very similar

experiment has been described by Vuister and Boelens (Vui87). In a similar manner,

Griesinger and coworkers have described a number of 3D pulse sequences created by the


combination of 2D sequences (Gri87a, Gri87b, Osc88). One of the justifications for these

experiments is that the use of more than two dimensions may help to separate peaks which

overlap even in two dimensions.


2.0 Introduction

One of the challenges of NMR imaging is the correlation of spatial and spectral

information. This chapter is not concerned with methods in which a single spatial point is

chosen as the source of a spectrum ("localized spectroscopy"); a review of these methods

may be found elsewhere (Aue86). The focus here is on methods in which image informa-

tion is correlated with spectral information. Many of these methods have been reviewed

by Aue (Aue86), Brateman (Bra86b), and Brady, et al. (Bra86a). The methods fall into

two broad categories. One category includes those techniques in which a single resonance

is chosen as the source of the spectral information contained in an image (Bot84, Dix84,

Haa85a, Hal84, Jos85, Ord85). These methods will be referred to as selective spectral

imaging techniques. Most require a separate experiment for each resonance of interest, an

obvious disadvantage if several resonances are to be examined and time is at a premium.

Other methods circumvent this problem by obtaining spatial information simultaneously

for every spectral resonance (Bro82, Cox80, Man85, Sep84). These will be referred to as

nonselective spectral imaging techniques. The majority of these are Fourier imaging

methods. Most encode the entire spectrum, which may include spin-spin coupling infor-

mation, by allowing free precession in the absence of applied gradients. Fourier transfor-

mation with respect to the precession period produces the spectrum which can be corre-

lated with the spectral information encoded during other time periods in the pulse

sequence. A potential disadvantage of nonselective spectral imaging methods is that in

acquiring data from the entire spectrum, regions which do not contain resonances must be

sampled, resulting in some amount of unused data matrix. Also the sampling of the spec-

tral information in addition to the spatial information necessitates a longer total

acquisition time. Both selective and nonselective spectral imaging methods suffer from

sensitivity to Bo inhomogeneity. Variations in the static field can cause separate reso-

nances to broaden to the extent that overlap of the spectral lines occurs. Clean separation

of spatial images as a function of the resonance frequency then becomes difficult or


The focus of this chapter is the use of spectral imaging to study dynamic systems

consisting of a solid polymer dissolving in liquid solvents. The use of NMR imaging to

study solid or near solid materials has not been widely applied. The resonance linewidths

of solids are usually very broad and the application of pulsed field gradients cannot dom-

inate the chemical shift dispersion without severe penalties in signal-to-noise. In addi-

tion, the T2 relaxation times of solids are often very short, precluding the use of

spin-echo imaging methods. Thus most studies of solids using NMR imaging methods

have been observations of an NMR-detectable liquid, usually water, which has become

distributed within the solid either by diffusion or force. Studies of woods (Hal86b), oil

cores (Rot85), glass-reinforced epoxy resin composites (Rot84), nylon (Bla86), rock

(Vin86), various building materials (Gum79), and ceramics (Ack88) have been done in this

way. Virtually no work has been done in which spectra are correlated with spatial posi-

tions in solids, although Hall and coworkers have examined a piece of sandstone soaked

with n-dodecane and water using a spectral imaging method (Hal86c).

This chapter describes the application of a nonselective spectral imaging method to

the observation of polymethylmethacrylate (PMMA) dissolving in deuterated chloroform

and in perdeuterated acetone. The work was based in part on the experiments of Mareci,

et al. in which two-dimensional proton NMR images of PMMA dissolving in chloroform

and deuterated chloroform were obtained at staggered time intervals during the solvation

process (Mar88). These authors used an imaging pulse sequence which produced images

whose contrast depended on the nuclear relaxation times, T1 and T2. It was found that

the spin-lattice relaxation times of the protons of the dissolved polymer were essentially

A_ *


constant during solvation. Because changes in relaxation times can be related to changes

in nuclear mobility, Mareci, et al. were able to deduce that polymer solvation in the

PMMA-chloroform system is a first-order phase transition. During the course of their

work, Mareci, et al. were able to obtain a well-resolved proton spectrum of the dissolved

PMMA at a static field strength of 2 tesla. This indicated that it might be possible to

observe chemical changes occurring at the polymer-solvent interface by using a spectral

imaging method to observe the spectra corresponding to various spatial positions in the

polymer-solvent system. This chapter describes such studies. These differed from most

other NMR imaging studies of solid-liquid systems in two ways:

(1) Sample spectra were correlated with their spatial position.

(2) The spectra observed were from the dissolved polymer, not from the solvent.

The results of these experiments showed large frequency shifts of the spectral resonances

of PMMA in the vicinity of the polymer-solvent interface. The magnitudes and direc-

tions of the frequency shifts could be explained by a simple theoretical model based on

changes in magnetic susceptibility at the polymer-solvent interface. Thus the shifts in the

resonance frequencies were probably not the result of chemical changes.

2.1 Analysis of a Spectral Imaging Method

The spectral imaging method chosen for these studies was a variation of that of

Maudsley, et al. (Mau83). This section describes a density operator analysis of the

preparation and evolution periods of the pulse sequence. The sequence has no mixing

period. The preparation and evolution periods of the method are described by the pulse

sequence shown in Fig. 2-1. Consider the application of this sequence to a system of two

coupled spins, labeled 1 and 2, with coupling constant J and located at two spatial posi-

tions, r1 and r2. The spins are assumed to be at thermal equilibrium initially so the den-

sity operator is

o(0) = (1l, + 12,). [2-1]





-t11/2-H--T t,/2 --

FIG. 2-1. The pulse sequence corresponding to the evolution period of a spectral imaging
method. The timing of the sequence and the labels given to various time intervals are
shown at the bottom. RF radiofrequency transmitter, g gradient.


i L~

For the rest of the analysis, the f term will be implied, as described in Chapter I, section
1.1.3. The preparation period begins with the application of the ir/2 pulse with phase
S=- 0, which produces transverse terms in the density operator expression. The evolution
period starts immediately after the first pulse. The tranverse terms evolve under the com-
bined effects of the static field and linear field gradient Hamiltonians during the time
period ti = r + (tj/2). For spin 1, the result is

o(ti,spin ) = cos(wjlt) cos(rJti) [Ily cos(2xkr r) 1x sin(2irk,.rl)] [2-2]

+ cos(wlti) sin(rJti) [211x12s cos(2wkrri) + 211y'2s sin(2ikr rl)]

+ sin(wlti) cos(rJti) [Ix cos(2k, r1) + Ily sin(2rkrr)]

+ sin(witi) sin(irJt) [211y2s, cos(2rkrri) 211xI12 sin(2rkrl)],

where w, is the angular precessional frequency of spin 1 and kr -= TGrr/(21r), where G, is

the gradient amplitude. After application of a r pulse with phase = 0, the density
operator evolves in the absence of gradients during t' = T (tj/2) to produce

o(t',spinl) = 11 sin(2wrkrr + w1(tj t;)) cos(rJ(t' + ti)) [2-3a]

+ Iy cos(2rkrri + wl(t[ t')) cos(arJ(tj + t;))
211xi,2 cos(27rkrri + w1(tj tj)) sin(rJ(ti + tl))

+ 211y2, sin(2wkrr, + w1(tj t')) sin(rJ(ti + t;)) .

Finally, this expression can be rewritten in the single-element basis by applying Eq. [1-27]
to produce

o(t;,spin I) = cos(rJ(tz + t')) ei(2rrl + wl(tj t) [2-3b]

+ 2I cos(rJ(ti + t;)) ei(2rl + l(tl -tj))

+ sin(rJ(t + t)) ei(2*krrl +(t -t))

+21iIf sin(irJ(t; + t")) e'(2krrl + w(tj tj))

1 in e-i(2 rkrrl + l(ti t))
1- l sin(rJ(tz + t1)) e

+1- f sin(J(t + t;)) e-i(2-rrl + -(tl tl))
+ 2-I{Is sin(1J(tz + tf))

This analysis shows that modulation produced by the pulse sequence of Fig. 2-1 is a

function of three properties of a spin: spatial position, spectral frequency, and coupling

constant. All of the terms of Eq. [2-3] are phase-modulated as a function of the spatial

position and the spectral frequency of spin 1. The terms are amplitude-modulated as a

function of the coupling constant, J. The modulation functions can be mapped out by

changing the functions in a stepwise fashion, as described in Chapter 1, section 1.2.2.

This process is often referred to as "phase-encoding". The spatially-dependent modula-

tion functions can be mapped out by changing the value of kr, which can be done by

altering the gradient amplitude and holding the time delays, t[ and tl, constant. The value

of k, could be altered by changing the duration of the gradient, but because ti and tj" are

constants, r must be constant, and so to allow a change in the spatially-dependent modula-

tion functions, the gradient amplitude must be varied. The modulation functions which

depend on spectral frequency are not eliminated except in the special case where tj = t',

as shown by Eq. [2-3]. For the spectral imaging method of this chapter, experimental

conditions were chosen such that t =- ti. In an alternative method, the terms of Eq. [2-3]

which depend on spectral frequency can be mapped out by altering the time delays, t; and

tj, in a stepwise fashion and holding the value of kr constant. This is the basis of a spec-

tral imaging method which is described in Chapter 3. This type of phase-encoding does

not eliminate the modulation terms which depend on spatial position. For either

phase-encoding method, the modulation terms which are dependent on the coupling con-

stant cannot be removed by adjusting the time delays. Thus, coupling always affects the

signal amplitudes.

The pulse sequence used for the studies described in this chapter is shown in

Fig. 2-2. It is a 2D spectral imaging method whose evolution period is identical to the





I *




o *Co
SE "
E c "

a *O

gso 6
4. C 000M
(U L

0 ;o

m -a c w

a 2
a .I

- 5 a a
<0 gL~q
E r2SI


evolution period of the pulse sequence of Fig. 2-1. The time delays r, t1/2, and T t1/2

are held constant. For the experiments of this chapter, the values of these delays were

chosen such that, in the notation of Eq. [2-3], the sum of ti and one-half the duration of

the selective 7r/2 pulse equalled t'. Under the assumption that the duration of the selec-

tive pulse is negligible relative to the durations of the delays, r and t1/2, the terms of

Eq. [2-3] which are dependent on the spectral frequencies of the spins vanish. The cou-

pling terms remain but can vanish when

t + t; [2-4]

where n = 0, 1/2, 1, 3/2,.... Thus for studies of coupled spin systems using the pulse

sequence of Fig. 2-2, the delays must be chosen with care. Since r is fixed,

phase-encoding of the spatial information is accomplished by altering the gradient ampli-

tude. This is indicated in Fig. 2-2 by the multiple bars describing the different ampli-

tudes of the gradient, gp. The signal is sampled in the absence of gradients during the

time period, t2. Modulation with respect to the detection period is a function of the pre-

cession frequencies of the spins and not of their spatial position. Thus the pulse sequence

is a method for correlating spectra with spatial position.

The preparation periods of the pulse sequences of Figs. 2-1 and 2-2 are different.

The former consists of a nonselective r/2 RF pulse which excites the portion of the sam-

ple which lies within the RF transmitter coil. The preparation period of the pulse

sequence of Fig. 2-2 is a "slice-selective" r/2 pulse. The combination of a

frequency-selective, "soft" RF pulse (indicated by the diamond) and a field gradient (the

"slice" gradient, g,i) allows the excitation of a plane of sample spins. Only spins in the

plane experience a r/2 tip angle. The slice thickness of the plane is controlled by the

duration of the RF pulse, the pulse shape, and the amplitude of the slice gradient.

Because the Hamiltonians describing the effects of a static field and a linear field gradient

both act during the slice-selection process, different but adjacent slices are selected for

each chemically-shifted species in the sample. The result is slice misregistration. This

effect can be overcome by applying a strong gradient, but this restricts the choice of slice

widths. This effect is an example of the convolution of spatial and spectral information

discussed in Chapter 3.

A final aspect of the pulse sequence of Fig. 2-2 is the effect of the time periods, TE

and TR, on the signal amplitude. The echo time, TE, is the period from the center of the

soft RF pulse to the center of the spin echo. The period between successive initiations of

the pulse sequence is the repetition time, TR, and includes a delay to allow the perturbed

spin system to relax toward thermal equilibrium. The amplitude of the echo is weighted

by the product eI/T2 (1 e TR T), where T1 and T2 are the nuclear spin-lattice and

spin-spin relaxation times, respectively. Thus two conditions must exist for the echo to

have appreciable amplitude. The TE must be short relative to T2 and the TR must be on

the order of or greater than T1. These conditions are not so easily met when obtaining

images of solid or semisolid materials, because the T2 values can be short.

2.2 Experimental

All experiments were carried out using a General Electric CSI-2 NMR imaging spec-

trometer equipped with a 2 T Oxford Instruments superconducting magnet with a 31 cm

clear bore diameter. With shim and gradient coils installed, the working clear bore was

reduced to a 23 cm diameter. The maximum attainable gradient strength was 0.03

mT mm-1. The RF coil was a slotted tube resonator built in house by the author; it was

tunable to both 'H and '9F frequencies. For these studies, only protons were detected.

The RF coil had a length of 80 mm and a diameter of 56 mm with an effective RF mag-

netic field over a length of 40 mm.

The pulse sequence used is shown in Fig. 2-2. Quadrature detection was used for all

experiments. Table 2-1 shows the phase cycle which was used; it performed several func-

tions. Signals which had not felt the effects of phase-encoding were moved to the edges

of the spatial axis by phase alternation of the a pulse with every other phase-encode step
(Gra86). These signals arose because of imperfect r/2 and r pulses and spin relaxation
during ti. The phase alternation of the r/2 pulse and receiver with signal averaging can-
celled out the effects of imperfect r/2 pulses and imbalance in gain between the two qua-
drature detection channels. Slice misregistration due to the chemical shift effect was
assumed to be negligible and a frequency-selective sinc-shaped r/2 RF pulse and a gra-
dient normal to the imaging plane were used to select slices for all images. All i pulses
were nonselective. Only the second half of the echo was acquired to circumvent problems
of centering the echo in the acquisition window and to allow digitization of the signal
until it decayed fully.
For all images, the soft r/2 pulse duration was 1 millisecond and the r pulse duration
was 86 microseconds. The spectral width was 2000 Hz and 256 complex points were
acquired for each phase-encode step, producing a spectral resolution of 7.8 Hz per point.
Thirty-two phase-encode steps of two signal averages each were performed. The
field-of-view along the spatial axis was 64 mm and so the resolution was 2 mm per data
point. The slice width was 5 mm. The phase-encode gradient duration was 4 mil-
liseconds. The repetition time, TR, was 15 seconds and the echo time, TE, was 20 mil-
An experiment was initiated by pouring approximately 4 ml of deuterated chloro-
form or perdeuterated acetone into a circular Pyrex dish (5 cm diameter, 1.5 cm deep),
then centering a polymethylmethacrylate (PMMA) block in the solvent. The polymer and
solvents were at ambient temperature. This point was time zero for the solvation process
which ensued. The blocks were made of commercial grade PMMA, each approximately I
cm on each side and 0.5 cm thick. The solvent covered approximately the lower half of
the block. The amount of solvent was based on the amount of chloroform empirically
found sufficient to dissolve the block slowly. Immediately after positioning the block, the
dish was then covered with a tight-fitting nylon cap to retard solvent evaporation. This


Table 2-1
2D Spectral Imaging:
Relative Pulse and Receiver Phases


Phase-Encode Signal Pulses Receiver
Step Average

7r/2 r

1 1 0 90 0
1 2 180 90 180
2 1 0 270 0
2 2 180 270 180

M 1 0 270 0
M 2 180 270 180

Refer to Fig. 2-2 and its legend for the notation corresponding to this table.


assembly was placed on a plexiglas support and inserted into the RF coil. The coil was

then placed in the magnet bore. The sample and coil could be positioned reproducibly

and imaging could be started within a few minutes after placing the block in the solvent.

The sample orientation relative to the gradient axes is shown in Fig. 2-3. The thinness of

the sample along the y axis meant that slice selection along this axis was not necessary.

Imaging was initiated at approximately 20-minute intervals, each image requiring 16

minutes to acquire. The slow acquisition was necessary to reduce ridges parallel to the

phase-encode axis. These appeared if TR was so short that the transverse magnetization

could not relax to near equilibrium before the pulse sequence was applied again. Thus

some time-averaging of the solvation process was unavoidable. Initially, the x gradient

was used for choosing the slice and phase-encoding was done with the z gradient. For the

next image, the x and z gradients were interchanged, the z becoming the slice axis and the

x the phase-encode axis. This interleaving of images was continued until no more shifts

in the spectral resonances were observed or until the sample lines broadened considerably

due to solvent evaporation and subsequent sample solidification.

Because static field inhomogeneity could possibly cause frequency shifts of reso-

nance peaks, care was taken to ensure that this did not occur. The static field inhomo-

geneity was reduced by adjusting the electronic shim coils, using the same Pyrex dish,

nylon cap, dish support, and RF coil as those used for the PMMA-solvent imaging experi-

ment, but replacing the sample with a few milliliters of CuSO4-doped water. Crude

adjustments were made using a one-pulse sequence, attempting to increase the time con-

stant of the signal decay. The spectral imaging sequence of Fig. 2-2 was then used to

assess the field homogeneity. Because water has a single 1H resonance line, the homo-

geneity was considered good when the frequency of the line did not change with position.

Several iterations of this procedure were adequate to set the shim currents. These current

settings were then used without modification during spectral imaging of PMMA solvation.

Examples of the spectral images obtained after swimming are shown in the contour plots

:' ,

















E .o
.(1. 4



of Fig. 2-4. There was some shifting of the water resonance frequency with position

along the x axis. Shifts of the resonance frequency with z axis position were less notice-

able. The falloff of signal intensity along the z axis was the result of the sample extending

slightly outside of the active region of the RF coil.

2.3 Results

The proton spectrum of a piece of PMMA partially dissolved in deuterated chloro-

form is shown in Fig. 2-5 along with a diagram of the methylmethacrylate monomer unit

and the assignments of the three peaks. Because the amplitudes of the peaks in the 2D

spectral images were weighted by the T1 and T2 values of their corresponding protons, it

was important to know these before imaging commenced. Based on the measurements of

Mareci, et al., estimates of the T1 values were 215 milliseconds for the ester methyl pro-

tons, 83 milliseconds for the methylene protons, and 52 milliseconds for the methyl pro-

tons (Mar88). Because the TR of the spectral imaging sequence was set to 15 seconds, TR

had virtually no effect on the amplitudes of the spectral peaks. However, since the T2

values of the protons could not be greater than their T1 values and since the pulse

sequence TE was set to 20 milliseconds, all of the peaks were attenuated by spin-spin

relaxation. The degree of attenuation relative to the case where TE was infinitely short

was estimated using the formula 100 x ( e'TE/T). Assuming that each proton's T2

equalled its T1, the ester methyl, methylene, and methyl peaks should have been reduced

by at least 9%, 21%, and 32%, respectively. This was a best case estimate; in the actual

system the T2's could have been much shorter than the Tj's, resulting in a greater percen-

tage of attenuation.

A spectral image of a PMMA block dissolving in deuterated chloroform is shown in

Fig. 2-6a as a stacked plot. Data acquisition was begun after 61 minutes of solvation. The

spatial dimension was defined by the x axis. The three spectral peaks of PMMA were

visible and changed in amplitude, linewidth, and resonance frequency along the spatial




* ^a

5 U

a aas

oa a

E .a o
*'< 5

i II
"*o 008
Ig-S a
oT '2
.E 8.
^ >E -

IC ..




I ,



00,o a
6 U

as &.
c c



** B
uE *>


I _ad
6 ,

* **

4-. tt

C,, 0 S


* I



I 1I
0 .. a0
oo or

0 0
g .S

o .~ .. o

0' c
E ^ E

o se
*" Ei E


o C
*- a e *

0 1-

o AH0 crs


>4 i E
=0 W0 "

-o2 <

.,o ,o

0 < 0 *-..
C 1 U.*Go,
E 'o

C.) 0 & .%


A .0 ;
0 'IA2

( O a-
E \ 0

i 8 g *"
; ,* :ff aj
i i c C
c~ -o
i e .
a t8 '

axis. The changes in linewidth and amplitude arose in part from the different degrees of

sample solidness. In the vicinity of the undissolved PMMA, near x = 0 mm, linewidths

were very broad and spread into the baseline. At the polymer-solvent interfaces where

solvation began, the mixture was gel-like and the linewidths began to narrow. Further

away from the polymer block a higher concentration of solvent was present and so the

three spectral lines were resolved. Because the polymer and solvent were clear, the degree

of solvation could not be followed visually. However, the spectral image showed thai

some polymer had diffused to the edges of the sample dish. The data of the stacked plot

of Fig. 2-6a is shown as a contour plot in Fig. 2-6b to emphasize the changes in the reso-

nance frequencies with spatial position. An overall curvature was seen which was due to

static field inhomogeneity as shown by comparison of Fig. 2-6b with Fig. 2-4a. In addi-

tion, the resonance frequencies of the spectral peaks all shifted strongly downfield

(toward positive frequency) near the polymer-solvent interface. The maximum shift was

roughly 70 Hz. This number could not be measured exactly because of the contributions

of the static field inhomogeneity and the varying linewidths.

A second spectral image of the same sample was obtained by initiating data acquisi-

tion after 81 minutes of solvation. The z axis defined the spatial dimension, and the

stacked plot result is shown in Fig. 2-7a. As in Fig. 2-6a, the three spectral lines

broadened as the concentration of solvent decreased near the polymer-solvent interface.

However, the contour plot corresponding to Fig. 2-7a, shown in Fig. 2-7b, shows that the

resonance frequencies shifted upfield near the interface then sharply downfield at the

interface. The shifts were about equal to those found for Fig. 2-6.

Spectral images of much later stages of solvation are shown in Fig. 2-8. At this

point, the polymer-solvent interface no longer existed and the mixture was distributed

fairly evenly throughout the sample dish. This is seen most clearly in Fig. 2-8a where the

spatial axis is the x. The rolloff at the ends of the z axis seen in Fig. 2-8b was due to the

sample not being completely inside the RF coil. Continued spectral imaging showed little


- 1o (

- 0 5


o e

- 06)-
- I ho

- S )g




- o 1.5







I I I I I I 1 1 A

SI 1 I I I I I I I I

8 N E 2

00 6

. -' 6
co CIO <


_CO ,. ] /g<
I a *51

"a C

*. c

*- o b.2
0 E an a.

I i I I I S I I i a




c rI

I N .5

0 :R


C -


- 0 I.S
- e

- Is






I I ''I' 'I ....

Go E

So x

S 1- *
o 0

0 40

o i

.o 0 o

0 a as


I.. a ...

w .006o



o E
I0 cM CN
Co o
0 a

n I

0S 0 0.

ou N

o) o. .E
G o o


=n c

O .0

o E..

change in the spatial distribution of the spectra. As the solvent evaporated and the sample

mixture hardened, the linewidths broadened to the extent that the peaks could no longer

be seen.

To test whether the appearance of the spatially-dependent frequency shifts was a

function of the solvent used, the previous experiments were repeated using perdeuterated

acetone in place of deuterated chloroform. The time required for solvation was much

longer since perdeuterated acetone was a poorer solvent. In addition, the solvent had

some water contamination which contributed a fourth peak to the three-peak PMMA

spectrum. A stacked plot spectral image and its corresponding contour plot are shown in

Fig. 2-9. Data acquisition was begun 180 minutes after placing the PMMA block in the

solvent. An anomalous water peak is seen in the third most downfield position. Despite

the differences in solvent and solvation duration, Fig. 2-9 agrees qualitatively with

Fig. 2-6. The slight curvature of the resonance lines due to static field inhomogeneity

along the x axis was present as well as the downfield shifts of the peaks near the

polymer-solvent interface. However, the magnitudes of the shifts were roughly half that

found when using deuterated chloroform as the solvent. The spectral image obtained 20

minutes later using the z gradient for phase-encoding is shown in Fig. 2-10. It agrees

qualitatively with Fig. 2-7 but again the magnitudes of the resonance shifts are smaller.

Spectral images obtained later in the solvation process are shown in Fig. 2-11. The

linewidths narrowed somewhat, indicating a more liquid-like sample mixture, but other-

wise the qualitative features of Fig. 2-10 remained. The amount of perdeuterated acetone

used was not sufficient to dissolve the polymer block and so spectral images of

near-homogeneous mixtures like that of Fig. 2-8 were never seen.

In summary, for a given solvent, the appearance of spatially-dependent resonance

shifts depended on the time of solvation. The signs of the shifts depended on the spatial

axis observed. Finally, the magnitudes of the shifts depended on the solvent used. Origi-

nally, it was suspected that the resonance shifts were the result of a chemical interaction at

0c S

.2 a

5*- >

- C0


.5 .

S .

vo. *A
.5 a ,

I- ,-

.c s g ^ e
an g

C .rE

O BI.5o (0

cu N
C ,






cm CM N
c, 0 C, I

I1111 s11(111 111 1 11111111) 0

V 0

o .
e g

) 'I
) 0 L


E 0




1 00


Sa .a


-. C N

Eo E



a *a c


*- 0 )

te. 8 --
M 5.u


- 0







2f Cm N
o 0 CN#
0O 1
I, I I, I Il II fI ,. I


(0 -
I) o

o 0
c o



* 0 a

L- 0


o E

0 3

o o
c0 0

o E a
.. CE N
I cO) 2"
* Z


0 E s1

-E .

0. E .S

E .S

o E
co E

I ! O cN

-- I

E, .

o s o E .-

C Ig

a s
0 Ia

x >

*E 0

< o

A o




the polymer-solvent interfaces. However, this possibility was eliminated when the signs

of the shifts were found to be dependent on the spatial axis. This behavior indicated

instead that the shift effect might be the manifestation of the differences in magnetic

field susceptibility between the polymer and solvent. Thus estimates of the shifts caused

by susceptibility differences were calculated using a crude model applied to each of the

samples studied.

2.4 The Susceptibility Model

The change in the static field which results when a sample of a particular geometry

and susceptibility is immersed in the field is given by

Bo'= Bo[ + ( -)xv], [2-5]

where Bo is the static field in the absence of the sample, c is a factor which depends on

the bulk sample geometry, and X, is the volume susceptibility, a dimensionless quantity

dependent on the sample molecule (Pop59). Using the Larmor equation, the frequency

difference between two chemical species possessing identical bulk sample geometries due

solely to differences in volume susceptibility is

A' = u2 U1- = 1- t)(Xv2 Xvi). [2-6]

This shift is not identical to the chemical shift phenomenon but behaves the same way in

terms of the evolution of product operators. Thus for the spectral imaging method used

here, a susceptibility effect appears as the addition of a constant frequency shift to each

of the chemical shift frequencies of a sample.

To analyze a particular problem using Eq. [2-6], the values of C are needed. These

have been described for several sample geometries (And69):

(1) For a cylinder whose length is infinitely greater than its diameter and oriented

transverse to the static field,


K = 2r [2-7a]

(2) For an identical cylinder oriented parallel to the static field,

S= 0. [2-7b]

The greatest susceptibility shifts are seen in cylinders oriented parallel to the static field.

It has been shown that for a cylinder oriented parallel to the static field and whose

length is about 10 times its diameter, n is not zero, but about 0.2 (Boz51). Thus the infin-

ite cylinder approximation is a good one for such a sample. In the spectral imaging

experiments of the dissolving polymer, the excited region was a bar about 5 mm wide (the

slice width), 2 mm deep (the solvent depth), and 50 mm long (the dish diameter). Thus

the length was at least 10 times the width. By considering the polymer-solvent bar to be

an infinite cylinder and using the analytical values of K in Eqs. [2-7], approximate suscep-

tibility shifts at the PMMA-solvent interface could be calculated if the volume suscepti-

bilities of the molecules of the system were known. These were estimated using Pascal

constants and the densities and molecular weights of the solute and solvents, according to

the procedure of Pople, Schneider, and Bernstein (Pop59). The results are shown in Table

2-2. Using the data of Table 2-2 and Eq. [2-6], the susceptibility shifts for coaxial

cylinders of various pairs of substances were calculated. The results are shown in Table

2-3. The polymer-solvent interfaces were considered parallel to the static field if the x

gradient was used to define the slice axis and transverse if the z gradient was used. Using

these criteria, the calculated susceptibility shifts of Table 2-3 were compared with the

frequency shifts seen in Figs. 2-6, 2-7, 2-9, and 2-10. The magnitudes and signs of the

frequency shifts were measured with reference to the peaks belonging to the ester methyl

protons, because these were the least-attenuated and their spatially-dependent frequencies

could be measured fairly accurately. At some spatial positions, the ester methyl peak was

so broad that its spectral frequency could not be measured. The spatially-dependent fre-

quencies of the methylene and methyl peaks could not be measured accurately because

they were quite broad and more attenuated than the ester methyl peaks. The frequencies

Table 2-2

Calculated Volume Susceptibilities

Molecule Xv x 106

Acetone -0.461

Chloroform -0.853

Methylmethacrylate (MMA) -0.527

Table 2-3

Calculated Susceptibility Shifts at 2 Tesla

System Orientation Shift (Hz)
(relative to Bo)

MMA, Acetone transverse -12

MMA, Acetone parallel 24

MMA, Chloroform transverse 58

MMA, Chloroform parallel -116

of the ester methyl peaks were corrected to remove the static field inhomogeneity contri-

bution. Using the data of Fig. 2-4, the difference between the frequencies of the water

peak at 0 mm and at some other position were calculated for each spatial position. These

differences were assumed to be due to static field inhomogeneity and were subtracted

from the spatially-dependent frequencies of the ester methyl peaks found from the poly-

mer solvation experiments. The data of Fig. 2-4a were used to correct Figs. 2-6 and 2-9,

and Figs. 2-7 and 2-10 were corrected using the data of Fig. 2-4b. The corrected fre-

quency shifts which could be measured were plotted versus spatial position relative to one

side of the polymer block and are shown in Fig. 2-12.

2.5 Conclusion

The susceptibility model successfully predicted a number of the features of the spec-

tral images of the polymer-solvent systems studied. In general, the observed frequency

shifts were in the range calculated. More specifically, for a particular slice axis, the mag-

nitude of the frequency shift observed when deuterated chloroform was used as the sol-

vent was greater than that observed when perdeuterated acetone was used. This observa-

tion was consistent for both slice axes, as seen by comparison of Fig. 2-6b with Fig. 2-9b

and Fig. 2-7b with Fig. 2-10b. For a particular solvent, the shift observed when the z

gradient defined the slice axis was of opposite sign from that observed when the slice axis

was defined by the x gradient. These observations are summarized for the ester methyl

peak of PMMA in Fig. 2-12.

Although the model was partially successful at explaining the observed frequency

shifts, several discrepancies existed between the model and the observed spectral images.

For a particular slice axis, the spectral images showed no difference in the signs of the

frequency shifts observed for the two solvents. The susceptibility calculations shown in

Table 2-3 predicted otherwise. The disagreement possibly was due to the close

equivalence of the calculated volume susceptibilities of acetone and methylmethacrylate.

I *0 g 6 i f i fl
.cagi.c Cuno

*So 2.)
-; > A i C > > >)

E'< .- S' .o g.5 BO
a 0)'d_. .oA
> a -

e'5 t4 0)O

. 0 .A

<." 0 0 -

S Eo a >-. *-
-W 0 c
cc a 0-gesa

A 000 C 0

0 C = c S
a 0- 0lll0S&j

19) c 0 > o "

o*a E l" a 0ir0

a.. 0. 0 1. R.ol
6are > .

CIO r i e3

Ea a 4 -. v




o- C


0 0

'0 O 0 0
SI )I 1b

(ZH) 4114S ,Aouenbejl

These calculations were too poor to say truly whether the susceptibility difference was

positive or negative. A second discrepancy existed between the magnitudes of the calcu-

lated shifts and the observed values. For a particular solvent, the magnitudes of the

observed shifts should have changed substantially with the slice axis according to the sus-

ceptibility model. Some support for this could be seen by comparing Fig. 2-6b with

Fig. 2-7b and Fig. 2-9b with Fig. 2-10b; the observed shift magnitudes appeared to be

greater when the x gradient was used to define the slice. This is consistent with the model

but is not conclusive because the observed shift magnitudes could not be accurately meas-

ured. The chief reason for this obstacle was that the observed shift magnitudes were on

the order of the linewidths in the vicinity of the polymer-solvent interfaces. This coupled

with the effect of static field inhomogeneity and the variations of peak amplitudes made

the exact measurement of frequency shifts impossible.

Thus far, the behavior of the frequency shifts which were observed in Fig. 2-7 has

not been explained. The shifts first moved upfield in the vicinity of the polymer-solvent

interface as predicted by the model, but then went unexpectedly downfield in the vicinity

of the as-yet-undissolved PMMA. The appearance of spectral peaks in the vicinity of

solid indicated that some solvent had entered the slice. Far more likely, though, was the

possibility that the polymer block had become loose and moved out of the slice region

along the phase-encoded spatial axis. This would explain two observations. First, the

slice would have encompassed both solid polymer and liquid solvent, which would explain

the appearance of peaks at spatial positions where none should have been seen. Secondly,

with the polymer slightly out of the slice, two interfaces would have been observable, one

parallel to the static field and the other transverse to it. This would explain the upfield

shift observed near the polymer-solvent interface and the downfield shift seen where the

polymer should have not yet dissolved. Figure 2-6 provides some evidence that the poly-

mer block was not centered at the x axis origin; the polymer-solvent interfaces are not

distributed symetrically with respect to the x = 0 mm position.

Although this work failed to observe any chemical changes at the polymer-solvent

interface, it has some implications for similar studies. A number of papers have described

the application of the basic spectral imaging technique described in this chapter to the

study of living systems (Bai87, Has83, Pyk83). One of the proposed uses is the measure-

ment of in vivo tissue pH at particular spatial locations by the measurement of the fre-

quency difference between the inorganic phosphate and phosphocreatine peaks detected

with 31P spectral imaging. However, very little has been said about the contribution of

susceptibility to the results of such studies. If an interface, such as one between a muscle

and an organ, exists along the phase-encoded spatial axis, then susceptibility changes

could cause anomalous frequency shifts of the corresponding spectral peaks. If both tis-

sues have the same orientation relative to the static field and differ only in their suscepti-

bilities, then all spectral peaks would be affected equally and the relative frequency shifts

would be unaffected. However, if the tissues do not have the same susceptibilities and

orientations relative to the static field, errors in the estimate of tissue pH could result. It

can be shown that this error is probably very small using the susceptibility model of sec-

tion 2.4. Equation [2-6] can be rewritten in units of parts per million (ppm) as

Av v2 v1= 1 106( C)(Xv2 Xv). [2-8]

In the worst case, two cylindrical samples with two different susceptibilities would be

oriented parallel and perpendicular to the static magnetic field. When K = 0 as for a

cylinder oriented parallel to the static magnetic field, then

AVparallel = 4.19 X 106(X,2 Xvl) [2-9]

Also, when c = 2r as for a cylinder oriented perpendicular to the static magnetic field,


AVperpendicular = 2.09 x 106(X,2 Xv) [2-10]

Subtracting AL/perpendicular from Avparallel produces


A' = 6.28 x 106(Xv2 Xvl) [2-11]

The value of Av is the frequency difference between two samples due to differences in

their volume susceptibility and sample orientation and is a source of error in the measure-

ment of the true frequency separation. As an example, the true frequency separation

between inorganic phosphate and phosphocreatine varies over about a 2.5 ppm range

between pH 6 and pH 7 (Gad82). If the maximum allowable error in the frequency

difference measurement is 0.25 ppm, and this is set equal to AW', then (Xv2 Xvl) must be

less than 4 x 10-8. This is very likely in biological tissues, since the inorganic phosphate

and phosphocreatine are in very dilute solution and their volume susceptibilities are prob-

ably very similar.

Finally, the spectral imaging experiments of this chapter have important implications

for materials science. In particular, spectral imaging could be used as a theological tool to

observe deformation and flow during the solvation process, possibly uncovering some

chemical process occurring at an interface. The experiments discussed in this chapter

have shown that it may be possible to observe semisolid materials directly, complementing

the observation of solvents as they penetrate solids. It may also be possible to observe the

curing process, observing spatially-localized chemical changes as a polymer hardens.


3.0 Introduction
In the previous chapter spectral imaging techniques were categorized as being either

selective or nonselective. In this chapter, a new set of nonselective spectral imaging tech-

niques is introduced. Like other methods in that class, they are sensitive to static field

inhomogeneity. However, they differ from the other techniques in that the number of

dimensions required for image acquisition and display equals the number required to

define the spatial image. For example, a three-dimensional experiment in which the spa-

tial information from each of two spatial dimensions and the spectral information are col-

lected separately may be compressed into a two-dimensional experiment. This is accom-

plished by including the spectral information with the spatial dimensions rather than let-

ting the spectrum comprise a separate dimension. Because the convolution theorem for

Fourier transformation is central to the applicability of these techniques, they have been

grouped under the term convolutionn spectral imaging". It is shown that under certain

conditions these methods drastically reduce the time required to obtain spectral and spa-

tial information without a loss in spectral or spatial resolution.

3.1 The Convolution Spectral Imaging Method

In Chapter 2, section 2.1, an expression for a density operator was given which

described the evolution period of a 2D spectral imaging method. This was derived by

using the density operator formalism to describe the effect of the pulse sequence of

Fig. 2-1 applied to a system of two weakly-coupled spins. The result appeared in

Eq. [2-3]. The method of Chapter 2 mapped out the spatial modulation functions of the

evolution period by changing the amplitude of a gradient with each pass of the pulse

sequence. The durations of the time intervals were fixed during an experiment and so the

modulation functions related to the spectral frequencies and coupling constants of the

spins were also constant. The spectral dimension, which was correlated with the spatial

dimension, was defined by allowing free precession during detection.

Using the same pulse sequence shown in Fig. 2-1, a different type of modulation

function can be mapped out by varying the time interval, ti; in effect the r pulse is

moved through the time window, T. The gradient amplitude, Gr, and its duration, r, are

fixed. Under these conditions, the density operator expression for two weakly-coupled

spins, given by Eq. [2-3], shows that the phase modulation function which is mapped out

is a function of the spectral frequencies of the spins. The spatial modulation functions are

constant. Also, the modulation functions related to spin-spin coupling are constant

because the start of detection occurs at a fixed time after the initial r/2 excitation pulse

(Bax79). The phase-encoded spectral width is given by the inverse of the amount, At1, by

which the ti interval is changed with each pass of the pulse sequence. The desired spec-

tral width and the number of phase-encode steps place strict limits on the TE value. This

can have serious consequences since the acquired signal amplitude depends on the TE and

the T2's of the sample, as shown in Chapter 2.

Phase-encoding of spectral information is the basis of the three-dimensional method

of Sepponen and coworkers (Sep84), from which convolution spectral imaging methods

are derived. With the Sepponen method, two spatial dimensions are correlated with a

spectral dimension. The spectral modulation functions produced during evolution are

mapped out by the process just described. In addition, the spatial modulation functions

produced during evolution are also mapped out by using the stepping of a phase-encode

gradient as described in Chapter 2. To maintain the independence of the phase-encoded

spectral and spatial information, for each step in the variation of the phase-encode gra-

dient, the time interval, t1, is stepped a number of times equal to the desired number of

points in the spectral dimension. The time interval is then reset to its initial value and the

phase-encode gradient is stepped to its next value. A complete cycle of time interval

stepping is repeated for each phase-encode gradient step.

The second spatial dimension of the three dimensions defined by the Sepponen tech-

nique is produced by the process of frequency-encoding, where a gradient is turned on

during detection to create spatial modulation. However, as mentioned in Chapter I,

modulation due to free precession also occurs during detection. The Hamiltonian describ-

ing the effect of the applied gradient must dominate the unwanted Hamiltonian which

describes the effect of the static field or spatial and spectral information will be mixed.

In contrast to the Sepponen method, for convolution spectral imaging techniques the

phase-encode gradient amplitude and the time interval are stepped simultaneously during

the evolution period. The result of this simultaneous stepping is a phase modulation of

the acquired signal which is a function of the spatial and spectral details of the object

being imaged. This phase modulation is the product of two phase modulations, one of

which is a function of the gradient stepping and the other a function of the time interval

stepping. From Eq. [2-3], which was derived for a single spin, the phase modulation

function corresponding to the gradient stepping has the form

h(kr,ri) e-12^rrl, [3-1]

where kr is a function of the applied gradient magnitude and r, is the position of the spin

along the r axis. The function kr has the form

'r Gr(t)dt
kr f 2 [3-2]

where is the gyromagnetic ratio, Gr(t) is a function describing the time-dependent

amplitude of the applied gradient, and r is the time during which the gradient is applied.

In general, for a distribution of spins along the r axis, S(ri), the modulated signal has the


s(kr) = fS(ri)h(kr,ri)dri [33

where R expresses the limits of integration imposed by the extent of the spin distribution

in r space. The phase modulation function corresponding to the stepping of the time

interval has the following form for a single spin not J-coupled to another:

h'(tl,l) = e-i2wtl'l, [3-4]

where vz = w1/2r is the resonance frequency of the spin and tz is the time during which

free precession occurs in the absence of gradients. In general, for a spectral distribution of

spins, S(vi), the modulated signal can be expressed by

s'(tl) = fS(ti)h'(ti,vi)dvi [3-5]

where N expresses the limits of integration imposed by the extent of the spectral distribu-

tion. If one takes FT to mean "the Fourier transform of", then ideally

FT[s(kr)] = S(ri) [3-6]

FT[s'(t)] = S'(Vi) [3-7]

In the convolution spectral imaging technique, the amplitude of the phase-encode

gradient and the time interval are stepped simultaneously. Thus kr and tz are related by a

constant, q:

tz = r7kr. [3-8]

The resulting doubly-modulated signal may be expressed by the product of Eqs. [3-3] and

[3-5]. By the convolution theorem (Bri74), the Fourier transform of this product is the

convolution of the spatial and spectral spin distributions. This transformed result may be

mapped into frequency space in which case the spatial spin distribution appears as a func-

tion of frequency scaled by r7. Denoting the convolution by an asterisk and using

Eq. [3-8], the Fourier transform can be written

FT[s(kr)s'(ti)] = ,[S(uIv)*S'(ui)] [3-9]

Alternatively, the Fourier transform of the product may be mapped into r space. The

spectral spin distribution then appears as a function of spatial position scaled by the

inverse of tj:

FT[s(kr)s'(ti)] (1/j)[S(ri)*S'(r1i/)] [3-10]

Each of the convolution functions shown in Eqs. [3-9] and [3-10] is the spatial distri-

bution of the sample spins offset by their spectral resonance frequencies or vice versa. A

plot of one of these functions for a hypothetical two-compartment sample containing two

different chemical species is shown in Fig. 3-la. The function is the convolution of the r

space information shown in Fig. 3-lb with the spectrum shown in Fig. 3-1c. The two

resonance lines are assumed to have widths much smaller than their chemical shift differ-

ence and Bo inhomogeneity is assumed to be negligible. Another way of viewing Fig. 3-la

is as a series of spectral frequencies, each of which has an identical spatial field-of-view

spread around it. The origin in spatial coordinates for a particular chemical species coin-

cides with its spectral frequency. Any spatial offset of a nucleus from the origin appears

as an offset from its spectral frequency. The idea of convolving two types of information

has been exploited in spectroscopy by the accordion experiment of Bodenhausen and

Ernst (Bod81) and the three-frequency experiment of Bolton (Bol82). The convolution of

a spatial axis and the zero-quantum spectrum has been accomplished recently in a similar

manner by Hall and Norwood (Hal86a).

Although Eqs. [3-9] and [3-10] were derived for phase-encoding, the mathematics

are similar for frequency-encoding. The two processes differ in that free precession,
which is the source of spectral information, occurs in the presence of a gradient during

frequency-encoding. The result is still a doubly-modulated signal, and Eqs. [3-9] and

[3-10] still hold. These equations represent the origin of the "chemical shift artifact"

(Bab85, Dwy85, Soi84) which is observed when a gradient applied during signal acquisi-

tion is too weak to obliterate the separation between spectral resonances.


---- .

0 ~i

-o -

g o

- 0 h- .

** .
So Zt

IJ- r
oS *i .
a &kg,
"t --o Eval


The pulse sequence for the acquisition of a two-dimensional convolution spectral

image is shown in Fig. 3-2a. The interval between the initiation of the pulse sequence at

successive phase-encode steps, TR, is kept constant. The time to the echo formation, TE,

is also kept constant; thus, the T2 relaxation weightings of the phase-encoding steps are

equivalent. Phase-encoding of spectral information is accomplished by moving the r

pulse through the window labelled T by incrementing t1 by an amount At1 in a stepwise

fashion. The spectral width of the phase-encoded axis is 1/At1.

The implementation of the pulse sequence of Fig. 3-2a with the frequency-encode

gradient, gfy, defined as the z gradient and the phase-encode gradient, g,, defined as the

x gradient, would result in an z,v, versus x,vx image where z and x define the spatial axes

and v, and vx define the impressed spectral information. Each planar zx image

corresponding to a spectral resonance will lie on a diagonal passing through the origin if

the sample is centered on the crossing point of the z and x axes as defined by the z and x

gradients. The planar images will be located on a diagonal parallel to that passing through

the origin if the sample is spatially offset from this crossing point. The center of each

image will be separated from that of its chemically shifted neighbor along the diagonal by

the chemical shift difference of the two species multiplied by a scaling factor. If the spec-

tral widths convolved with each spatial dimension were identical, this scaling factor would

equal vT.

Like other nonselective spectral imaging methods, convolution spectral imaging tech-

niques collect spectral data from all points of the spectrum including those which do not

contain resonances. The advantage lies in the ability to compress three dimensions of data

into two dimensions for planar spectral imaging or four dimensions into three for volume

spectral imaging with a corresponding saving in total acquisition and processing time. It

will be shown that the techniques are best suited for small samples whose frequency spec-

tra are composed of well-separated resonances and that these techniques optimize the

available signal-to-noise ratio. Thus convolution spectral imaging may be most applicable

L,,,- E -. II U

EX -'- M

am &a

a a o


Q a.
lI .*is .

S- .0 u t
_ .c g 4 i

.B 6o

> (

200 4)g
f4. as .-

c as C.s
C S u 00
0006 b.1V
Ei ai" O


0 -


t i W


wI I
r A
^ v F

flc r0







0 --'




t -



0.a 0.
0 e 0

to the field of NMR microimaging (Agu86).

3.2 Experimental

Experiments were carried out using the spectrometer and RF coil described in

Chapter 2, section 2.2. The pulse sequences used are shown in Fig. 3-2. Quadrature

detection was employed in all experiments. Slice selectivity was used if the spectral reso-

nances of the sample under investigation were sufficiently close that the application of a

strong gradient during slice selection could eliminate slice misregistation due to the chem-

ical shift effect. Otherwise, the entire sample volume was excited. Thus a

frequency-selective sinc-shaped ir/2 RF pulse and a gradient normal to the imaging plane

were used to select slices for all 'H images. The w/2 RF pulse was nonselective in all 19F

imaging experiments. All 7 pulses were nonselective.

The effects of imperfect r pulses and DC imbalance between the quadrature chan-

nels were corrected by using a two-step phase cycle during signal averaging impressed on

the 7/2 pulse and receiver. This phase cycle had the effect of cancelling out artifacts due

to magnetization which was not phase-encoded. However, as pulse repetition times were

decreased for more rapid image data acquisition, the ability of the cycle to suppress

artifacts was diminished. The origins of these residual artifacts were from

non-steady-state magnetization which was not phase-encoded and instrumental errors

causing DC offset. Therefore gradient proportional phase incrementation (GPPI) of the 7r

pulse was also added to the two-step phase cycle (Gra86). This had the effect of moving

artifacts which appeared at the midpoint of the phase-encoded axis to the edges of that

axis without increasing the image acquisition time by requiring signal averaging. The ini-

tial desire was to impress GPPI on the ir/2 pulse and the receiver to shift both types of

artifact. However, this was not allowed by the instrument design. Thus GPPI had to be

impressed on the x pulse. This could only shift artifacts resulting from magnetization

which had not been phase-encoded. Artifacts due to instrumental DC effects remained at

the midpoint of the phase-encoded axis. Fortunately, these were quite minimal. The phase

cycle for the pulse sequence of Fig. 3-2a is shown in Table 3-1, and that for the sequence

of Fig. 3-2b is shown in Table 3-2.

To reduce truncation artifacts, it was ensured that the echoes corresponding to the

middle steps in the phase-encoding process were centered in the acquisition window. This

was accomplished in a set-up mode before acquisition of the image data by turning all

phase-encode gradients off and placing the r pulse in the center of the time delay, T (see

Fig. 3-2). The amplitude of the frequency-encoding gradient on during the time, r, was

then carefully adjusted. This set-up mode was also used for pulse calibration and for

choosing transmitter and receiver gain settings. In most cases the values of TR and TE

were chosen such that very little relaxation attenuation was allowed.

3.3 Practical Aspects of the Method

Convolution spectral imaging is most effective for small objects containing molecular

species with a spectrum of well-separated single peak resonances, which implies that the

technique may find its greatest application in the field of NMR microimaging. The effec-

tiveness of the method is enhanced if short phase-encode gradient times are employed.

These conclusions become apparent if one considers the sources of signal modulation and

if one makes the restriction that the frequency separation between resonances is greater

than the frequency spread caused by the applied gradients. This restriction is necessary to

prevent the overlap of the spatial images comprising the convolution spectral image. How-

ever, it will be shown that this restriction may be relaxed for certain sample geometries.

Consider a single spin, not spin-spin coupled to another, which is subjected to the

two-dimensional technique of Fig. 3-2a, again letting z be the frequency-encode dimen-

sion and x be the phase-encode dimension. During the acquisition time, t2, the signal is

modulated in part by the function

h"(m,,t2) = e-i2~v + m-t2 ]

S"*' ;

Table 3-1

Two-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases


Phase-Encode Signal Pulses Receiver
Step Average


1 1 0 0 0

1 2 180 0 180

2 1 0 180 0

2 2 180 180 180

M 2 180 180 180

Refer to Fig. 3-2a and its legend for the notation corresponding to this table.

Table 3-2

Three-Dimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases


Phase-Encode Phase-Encode Signal Pulses Receiver
Step Step Average
(gp,2) (gpi)

wr/2 X

1 1 1 0 0 0

1 1 2 180 0 180

1 2 1 0 180 0

1 2 2 180 180 180

1 M 2 180 180 180

2 1 1 0 180 0

2 1 2 180 180 180

2 2 1 0 0 0

2 2 2 180 0 180

L M 2 180 0 180

Refer to Fig. 3-2b and its legend for the notation corresponding to this table.