NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
To my wife and best friend, Lisa
U OF F LIBRARIES
A; "*i
ACKNOWLEDGMENTS
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 coadvisor, Dr. Tom Mareci, who guided me virtually stepbystep 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 oneonone 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.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ....................................................................................................iii
LIST O F TA BLES ............................................................................................................... vii
LIST O F FIG U RES ........................................................................................................... viii
A BSTR A CT ........................................................................................................................... x
CHAPTER
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
1.1.5.1 The Effect of a Static Magnetic Field.................................13
1.1.5.2 The Effect of a Linear Field Gradient ...............................17
1.1.5.3 Application of Radiofrequency (RF) Pulses....................... 18
1.1.5.4 Application of PhaseShifted RF Pulses.............................19
1.1.6 Observable Magnetization...................... ........................................21
1.2 The TwoDimensional 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
2 SPECTRAL IMAGING AND APPLICATIONS TO THE STUDY
OF DYNAMIC POLYMERSOLVENT 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 SignaltoNoise Considerations...... ............................................126
3.4 Convolution Spectral Imaging at High Field................................................128
3.5 Conclusion ................................................................................................140
4 QUANTIFICATION OF EXCHANGE RATES
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
,.:
LIST OF TABLES
TABLE page
11 Effects of the SingleElement Operators...........................................................10
12 Solutions to the Equation of Motion of the
Density Operator Expressed as Cartesian Space Rotations ..................14
13 Effect of a Product Operator Hamiltonian on Terms of
a Cartesian Product Operator Basis Set............................................15
14 Transformations of Cartesian Operators by PhaseShifted
R F Pulses ..............................................................................................20
21 2D Spectral Imaging: Relative Pulse and Receiver Phases ...............................40
22 Calculated Volume Susceptibilities ................................................... ...........67
23 Calculated Susceptibility Shifts at 2 Tesla....................................................68
31 TwoDimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases.....................................................88
32 ThreeDimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases.....................................................89
41 Relative Pulse and Receiver Phases for RED NOESY.....................................160
42 Temperature Dependence of Relaxation Rates, Exchange Rates,
and Free Energies of Activation for DMF, DMA, and DMP.............169
43 Activation Parameters of DMF Found Using RED NOESY Data ..................70
44 Activation Parameters of DMF: Literature Values.........................................171
45 Activation Parameters of DMA: Literature Values ........................................172
46 Activation Parameters of DMP: Literature Values .........................................173
47 Relaxation Rates of DMF Methyl Protons........................................................ 182
vii
.' *'" 'J
LIST OF FIGURES
FIGURE page
21 The pulse sequence corresponding to the evolution period of
a spectral imaging method .....................................................................33
22 A pulse sequence for twodimensional spectral imaging ..................................36
23 Sample orientation for studies of PMMA solvation...........................................42
24 Spectral images used to observe the quality of the
static field hom ogeneity ................................... .....................................45
25 The 'H spectrum of a piece of PMMA partially
dissolved in chloroform at 2 T..............................................................48
26 Spectral images of PMMA in deuterated chloroform
after 61 minutes of solvation ................................................................49
27 Spectral images of PMMA in deuterated chloroform
after 81 minutes of solvation................................................................52
28 Spectral images of PMMA in deuterated chloroform
at late stages of solvation............................... ..........................................54
29 Spectral images of PMMA in perdeuterated acetone
after 180 minutes of solvation ..............................................................58
210 Spectral images of PMMA in perdeuterated acetone
after 200 minutes of solvation ..............................................................61
211 Spectral images of PMMA in perdeuterated acetone
at late stages of solvation............................... ..........................................63
212 PM M A solvation ............................................................................................... 71
31 The convolution of spectral and spatial information........................................80
32 Pulse sequences for convolution spectral imaging .............................................83
33 The effect of sample geometry...........................................................................93
34 The interaction of TE and the spatial resolution .............................................106
35 Threedimensional convolution spectral imaging........................................... 112
36 Convolution spectral imaging at high field ................................................130
viii .
!
41 The NOESY pulse sequence ............................................................................ 146
42 Peak intensity behavior as a function of
the mixing time and exchange rate ................................................152
43 The RED NOESY pulse sequence .....................................................................156
44 A plot of In(k') versus 1000/T for DMF ....................................................165
45 A plot of In(k') versus 1000/T (corrected) for DMF.......................................166
46 A plot of In(k'/T) versus 1000/T for DMF ................................................167
47 A plot of ln(k'/T) versus 1000/T (corrected) for DMF...................................168
48 Four NOESY spectra of DMF obtained using the RED NOESY sequence .....174
49 Behavior of peak intensities of NOESY spectra of
D M F at 347 K .........................................................................................178
I
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
NOVEL PULSE METHODS
FOR MULTIDIMENSIONAL NMR IMAGING AND SPECTROSCOPY
By
MICHAEL D. COCKMAN
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 spin1/2 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 twodimensional 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. Timedependent frequency shifts of the spectral frequencies of the polymer
were observed in the vicinity of the polymersolvent 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 threedimensional 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,Ndimethyl
amides to determine the exchange rates of the N,Nmethyl groups. For one of these
molecules, N,Ndimethylformamide, 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 mol1, 28, 19 kcal mol1, and
5 cal mol1 K', respectively.
CHAPTER 1
NMR THEORY AND MULTIDIMENSIONAL NMR
1.0 Introduction
Since the original concept of twodimensional 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.
Twodimensional 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, twodimensional 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 twodimensional 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 spin1/2 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* [11]
n
For a system of N spin1/2 nuclei, there are 2N eigenstates. In the Dirac notation, the
expectation value of an observable, Q, is described by
= < 1QI> [l2a]
= [l2b]
m n
L
= E CmCn [l2c]
m n
The products of the coefficients, cmCn, may be arranged to form a matrix which is called
the density matrix. If 0 is timedependent, then either the eigenstates or the coefficients
can carry the time dependence. In the "Schridinger representation", the eigenstates are
allowed to be timeindependent and the timedependence 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. [13]
With this definition, the expectation value for an observable can be written in terms of
the density operator. By Eqs. [12] and [13] and the orthonormality of the eigenstates un,
'E E [l4a]
m n
= Tr [p Q], [l4b]
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 timeindependent 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 timedependent Schr6dinger equation using the method described by
Slichter (Sli80). The timedependent Schridinger equation is
at
i(^ =HV>, [151
where h = h/2r and h is Planck's constant. Substituting Eq. [11] into Eq. [15] produces
acn
ixE un E cnHun [16]
n n
since only the can's are timedependent. Multiplying both sides of the equation from the
left by uk leads to
iE tUkun E CnUkHun. [17]
in n
Integrating and using the orthonormality of the basis functions produces
acn
i = n cEn. [18]
The equation of motion of the density operator may be derived by taking the deriva
tive of Eq. [13] with respect to time and substituting Eq. [18] into the result:
cm Cn 19a]
t c cn a + cm
= cJ[(i/)E Cm] + [(i/N)E Cn]cm [l9b]
m n
(i/N)[E CnC E CnCm] [ 19c]
m n
= (i/l)[E E ] [19d]
m n
(i/)[ ]. [ 9e]
Thus the time derivative, or the equation of motion, of the density operator is
I1
= (i/NI) [p,H] [110
When H is timeindependent, a solution of the equation of motion is
p(t) e(i/V)Ht p(0) e(i/i)n [111]
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, [112]
where the Larmor relationship,
w Bo, [113]
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,
HIum> = Em un> $Isumn> = Nmlumn> [114]
where m is one of 21 + 1 values in the range I, I1, 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 = [115]
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 [116]
Ee
n
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
S(Em/kT)
Pm E I(En/kT) [ 117]
n
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)). [118]
Finally, because HIum> = Emlum>, it follows that
p = (21+1)(1(H/kT)) = (21+1) (21+1)'() wl,/kT) [119)
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) [120]
the reduced density operator at thermal equilibrium may be written
a = PIS [121]
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
directly:
a = (i/) [,H] [122]
A solution of this equation for a timeindependent Hamiltonian is
o(t) = e(i/V)Ht o(0) e(i/)Ht [123]
and the expectation value of an observable is
= Tr [a Q] [124]
Equation [121] 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 [125]
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 sec1 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
I
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 sec1) at temperatures ranging from 293 to 363 K. For
all of these cases, the high temperature assumption holds and Eq. [121] 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 timeindependent
basis operators, B,, weighted by timedependent coefficients, b,(t):
o(t) = E b.(t)B [126]
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 singlespin, Cartesian angular momentum operators, I,,
ly, and I,, and the unity operator, E. The basis set for N spin1/2 nuclei consists of 4N
product operators. For a system of two spin1/2 nuclei the basis consists of:
The unity operator:
(1/2)E
Onespin operators:
1lx Ily 11 ', l 12x 2y 12
Twospin 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 wellsuited 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 singleelement 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
spin1/2 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 singleelement operator to an eigenstate can
produce a change in the spin state. The effects of these operators are shown in Table I 1.
Table 11 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 singleelement 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 singleelement operators by the
following:
2Ek .(la+ If), [l27a]
Ikx +Ik+ ), [127b]
Iky (I+ Ik) ,[127c]
Ik = (I If), [I27d]
where k indicates a particular spin. The Cartesian product operator basis may be rewrit
ten as a singleelement product operator basis by the use of these relationships.
,I
Table 11
Effects of the SingleElement Operators
Initial Spin State
Operator
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 singleelement 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. [123] 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 threedimensional Cartesian
space. The following analysis follows the method described by Slichter (Sli80). Recall
that for a timeindependent 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 1.1.5.1, this describes a spin, not at thermal equilibrium, whose
>7
state is changing under the effect of chemical shift precession. Define
f() = eil, 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'() = ei9ls IY ei91.,
and the second derivative of f(O) is
f"() = ei' x'I e'i1I
Thus the relationship of Eq. [128] 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),
[133]
which may be verified by substitution. Thus
[134]
The coefficient, a, may be found by finding the solution of Eq. [133] when 0 equals 0
and using Eq. [128]. Finding the solution of the first derivative of Eq. [133] when 4
equals 0 and using Eq. [130] gives the coefficient, b. The results are that
a = I, [135a]
b= l [135b]
Finally, the following is obtained:
ei41 Ix e i, = Ixcos(O) + lysin() .
[128]
[129]
[130]
[131]
[132]
[136]
:1
ei#IL Ix ei#L a cos(O) + b sin(O) .
Equation [136] 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 12. 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 onespin 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
is
e i# 11xs2 ei0 ffi ( 1xcos(o) + Ilysin(O))12,. [137]
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 13.
1.1.5.1 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
14
Table 12
Solutions to the Equation of Motion of the Reduced Density Operator
Expressed as Cartesian Space Rotations
o(t) ei#lx o(t) e'i#x ei'y o(t) eiol' eiIs 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 13
Effect of a Product Operator Hamiltonian on
Terms of a Cartesian Product Operator Basis Set
a(t) ei2 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
i=N
X'H = E wil=. .[139]
i=1
The spinspin coupling term is
j=(N ) k=N
NIH = 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
IH 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 12 where i = wit :
lx licos(wit) + Iiysin(wit) [141a]
Iy liycos(wit) lixsin(wit) [ 141 b]
Ii Ii,. [141c]
Equation [141] 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 13 where jk = rJjkt the evolutions of the onespin
operators under the coupling Hamiltonian are given by
1jx Ijxcos(lJjkt) + 21jylksin(rJjkt) [142a]
ix Jx Jr
Ijy Ijycos(rJjkt) 2IjxIksin(lJjkt) [142b]
lj. . l. [142c]
Twospin operators also evolve under the Hamiltonian which describes spinspin cou
pling. Examples are:
21jIks  2Ijxlkcos(wJjkt) + Ijysin(iJjkt) [143a]
21jylks . 2IjyIkcos(rJjkt) Ijxsin(rJjkt) [143b]
1.1.5.2 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
)IH = r, Yl [144]
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 [145]
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) [I46a]
Iy . iycos(7Grrit) lisin(yGrrt) [146b]
lis lis. [146c]
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. Twospin
operators also evolve under the effect of a linear field gradient. Examples are:
2IjxIks , 2Ijxlk.cos(Grrjt) + 21jylkssin(yGrjt), [147a]
.4
21jylks 21jyIk.cos(yGrrjt) 21xlk.sin(yGrrjt) [147b]
1.1.5.3 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
iH yBo,I yBl[Icos(not) + lysin(Not)] [148]
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 timedependent, but on transformation
to a frame rotating with angular frequency flo, the timedependence of the alternating
field vanishes and the Hamiltonian becomes
H = (fl wi)I, + x, [149]
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 offresonance I, term vanishes. Also, the term becomes negligible when the
applied alternating field is strong enough that 0f >> (flowi) (a "hard" pulse). For these
cases, the Hamiltonian becomes
IH = nlx [150]
In terms of Table 12, =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, [ I51a]
ly lycos() + Isin(O) [151b]
I, l,cos(O) lysin(0) [I51c]
.:1 ,
1.1.5.4 Application of PhaseShifted 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 "phaseshifted" relative to the x axis. In onedimensional NMR experi
ments, phaseshifting 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 phasecycling 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 phaseshifted RF pulse may be written
o(t) = e isi e inx eiIs oa(0) e ifI, eienx e i1. [152]
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)(lcos(0)) Is sin(e)sin(0) [153a]
ly lx cos(C)sin(()( cos(0)) + ly (cos2(f)cos(0) + sin2(f)) + 1I cos(C)sin(0) [153b]
I, Ix sin(f)sin(0) ly cos(f)sin(e) + I, cos(e) [153c]
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 14 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
offresonance effects are negligible. These assumptions are made throughout this work.
Table 14
Transformations of Cartesian Operators by PhaseShifted RF Pulses
C (degrees)
0 90 180 270
Initial Operator Final Operators
Ix Ix IxcOIsO Ix IxcO+I,sO
ly IycO+ISs ly lycOl,sO ly
1, lcOlyse 1,cO+lxse lIcO+lyse IScOlIse
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
operators:
I' = (Ix + ly) [154]
Thus the expectation value for the detected signal is
= Tr [o(1x + ily)], [155]
or transforming to the singleelement operator basis,
Tr [al] [156]
This implies that o(t) must consist of IY operators to give a nonzero trace and thus observ
able magnetization. In terns of the twospin 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 TwoDimensional 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
twodimensional 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 amplitudemodulated 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 twodimensional
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, spinspin 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) [157]
The amplitude of each of the transverse components oscillates during t according to the
angular frequency, w, of the spin. Thus the components are "amplitudemodulated". If
the components are combined using the singleelement operator basis, then
o(t) = (I+e i I ei) [158]
The phases of the I+ and I' terms oscillate according to w and the terms are said to be
"phasemodulated". This example shows that signal phase modulation may be produced
by properly combining amplitudemodulated 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 phasemodulated signal may be written as
'"4
FT [ei' e/T2] = A' + iD [159]
where A and D are absorption and dispersion Lorentzian functions, respectively. These
functions have the forms:
T,
A(w) (+( 2 ) [l60a]
(l+((w lo)2T,2))
(W' O)Ta
D(w) ( )T22 [l60b]
(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" + ei"*)etT2] [161a]
S[A+ + iD++ A + iD], [i61b]
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 [162b]
(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 amplitudemodulated 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 amplitudemodulated signals into
.4/
phasemodulated 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 doublymodulated 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 twodimensional 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 [163]
The detected signal is found by using the trace relation
= Tr[o(tl,t2)I [164]
This equation is simply the 2D case of the expression derived earlier for any a, Eq. [156].
Substituting for o(tl,t2),
<1+> Tr[e(i/g)Ht2(t) e(i/)Ht21+] [165]
Because the trace is invariant to cyclic permutation of the operators, the expression for
becomes
Tr[i(tl)e(i/g)Ht2I+e(i/$)Ht2] [166]
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
on.
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 spinspin 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 Ndimensional
space could be described by a density operator of the form o(tl,t12, tl(Nl),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 Ndimensional (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 timedomain 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 timedomain sig
nals, each of which is digitized into 1K, 16bit words. Thus this matrix requires onehalf
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
offline 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 smallmemory 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 threedimensional spatial image, but technological limits at the
time precluded its implementation. True threedimensional 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 twodimensional 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 pseudo3D 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 threedimensional data is compressed into two dimen
sions. True 3D spectroscopy was first implemented by Plant, et al. by combining the 2D
COSY and Jresolved 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
29
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.
CHAPTER 2
SPECTRAL IMAGING AND APPLICATIONS
TO THE STUDY OF DYNAMIC POLYMERSOLVENT SYSTEMS
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 spinspin 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
impossible.
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 signaltonoise. In addi
tion, the T2 relaxation times of solids are often very short, precluding the use of
spinecho imaging methods. Thus most studies of solids using NMR imaging methods
have been observations of an NMRdetectable liquid, usually water, which has become
distributed within the solid either by diffusion or force. Studies of woods (Hal86b), oil
cores (Rot85), glassreinforced 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 ndodecane 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 twodimensional 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 spinlattice relaxation times of the protons of the dissolved polymer were essentially
A_ *
32
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
PMMAchloroform system is a firstorder phase transition. During the course of their
work, Mareci, et al. were able to obtain a wellresolved 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 polymersolvent interface by using a spectral
imaging method to observe the spectra corresponding to various spatial positions in the
polymersolvent system. This chapter describes such studies. These differed from most
other NMR imaging studies of solidliquid 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 polymersolvent 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 polymersolvent 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. 21. 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,). [21]
*
n/2
RF
g
t11/2HT t,/2 
[+Tt~T
FIG. 21. 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.
j'
.eq
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)] [22]
+ 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)) [23a]
+ 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 singleelement basis by applying Eq. [127]
to produce
o(t;,spin I) = cos(rJ(tz + t')) ei(2rrl + wl(tj t) [23b]
2
+ 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 ei(2 rkrrl + l(ti t))
1 l sin(rJ(tz + t1)) e
+1 f sin(J(t + t;)) ei(2rrl + (tl tl))
+ 2I{Is sin(1J(tz + tf))
This analysis shows that modulation produced by the pulse sequence of Fig. 21 is a
function of three properties of a spin: spatial position, spectral frequency, and coupling
constant. All of the terms of Eq. [23] are phasemodulated as a function of the spatial
position and the spectral frequency of spin 1. The terms are amplitudemodulated 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 "phaseencoding". The spatiallydependent 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 spatiallydependent 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. [23]. For the spectral imaging method of this chapter, experimental
conditions were chosen such that t = ti. In an alternative method, the terms of Eq. [23]
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 phaseencoding does
not eliminate the modulation terms which depend on spatial position. For either
phaseencoding 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. 22. It is a 2D spectral imaging method whose evolution period is identical to the
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evolution period of the pulse sequence of Fig. 21. 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. [23], the sum of ti and onehalf 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. [23] which are dependent on the spectral frequencies of the spins vanish. The cou
pling terms remain but can vanish when
t + t; [24]
where n = 0, 1/2, 1, 3/2,.... Thus for studies of coupled spin systems using the pulse
sequence of Fig. 22, the delays must be chosen with care. Since r is fixed,
phaseencoding of the spatial information is accomplished by altering the gradient ampli
tude. This is indicated in Fig. 22 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. 21 and 22 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. 22 is a "sliceselective" r/2 pulse. The combination of a
frequencyselective, "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 sliceselection process, different but adjacent slices are selected for
each chemicallyshifted 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. 22 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 spinlattice and
spinspin 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 CSI2 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 mm1. 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. 22. Quadrature detection was used for all
experiments. Table 21 shows the phase cycle which was used; it performed several func
tions. Signals which had not felt the effects of phaseencoding were moved to the edges
of the spatial axis by phase alternation of the a pulse with every other phaseencode 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 frequencyselective sincshaped 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 phaseencode step, producing a spectral resolution of 7.8 Hz per point.
Thirtytwo phaseencode steps of two signal averages each were performed. The
fieldofview along the spatial axis was 64 mm and so the resolution was 2 mm per data
point. The slice width was 5 mm. The phaseencode gradient duration was 4 mil
liseconds. The repetition time, TR, was 15 seconds and the echo time, TE, was 20 mil
liseconds.
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 tightfitting nylon cap to retard solvent evaporation. This
"4
Table 21
2D Spectral Imaging:
Relative Pulse and Receiver Phases
Phases
PhaseEncode 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. 22 and its legend for the notation corresponding to this table.
I
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. 23. The thinness of
the sample along the y axis meant that slice selection along this axis was not necessary.
Imaging was initiated at approximately 20minute intervals, each image requiring 16
minutes to acquire. The slow acquisition was necessary to reduce ridges parallel to the
phaseencode 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 timeaveraging of the solvation process was unavoidable. Initially, the x gradient
was used for choosing the slice and phaseencoding 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 phaseencode 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 PMMAsolvent imaging experi
ment, but replacing the sample with a few milliliters of CuSO4doped water. Crude
adjustments were made using a onepulse sequence, attempting to increase the time con
stant of the signal decay. The spectral imaging sequence of Fig. 22 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
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of Fig. 24. 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. 25 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 spinspin
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. 26a 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
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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 polymersolvent interfaces where
solvation began, the mixture was gellike 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. 26a is shown as a contour plot in Fig. 26b 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. 26b with Fig. 24a. In addi
tion, the resonance frequencies of the spectral peaks all shifted strongly downfield
(toward positive frequency) near the polymersolvent 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. 27a. As in Fig. 26a, the three spectral lines
broadened as the concentration of solvent decreased near the polymersolvent interface.
However, the contour plot corresponding to Fig. 27a, shown in Fig. 27b, 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. 26.
Spectral images of much later stages of solvation are shown in Fig. 28. At this
point, the polymersolvent interface no longer existed and the mixture was distributed
fairly evenly throughout the sample dish. This is seen most clearly in Fig. 28a where the
spatial axis is the x. The rolloff at the ends of the z axis seen in Fig. 28b was due to the
sample not being completely inside the RF coil. Continued spectral imaging showed little
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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 spatiallydependent 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 threepeak PMMA
spectrum. A stacked plot spectral image and its corresponding contour plot are shown in
Fig. 29. 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. 29 agrees qualitatively with
Fig. 26. 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
polymersolvent 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 phaseencoding is shown in Fig. 210. It agrees
qualitatively with Fig. 27 but again the magnitudes of the resonance shifts are smaller.
Spectral images obtained later in the solvation process are shown in Fig. 211. The
linewidths narrowed somewhat, indicating a more liquidlike sample mixture, but other
wise the qualitative features of Fig. 210 remained. The amount of perdeuterated acetone
used was not sufficient to dissolve the polymer block and so spectral images of
nearhomogeneous mixtures like that of Fig. 28 were never seen.
In summary, for a given solvent, the appearance of spatiallydependent 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
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the polymersolvent 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], [25]
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). [26]
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. [26], 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,
I
K = 2r [27a]
(2) For an identical cylinder oriented parallel to the static field,
S= 0. [27b]
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 polymersolvent bar to be
an infinite cylinder and using the analytical values of K in Eqs. [27], approximate suscep
tibility shifts at the PMMAsolvent 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
22. Using the data of Table 22 and Eq. [26], the susceptibility shifts for coaxial
cylinders of various pairs of substances were calculated. The results are shown in Table
23. The polymersolvent 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 23 were compared with the
frequency shifts seen in Figs. 26, 27, 29, and 210. 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 leastattenuated and their spatiallydependent 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 spatiallydependent 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 22
Calculated Volume Susceptibilities
Molecule Xv x 106
Acetone 0.461
Chloroform 0.853
Methylmethacrylate (MMA) 0.527
Table 23
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. 24, 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 spatiallydependent frequencies of the ester methyl peaks found from the poly
mer solvation experiments. The data of Fig. 24a were used to correct Figs. 26 and 29,
and Figs. 27 and 210 were corrected using the data of Fig. 24b. 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. 212.
2.5 Conclusion
The susceptibility model successfully predicted a number of the features of the spec
tral images of the polymersolvent 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. 26b with Fig. 29b
and Fig. 27b with Fig. 210b. 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. 212.
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 23 predicted otherwise. The disagreement possibly was due to the close
equivalence of the calculated volume susceptibilities of acetone and methylmethacrylate.
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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. 26b with
Fig. 27b and Fig. 29b with Fig. 210b; 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 polymersolvent 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. 27 has
not been explained. The shifts first moved upfield in the vicinity of the polymersolvent
interface as predicted by the model, but then went unexpectedly downfield in the vicinity
of the asyetundissolved 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 phaseencoded 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 polymersolvent interface and the downfield shift seen where the
polymer should have not yet dissolved. Figure 26 provides some evidence that the poly
mer block was not centered at the x axis origin; the polymersolvent interfaces are not
distributed symetrically with respect to the x = 0 mm position.
Although this work failed to observe any chemical changes at the polymersolvent
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 phaseencoded 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 [26] can be rewritten in units of parts per million (ppm) as
Av v2 v1= 1 106( C)(Xv2 Xv). [28]
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) [29]
Also, when c = 2r as for a cylinder oriented perpendicular to the static magnetic field,
then
AVperpendicular = 2.09 x 106(X,2 Xv) [210]
Subtracting AL/perpendicular from Avparallel produces
I
A' = 6.28 x 106(Xv2 Xvl) [211]
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 108. 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 spatiallylocalized chemical changes as a polymer hardens.
CHAPTER 3
CONVOLUTION SPECTRAL IMAGING
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 threedimensional 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 twodimensional 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. 21 applied to a system of two weaklycoupled spins. The result appeared in
Eq. [23]. 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. 21, 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 weaklycoupled
spins, given by Eq. [23], 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 spinspin coupling are constant
because the start of detection occurs at a fixed time after the initial r/2 excitation pulse
(Bax79). The phaseencoded 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 phaseencode 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.
Phaseencoding of spectral information is the basis of the threedimensional 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 phaseencode
gradient as described in Chapter 2. To maintain the independence of the phaseencoded
spectral and spatial information, for each step in the variation of the phaseencode 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
phaseencode gradient is stepped to its next value. A complete cycle of time interval
stepping is repeated for each phaseencode gradient step.
The second spatial dimension of the three dimensions defined by the Sepponen tech
nique is produced by the process of frequencyencoding, 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
phaseencode 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. [23], which was derived for a single spin, the phase modulation
function corresponding to the gradient stepping has the form
h(kr,ri) e12^rrl, [31]
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 [32]
where is the gyromagnetic ratio, Gr(t) is a function describing the timedependent
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
form
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 Jcoupled to another:
h'(tl,l) = ei2wtl'l, [34]
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 [35]
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) [36]
FT[s'(t)] = S'(Vi) [37]
In the convolution spectral imaging technique, the amplitude of the phaseencode
gradient and the time interval are stepped simultaneously. Thus kr and tz are related by a
constant, q:
tz = r7kr. [38]
The resulting doublymodulated signal may be expressed by the product of Eqs. [33] and
[35]. 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. [38], the Fourier transform can be written
FT[s(kr)s'(ti)] = ,[S(uIv)*S'(ui)] [39]
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/)] [310]
Each of the convolution functions shown in Eqs. [39] and [310] 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 twocompartment sample containing two
different chemical species is shown in Fig. 3la. The function is the convolution of the r
space information shown in Fig. 3lb with the spectrum shown in Fig. 31c. 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. 3la
is as a series of spectral frequencies, each of which has an identical spatial fieldofview
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 threefrequency experiment of Bolton (Bol82). The convolution of
a spatial axis and the zeroquantum spectrum has been accomplished recently in a similar
manner by Hall and Norwood (Hal86a).
Although Eqs. [39] and [310] were derived for phaseencoding, the mathematics
are similar for frequencyencoding. The two processes differ in that free precession,
which is the source of spectral information, occurs in the presence of a gradient during
frequencyencoding. The result is still a doublymodulated signal, and Eqs. [39] and
[310] 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.
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The pulse sequence for the acquisition of a twodimensional convolution spectral
image is shown in Fig. 32a. The interval between the initiation of the pulse sequence at
successive phaseencode steps, TR, is kept constant. The time to the echo formation, TE,
is also kept constant; thus, the T2 relaxation weightings of the phaseencoding steps are
equivalent. Phaseencoding 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 phaseencoded axis is 1/At1.
The implementation of the pulse sequence of Fig. 32a with the frequencyencode
gradient, gfy, defined as the z gradient and the phaseencode 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 wellseparated resonances and that these techniques optimize the
available signaltonoise ratio. Thus convolution spectral imaging may be most applicable
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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. 32. 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
frequencyselective sincshaped 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 twostep 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 phaseencoded. 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
nonsteadystate magnetization which was not phaseencoded and instrumental errors
causing DC offset. Therefore gradient proportional phase incrementation (GPPI) of the 7r
pulse was also added to the twostep phase cycle (Gra86). This had the effect of moving
artifacts which appeared at the midpoint of the phaseencoded 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 phaseencoded. Artifacts due to instrumental DC effects remained at
the midpoint of the phaseencoded axis. Fortunately, these were quite minimal. The phase
cycle for the pulse sequence of Fig. 32a is shown in Table 31, and that for the sequence
of Fig. 32b is shown in Table 32.
To reduce truncation artifacts, it was ensured that the echoes corresponding to the
middle steps in the phaseencoding process were centered in the acquisition window. This
was accomplished in a setup mode before acquisition of the image data by turning all
phaseencode gradients off and placing the r pulse in the center of the time delay, T (see
Fig. 32). The amplitude of the frequencyencoding gradient on during the time, r, was
then carefully adjusted. This setup 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 wellseparated 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 phaseencode 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 spinspin coupled to another, which is subjected to the
twodimensional technique of Fig. 32a, again letting z be the frequencyencode dimen
sion and x be the phaseencode dimension. During the acquisition time, t2, the signal is
modulated in part by the function
h"(m,,t2) = ei2~v + mt2 ]
S"*' ;
Table 31
TwoDimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases
Phases
PhaseEncode Signal Pulses Receiver
Step Average
ir/2
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. 32a and its legend for the notation corresponding to this table.
Table 32
ThreeDimensional Convolution Spectral Imaging:
Relative Pulse and Receiver Phases
Phases
PhaseEncode PhaseEncode 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. 32b and its legend for the notation corresponding to this table.
