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 Title:
 Diffusion tensor imaging and the measurement of diffusion tensors in biological systems
 Creator:
 Yang, Lei, 1967
 Publication Date:
 1999
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 English
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 xi, 114 leaves : ill. ; 29 cm.
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 Subjects / Keywords:
 Anisotropy ( jstor )
Diffusion coefficient ( jstor ) Diffusion tensor imaging ( jstor ) Hydrocephalus ( jstor ) Imaging ( jstor ) Rats ( jstor ) Spinal cord ( jstor ) Tensors ( jstor ) Term weighting ( jstor ) White matter ( jstor ) Diffusion ( lcsh ) Dissertations, Academic  Physics  UF ( lcsh ) Physics thesis, Ph. D ( lcsh )
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 bibliography ( marcgt )
nonfiction ( marcgt )
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 Thesis:
 Thesis (Ph. D.)University of Florida, 1999.
 Bibliography:
 Includes bibliographical references (leaves 109112).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Lei Yang.
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DIFFUSION TENSOR IMAGING AND THE MEASUREMENT OF DIFFUSION TENSORS IN BIOLOGICAL SYSTEMS
By
LEI YANG
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
1999
Copyright 1999
By
Lei Yang
ACKNOWLEDGEMENTS
I would like to thank my supervisor, Dr. Andrew, for his kindness, patience, and encouragement of my graduate career. Without him, this dissertation would not have been finished. I learned much from his deep insights into Physics. I would also like to thank my research supervisor, Dr. Mareci, for his generosity and guidance. He led me and the whole research group into the diffusion imaging area, which has produced one paper, two SMRM conference oral presentations, three conference abstracts, and three more submitted papers. His excellent idea of introducing coherence transfer pathway into the diffusionweighting calculation was the starting point of the new formula and new diffusion tensor imaging methods. Also thanks to Dr. Inglis for teaching me every detail about NMR, giving me invaluable suggestions about my research. I would also like to give special thanks to Dr. Sullivan and to Dr. Yelton for taking care of international students like me who was thousands of miles away from home and struggling to survive in a foreign country.
I express my sincere thanks to many colleagues who have made my stay at the University of Florida something that I will remember fondly for years. I appreciate Dr. Wirth, Dr. Jones, Dr. Harris, Xeve Silver for their help in surgery, Dr. Briggs for letting me use his office, Dr. Fitzsimmons for helping me build the rf coil, Dan Plant for hardware support, and Haiquan Dai for computer support.
iii
This thesis is dedicated to my father, Yuhui Yang, who survived the disaster of
the cultural revolution since 1967. It was his tolerance and love that supported me and my whole family through the dark ages.
All these achievement belong to Jesus Christ, my lord and savior. He is always with me, no matter what happens. May all praises go to God!
iv
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS...................................... ...... .......... ii
ABSTRACT ............................................................ ix
CHAPTERS
1 INTRODUCTION AND BACKGROUND................................................. 1
Intro du ctio n ..................................................................................... 1l
B ackground ............................................................................... . .. 4
Brownian Motion and Diffusion Distance......................................... ...... 4
Diffusion Effects on NMR...............................................................5
Methods for Apparent Diffusion Coefficient(ADC) Measurement.................6
Bipolar gradient pulse spinecho sequence..........................................6
Bipolar gradient pulse stimulatedecho sequence......................................7
Problems for these two techniques...................................................7
Diffusion Coefficient as an Important Parameter in Biomedical Systems............8
Clinical application of diffusion MR imaging.......................................9
Unsolved Problem: bValue Calculation for Any Order of Quantum Coherence....9
bValue calculation problem in different sequence................................9
bValue calculation problem in multiple quantum coherence......................11
C onclusion ................................................................................... . 11
2 A COHERENCE TRANSFER APPROACH TO GRADIENT
WEIGHTING FACTOR CALCULATION.......................................... ...... 17
T h eory ....................................................................................... .... 17
Spin Evolution in the Presence of Anisotropic Diffusion...................... ...... 18
Explicit Effect of RF Pulses...............................................................21
Application to Different Pulse Sequences....................................................23
Applying the Theory to Single Quantum Coherence Sequences
in Isotropic Condition................................................................... 24
Gradient echo sequence............................................................. 24
Spinecho sequence................................................................. 24
Stimulcatedecho sequence.......................................................... 25
Double Quantum Coherence..............................................................25
3 DIFFUSION TENSOR METHODS..........................................................28
V
Data Acquisition and Processing...... .............................................28
Formulas for Data Acquisition........................................................ 28
Data Processing............................................... ... ........ 30
Spin Echo and Stimulated Echo Diffusion Tensor Imaging............................ 30
bMatrix of Spin Echo Sequence.................................................... 30
bMatrix of Stimulated Echo.......................................................... 32
Experim ents............................................. ............ ....... ... ... 34
Conclusion............................................................ ...36
4 FAST DIFFUSION TENSOR IMAGING USING A MULTIPLE
STIMULATED ECHO SEQUENCE..................................................... 41
Introduction................................................................................. 4 1
Methods..................................................................... ............41
bMatrix Calculation for Stimulated Echo Sequence....................................41
General Library for bMatrix Calculation for Stimulated Echo................... 43
Multiple Stimulated Echo Sequence..................................................... 44
R esu lts....................................................................................... . . 46
Water and Celery Experiments........................................................46
Rat Brain in vivo Experiments..........................................................51
5 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED
TRACE IMAGING OF RAT SPINAL CORD IN VITRO.............................. 53
Introduction .................................................................................... 53
Diffusion Tensor Imaging............................................................... 53
A nisotropy Index ............................................................................54
Anatomic Structure of Spinal Cord..................................................... 55
Gray matter of spinal cord............................................................... 55
White matter of spinal cord..............................................................56
M ethods................................................................................... . . . 56
Fixation of Rat Spinal Cord................................................................ 56
Diffusion Tensor Imaging Experiments................................................56
D ata A nalysis............................................................................. 58
R esu lts....................................................................................... . . 58
Diffusion Tensor Values of Normal Spinal Cord............................... ......58
Diffusion Tensor Values of Injured Spinal Cord ....................................59
Colors in Gray Matter Region..........................................................59
C olor Trace Im ages.................................................................. ......59
Colors in Gray Matter Region of Normal Rat Spinal Cord.........................59
Colors in White Matter Region of Normal Rat Spinal Cord...........................60
Color Trace Images of Injured Spinal Cord...........................................60
Trace is an Important Parameter for Different Nerve Tissues...................... 60
Anisotropy Index of Normal Rat Spinal Cord......................................... 60
Vi
Anisotropy Index of Normal and Injured White Matter Tissues ................61
Diffusion Tensor Imaging of Human Spinal Cord............................... ..61
C onclu sion .................................................................... .... .......6 1
6 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF NORMAL, HYDROCEPHALUS AND SHUNTED RAT B RA IN IN VITR O ..................................................................................75
Introduction ......................................................................................75
Infantile Hydrocephalus.................................................................75
Shunt Treatment of Hydrocephalus...................................................76
White Matter Pathology in Hydrocephalus.......................................... 76
MR Imaging Techniques.................................................................77
Anatomy Structure of Rat Brain.......................................................78
M eth o d s..........................................................................................7 8
Fixation of the Rat Brain...............................................................78
Shunt Surgery .......................................................................... ...78
M icroim aging ............................................................................. 78
D ata Processing.......................................................................... 80
R esults....................................................................................... . . 8 1
Diffusion Tensor Values of Normal Rat Brain.......................................81
Diffusion Tensor Values of Hydrocephalus Rat Brain..............................81
Diffusion Tensor Values of Rat Brain...................................................81
Color Trace Im ages...................................................................... 81
Color in Cerebral Cortex.................................................................81
Color in Corpus C allosum .................................................................82
Anisotropy Index of White Matter in Normal and Hydrocephalic Rat Brain .......82 Diffusion Tensor Trace of Lateral and Dorsal Cortex..............................82
Stimulated Echo Diffusion Tensor Imaging.............................................83
C onclusion .............................................................................. . . . .. 83
7 SUMMARY, INTERPRETATION, AND CONCLUSION..........................98
Introduction ..................................................................................... 9 8
Achievements and Signification..............................................................98
A New Formula for Evaluating Diffusion Weighting ................................. 98
Two Diffusion Tensor Imaging Techniques..........................................99
Diffusion Tensor Imaging of Normal and Injured Spinal Cord.................... 99
Diffusion Tensor Imaging of Normal, Hydrocephalus and Shunted Rat
B ra in .........................................................................................10 0
Diffusion Model in Molecular Level.......................................................100
Future Studies..................................................................................107
Double Quantum Diffusion Tensor Imaging and Spectroscopy...................107
Spinal Cord Injury Model............................................................... 107
In Vivo Diffusion Tensor Imaging......................................................107
S um m ary ........................................................................................107
vii
R E F E R E N C E ..................................................................................109
BIOGRAPHICAL SKETCH................................................................ 113
viii
Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
DIFFUSION TENSOR IMAGING AND THE MEASUREMENT OF DIFFUSION TENSORS IN
BIOLOGICAL SYSTEMS
By
Lei Yang
May 1999
Chairman: E. Raymond Andrew
Major Department: Physics
This dissertation revisits the theory of diffusionweighting factor of
gradients for any sequence and presents a new comprehensive formula which not only unifies all former results, but also predicts new results for unsolved pulse sequences. Then, on the basis of this new formula, two diffusion tensor imaging techniques were built, tested, and applied to the rat spinal cord and brain, thus showing the biological and clinical significance of diffusion tensor imaging.
There have been many previous attempts to derive an expression of diffusionweighting factor of gradients in different pulse sequences, but until now, nobody has been able to derive a formula for any doublequantum and multiplequantum coherence sequences, only under in case had this problem been solved experimentally. This dissertation introduces a new method which not only unifies all different expressions for single quantum coherence, but also predicts the expressions for doublequantum and
ix
multiplequantum coherence sequences. Next, using the new formula for diffusionweighting factor, two diffusion tensor imaging techniques were built: spin echo diffusion tensor imaging and multistimulated echo diffusion tensor imaging. The EchoPlan Imaging (EPI) spin echo diffusion tensor imaging had been built in 1994, but the multiecho stimulated echo diffusion tensor imaging technique, along with Basser's EPI stimulated echo diffusion tensor imaging technique, were the first two techniques in the world that used stimulated echo sequence in diffusion tensor imaging.
In this dissertation, diffusion tensor imaging techniques were first applied to
normal and injured rat spinal cords and brain, color trace images were presented which clearly show the fiber orientations of the spinal cord that were not accomplished by any other MRI. We found that the injury of the spinal cord causes the loss of anisotropy of white matter, which is consistent with other results using apparent diffusion coefficient (ADC). It is interesting that there were substantial changes in diffusion characteristics in the injured area which appeared normal by conventional imaging. These results imply that there are consequences of spinal cord injury which dramatically alter axon structure but do not change water content, so are not detected by conventional imaging. We found that the hydrocephalus of the rat brain causes the loss of anisotropy of white matter, and hydrocephalus results in increased water content in the brain surrounding the ventricles. This has been confirmed by independent measurements.
This is a new direction in the noninvasive investigation of the normal and abnormal brain and spinal cord using a new imaging technique to study tissue organization, pathological changes, and recovery with treatment. The technique could ultimately be applied to clinical situations (in the same way as ADC is now being applied
X
to measure the progress of clinical stroke). The dissertation concludes with a theoretical interpretation of the measured diffusion constants.
xi
CHAPTER 1
INTRODUCTION AND BACKGROUND Introduction
The discovery of nuclear magnetic resonance (NMR) in 1946 (Bloch et al, 1946, Purcell et al, 1946) brought the world a powerful investigative technique which can be used in many different scientific fields. The publication of the first magnetic resonance imaging in 1973 (Lauterbur, 1973) created a big impact and opened the door for NMR in medical science. Since then, MRI has become one of the best noninvasive imaging methods for providing high quality images of soft tissue in the normal and injured condition. Gillies (1994) gives a general introduction on NMR in biomedicine.
The repair of spinal cord injury requires that doctors be able to see the difference between normal and injured white matter as a first step, because the extent of white matter spared following spinal cord injury (SCI) may be related to the degree of function that is eventually recovered in the limbs caudal of the lesion (Schrimsher, 1993). However, conventional magnetic resonance imaging of the spinal cord cannot provide such explicit information. It has been found that water diffusion along the white matter tract is faster than that perpendicular to it, which means that the water diffusion of white matter is anisotropic (LeBihan, 1991). When the white matter has been injured, this anisotropy is also changed (Ford, 1994). Diffusionweighted imaging has been an effective technique for identifying nerve tissue pathology since it permits the discrimination of gray and white matter structures (Moseley, 1990) and normal and
I
edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (TI, T2 ). Diffusionweighted imaging is, however, only an approximation of the actual diffusion sensitivity of an anisotropic sample. A muchimproved estimation can be achieved by measuring an apparent diffusion tensor (ADT) (Basser, 1994), which is based on the diffusion coefficients in various directions. In this way it should be possible to identify regions with varying degrees of injury of the spinal cord .
The Dr. Mareci research group in the University of Florida Center for Structural Biology has successfully pinpointed diseased white matter tissue. As an extension of this pioneering activity, this dissertation will present a set of methods to advance our ability to determine the exact location of injured white matter. The objectives are listed below:
A General formula for the diffusionweighting factor will be given, its application to the special pulse sequences ( spin echo, stimulated echo, double quantum COSY, etc.) will be presented, and the results will be compared with other people's results, which shows consistency.
Methods for producing ADT images will be presented and rat spinal cord and
brain ADT images in vitro will be acquired and analyzed in view of pathological changes between normal and injured conditions.
Fast ADT imaging methods will be presented, which will be able to reduce the acquisition time to perform ADT imaging of rat spinal cord in vivo. The ADT data for normal and injured spinal cord in vivo will be acquired and analyzed.
This dissertation consists of seven chapters. In the first chapter, work that has been done in diffusion will be reviewed, some unsolved problems will be pointed out, and the methods to solve these problems will be discussed.
The second chapter will discuss the calculation of the diffusionweighting factor from gradients, and its application to the ADT. Calculating the diffusionweighting factor is the key to diffusioncoefficient measurements, but calculating these factors is different for different sequences under different orders of quantum coherence. This alculation is also very difficult when there are many gradients. Here the application of the concept of coherence transfer pathway to calculating the diffusionweighting factor will be introduced, which will give a general expression for the diffusionweighting factor calculation for any sequence with an arbitrary order of quantum coherence.
There are basically two methods to do diffusion tensor imaging, spin echo and stimulated echo. In the third chapter the use of these two methods will be described in detail, and their advantages and disadvantages will be discussed. In Chapter 4, fast diffusion tensor imaging techniques which have been used in in vivo experiments will be presented and compared. In Chapters 5 and 6 these two methods will be applied to the rat spinal cord and rat brain. Data from normal and injured spinal cords and from normal, hydrocephalic, and shunted rat brains will be presented. The biological significance of these results will be discussed. In Chapter 7 the results obtained in this dissertation are summarized, the values of measured diffusion coefficients are analyzed, and indications for future work are given.
4
Background
In this chapter, several topics listed below will be discussed in order to provide basic knowledge about diffusion tensor imaging.
Brownian motion and diffusion distance.
Diffusion effects on NMR.
Methods for apparent diffusion coefficient measurement.
Diffusion coefficient as an important parameter in biomedical systems.
Diffusion time and tissue structure.
Unsolved problems.
Brownian Motion and Diffusion Distance
In a nonuniform system where species do not distribute uniformly, the selfdiffusion of a particle is described by Fick's Law (Callaghan, 1991):
ZIP'= DV'2p,(j: 'i: ,t) [1.1]
Here, Ps(j'l' , t) is the probability of finding a molecule, originally at a position i', at a position j at time t, D is the diffusion coefficient.
Under equilibrium conditions, the diffusion of the molecules is called Brownian motion (Callaghan, 1991), and Ps has the initial condition:
PJ(Fi ,Ft) = 9W i F [1.2]
For unrestricted diffusion, P, > 0 as i' coo. This combined with Eq. [1] yields:
Ps(f'I ,J) =(47Dt) 32//4Dt] . [1.3]
This equation means the probability of finding a molecule originally at a position f' at a
position j at time t , is a Gaussian distribution.
5
The diffusion distance (RMS, root mean square displacement) < (F F ')2 > is given by:
< (F F )2 >= fdF .(F F ')2 .e(i' )214D / (4=Dt) 12 (6Dt) [1.4]
This, socalled Einstein equation, gives direct interpretation of the diffusion coefficient. In one dimensional case:
<(FF') >= 2Dt [1.5]
Diffusion Effects On NMR
Molecular selfdiffusion was first observed to have an effect on NMR in 1954 (Hahn, 1954). In Hahn's spin echo experiment, measured T2 values were found to be abnormally low for liquids, such as water, an effect exacerbated by the poor homogeneity of the magnet at that time.
The diffusion effect on magnetization in a magnetic field can be solved using the Bloch equation (Torrey, 1956, Stejskal and Tanner, 1965). In a single quantum coherence spin echo experiment, one gets:
M= 4 *eT eW [1.6]
te t,
where b = y2 ( G(t")dt",)2dt' [1.7]
0 0
in which G(t) is the magnetic field gradient. In the pulsed gradient spin echo sequence (Table 1.2), b=y2G232 (A 5 /3) [1.8]
6
This is called the Stejskal and Tanner equation. Here 6 is the gradient duration, and A is the separation between two pulsed gradients.
Methods for Apparent Diffusion Coefficient (ADC) Measurement
In biological systems, the diffusion of water molecules is not only hindered by other molecules, but also restricted by some obstacles such as cell membranes, which cause the mean square displacement to be shortened. The measured diffusion coefficient, which is based on the signal attenuation of NMR signals ( Eq. 1.6), is called the Apparent Diffusion Coefficient (ADC).
Based on the basic principles discussed in the previous section, several NMR techniques have been proposed to study the diffusion. The most popular techniques are the bipolar gradient pulse spin echo sequence and the stimulated echo sequence.
Bipolar gradient pulse spinecho technique
The effect of diffusion on spin echo signals in the presence of a pair of gradients ( Table 1.2) is given by:
y2G25A , 13)D
M= (4 e , [1.9]
where S is the duration of each pulse, and A is the pulse interval. This is usually called Spin Echo ADC Method (Table 1.2).
Compared with the technique using a constant gradient pulse, the advantages of using bipolar gradients are (1) It is better suited for imaging, and (2) It has a welldefined diffusion time A.
7
Bipolar gradient pulse stimulatedecho technique
A stimulated echo is generated from a sequence comprised of three RF pulses
separated by time intervals tc and r, between the pulses (Table 1.2). After the end of the second RF pulse, part of the transverse magnetization (exactly half in the case where 900 pulses are used) is stored as longitudinal magnetization, which becomes insensitive to field inhomogeneities. The third RF pulse returns the stored magnetization to the transverse plane at time tc after the third pulse. The amplitude of the stimulated echo is, in the case where the three RF pulse are 900 pulses ( Hahn, 1950):
r2 2r
M= e T 2 [1.10]
M=Z 3*ele2 *e G2] /D
2
This is called the stimulated echo ADC method (Table 1.2).
The advantages of using bipolar gradients are:
This technique is particularly useful for qspace diffusion experiments where a long diffusion time is required. Because T, is usually much larger than T2 in tissues, longer diffusion times can be achieved with a stimulated echo sequence than with a spin echo sequence without the usual signal loss due to T2 decay.
When the tissue has very short T2, the spin echo cannot be used, while the stimulated echo is useful due to the signaltonoise consideration.
When the gradient system has large eddy current, the stimulated echo will eliminate most of the effect due to the eddy current, while spin echo cannot. Problems for these two techniques
These two methods have been used (Stejskal, 1965, Cleveland, 1976, Cooper, 1974, Ford, 1994) to calculated ADC in spectroscopy and imaging experiments.
8
However, the Stejskal and Tanner equation does not take into account the diffusionweighting factor contributed by imaging gradients, which includes bvalue contributed by each imaging gradient and bvalue contributed by the interaction between imaging and diffusion gradients, which are called cross terms.
Thus the ADC methods only measure approximate diffusion coefficients ( Mattiello, 1994) and need to be improved.
Diffusion Coefficient as an Important Parameter in Biomedical Systems
MRI and spectroscopy are the only ways to study the molecular process in vivo noninvasively. The Brownian motion of molecules in liquids is restricted by the obstruction inside the biological systems, such as fibers, intracellular organelles (Nicholson, 1981) and membranes. So, water diffusion coefficients vary according to tissue microstructure, physiological or pathological state. This is a potential source of tissue contrast for characterization of functional studies.
Using Eq. 1.6, one will be able to measure the diffusion coefficient by varying the b values and keeping diffusion time the same. If the measured diffusion coefficient decreases with diffusion time t, this would indicate that the molecules are experiencing an impermeable or semipermeable barrier and can not diffuse beyond this barrier freely.
In normal diffusion experiments, typical diffusion times lie in the range 10200ms. For water in room temperature (20 C) with D~2* 10 mm2/s, this range of diffusion time yields RMS path length of 630 pm, which is of the same order as nerve cell diameter(1 030 im) and larger than nerve fiber diameter (1p jm). So water diffusion
9
allows microdynamic studies on a scale that is much smaller than the resolution of macroscopic NMR images.
Clinical application of diffusion MR imaging
It is found that water diffusion in different tissues vary (about 2 to 10 times less than that of pure water (table 1.1)). Also, even in the same tissue like white or gray matter, diffusion coefficient is not the same in different directions (LeBihan, 1990, Inglis et al., 1996), in different pathological conditions (Ford, 1994). Diffusion thus appears as a new source of contrast for MR imaging. It has been shown that there is no correlation between the diffusion coefficient and the relaxation times T1,T2. It has been found that T,, may be normal in the diseased state, while diffusion is changed such as in early brain ischemia (Moseley, 1990).
Unsolved Problem: bValue Calculation for Any Order of Quantum Coherence bValue calculation problem in different sequence
As we have discussed before, bvalue calculation is the key to diffusion coefficient measurement, but it is also very complex because it consists of:
1. bvalue contributed by each gradient.
2. bvalue contributed by the interaction between two gradients, which are called cross terms.
The Stejskal and Tanner equation ( Eq. 1.8) for a pair of rectangular gradients is unable to calculate the diffusion weighting factor of the following cases:
1. In normal experiments there are always background gradients.
2. In imaging experiment, there are many imaging gradients.
3. There are different shapes of gradients, such as trapezoidal gradient, sinusoidal gradient.
10
4. In multiple quantum coherence (MQ) experiments, there are gradients in different coherence pathways (Table 1.2).
Some scientists try to use different pulse sequences to get crossterm free images. Noeman (1990) ran the experiment twice with the diffusion gradient in opposite directions. The geometric average of the signal values has no cross term effect from imaging and diffusion gradients, but the cross terms between imaging and imaging gradients are still existing. The multiecho pulse method proposed by Van (1991) and Williams (1978) reduces medium and longrange gradients. The combination of alternating field plus multiple RF pulse method proposed by Karlicek and Lowe (1980), and Hong (1992) for spin echo and Cotts (1989) for stimulated echo eliminate medium and longrange localgradient completely. All these methods are unable to derive offdiagonal terms of the diffusion tensor.
On the other hand, Mattiello ( 1994) successfully calculated all the crossterms and self terms of the gradients by a pairwised method in single quantum coherence in 2D FT spin echo sequence, but it is not clear how to solve this crossterm problem in gradient echo, stimulated echo, and in MQ conditions. According to Torrey (1956), the work by Stejskal and Tanner's (1965) and Tanner (1970), the bvalue expression for GE, SE, and STE are quite different. In a review paper (LeBihan, 1991), one can find that, when a pair of rectangular gradients apply in STE, one between first and second 900, one after third 900, the StejskalTanner equation still applies. This might seem to imply the bvalue calculation for spin echo and stimulated echo are actually the same, but this conclusion is not supported by either experiment or theoretical analysis.
I I
bValue calculation in multiple quantum coherence
The bvalue calculation of multiple quantum coherence has been studied (Martin, 1978; D.Zax, 1983) in a special case, when a pair of rectangular gradients is applied in the double quantum evolution time (Table 1.2), where
b =y22 2 G 22 (A 5/3),[1.11]
which is 22 times larger than that of the single quantum case. This relation was supported by a set of experiments. However, it is not clear what the crossterm is between gradients in single quantum evolution time and gradients in multiple quantum revolution time, for example, in double quantum COSY experiments (Fig 1.1)
Conclusion
In summary, diffusion spectroscopy and imaging have been very powerful methods for studying tissue structure in different biological states. However, these methods need to be improved in order to show more accurate diffusion sensitivity. In this dissertation, a new theory and methods and their applications to the study of the spinal cord and brain will be presented.
12
900 900 900
G G G3
2
P0
2
Fig 1.1 Double Quantum COSY experiments
Table 1.1: Diffusion coefficient of different tissue under different condition. (unit: 10 ' mm2/s)
control pathology temperature source of data
and method used
orientation parallel to fibers perpendicular to parallel to fibers perpendicular to
free water 2.34 25.5 0C Stajekal (1965)
spectroscopy
muscle 1.40 1.03 25 0C Cleveland (1976)
spectroscopy
liver 0.58 0.70 25 C Copper(1974)
spectroscopy
dosal column 1.00 0.15 0.55 0.40 Ford (1994)
white matter ADC
(spinal cord)
lateral column 1.00 0.20 0.50 0.45 25"C Ford (1994)
white matter ADC
(spinal cord)
Table 1.1: continued
control pathology temperature source of data
and method used
orientation parallel to fibers perpendicular to parallel to fibers perpendicular to white matter 0.9 0.45 37"C Horsfield (1994)
(human brain) ADC
gray matter 0.75 0.55 0.55 0.50 25 0C Ford (1994)
(spinal cord) ADC
white matter 0.92 0.34 0.73 0.60 20 C Yang (1996)
(rat brain) ADT
dorsal cortex 0.99 0.70 1.20 0.71 200C Yang (1996)
(rat brain) ADT
Table 1.2 Pulse sequences and their bvalue expression.
Pulse sequence bvalue expression bvalue expression source
(general) (a pair of rectangular
gradients)
Gradient echo (SQ)
9(r
tte
42 te C' d 2G 2 2(A / 3) Torrey
L A  f y (f G(t ")dt ")2dt' 1956
Spin echo (SQ) (F (t)2* *f (t))*
90IO 0 Stejskal
(F (t)2* *f (t)t y2G2 2(A  / 3) and
A J Tanner,
G 1965
Stimulated echo (SQ) 2
(fj dt"g (k +
, , (~~~~2T1I+T2 22217
2 1) ( J dt"g (k ,t") 2 G 2 2(A /3) Tanner,
1r2 r1+(2
+ fdtf f dt"g (k ,t)l)
Table 1.2 Continued
pulse sequence bvalue expression bvalue expression source
(general) (a pair of rectangular
gradients)
Spin echo (DQ)
9 0w 180 9w
N/A y 2 22 G2,2(A  5 / 3) Martin,
1978
G A
[ 1
Spin echo ( Multiple Quantum of order n) N/A y2n2G252(A 5/3) Bax,
1982
General format
G tMareci ,
t dt g(k t G _  / 3) Yang, and
6 0Inglis,
" J d g .* 19 9 5
f dt g (1 ,t)
CHAPTER 2
A COHERENCE TRANSFER APPROACH TO GRADIENT WEIGHTING FACTOR
(bVALUE) CALCULATION
As we have discussed in Chapter 1, calculating the diffusion weighting factor (bvalue) is the key to diffusion coefficient measurements, but calculating these factors is different for different sequences under different orders of quantum coherence. Calculation is also very difficult when there are many gradients. Here the application of the concept of coherence transfer pathway to diffusion weighting factor calculation will be introduced, which will give a general expression for the diffusion weighting factor calculation for any sequence with an arbitrary order of quantum coherence.
Theory
Scientists have been trying to derive bvalue expressions for different pulse
sequences. In 1956, Torrey obtained a general expression (Torrey, 1956) for the Gradient Echo sequence ( Fig 1.1) using the Bloch equation. Nine years later, Stejskal and Tanner derived a general expression (Stejskal, 1965) for the Spin Echo sequence using Bloch equation and a step function. In 1970, Tanner successfully used the treatment of Carr and Purcell (Carr, 1954) to get a general expression for the stimulated echo sequence (Tanner, 1970). The bvalue calculation of multiple quantum coherence sequences has been so difficult that nobody has been able to use the old methods mentioned above to solve it.
17
18
The bvalue calculation of multiple quantum coherence has been studied (Martin, 1982) in a special case when a pair of rectangular gradients applied in the double quantum evolution time (Fig. 1.1), where b = y 2 G 22 (A ,5 / 3) [2.1]
which is 22 times larger than that of the single quantum case. This relation was supported by a set of experiments.
Spin Evolution in the Presence of Anisotropic Diffusion
By measuring the signal attenuation caused by diffusion, scientists will be able to measure the diffusion coefficient. The diffusion effect can be described by the BlochTorrey equation of motion for transverse magnetization (Callaghan, 1991):
=Miy + M+V D  VM+, [2.2]
where M+ = M, + i Mv , D is the diffusion tensor, G is the gradient, and VT is the transpose of V. The solution to this equation has the general form:
M+(r, t) = A(t)e'yrF(t) [2.3]
where F(t) = dt' G(t'). A(t) is the timedependent amplitude of the magnetization and
0
P(t) can be visualized as the contribution of an applied gradient to the precessional phase, p(t) = y r  P(t), of the magnetization at time t .
Information on the diffusion process is contained in the timedependent amplitude coefficient, A(t). Expanding Eq.[2.3] in terms of this general solution, given in Eq.[2.3],
19
results in the following definition of this coefficient,
Y dt'F (t')D.F(t')
A(t) = M0 e 0 [2.4]
Expanding the diffusion dependent portion of the exponential factor in terms of Cartesian coordinates results in,
T
F (t)  D  F(t)= X X F(t)Fj (t)Di l=x,y,zJ =x,y, z
Then the timedependent coefficient becomes,
 Y Y b;;(t)Di
A(t) = MO e 'z [2.5]
where MO is the equilibrium magnetization and the diffusion weighting factor, b7;, can be written as the following,
by(t) = y2 dt'i (t')F(t') [2.6]
0
In general, a pulse sequence can be considered as a series of RF pulses and timedependent gradient pulses. If each gradient pulse is described by a timedependent function, g(k,t) for the kth gradient pulse, the general form of all gradients can be written as,
n
G(t) = g(k, t) [2.7]
k=1
where,
20
0 , tk :! t :! tk + '5k g(k, t)=0 , 0 tk t < tk + 4 [2.8]
Using this timedependent form of the gradient,
n t
F(t) = Y fdt' g(k,t'), [2.9]
k=1 0
the diffusion weighting factor becomes, n n t
bU(t) = 'Y2 f dt' iQ (k, t')F {t' k=1l=1 0
n7 n dt It' Ift'
= d dt" gi(k, t" dt" gjQ,t") [2.10]
k=11=1 0 x0 / 0
n n
I bi.(t))kI
k=11=1
where
(b j(t)) kI 2 fdt'(F(k,t'))(Fj (l,t')) [2.11]
0
Here k, 1 are series number of gradients, and i, j are the directions that the gradient applies.
Eq.[11] gives the diffusion effect of a pair of gradients in the pulse sequence, and the sum of such pairwise combination of all gradients gives the total diffusion effect of all gradients in the pulse sequence (Eq. [10]), so Eq. [11] is the start point of bvalue calculation for any specific pulse sequence. Recently, Mattiello (1994) introduced analytical expressions for the evaluation of the diffusion weighting matrix. Since they
21
explicitly include the effect of a spin echo RF pulse sequence in their analysis, it is only applicable to spin echo imaging sequences. By contrast, our analysis is completely general in terms of RF, for known but arbitrary shaped gradients. Explicit Effect of RF Pulses
The method of calculating diffusion weighting factors developed above has been done by Torrey (Torrey, 1956) in the case of isotropic diffusion. It can be applied directly to relatively simple sequences, such as the gradient echo sequence which has only a single RF excitation pulse followed by a series of gradient pulses. However, in the more complicated sequence with more than one RF pulse, Eq. [2.10] can not be applied directly. Using the step function to deal with the effect of second 180' RF pulse (Fig. 1.1), Stejskal and Tanner derived a general expression (Stejskal, 1965) for Spin Echo sequence using the Bloch equation. Later, Tanner successfully used the treatment of Carr and Purcell (Carr, 1954) to deal with the second and third 900 RF pulse (Fig 1.1) and obtained a general expression for Stimulated Echo sequence (Tanner, 1970) . The bvalue calculation of multiple quantum coherence sequences has been so difficult that nobody can use the old methods mentioned above to solve it.
The effect of any sequence of RF pulses can be understood in the context of
coherence transfer pathways (Bodenhausen). The coherence transfer pathway is defined by the value of the coherence order, pi, during the interval following the ith RF pulse. For systems of isolated spin1/2, pi can be 1, 0, or 1. The magnetization will form an echo during interval n, due to precession in the main field, whenever
22
n
Zpgr =0. [2.12]
i=1
Here r is the duration of the ith interval. Using the same formulation, an echo is formed, due to precession in the applied gradient field, when the following condition is satisfied,
n
pi 0, [2.13]
i=1
where F(i,81)= fdt' j(i, t').
ti
Since the effect of the RF pulse sequence on the applied gradients is contained in the coherenceorder pathway (through Eq. [2.13]), this can be incorporated into the calculation of the diffusion weighting factors in the following fashion. The RF pulse sequence can be included through the definition of an effective gradient, which includes the coherence order, by modifying Eqs. [2.7] and [2.8] to be,
n
Y (t) (k, t) [2.14]
k=1
where
g (k,t)= [2.15]
0 ,0 t < tk or t> tk + 4k
23
This modified form of the gradient can be substituted directly into Eqs [2.911], resulting in the final diffusionweighting factor for any sequence of any order of quantum coherence.
Application to Different Pulse Sequences Now we apply the concept of coherence pathway to the bvalue calculation for
some specific sequences, consider a general sequence (Fig 2.1), the general equation of btensor becomes
bij(t )= YI Zdt' Sdt" g(k t") * dt" *(1 t)
k=lk=1O 0 9 t) 0t9( [2.16]
The self and cross terms of gradient G, G, are:
(bij(t) ) = 2PkpGkGI(i +12ti61/2  /2) [2.17]
(by(t)) y2*pG2(s(tt,2*s,/3g/2)*2/2+ [2.18]
where p,, Pk is the coherence order, and v = k or 1 , s is the raising time of the gradient.
The bvalue of any sequence (including GE, SE, STE for any order of quantum coherence) are the sum of different combinations of these two buildingblock terms. So this provides people a straightforward way to analyze and create crosstermfree sequences or doing diffusiontensor calculation. The above formulas are tested by applying them to some wellknown sequences (such as single quantum coherence spinecho and stimulatedecho experiments) and getting same results as what has been done by other scientists using other methods, also they give predictions about multiquantum coherence gradient diffusion weighting which has not been done before.
24
Applying the Theory to Single Quantum Coherence Sequences in Isotropic Condition
Gradient echo sequence
For the general sequence (Fig 2.1), with a = 90,8p = 0, r = 2, the sequence reduces to Gradient echo sequence, where Pk=Pm.=PI=1. Consider only two gradients in isotropic diffusion: Gk=  G1, 1k = 51
b (t)= (b (t) +b(t) )+2* (b (t))k
(b kk li k
=72* G (S ((ti tk)Sk / 3)9k* E2 / 6+ 3 / 30) [2.19]
which is consistent with the results in Callaghan's book (Callaghan, 1991). Spinecho sequence
For the general sequence (Fig 2.1), with a = 90,8 = 180, r = r2 te / 2 , the sequence reduces to Spinecho sequence, where Pk=1, p,=.
Using the condition that the echo is formed:
, r] I te
f dtg (k t)= fdt g (k ,t ) [2.20]
0 =1
The general equation of bvalue becomes
b(t) = 72 d Y dt" (g (k ,t" < r1)  g (k , t"> r 1))
=J (F (t)2* *f(t) *(F (t)2* *f(t) dt [2.21]
0 2.1
25
where (t) = Gi(t)dt, = 0,when t < r I, = 1 when t > r l, which yields the Stejskal Tanner's equation (Stejskal, 1965). Stimulatedecho sequence
For the general sequence (Fig 2.1), with a = 90,,8 = 90, r I w r 2, another RF pulse 0= 90 at t = r 1, the sequence reduces to stimulatedecho sequence, where Pk= 1, pm=0, pi=IUsing the condition that the echo is formed:
n 1 1 n r 1 2
fdt" g (k ,t") 1'dt"g (k t)
0 = r 2
The general equation of bvalue becomes nnrf]+ r2+ry2 t
bij(r1+r2)= y2 I + + dt' idt"fg,(k ,t")
== 0 rl r2 ) 0
n ri rrl+r2 rlr (rl+r2
=Y( (JdtJdt"g (k ,t) +(r2r1 ) f dt"g (k t")) + Jdt" .f dt"g (k ,t)J
k= 0 r 2 r 2 t"
[2.22]
which yields J.E Tanner's equation (Tanner, 1970). Double quantum coherence
In doublequantum COSY experiment (Fig 2.2): (bijt) )ki=r2GkG,(r+r2tlsI2 E /2 ) (biJt)) =y24 * m(ttm2*5/ 31/2)sm*e2 /12+c /60) etc.
26
These formulas are still waiting for experimental verification.
If Gk, Gm are both between 0 and P, and Gk = Gm = G, E=O:
b= 2 G 2g2(t tk./3)
Which yields to Martin's result (Martin, 1982).
It will be seen that the general formula developed in this chapter not only includes special cases previously used, but is also valuable for the experimental work documented in this thesis and for other new sequences.
27
Gk
t k t k+ 6k + 8
P
t
t I + 1 + E
m
t
coherence pathway
Fig 2.1 General pulse sequence
0
'~ P
v Gm
ti
Pm=2
coherence pathway
t
for doublequantum coherence COSY experiment.
Fig 2.2 Double quantum coherence sequence and coherence pathway
Gk
tk
tk+ 6k
GI
tI+ 61
+P
G
61 rI
CHAPTER 3
DIFFUSION TENSOR METHODS
As we have discussed in Ch. 2, the signal attenuation due to the diffusion of molecules is described by Eq. [2.5], and diffusion weighting due to the gradient is expressed by Eq. [2.16]. In this chapter, we will apply these equations on the spin echo and the stimulated echo sequence and build up methods for diffusion tensor imaging.
Data Acquisition and Processing
Formulas for Data Acquisition
Taking the logarithm of Eq. [2.5], we obtain:
InS = Ybij*DijS(0) [3.1]
ii
where S is the NMIR signal intensity with diffusion gradient, and S(0) is the signal intensity without diffusion gradient. This is a 6variable linear function where bij=bji. We use multivariable linear regression (Rao, 1965) of Eq. [3.1] to calculate Dij in a voxel. All echo intensities are measured for a series of gradient pulse sequence in which gradients are applied in seven nonlinear directions. In each direction, m measurements of S are made at different gradient strength. These 7*m observations are stored as an (7*m)*1 column vector.
28
29
InSI
inS2 Y= InS3
inS7 *
We also define the column of vector of parameters to be determined:
DiI D12 D13 D= D22 D23 D33
,inS(0)
Then Eq. [3.1] can be written as:
X*D=Y
where X is the bvalue matrix:
2*b41 2*b31 b42 2*b 1 2*b31 b 2 2*b2 2*b31 b2
2*b21 2*b, b322
2 *bl2 2 *b 2 2 *b32
bi b33 b3
b33
2*b32 b33
1
1
1
1,
Then the diffusion tensor b is (Roa, 1965):
[3.6]
D=(X'X)'X'Y
[3.2]
[3.3]
[3.4]
b2
b2 X b3
Ibn
[3.5]
30
Define f and P as:
Y = PY [3.7]
P = X (X'Xy) IX [3.8]
and the square root of standard error is : SSE =(Y Y )=Y'PY' [3.9]
Data Processing
Eq. [3.5] is the start point to design the experiment and data processing of the diffusion tensor imaging. In Fig. 3.1 and 3.2, flow chart of data processing are shown. The value of each b element was calculated using Mathematica (Wolfram Research, Inc.) (Fig. 3.1) and matrix linear regression was performed using Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign) (Fig. 3.2).
Spin Echo and Stimulated Echo Diffusion Tensor Imaging bMatrix of Spin Echo Sequence
For the diffusionweighted spin echo ("Sems.c") sequence (Fig 3.3), the bMatrix can be derived from Eq. [2.16]. Here r = read, p = phase, s = slice, and the number i denotes the time interval of the pulse sequence. For example, gdro * gror * m231 gives the cross terms between gradients gdro and gror (Fig 3.3).
31
brr=72{gdro2*m22+2*gdro*gror*m231 +gror2*m331+ 2*gror*gro*m371 +gro2 *m77} bpr = 2{gdro*gdpe *m22 +gdpe*gror*m231 +gdro*gpe *m232 + gpe*gror*m3321 +gpe*gro *m372 }
b ppy2{gdpe2 *m22 +2*gdpe*gpe*m232 +gpe2*m332}
bsrY 2{gss*gdro *(ml2m233/2+m24)+gss*gro*(ml7m373 /2) + gss*gror *(m131m3331/2 +m341)+gdss*gdro*mn22 +gdss*gror *m231}
bsp =y2 {gss*gdpe*(m2 m233/2+m24) +gss*gpe*(m132m3332/2+m342) +gdpe*gdss *m22+gpe*gdss*m232 } bss = 2{ gss2 _(M m133+2*m14+m333/4m343+m44)+ gss*gdss(2*m12 m233+2*m24)+gdss2*m22}
here:
mll
m12 m22
m13 m23 m33
m14 m24 m34 m44
,m17 m27 m37 m47 m77j
I,
1/ 4 *(2 *(TE , 51/ 3) , * gE / 6
+ 6/ 30)
[1/2*6 *( 2 t2)] g2 (A2 152/3)6*/6+/301
(5 52 (t,  t3 (52 (3,(U22t3  TE T
1 2T 3  1)
133/2S/2)/2] [(53 2i  )*/2) 2 / E
[ '53/ 82*/5,2 +23 , g/ 1 2 + / 6 0 s/8*(s8+e/3)] [,2/4* +/3 33/4*(3)+ /3]) [1/2(,5/6+4/34)
J1/16*5,( + ,2/3)] 0 33/8*( 7+ /3) 0 1/4(,5/6+ /30)
if g3i # 53jwhen i j, then
.. 3 833 E2*33 53, d 3P2i '3i93jm33y= 6 2 ++ 3i53j *(te  t3)  2 2
6 12 120 2 2
bMatrix of Stimulated Echo
For the Diffusionweighted Stimulated Echo ("Stems.c") sequence (Fig 3.4), the bMatrix can be derived from Eq. [2.16]. Here r = read, p = phase, s = slice, and the number i denotes the time interval of the pulse sequence. For example, gdro *gror *m231 gives the cross terms between gradients gdro and gror (Fig 3.4).
b= gro2(k m221 + m77) +gro * gdro(2m67 + 2klm231) +gdro 2(m33 + m66)
b22 gpe2 (m222) + gpe * gdpe * (2m232) + gdpe2 (m33 + m66)
b gss2(ml1 +2ml4+ m44+k m223  2kam123  2k3m243 + m55) + gss * gdss(2m1 3+ 21n34  2k3m233 + 2m56) + gdss2 (n33 + m66)
b1 = gpe * gss(m122 + m242  k3m2232) + gdpe * gss(ml 3 + m34  k3m233 + m56) + gpe * gdss(m232) + gdpe * gdss(m33 + m66)
b12 = gro * gpe(kim222 1) + gro * gdpe(m231 + m67) + gpe * gdro * m232 +gdpe*gdro *(m33 +m66)
b1 = gro * gss(km121+klm241  kk3m2231+ m57) + gdro * gss(m13 + m34  k3m233 + m56) + gro * gdss(kim231 + m67) + gdro * gdss(m33 + m66)
Here mij= mji, 1 is gro direction, 2 is gpe, 3 is gss.
mll
m12 m22
m13 m23 m33 m14 m24 m34 m44,
 /12 3g /24+ / /480 + Z16 /4
1 / 2 * , (2 * ( ,1 tl 92/2c/2)
1 / 2*Q, (rt3 3 3/2  e/2) ,5,/16 *( +4 /3)
2/3* t' (5 12+ g/60+6 (2  t2 /2)
32, (5 ( l t3 9,/26/2)
S/8*(3 + g/3)
g ,1 2/3*3 + '3 /60+6 (T2   /2)
33/8*(3 +g /3)
, /24+ g /120
if 2, when i j, then
m22ij = 2i " 2i + C+S1 *( t2)  jL
6 12 120 2 2
'm55
m56 m66= ,m57 m67 m77,
34
[(553 24 + E'/480 +
l/2*Ss33*(r2rl+]1/3*3Cs23/12+ 3/660
,55/4)+3c2/24 +15 2+ it i
1/4*8587(r r(+5 5,/2s3(r  t6 51/72*5/ 2s,/24+
55/4)+(5762/48 ]3/2e/2 j) _ s3120+sir2/4
Experiments
In the following experiments, a diffusionsensitizing gradient pair was included in a spin echo imaging sequence (Fig. 3.3); this is the conventional pulsedgradient spin echo (PGSE) sequence for diffusion imaging. To obtain maximum diffusion sensitivity, the first gradient pulse was applied immediately following the 90 excitation pulse and the second just before acquisition. Seven noncollinear directions are required to determine the six independent elements of the diffusion tensor. An unbiased tensor measurement, i.e. one that is independent of the sample orientation with respect to the gradient axes, was obtained by applying diffusionweighting gradients along polar angles, (0, 4), as follows: (00, 00), (450, 00), (54.740, 450), (450, 900), (900, 00), (900, 45), and (900, 900). Multiple images having an incremented diffusion gradient magnitude were acquired for each diffusion weighting direction. The image acquired with no diffusion gradients was used to normalize the data prior to matrix linear regression (see below).
The accuracy of diffusion weighting factors deduced using our formalism was
tested by applying the diffusion tensor imaging method to a sample of pure water at 12.5
2.0' C. The water was stored in a glass tube (2 cm in diameter and 4 cm in length).
35
Small amount of CuSO4 (about 0.01 ml) was added to water to decrease T,. A pulsed field gradient spin echo imaging sequence was implemented on a SISCO imaging spectrometer (Varian NMR Instruments, Palo Alto, California) equipped with a 4.7 T, 33 cm bore magnet (Oxford Instruments Ltd., Oxford, England) and an actively shielded gradient/shim system (Nalorac Cryogenics Corp.) capable of 5 G/cm along x, y and z. Ten accurately calibrated gradient amplitudes from 0.5 to 5 G/cm in 0.5 G/cm increments were applied along the seven directions, yielding maximum bmatrix values of 1200 s/mm2. Each image was acquired as a 128 x 128 matrix and the overall experiment time was 5 hours. On a pixelbypixel basis, a sevenparameter matrix linear regression (Rao, 1965) was used to perform a fit for each element of the diffusion tensor (Dy). This was accomplished by determining the natural logarithm of the ratio of measured signal to the signal with no diffusionweighting gradients, then determining the diffusion tensor using the known values of b / according to Eq. [3.1]. The value of each by element was calculated using Mathematica (Wolfram Research, Inc.) and matrix linear regression was performed using Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign). The calculated apparent diffusion tensor is shown below along with its standard error matrix (Rao, 1967):
1.54 0.04 0.05 r 0.01 0.01 0.01
D [0.04 1.54 0.03 + 0.01 0.01 0.01 X 103 ""'" [3.10]
,0.05 0.03 1.52 0.01 0.01 0.01,
The correlation coefficient, r2 = 0.9993. As expected for a sample exhibiting isotropic translational diffusion the diagonal terms of the diffusion tensor are equivalent within experimental error, while the offdiagonal terms are nearly zero. The offdiagonal terms
36
are a little bit out of error value, this might be due to the nonuniform magnetic field. In this case the diffusion tensor correctly reflects the scalar diffusion of the water.
The accuracy of diffusion weighting factors deduced using our formalism was also tested by applying the diffusion tensor imaging method to pure water at 20.5 L.00C using the multiple stimulated echo sequence (Mareci, 1986), two gradient amplitudes (2.5G/cm and 4.5 G/cm) were applied along 7 directions (x, y, z, xy, xz, yz, and xyz), yielding maximum bmatrix values of 1300 s/mm2 please refer to Ch. 2 about the bvalue calculation). 6 x 7 diffusionweighted images were acquired in a total acquisition time of 56 minutes.
The calculated apparent diffusion tensor is shown below along with its standard error matrix (Rao, 1967):
2.12 0.12 0.19 ( 0.08 0.0 +0.11
D = 0.12 2.10 0.25 + 0.10 0.12 0.10 x 10',"/Q [3.11]
D ,{0.19 0.25 2.20, 0.10 0.10 0.10[
The correlation coefficient, r2 = 0.9788. Again as expected for a sample exhibiting isotropic translational diffusion the diagonal terms of the diffusion tensor are equivalent, within experimental error, while the offdiagonal terms are nearly zero. Errors are ten times larger than those in spin echo experiments, that might be because of the less number of points used in linear square fit. The values of D are significantly higher in the second experiment compared with the first experiment; this is attributed to the difference of temperature in the two experiments (12.5'C and 20.50C respectively)"
37
Conclusion
In this chapter, methods for diffusion tensor imaging using spin echo and
stimulated echo sequence are built and successfully tested by the experiments. Now we can apply these method to study biological structures.
I Start processing
Input the parameter from the image file
Calculate the time for
each gradient
Calculate bvalue for
all gradient
combination
Calculate total bvalue
for all bmatrix
Multiple variable linear Output regression
regression result
Fig 3.1 Flow chart of data processing for multiple linear regression
38
Start image processing
Input multiple linear regression result
Input images, do Fourier
Add weight for each image from multiple linear regression results
Add all images together including weighting
Output diffusion tensor images End:t ::
Fig 3.2 Flow chart for image processing.
39
9(00
slice
1800
A
gss gdss gss gdss
AL 4aLAL
gror gdro read
pe gdpe phase
time period
12 3 4
I
5
Fig 3.3 Spin echo sequence
gdro gro gdpe
I
I
6
7
I I
40
900
gss gdss
900 A
gss
gdss
A& AL AL LX
gror gdro gdro gdro gro
AT\ 7\
gpe gdpe gdpe gdpe
/IN f7\\
wI
1 UJ
12 3 4
5
6 7
Fig 3.4 Stimulated echo sequence
90v
gss gdss
slice
read phase
time period
CHAPTER 4
FAST DIFFUSION TENSOR IMAGING USING A MULTIPLE STIMULATED ECHO SEQUENCE
Introduction
Diffusionweighted imaging is an effective technique to identify brain tissue
pathology since it permits discrimination of grey and white matter structures (Moseley, 1990) and normal and edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (TI, T2 ). Diffusionweighted imaging, however, is only an approximation to the actual diffusion sensitivity of an anisotropic sample. A much improved estimation can be achieved by measuring an apparent diffusion tensor. In this way it should be possible to identify regions with varying degrees of edema. Diffusion tensor imaging has been previously performed using a spinecho sequence (Basser, 1994). However, the total acquisition time is typically more than 3 hours, which is prohibitive for in vivo experiments. Here, we introduce a faster diffusiontensor imaging method using stimulated echoes which reduces the acquisition time to less than an hour and which is successfully used to obtain in vivo images from the rat brain.
Methods
bmatrix Calculation for Stimulated Echo Sequence
The bvalue expression (Eq. 1.7) for a gradient echo is given by (Torrey, 1956):
2
b(te) =r t 'dt"f G(t") dt' t
* 0
41
42
Here we substitute G(t") by including the coherence order Pi: (Bodenhausen, 1984, Bain, 1984, Mareci, 1988)
n
G*(t) = g(k,t) [4.2]
k=1
where,
, t) Pk , tk t tk + (k
0 ,0 !5 t tk + (k
Then we get b expression for stimulated echo sequence:
b(rl+r2) = 0 jdt" G*(t") dt' [4.3]
0
This expression is equivalent to Tanner's expression (Tanner, 1970), this is shown as follows:
For stimulated echo:
Pk= 0 rl I < r2 [4.4]
1 ,r2 ti
Using the condition that a stimulated echo is formed: fdt G (t) = dt G (t") [4.5]
The general equation of btensor becomes: b (rl+r2)= v2y + + fdt"G* (t') C r 2 2 ( rI+r2 [4.6]
=7 (Jdt fdt G(t) +(72z1) jdt"G(t )+fdt Jdt"G(t)
(0r2 )) t
which agrees with J.E Tanner's equation (Tanner, 1970).
43
General Library for bMatrix Calculation for Stimulated Echo
From eq [4.6], rewrite it in matrix form: b(r +r2)=y2 t dt g *(k,t" 1) dtlg,*(t"
k= 1 0 [4.7]
n Z
b (t))k
k= 1=1
where
(b ) (t)  Y t' dt" ,*(k,") dt" g9*(lt") [4.8]
This formulation of the diffusion weighting is central to the calculation of
diffusionweighting factors and forms the starting point for the explicit application to specific pulse sequences. The indices k and 1 refer to a pair of gradient pulses applied in the i andj directions, respectively. Therefore, the calculation involves the explicit evaluation of pairwise autoterms (k = 1) and crossterms (k w 1) in the diffusionweighting factors, where the total diffusion weighting is the sum of these pairwise terms.
Here we give an example for bij term for a pair of trapezoidal pulse shape gradient (other shape gradients can be calculated in the same way):
(bh(t)) k = y p4GGj(6 (t tk 231k/3/2)  k 2 'e3 /6 0) [4.9]
and the cross term as,
(b,(t))=  y p pGikGj, 5(t t, 8, /2  e / 2) [4.10]
The advantage of using Eq 4.8 rather than Tanner's expression (Eq. 4.6) is, we have the library for b value calculation for stimulated echo sequence, which is the same
44
as the library for gradient echo and spin echo sequence, for any order of quantum coherence.
Multiple Stimulated Echo Sequence
Stimulated echo and multiple stimulated echoes method have been used to
measure T, relaxation time (Haase, 1986) (Mareci, 1986)and diffusion cofficient (Tanner, 1970, Merboldt, 1985), diffusion coefficient in inhomogeneous fields (Latour, 1993) and in Heterogenous System(Cotts, 1989). In a system with long T, and short T2 (such as in injured part of spinal cord, T, ~ 25ms(Ford, 1994)), the advantage of using stimulated echo (MacFall, 1988, Lebihan, 1991) is that, the echo time TE can be decreased, and TL can be increased so as to have enough signalto noise ratio, while with long diffusion time(A), the diffusion weighting factor b will still be large enough to ensure a good fit for diffusion tensor calculation.
Stimulated echo is also very useful in studying the timedependant restricted diffusion (Gates, 1994)).
The multiplestimulated echo sequence (Fig. 4.1) (Frahm, 1985, Mareci, 1986) is
used to acquire three diffusionweighted echoes (one spinecho and two stimulatedecho) per transient. Crush gradients gcrush (Fig. 4.1) are used to get rid of unused spinecho.
The signal intensity of the three echoes is given by(Mareci, 1986):
n_1  1 1: h,() tD,
A (t)=Mosinan* cosai *e TEJ6 TLne eT, e [4.11]
Here n is the number of echoes in the sequence, if we treat first spin echo as a special stimulated echo where TL=0, then TLO=0, a0=900 for the first spin echo, TLI, a =450 for
the first stimulated echo, and TL2, cx2=900 for the second stimulated echo. The diffusionweighting factor(bij) caused by the diffusion gradient is given by: b dr =22Gk (Af 5 / 3) [4.12]
Here A0= Ase, A,= As,i, A2= A ste2 Gdiff and 6 is the strength and time duration of the diffusion gradient.
The diffusion weighting factor increases as the time (A) between the first
(dephasing) diffusion gradient and each rephasing diffusion gradient increases, so the three echoes in the pulse sequence have different bij values.
When the data are processed, each diffusion weighted image is divided by its corresponding nondiffusionweighted image, thus A,(Gd,ff,t)1An(0, t)= e ''''[4.13] So each echo is used as one single point to perform linear regression for each element of tensor according to eq (3), having taken natural logarithms.
For six elements of diffusion tensor, seven different gradient weighting directions are required. We typically use six bij values (two different diffusion gradient amplitudes) per direction to ensure a good fit to the data. One set of nondiffusionweighted images were also collected to normalize the data by removing the differential Ti effects produced by the different diffusion weighting times (A).Therefore, a total of eight experiments are required: one nondiffusionweighted and seven differently weighted, Gdiff oriented experiments.
46
Using the generalized library for calculating the diffusion weighting factor (bfactor) used for all sequences (Yang, 1995), the bij of stimulatedecho sequence were calculated, including the crossterms between the diffusion and imaging gradients.
On a pixelbypixel basis,a sevenparameter matrix linear regression(Rao, 1967) was used to perform a fit for each element of tensor according to eq. [3.6], having taken natural logarithms. Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign) was used to perform matrix linear regression and to process the 42 diffusionweighted images to obtain the diffusion tensor images.
Results
The fast diffusion tensor imaging method using multiple stimulated echo
sequence was first tested using water and celery samples, then applied to the rat brain in vivo experiments.
Water and Celery Experiments
Using the multiple stimulated echo sequence described above, 42 diffusionweighted images were acquired in a total acquisition time of 56 minutes. Diffusion gradients were applied along x, y, z, xy, xz, yz, and xyz, while varying bij between 40 and 1300 s/mm2 The accuracy of bij was measured using the general formalism Eq. [4.7] and this was tested by applying the fast diffusion tensor imaging method to a sample of pure water and a piece of celery at 20.5 1.00C. The water is contained in a cylindered glass vessel (2 cm in diameter and 3 cm in length). Small amount of CuSO4 (about 0.01 ml) was put into the water inorder to decrease T,. A piece of fresh celery (2.5 cm in length, 2 cm in width) was
47
put into the vessel, with its fiber orienting along vessel. The vessel was put into the 4.7 T magnet, with its axis along the magnetic field.
The calculated apparent diffusion tensor is shown below along with its standard error matrix. The diffusion coefficient of water surround the celery is (Fig 4.2):
2.12 0.12 0.19 0.08 0.11 0.10
= 0.11 2.10 0.24 + 0.10 0.12 0.10 xo3'"'" [4.14]
0.19  0.24 2.20 0.10 0.10 0.10
The diffusion coefficient of the celery fiber is (Fig 4.2):
0.45  0.27 0.14' 0.21 0.05 0.06
D= 0.27 0.43 0.06 + 0.05 0.15 0.05 x1"""/ [4.15]
0.14 0.06 1.37 , 0.06 0.05 0.147
The diffusion coefficient of the celery (nonfiber region) is (Fig 4.2):
0.98  0.06  0.11 0.10 0.10 0.11
D= 0.06 1.01 0.16 + 0.05 0.12 0.07 x10,""" [4.16]
0.11 0.16 0.90 i0.11 0.07 0.13,
Each value of Di is obtained from a plot of ln(S/SO) against b(t). The correlation coefficient of this process is r2 = 0.9788. As expected for a sample exhibiting isotropic translational diffusion, the diagonal terms of the diffusion tensor in [4.14] are equivalent, within experimental error, while the offdiagonal terms are nearly zero. In this case the diffusion tensor correctly reflects the scalar diffusion of the water.
In the region of the fiber inside celery, Dzz (1.37 0.14 *10 3 mm2/s ) is much larger than Dxx (0.45 0.21*10 3 mm2/s) and Dyy (0.43 0.15 *10 3 mm2/s), which shows diffusion anisotropy inside the celery fibers. In the nonfiber region inside the celery, the diagonal terms of the diffusion tensor are equivalent (Eq. 4.16), but less than
48
those of pure water (Eq. 4.14), while the offdiagonal terms are nearly zero, which shows isotropic but restricted diffusion inside celery (nonfiber region). Rat Brain in vivo Experiments
The fast diffusion tensor imaging method was used to acquire in vivo brain images from 21 dayold rats. A diagram of rat brain anatomical structure is shown in Fig. 4.3. The rat cerebral cortex at 21days after birth has a long T1 (550 ms) and a short T2 (60 ms) (Harris, 1992) and therefore TE and TL were optimized to 33 ms and 210 ms, respectively. This enabled us to attain b values up to 1300 s/mm2 and to acquire a good signaltonoise ratio, both of which are critical for the tensor calculation. The in vivo diffusion tensor data is shown in the table below (Table 4.1, Fig 4.3). As expected the corpus collosum exhibits anisotropic diffusion; greater in the x plane (parallel to the white matter fiber) than in the perpendicular y or z directions while the diffusion in the cortex is almost isotropic. The diffusion tensor values are in reasonable agreement to measurements made using localized spectroscopy in human brain (Gates, 1994).
These experiments on in vivo rat brain were carried out as an example of the method. More deatailed experiments on rat brain are reported in Chapter 6.
49
Table 4.1 Diffusion tensor values of rat brain.*unit: 103 mM2 /s
Rat Human
(Gates, 1994)
corpus cerebral corpus
collosum cortex collosum
temp. ("C) 37.0 37.0 ~37.3
Dxx 1.07(0.29) 0.64(0.26) 1.15
Dyy 0.43(0.14) 0.74(0.16)
Dzz 0.46(0.23) 0.65(0.23) 0.71
TL2
TLI
909 450
A f~A
9(r
gss gssr gdss gss gdss gcrush gss gdss gcrush gss gdss
gror gdro gdro gro gcrush gdro gro gcnsh gdro gro
AL /\m A EV
gpe gdpe phase
gdpe
gcrush
gdpe
crush
gdpe
Ase
I  Aste I I I Aste2
1  P 0
1 first spin echo
first stimulated echo
_____ second stimulated echo Fig 4.1 Multiple stimulated echo sequence.
slice readc
Ir
15,~
K;
t4
.. p .  U
AJIL
4 a ia ''
 a
e
L*. ~  a.
~ '' a,.*
Fig 4.2 Water and Celery diffusion tensor maps.
cerebral cotez...  
diffusion tensol measurement region I diffusion tens or measurement region 2
Fig 4.3 Basic anatomical structure of the rat brain illustrating the regions used for the diffusion tensor measurements in the table 4.1.
corus collosum
CHAPTER 5
HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE
IMAGING OF RAT SPINAL CORD IN VITRO Introduction
Diffusion Tensor Imaging
Diffusionweighted imaging (DWI) is an effective technique to identify nerve tissue pathology, since it permits discrimination of grey and white matter structures (Moseley, 1990) and between normal and injured regions (Ebisu, 1993) with higher specificity than conventional contrast (Ti, T2 ). Apparent diffusion coefficient (ADC) images have also been obtained for spinal cord by using a series of diffusion weighted images. However, when the rootmeansquared (RMS) path length of the diffusion of the molecules is so long that the molecules hit the boundary, diffusion becomes anisotropic and is a tensor rather than a scalar quantity. So the ADC value measured for a specific structure depends on the directions of the applied gradients and is only an approximation to the accurate anisotropic diffusion coefficient measurement. Diffusiontensor images (Basser, 1994) and color trace imaging has been used successfully to show neuronal orientation and pathological changes of rat spinal cord using microimaging (Inglis, 1996). Here we use the same method to study grey and white matter organization and tissue injury in the spinal cord of young rats. Anisotropic apparent diffusion tensors are computed for data on neuronal orientation, and the 'trace' is computed as a measure of
53
54
orientationindependent tissue space available for water diffusion, which shows neuronal damage in rat spinal cord.
Anisotropy Index
Diffusion approaches have been used to measure white matter tissue anisotropy. The most widely used is the ratio of ADCs measured with diffusion gradients presumed to be parallel and perpendicular to the fiber direction ( ADC(90)/(O) ) (Douek, 1991, van Geldren, 1994, Basser, 1994). Also another scalar index of anisotropy was proposed using ADCs measured with diffusion gradients applied in three orthoganal directions [ Standard Deviation Index]. On theoretical ground, ADCs are just approximation of the tissue diffusion nature. Pierpaoli et al (Pierpaoli, 1995) proposed another index: ratio of the principal diffusivities (eigenvalues X, )2, X3) of diffusion tensor X/X3, which are invariant to rotations of tissue within the NMR magnet. However, due to the complexity of brain and spinal cord tissue structure, diffusion tensor of each image pixel has different principal axis and need to be normalized seperately. Practically this requires huge amount of data processing and still unable to give a general overview of whole brain's tissue structure. Here we use color trace and anisotropy index:
Anisotropyindex = 2*D, /(D,,+DYY)
Which clearly show tissue structure and detail informations about tissue damage of brain and spinal cord. Here zz is parallel, yy and xx is perpendicular to the fiber direction.
Anatomic Structure of Spinal Cord
Gray matter of spinal cord
The central cellular region of the spinal cord is referred to as the gray matter.
Afferent fibers of peripheral nerves form the dorsal root and enter the spinal cord at the dorsal margin (Fig 5.1). The Dorsal horn and the intermediate zone contain the posterior marginal nucleus, the substania gelatinosa, the nucleus proprius, Clarkes's nucleus and the intermediolateral nucleus.
Substania gelatinosa locates on the top of dorsal horn. Fibers from peripheral
nervous system ( medial division and lateral division of dorsal root) enter the spinal cord here, and send branches that ascend and descend in the cord and some of them arborize in the spinal gray matter, some of them terminate on the top of dorsal horn, or pass through the superficial dorsal horn to terminate at the bottom of dorsal horn. Largediameter fibers have been observed to pass through intermediate zone and terminate directly in motor nuclei which are at bottom of ventral horn. Also inside gray matter there are neuronal processes that interconnect sensory neurons in the dorsal horns with motor neurons in ventral horns. In summary, at Substania gelatinosa fibers travel in the anterposteral direction and lateral direction, while in lower part of dorsal horn, intermediate zone and ventral horn, the predominant fiber orientation is lateral.
Ventral horn contain largest nerve cells (motor nuclei), their axons form ventral root which go across white matter into muscles and directly innervate them. Inside white matter ventral root travels mainly in lateral direction.
56
In ventral and dorsal gray commissures there are a large number of primary
afferent axons that are transverse to the colateral gray matter, so in this region the fibers travel in dorsoventral direction.
White matter of spinal cord
The area surrounding the gray matter is called the white matter of the spinal cord. Located in the white matter are the major ascending and descending nerve fiber tracts as well as pathways made up of the short axons that connect local regions of the cord.
Methods
Fixation of Rat Spinal Cord
Injuries are performed at the T 13 vertebral level 90 minutes after a single level laminectomy. An impounder is dropped from 25mm to generate a moderate to severe injury model. After injury the tissue is closed in layers. At the desired time intervals the injured animals are euthanized and perfused with a fixative. After a deep level of anesthesia is obtained the thoracic cavity is opened and the animal perfused via transcardial exsangination with 250ml of saline followed by 250ml of 4% paraformaldahyde. The animal is kept in a refrigerator for 24 Hrs for complete tissue fixation. The spinal cord is removed by a full laminectomy from the level of the cauda equina to the cervical cord or other levels depending on the length of cord desired. Diffusion Tensor Imaging Experiments
Three normal and two injured rat spinal cords were imaged on a 600 MHz Varian Unity microimaging system using a diffusionweighted spin echo sequence. Sample temperature was maintained at 20'C throughout. Acquisition parameters were:
TR/TE=3000/44 ins, slice thickness = 0.75 mm, 4 averages, 0.5x0.5 cm FOV ( 39x39 pm in plane resolution), diffusion weighting duration 6 = 10 ins, diffusion weighting gradient pair separation A = 26 ms, data matrix = 128*128. At first, routine imaging was performed, which included sagittal imaging and multislice imaging over region of interest. Second, diffusionweighted images were acquired with diffusion weighting gradients aligned at polar angles, (0, (p) as follows: (90', 00), (900,900), (00, 0 ), (45 , 0 ), (45 0, 90 ), (90 , 450), and (54.70, 45 ) These angles corresponding to the directions x, y, z, x = z, x = y, y = z, and x = y = z. In each direction, images were acquired with diffusion gradient amplitude of 3, 9, 15, 27 and 27 G/cm, giving 35 separate image acquisitions. One further image was acquired without diffusion weighting gradients and was used to normalize the signal intensity of all diffusionweighted images before calculation of the diffusion tensor. The overall measurement time was 15 hours.
For each image acquisition, a new bmatrix was determined, taking into account all imaging and diffusion gradient auto and crossterms, except the phase encoding gradient that was considered to be zero (i.e., its central value, for simplicity). Trapezoidal gradient shapes were assumed and bmatrix were found numerically using Mathematica ( Wolfram Research, Inc.). At the largest diffusion weighting gradient of 27 G/cm, maximum values of 11583, 13914 and 11979 s/mm2 were obtained for b,, by, and b, respectively, for phase encoding along x, readout along y, and slice selection along z. These values were selected to optimized the experiment for the spinal cord tissue. At 27G/cm, a signaltonoise ratio of approximately 3 was obtained for spinal cord tissue. However, due to the inherent diffusion weighting of the microscopy sequence, no signal
58
was observed for free water at diffusionweighting gradient strength above 3G/cm for readout direction and above 9 G/cm for all other directions, which lead to poor linear square fit for water.
From the series of images, on a pixelbypixel basis, multivariate matrix linear regression was used to find the six unique elements of the diffusion tensor (Rao, 1965), each tensor element being an image displaying anisotropic diffusion coefficients (Moseley, 1990). The complete ADT were calculated using Viewit software. By adding the diagonal tensor elements as images with separate monochromatic (8bit) color scales, a color image was constructed that depicts tissue organization, using red, green and blue for D, D, and D, respectively (Ebisu, 1993). An anisotropic index map was also made by dividing D, by the average of Dx and D~Y as images. Data Analysis
The color trace images greatly enhances the recognition of anatomical features, for which the diffusion tensors can then be extracted ( Table I). Two color trace images are presented. Fig. 5.2 shows a color trace image of a normal rat spinal cord, Fig. 5.3 shows four color trace images of a injured spinal cord, Fig. 5.4 shows the full tensor of a normal spinal cord. Comparisons are made in Fig 5.6, 5.7, 5.8.
Regions of interest were marked using Dispunc software and a mean ADT calculated.
Results
Diffusion Tensor Values of Normal Rat Spinal
The diffusion tensor values of three normal rat spinal cord are pretty consistent, within experiment error. For example, D,, of Dorsal funiculus are 0.22+/0.08, 0.27+/
59
0.10, 0.32+/0.09. The variation of diffusion values may also due to the biological difference between the rat spinal cord samples. Diffusion Tensor Values of Injured Spinal Cord
The diffusion tensor values of two injured rat spinal cord are different. For
example, DYY of Ventrolateral funiculus are 0.48+/0.13, 0.77+/0.13. The variation of diffusion values may due to the pathological difference between the rat spinal cord samples, i.e. more serious injury cause more white and gray matter damage, which also increase the diffusion tensor value (Table 5.15). Diffusion Tensor Values of Rat Spinal Cord
All the diffusion values of rat spinal cord are much less than that of free water at 20'C (2.5 *101 mm2/s), which are as predicted. The restrictions and obstacles inside the tissue slow down the diffusion.
Color Trace Images
Color trace images of rat spinal cord clearly indicate the predominant diffusion direction of the spinal cord, which is also the predominant nerve fiber tracts orientation. Colors in Gray Matter Region of Normal Rat Spinal Cord
As it has been discussed in anatomy of spinal cord (Fig. 5.1), at substania
gelationosa fibers travel in the anterposteral direction and in plane, this region is expected to show orange color ( red+blue or red+green) in color trace images (Fig. 5.2, 5.1). In lower part of dorsal horn , intermediate zone and ventral horn, the predominent fiber orientation is lateral, so we expect to see mainly green in these area. Ventral root goes across the white matter laterally, then it is supposed to show green among the red
60
backgrounds ( white matter) (Fig. 5.2, 5.1). At last, fibers travel dorsoventrally in gray commissures, so blue color would be the right color (Fig. 5.2, 5.1). Color of the White Matter of Normal Rat Spinal Cord
In white matter fibers mainly ascend and descend ( anterposteral), so it should all be red (Fig. 5.2, 5.1).
Therefore, the observed diffusion anisotropy is consistent with the prevailing orientation of fibers in the spinal cord.
Color Trace Images of Injured Rat Spinal Cord
The red color of white matter turns into gray in injured area, and the color of gray matter changes to gray or white (Fig 5.3), which shows different degree of injury. Trace is an Important Parameter for Different Nerve Tissues
In normal rat spinal cord, the average trace among different white matter regions (DF, VLF, VF, DLF) is 0.82 0.12 x104 mm2/s (table 5.15.5, fig 5.6), while the average trace among different gray matter regions (SG, DH, VH, GC) is 1.64 0.14 x104 mm2/s (table 5.15.5, fig 5.6). These data suggest that trace value can be used to differentiate white matter and gray matter tissues.
We also note that all the diffusion values of the spinal cord are much smaller than that of pure water (25* 104 mm2/s) due to the restriction, tortuosity, reflection, etc. Anisotropy Index of Normal Rat Spinal Cord
The anisotropy indexes of all normal samples in different white matter regions
(DF, VLF, VF, DLF) are ranging between 3.2 to 7.1 (table 5.15.5, fig 5.7), which shows high diffusion anisotropy among white matter tissues. The anisotropy indexes of all
61
normal samples in different gray matter regions regions (SG, DH, VH, GC) are close to 1 (table 5.15.5, fig 5.7), which shows isotropic diffusion in gray matter. In Substantia gelatinosa(SG), anisotropy indexes of all samples are around 2, which suggests that both white matter and gray matter are coexisting in this area. This is is consistent with the prevailing anatomic structure of the spinal cord. Anisotropy Index of Normal and Injured White Matter Tissues
In all white matter regions of injured sample, the anisotropy indexes are all decreased (table 5.15.5, fig 5.8), which suggests that anisotropy index is related to white matter damage. However, it is still unknown about how the degree of injury is related to anisotropy index.
Diffusion Tensor Imaging of Human Spinal Cord
Diffusion tensor imaging was also applied in human spinal cord, the result was published in SMRM conference in 1997 (Inglis, 1997)(Fig. 5.5).
Conclusion
Diffusion tensor imaging and color trace are very powerful methods in studying gray and white matter organization and tissue injury, and become an important reference in future spinal cord study.
62
Table 5.1 Diffusion coefficient of normal spinal cord (sample #1).
' '
Table 5.1  5.5. and Fig. 5.1
5.2. Apparent diffusion coefficients (x104 mm2/s) recorded
at 20 0C in 8 regions from fixed rat cervical spinal cord. Dyy is the lateral, Dxx is the dorsoventral, and Dzz is the anteroposteral direction. Different values between the three directions indicate tissue anisotropy. Values were ADT calculated in the region of interested.
Dzz/(O. 5 *(D
trace
Dzz
Dxx
Dyy
Anatomical
xx+Dyy))
Region
Dorsal 0.22(0.08) 0.30(0.09) 1.46(0.02) 0.66 5.6
funiculus(DF)
Ventrolateral 0.30(0.12) 0.33(0.12) 1.44(0.30) 0.69 4.5 funiculus(VLF)
Ventral 0.23(0.08) 0.33(0.09) 1.23 (0.24) 0.64 4.9
funiculus(VF)
Dorsolateral 0.26(0.08) 0.29(0.11) 1.30(0.21) 0.62 4.7
funiculus(DLF)
Substantia 1.01(0.17) 1.22(0.18) 2.27(0.39) 1.50 2.0
gelatinosa(SG)
Dorsal hom(DH) 1.40(0.18) 1.51(0.17) 1.83(0.19) 1.58 1.25
Ventral 1.44(0.18) 1.42(0.17) 1.54(0.24) 1.47 1.08
horn(VH)
Gray 1.92(0.15) 1.66(0.24) 2.29(0.32) 1.96 1.28
commissure(GC)
63
Table 5.2 Diffusion coefficient of normal spinal cord (sample #2). Anatomical Region Dy Dxx Dzz trace Dzz/(O. 5 *(D
xx+Diy))
Dorsal 0.27(0.10) 0.30(0.07) 2.00(0.22) 0.86 7.1
funiculus(DF)
Ventrolateral 0.40(0.09) 0.50(0.07) 1.91(0.18) 0.94 4.2
funiculus(VLF)
Ventral 0.34(0.11) 0.42(0.08) 1.90(0.25) 0.88 5.0
funiculus(VF)
Dorsolateral 0.53(0.09) 0.47(0.06) 1.92(0.21) 0.97 3.88
funiculus(DLF)
Substantia 0.87(0.06) 1.35(0.08) 2.63(0.24) 1.62 2.36
gelatinosa(SG)
Dorsal horn(DH) 1.35(0.21) 1.60(0.14) 1.89(0.20) 1.61 1.1 Ventral hom(VH) 1.80(0.12) 1.68(0.12) 1.45(0.16) 1.64 0.8 Gray 2.26(0.20) 1.28(0.16) 2.06(0.23) 1.86 1.2
commissure(GC)
64
Table 5.3 Diffusion coefficient of normal spinal cord (sample #3). Anatomical Dyv Dxx Dzz trace Dzz/(0.5*(D
Region xx+Dyy))
Dorsal 0.32(0.09) 0.49(0.07) 1.80(0.46) 0.87 4.5
funiculus(DF)
Ventrolateral 0.46(0.09) 0.53(0.13) 1.58 (0.38) 0.86 3.2
funiculus(VLF)
Ventral 0.39(0.07) 0.67(0.18) 1.89(0.37) 0.98 3.6
funiculus(VF)
Dorsolateral 0.46(0.09) 0.38(0.09) 1.58(0.43) 0.81 3.8
funiculus(DLF)
Substantia 0.93(0.16) 1.26(0.10) 2.19(0.36) 1.46 2.0
gelatinosa(SG)
Dorsal horn(DH) 1.53 (0.22) 1.90(0.29) 1.84(0.29) 1.76 1.1 Ventral hom(VH) 1.51(0.26) 1.67(0.21) 1.72(0.32) 1.63 1.1 Gray 1.73(0.38) 1.48(0.20) 1.84(0.29) 1.68 1.2
commissure(GC)
65
Table 5.4 Diffusion coefficient of injured spinal cord (sample #4538). Anatomical Dyy Dxx Dzz trace Dzz/(0.5*(D
Region xx+Dvy))
Dorsal 1.60(0.63) 1.54(0.65) 1.91(0.60) 1.7 1.2
funiculus(DF)
Ventrolateral 0.48(0.13) 0.57(0.13) 1.30(0.33) 0.8 2.5
funiculus(VLF)
Ventral 0.58(0.17) 0.66 (0.11) 1.05(0.20) 0.8 1.7
funiculus(VF)
Dorsolateral 0.46(0.12) 0.35(0.13) 1.28(0.24) 0.7 3.2
funiculus(DLF)
Substantia 0.54(0.10) 0.75(0.14) 1.25(0.13) 0.9 1.9
gelatinosa(SG)
Dorsal hom(DH) 2.23(0.40) 2.21(0.43) 2.21(0.44) 2.2 1.0 Ventral horn(VH) 1.54(0. 63) 1.45(0.64) 1.74(0.31) 1.6 1.2
Gray
commissure(GC)
Unable to locate
Unable to locate
Unable to locate
I I ___________
66
Table 5.5 Diffusion coefficient of injured spinal cord (4529). Anatomical Dyy Dxx Dzz trace Dzz/(0.5*(D
Region xx+Dyvy))
Dorsal 1.49(0.54) 1.64(0.55) 1.77(0.52) 1.6 1.1
funiculus(DF)
Ventrolateral 0.77(0.13) 0.82(0.11) 1.05(0.15) 0.9 1.3 funiculus(VLF)
Ventral 0.48(0.14) 0.50(0.14) 1.24(0.21) 0.7 2.5
funiculus(VF)
Dorsolateral Unable to Unable to Unable to
funiculus(DLF) locate locate locate
Substantia Unable to Unable to Unable to
gelatinosa(SG) locate locate locate
Dorsal horn(DH) Unable to Unable to Unable to locate locate locate
Ventral horn(VH) Unable to Unable to Unable to locate locate locate
Gray Unable to Unable to Unable to
commissure(GC) locate locate locate
67
Lateral direction
Dorsal root Sibstantia gelationsa
Dorsal homn Inte mediate Gray atter zone
Ventral Hom White matte
Ventral FRot
Dorsoventral direction Anterposteral direction
Fig 5.1 Anatomic structure of spinal cord. Color legend in the color trace images: Dxx (Dorsoventral, blue), Dyy (Lateral, green), Dzz (Anterposteral, red).
68
Fig 5.2 Diffusion tensor (color trace) of normal rat spinal cord
69
Fig 5.3 Diffusion tensor (color trace) of injured rat spinal cord
.a~
1. ?~. !.~ 
4
 *
.CI W>
ft. 4.4 .4
ft 1
.utt
'4
It.
V 
 t
   n
C
V.
'2
'
0
ft ft ~1W'
4'
C
~,24'' ~ t ft 'ft
'it'
1*" .. WV,'
Fig. 5.4 Full dii fIsion tensor of normal rat spinal cord.
70
p
Al!
ft. r~.
ft. ~ *4.4ft
ft ft
"'
ft.'
~44t~ ~ XI'?
2' t,.2
~ ~
* ft., , '
4$
'4> C44,
 , St>'
i
b
71
Fig 5.5 Diffusion tensor (color trace) of Normal human spinal cord
2.5
2 1.5
0.5
0
sample #1 sample #2 sample #3
(DF) (VLF) (VF)
(DLF) (SG)
(DI) (VH) (GC)
Anatomical region
Fig 5.6 Diffusion tensor trace of rat spinal cord
k~)
6
8
7
6
5
3
2
(DF) (VLF) (VF) (DLF) (SG) (DH) (VH) (GC) Anatomical Region
Fig 5.7 Anisotropy Index of normal rat spinal cord
* sample#1
 Sample#2
Sample#3
1
*Samiple# I
sample #2 sample #3
sample #4538 )Ksample #4529
Anatomical Region
Fig 5.8 Anisotropy Index of white matter (normal and injured)
8
7
6
5
4
3
2
1
0
0 2~
0
(DF)
(VLF)
(VF)
(DLF)
3
CHAPTER 6
HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF
NORMAL, HYDROCEPHALUS, AND SHUNTED RAT BRAIN IN VITRO Introduction
Infantile Hydrocephalus
Hydrocephalus is a serious brain disease encountered in some infants at birth or shortly after. It is caused by the accumulation of fluid in the brain, especially in young children, which makes the head enlarge and cause mental handicap. It can be treated by inserting a shunt tube which drains the excess fluid.
One hypothesis is that the primary cause of neuronal injury in hydrocephalus is the damage in the periventricular white matter and the corpus callosum. Abnormal transependymal absorption of cerebrospinal fluid (CSF) results in increased white matter water content, causing axonal damage which leads to subsequent dendritic and synaptic degeneration in grey matter. Pathological changes in the periventricular white matter have been reported for hydrocephalus in human infants (Weller & Shulman, 1972) in rabbits (Del Bigio & Bruni, 1988) and in the kitten model of infantile hydrocephalus (Del Bigio et al, 1994). In the kitten, the changes such as delayed myelination and gliosis persisted despite shunt treatment (Del Bigio et al ,1994, Chumas et al, 1994). Hence a comparison between hydrocephalic and agematched control rats will provide information on the timing and nature of the pathological changes due to the hydrocephalus. The HTx rat is
75
76
an ideal model for this study. It has inherited hydrocephalus due to aqueduct stenosis with an onset in late gestation, a stage which is equivalent to pre midgestation in the human. Ventricular enlargement starts prenatally and progresses rapidly after birth, with death occurring at 46 weeks (Jones and Bucknall, 1988). Ventriculomegaly is associated with severe thinning of the cerebral cortex, disruption of the grey matter laminar structure by 10 days after birth (Jones et al, 1991) and abnormal pyramidal cell morphology in layer V by 21 days (Boillat et al, 1993, Harris et al, in preparation). White matter pathology has not been studied in this model.
Shunt Treatment of Hydrocephalus
In Dr. Jones' recent studies, the hydrocephalus has been treated by inserting a ventriculosubcutaneous shunt. Conventional T,weighted spinecho MR imaging was then used to measure the extent of ventriculomegaly in both untreated and treated rats. The effect of the shunt was to reduce the ventriculomegaly, prevent further dilation and to restore cortical thickness (Harris et al, 1994, Jones et al, 1995). The effect of shunt treatment at 36 days after birth was indistinguishable from treatment at 812 days after birth when the rats were examined at 21 days, suggesting that shunt timing may not be important. However, using other techniques we have shown that, although early treatment is more effective than later treatment, neither cortical ultrastructure, neuronal dendritic organization or cortical metabolism are completely normalized by shunting (Boillat et al 1993, Harris et al 1995, and in preparation). White Matter Pathology in Hydrocephalus
Hitherto, Dr. Jones' studies have concentrated on cortical grey matter changes,
when the primary cause of cortical damage may be pathology in the periventricular white
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matter. Dr. Jones et al. have observed edematous white matter in the periventricular regions of HTx rats with advanced hydrocephalus by conventional histology and also thinning of the corpus callosum and an absence of the septum pellucidum, which can be seen on T,weighted MR images. A recent finding by us is that the concentration of many metabolites in the cerebral cortex is reduced in hydrocephalic HTx rats (Harris et al, 1995 and in preparation) and that tissue water content is increased. These changes may indicate mild cytotoxic edema, when intracellular water increases and the tissue would be expected to become less anisotropic. In kaolininduced hydrocephalus, however, there was a large increase in anisotropy, similar to that seen in vasogenic edema, due to an increase in extracellular water content (Ebisu et al, 1993). This discrepancy may be due to differences between infant and adult hydrocephalus and is one that can be resolved with the proposed tensor mapping experiments. MR Imaging Techniques
Diffusionweighted imaging (DWI) is an effective technique to identify brain tissue pathology, since it permits discrimination of gray and white matter structures (Moseley, 1990) and between normal and edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (TI, T2 ). Diffusiontensor images (Basser, 1994) and color trace imaging has been used successfully to show neuronal orientation and pathological changes of rat spinal cord using microimaging (Inglis, 1997). The apparent diffusion coefficient (ADC) can be obtained using DWI but it provides a quantitative measure of diffusion along the diffusion weighted axis only. Conversely, DTI provides quantitative diffusion coefficients (in mm2 /s) in all directions. Another parameter which can be extracted is the trace, which is invariant to the orientation of the
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brain with respect to the scanner axes. This information can be relayed in the form of pseudothree dimensional images, two spatial and one color, thus illustrating fiber orientation within the image plane (Inglis, 1997). Here we use the same method to study grey and white matter organization and tissue edema in young rats with infantile hydrocephalus and in shunttreated hydrocephalic rats. Anisotropic apparent diffusion tensors are computed for data on neuronal orientation, and the 'trace' is computed as a measure of orientationindependent tissue structure, which shows neuronal damage in infantile hydrocephalus.
Anatomy Structure of Rat Brain
Organization of the cerebral cortex: cerebral cortex contains three type of neurons: pyramidal, stellate and fusiform cells (Fig. 6.1).
Pyramidal cell are the most prominent of three types of cells. They have two types of dendrites, basilar dendrites that arborize near the layer of origin of the cell, and apical dendrites that ascend perpendicularly toward the surface of the cortex.
Stellate cells are starshaped, some of their axons project horizontally through a local cortical region.
Fusiform cells are spinal shaped and their axon often project out of the cortex (Fig
6.1).
Methods
Fixation of the Rat Brain
HTx rats have inherited infantile hydrocephalus which develops in late gestation. 21day old rats were perfused intravascularly with a fixative and the brain excised from the skull. The fixative procedure is as follows: rat brains were anesthetized with 60mg/kg
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pentobarbitone. The rat chests were opened and a needle was put into the left cardiac ventricle. We perfuse 5ml saline and 1020 ml fixative:
Fixative = 4% paraformaldehyde in 0.1 M phosphate buffer, PH 7.3 Shunt Surgery
The shunts were prepared from Teflon tubing( OD 0.76 mm, ID 0.3 mm) which was preshaped in a 900 bend to form the proximal end to insert into the lateral ventricle. The distal end was attached to an openended 1 to 1.5 cm silicone tube ( OD 1.5mm, ID 0.5mm) and the shunts were stored in Cidex solution. Prior to use, shunts were rinsed in sterile water and filled with sterile artificial cerebrospinal fluid containing 0.1 mg/ml Gentamicin antibiotic.
Microimaging
Brains from 3 control rats, 3 hydrocephalic rats, and 3 rats shunttreated at 10 days after birth were imaged on a 300 MHz Varian Unit microimaging system using a diffusionweighted spin echo sequence. A threeaxis gradient set was constructed in our laboratory to fit onto a broadband X{1H} Nalorac probe in place of the variable temperature dewar. The X nucleus coil was removed to leave the 'H decoupler coil of internal diameter 18 mm. This coil served as the transmit/receive coil and was retuned for optimal signaltonoise. Gradient strengths of 10.7, 11.4 and 25.6 G/cm along x, y and z, respectively, were attained with 20 A Highland power supplies. The brains were approximately 15 mm in cross section. Acquisition parameters were: TR/TE=3000/70 ms, slice thickness = 1.0 mm, 4 averages, 1.5x1.5 cm FOV (107x107 ptm in plane resolution). 35 diffusionweighted images were acquired in a total acquisition time of 15
80
hours. Five different values of diffusion gradient were applied along x, y, z, xy, xz, yz, and xyz, varying bij (diffusionweighting factor) between 20 and 1846 s/mm2 Conventional T2 weighted images were also acquired for comparison. Data Processing
From the series of images, the apparent diffusion tensor was computed, each tensor element being an image displaying anisotropic diffusion coefficients (Basser, 1994). Matrices of diffusionweighting factors were computed using Mathematica software (Fig 3.1) and the complete ADT by a sevenparameter matrix linear regression using Viewit software (Fig 3.2). By adding the diagonal tensor elements as images with separate monochromatic (8bit) color scales (Figs 6.3, 6.4, 6.5), a color image was constructed that depicts tissue organization, using red, green and blue for DYY, D, and D, respectively (Inglis, 1997).
The color trace image greatly enhances the recognition of anatomical features, for which the diffusion tensors can then be extracted ( Table I). Color trace images are presented (Fig 6.2, 6.3, 6.4). Regions of interest were outlined using Dispunc software (Fig 3.1) and a mean ADT calculated.
Results
Diffusion Tensor Values of Normal Rat Brain
The diffusion tensor values of three normal rat brains are pretty consistent, within experiment error. For example, DYY of white matter are 0.92+/0.25, 1.06+/0.15, 1.07+/0.10. The variation of diffusion values may also due to the biological difference between the rat brain samples.
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Diffusion Tensor Values of Hydrocephalic Rat Brain
The diffusion tensor values of three hydrocephalus rat brain are different. For example, D, of white matter are 0.50+/0.08, 0.77+/0.12, 1.05+/0.28. The variation of diffusion values may due to the pathological difference between the rat brain samples, i.e. more serious hydrocephalus cause more white and gray matter damage, which also increase the diffusion tensor value (Table 6.1, 6.2, 6.3). Diffusion Tensor Values of Rat Brain
All the diffusion values of rat brain are much less than that of free water at 20'C (2.5 * 103 mm2/s), which are as predicted. The restrictions and obstacles inside the tissue slow down the diffusion.
Color Trace Images
The color trace images of the rat brains clearly indicate the predominant diffusion direction in each pixel of the brain, which also corresponds to the predominant orientation of the nerve fiber tracts (Fig 6.3, 6.4, 6.5, 6.7). Color in Cerebral Cortex(CX)
As it has been discussed in 'anatomy of rat brain' (Ch6), in cerebral cortex the predominant fiber orientation is perpendicular to the pia surface, i.e., in lateral cortex, fibers travel along dorsoventral direction, so green color is expected in this region; in dorsal cortex, fibers travel along lateral direction, so red color is expected in this region (Fig. 6.3).
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Color in Corpus Callosum
Near lateral cortex, the fibers of corpus callosum is along lateral direction, so red color should been seen, while near dorsal cortex, the fibers of corpus callosum is along lateral direction, so green color should been seen.
All these color expectations match the diffusion tensor color images of rat brain. (Fig 6.3).
Anisotropy Index of White Matter in Normal and Hydrocephalic Rat Brain
The anisotropy indexes of the white matter in normal rat brain are above 2 (table 6.16.4, fig 6.10), in the hydrocephalic cases, they drop to near 1, in the shunted case, it increases back to normal. All these show that white matter is damaged in the hydrocephalic conditions and may be recovered after shunted treatment. Diffusion Tensor Trace of Lateral and Dorsal Cortex
The diffusion trace of lateral cortex in the hydrocephalic cases are higher (0.83 + 0.04) than that of normal cases (0.80 0.01 x103 mm2/s) (table 6.16.4, fig 6.10). The diffusion trace of dorsal cortex in the hydrocephalic cases are also higher (0.85 0.03 x103 mm2/s) than that of normal cases (0.79 0.02) (table 6.16.5, fig 6.9). The changes in hydrocephalus and in shunted case are much less than were found in chapter 5 for spinal cord injury, which suggest a much less dramatic change in anatomic structure.
It is concluded that hydrocephalus results in increased water content in the brain surrounding the ventricles. This has been confirmed by independent measurements (Jones, 1996). It is found that, at 21 days, cortical H0 (mi/kg wet weight) increased by
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1.5%, and H20 (ml/kg dry weight) increased by 9%. Shunt treatment normalized the gray matter but there remained an increase of water content in the white matter. Stimulated Echo Diffusion Tensor Imaging
Stimulated echo diffusion tensor imaging was also applied in rat brain (Fig.6.6), which seem to show more anatomical structure of rat brain.
Conclusion
Diffusion tensor imaging and color trace are very powerful methods in studying gray and white matter organization and tissue edema of rat brain, also in studying the effect of shunttreatment, and become an important reference in future rat brain study.
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Table 6.1: Diffusion coefficient of rat brain (normal).
Control
direction
Dxx(red)
Dyy(green)
Dzz(blue)
Trace
color in color map
white matter 0.34 (0.1) 0.92 (0.25) 0.40 (0.11) 0.55 green dorsal CX 0.99 (0.23) 0.70 (0.19) 0.63 (0.16) 0.77 red
lateral CX 0.75 (0.14) 0.91 (0.19) 0.70 (0.17) 0.79 green striatum 0.75 (0.15) 0.64 (0.13) 0.64 (0.14) 0.67 red
Control
white matter 0.38(0.09) 1.06(0.15) 0.42(0.10) 0.62 Green dorsal CX 1.15(0.17) 0.68(0.11) 0.59(0.09) 0.79 red
lateral CX 0.74(0.07) 0.99(0.13) 0.65(0.08) 0.81 green striatum 0.83(0.11) 0.86(0.12) 0.65(0.11) 0.78 red
control
white matter 0.50(0.06) 1.07(0.10) 0.57(0.11) 0.71 green dorsal CX 1.17(0.19) 0.66(0.11) 0.60(0.09) 0.80 red
lateral CX 0.77(0.07) 0.94(0.09) 0.70(0.07) 0.81 green striatum 0.78(0.11) 0.89(0.13) 0.65(0.10) 0.77 red
Table 6.2: Diffusion coefficient of rat brain(hydrocephalic3).
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Hydrocephalic 1 Dxx(red)
Dy(green)
Dzz(blue)
Trace
color in color map
white 0.60 (0.08) 0.73 (0.02) 0.50 (0.08) 0.61 green
matter
dorsal CX 1.20 (0.02) 0.71 (0.11) 0.73 (0.11) 0.88 red
lateral CX 0.84 (0.12) 0.94 (0.16) 0.84 (0.13) 0.87 green
striatum 0.94 (0.14) 0.72 (0.10) 0.78 (0.10) 0.81 red
Hydrocephalic2
white 0.71 (0.07) 0.75 (0.12) 0.77 (0.12) 0.74 green
matter
dorsal CX 1.08 (0.13) 0.72 (0.08) 0.70 (0.07) 0.83 red
lateral CX 0.70 (0.14) 1.00 (0.18) 0.72 (0.15) 0.80 green
striatum 0.90 (0.11) 0.85(0.11) 0.68(0.10) 0.81 red
Hydrocephalic3
white 0.64(0.14) 0.90(0.23) 1.05(0.28) 0.86 light green
matter
dorsal CX 0.95(0.13) 0.79(0.09) 0.74(0.16) 0.84 red
lateral CX 0.77(0.08) 1.01(0.14) 0.74(0.09) 0.83 green
striatum 0.86(0.11) 0.85(0.11) 0.66(0.09) 0.79 red
Table 6.3: Diffusion coefficient of rat brain (shunted).
direction
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shunted
I I
white matter
0.47(0.02) 1.21(0.32)
0.46(0.07)
I 0.71
color in
color map
dorsal CX 1.10(0.18) 0.71(0.11) 0.55(0.10) 0.79 green
lateral CX 0.70(0.09) 0.95(0.12) 0.66(0.12) 0.77 red striatum 0.81(0.12) 0.68(0.10) 0.65(0.09) 0.71 green
Table 6.16.3. Apparent diffusion coefficients (x103 mm2/s) recorded at 20 0C in 4
regions (periventricular white matter, dorsal and lateral 21day rat brains. Dxx is the lateral, Dyy is the anteroposteral direction. Different values between the anisotropy. Values were ADT calculated in the region type are the largest of the components.
cortex and striatum) from fixed dorsoventral, and Dzz is the three directions indicate tissue of interest. The figures in bold
Table 6.4: Degree of anisotropy of white matter.
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Sample degree of anisotropy (
Dyy/0.5(Dxx+Dzz)) Control 2.43
Control2 2.65
Control3 2.00
Hydrocephalic 1 1.32
Hydrocephalic2 1.01
Hydrocephalic3 1.07
Shunted 2.60
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pia surface of cerebral cortex
3
cerebral cortex
White matter
Fig 6.1 Neurons' shapes and their dendritic and axonal branching patterns of the cerebral cortex. 1, Pyramidal cell 2, Stellate cell 3, fusiform cell.
2
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lateral CX
. dorsal CX
white matter
Fig 6.2 Rat brain T2weighted image with region of interest Labelled.(0.5cm * 0.5cm)
stn atum
AL A

Full Text 
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DIFFUSION TENSOR IMAGING AND THE MEASUREMENT OF DIFFUSION TENSORS IN BIOLOGICAL SYSTEMS By LEI YANG 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 1999
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Copyright 1999 By Lei Yang
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ACKNOWLEDGEMENTS I would like to thank my supervisor, Dr. Andrew, for his kindness, patience, and encouragement of my graduate career. Without him, this dissertation would not have been finished. I learned much from his deep insights into Physics. I would also like to thank my research supervisor, Dr. Mareci, for his generosity and guidance. He led me and the whole research group into the diffusion imaging area, which has produced one paper, two SMRM conference oral presentations, three conference abstracts, and three more submitted papers. His excellent idea of introducing coherence transfer pathway into the diffusionweighting calculation was the starting point of the new formula and new diffusion tensor imaging methods. Also thanks to Dr. Inglis for teaching me every detail about NMR, giving me invaluable suggestions about my research. I would also like to give special thanks to Dr. Sullivan and to Dr. Yelton for taking care of international students like me who was thousands of miles away from home and struggling to survive in a foreign country. I express my sincere thanks to many colleagues who have made my stay at the University of Florida something that I will remember fondly for years. I appreciate Dr. Wirth, Dr. Jones, Dr. Harris, Xeve Silver for their help in surgery, Dr. Briggs for letting me use his office, Dr. Fitzsimmons for helping me build the rf coil, Dan Plant for hardware support, and Haiquan Dai for computer support. iii
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This thesis is dedicated to my father, Yuhui Yang, who survived the disaster of the cultural revolution since 1967. It was his tolerance and love that supported me and my whole family through the dark ages. All these achievement belong to Jesus Christ, my lord and savior. He is always with me, no matter what happens. May all praises go to God! iv
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TABLE OF CONTENTS Page ACKNOWLEDGMENTS iÂ« ABSTRACT ix CHAPTERS 1 INTRODUCTION AND BACKGROUND 1 Introduction 1 Background 4 Brownian Motion and Diffusion Distance 4 Diffusion Effects on NMR 5 Methods for Apparent Diffusion Coefficient(ADC) Measurement 6 Bipolar gradient pulse spinecho sequence 6 Bipolar gradient pulse stimulatedecho sequence 7 Problems for these two techniques 7 Diffusion Coefficient as an Important Parameter in Biomedical Systems 8 Clinical application of diffusion MR imaging 9 Unsolved Problem: bValue Calculation for Any Order of Quantum Coherence. ...9 bValue calculation problem in different sequence 9 bValue calculation problem in multiple quantum coherence 11 Conclusion 11 2 A COHERENCE TRANSFER APPROACH TO GRADIENT WEIGHTING FACTOR CALCULATION 17 Theory 17 Spin Evolution in the Presence of Anisotropic Diffusion 18 Explicit Effect of RF Pulses 21 Application to Different Pulse Sequences 23 Applying the Theory to Single Quantum Coherence Sequences in Isotropic Condition 24 Gradient echo sequence 24 Spinecho sequence 24 Stimulcatedecho sequence 25 Double Quantum Coherence 25 3 DIFFUSION TENSOR METHODS 28 v
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Data Acquisition and Processing 28 Formulas for Data Acquisition 28 Data Processing 30 Spin Echo and Stimulated Echo Diffusion Tensor Imaging 30 bMatrix of Spin Echo Sequence 30 bMatrix of Stimulated Echo 32 Experiments 34 Conclusion 36 4 FAST DIFFUSION TENSOR IMAGING USING A MULTIPLE STIMULATED ECHO SEQUENCE 41 Introduction 41 Methods 41 bMatrix Calculation for Stimulated Echo Sequence 41 General Library for bMatrix Calculation for Stimulated Echo 43 Multiple Stimulated Echo Sequence 44 Results 46 Water and Celery Experiments 46 Rat Brain in vivo Experiments 51 5 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF RAT SPINAL CORD IN VITRO 53 Introduction 53 Diffusion Tensor Imaging 53 Anisotropy Index 54 Anatomic Structure of Spinal Cord 55 Gray matter of spinal cord 55 White matter of spinal cord 56 Methods 56 Fixation of Rat Spinal Cord 56 Diffusion Tensor Imaging Experiments 56 Data Analysis 58 Results 58 Diffusion Tensor Values of Normal Spinal Cord 58 Diffusion Tensor Values of Injured Spinal Cord 59 Colors in Gray Matter Region 59 Color Trace Images 59 Colors in Gray Matter Region of Normal Rat Spinal Cord 59 Colors in White Matter Region of Normal Rat Spinal Cord 60 Color Trace Images of Injured Spinal Cord 60 Trace is an Important Parameter for Different Nerve Tissues 60 Anisotropy Index of Normal Rat Spinal Cord 60 vi
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Anisotropy Index of Normal and Injured White Matter Tissues 61 Diffusion Tensor Imaging of Human Spinal Cord 61 Conclusion 61 6 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF NORMAL, HYDROCEPHALUS AND SHUNTED RAT BRAIN IN VITRO 75 Introduction 75 Infantile Hydrocephalus 75 Shunt Treatment of Hydrocephalus 76 White Matter Pathology in Hydrocephalus 76 MR Imaging Techniques 77 Anatomy Structure of Rat Brain 78 Methods 78 Fixation of the Rat Brain 78 Shunt Surgery 78 Microimaging 78 Data Processing 80 Results 81 Diffusion Tensor Values of Normal Rat Brain 81 Diffusion Tensor Values of Hydrocephalus Rat Brain 81 Diffusion Tensor Values of Rat Brain 81 Color Trace Images 81 Color in Cerebral Cortex 81 Color in Corpus Callosum 82 Anisotropy Index of White Matter in Normal and Hydrocephalic Rat Brain 82 Diffusion Tensor Trace of Lateral and Dorsal Cortex 82 Stimulated Echo Diffusion Tensor Imaging 83 Conclusion 83 7 SUMMARY, INTERPRETATION, AND CONCLUSION 98 Introduction 98 Achievements and Signification 98 A New Formula for Evaluating Diffusion Weighting 98 Two Diffusion Tensor Imaging Techniques 99 Diffusion Tensor Imaging of Normal and Injured Spinal Cord 99 Diffusion Tensor Imaging of Normal, Hydrocephalus and Shunted Rat Brain 100 Diffusion Model in Molecular Level 100 Future Studies 107 Double Quantum Diffusion Tensor Imaging and Spectroscopy 107 Spinal Cord Injury Model 107 In Vivo Diffusion Tensor Imaging 107 Summary 107 vii
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REFERENCE 109 BIOGRAPHICAL SKETCH 113 viii
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DIFFUSION TENSOR IMAGING AND THE MEASUREMENT OF DIFFUSION TENSORS IN BIOLOGICAL SYSTEMS By Lei Yang May 1999 Chairman: E. Raymond Andrew Major Department: Physics This dissertation revisits the theory of diffusionweighting factor of gradients for any sequence and presents a new comprehensive formula which not only unifies all former results, but also predicts new results for unsolved pulse sequences. Then, on the basis of this new formula, two diffusion tensor imaging techniques were built, tested, and applied to the rat spinal cord and brain, thus showing the biological and clinical significance of diffusion tensor imaging. There have been many previous attempts to derive an expression of diffusionweighting factor of gradients in different pulse sequences, but until now, nobody has been able to derive a formula for any doublequantum and multiplequantum coherence sequences, only under in case had this problem been solved experimentally. This dissertation introduces a new method which not only unifies all different expressions for single quantum coherence, but also predicts the expressions for doublequantum and ix
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multiplequantum coherence sequences. Next, using the new formula for diffusionweighting factor, two diffusion tensor imaging techniques were built: spin echo diffusion tensor imaging and multistimulated echo diffusion tensor imaging. The EchoPlan Imaging (EPI) spin echo diffusion tensor imaging had been built in 1 994 , but the multiecho stimulated echo diffusion tensor imaging technique, along with Basser's EPI stimulated echo diffusion tensor imaging technique, were the first two techniques in the world that used stimulated echo sequence in diffusion tensor imaging. In this dissertation, diffusion tensor imaging techniques were first applied to normal and injured rat spinal cords and brain, color trace images were presented which clearly show the fiber orientations of the spinal cord that were not accomplished by any other MRI. We found that the injury of the spinal cord causes the loss of anisotropy of white matter, which is consistent with other results using apparent diffusion coefficient (ADC). It is interesting that there were substantial changes in diffusion characteristics in the injured area which appeared normal by conventional imaging. These results imply that there are consequences of spinal cord injury which dramatically alter axon structure but do not change water content, so are not detected by conventional imaging. We found that the hydrocephalus of the rat brain causes the loss of anisotropy of white matter, and hydrocephalus results in increased water content in the brain surrounding the ventricles. This has been confirmed by independent measurements. This is a new direction in the noninvasive investigation of the normal and abnormal brain and spinal cord using a new imaging technique to study tissue organization, pathological changes, and recovery with treatment. The technique could ultimately be applied to clinical situations (in the same way as ADC is now being applied X
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to measure the progress of clinical stroke). The dissertation concludes with a theoretical interpretation of the measured diffusion constants. xi
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CHAPTER 1 INTRODUCTION AND BACKGROUND Introduction The discovery of nuclear magnetic resonance (NMR) in 1946 (Bloch et al, 1946, Purcell et al, 1946) brought the world a powerful investigative technique which can be used in many different scientific fields. The publication of the first magnetic resonance imaging in 1973 (Lauterbur, 1973) created a big impact and opened the door for NMR in medical science. Since then, MRI has become one of the best noninvasive imaging methods for providing high quality images of soft tissue in the normal and injured condition. Gillies (1994) gives a general introduction on NMR in biomedicine. The repair of spinal cord injury requires that doctors be able to see the difference between normal and injured white matter as a first step, because the extent of white matter spared following spinal cord injury (SCI) may be related to the degree of function that is eventually recovered in the limbs caudal of the lesion (Schrimsher, 1993). However, conventional magnetic resonance imaging of the spinal cord cannot provide such explicit information. It has been found that water diffusion along the white matter tract is faster than that perpendicular to it, which means that the water diffusion of white matter is anisotropic (LeBihan, 1991). When the white matter has been injured, this anisotropy is also changed (Ford, 1994). Diffusionweighted imaging has been an effective technique for identifying nerve tissue pathology since it permits the discrimination of gray and white matter structures (Moseley, 1990) and normal and 1
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2 edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (Ti, T2 ). Diffusionweighted imaging is, however, only an approximation of the actual diffusion sensitivity of an anisotropic sample. A muchimproved estimation can be achieved by measuring an apparent diffusion tensor (ADT) (Basser, 1994), which is based on the diffusion coefficients in various directions. In this way it should be possible to identify regions with varying degrees of injury of the spinal cord . The Dr. Mareci research group in the University of Florida Center for Structural Biology has successfully pinpointed diseased white matter tissue. As an extension of this pioneering activity, this dissertation will present a set of methods to advance our ability to determine the exact location of injured white matter. The objectives are listed below: A General formula for the diffusionweighting factor will be given, its application to the special pulse sequences ( spin echo, stimulated echo, double quantum COSY, etc. ) will be presented, and the results will be compared with other people's results, which shows consistency. Methods for producing ADT images will be presented and rat spinal cord and brain ADT images in vitro will be acquired and analyzed in view of pathological changes between normal and injured conditions. Fast ADT imaging methods will be presented, which will be able to reduce the acquisition time to perform ADT imaging of rat spinal cord in vivo. The ADT data for normal and injured spinal cord in vivo will be acquired and analyzed .
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3 This dissertation consists of seven chapters. In the first chapter, work that has been done in diffusion will be reviewed, some unsolved problems will be pointed out, and the methods to solve these problems will be discussed. The second chapter will discuss the calculation of the diffusionweighting factor from gradients, and its application to the ADT. Calculating the diffusionweighting factor is the key to diffusioncoefficient measurements, but calculating these factors is different for different sequences under different orders of quantum coherence. This alculation is also very difficult when there are many gradients. Here the application of the concept of coherence transfer pathway to calculating the diffusionweighting factor will be introduced, which will give a general expression for the diffusionweighting factor calculation for any sequence with an arbitrary order of quantum coherence. There are basically two methods to do diffusion tensor imaging, spin echo and stimulated echo. In the third chapter the use of these two methods will be described in detail, and their advantages and disadvantages will be discussed. In Chapter 4, fast diffusion tensor imaging techniques which have been used in in vivo experiments will be presented and compared. In Chapters 5 and 6 these two methods will be applied to the rat spinal cord and rat brain. Data from normal and injured spinal cords and from normal, hydrocephalic, and shunted rat brains will be presented. The biological significance of these results will be discussed. In Chapter 7 the results obtained in this dissertation are summarized, the values of measured diffusion coefficients are analyzed, and indications for future work are given.
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Background In this chapter, several topics listed below will be discussed in order to provide basic knowledge about diffusion tensor imaging. Brownian motion and diffusion distance. Diffusion effects on NMR. Methods for apparent diffusion coefficient measurement. Diffusion coefficient as an important parameter in biomedical systems. Diffusion time and tissue structure. Unsolved problems. Brownian Motion and Diffusion Distance In a nonuniform system where species do not distribute uniformly, the selfdiffusion of a particle is described by Fick's Law (Callaghan, 1991): dP s (r y \r ,0 = Â£)Y 2 p s (r '\p ,t) [1.1] a Here, P s (f'r ,t) is the probability of finding a molecule, originally at a position ? , at a position f at time t, D is the diffusion coefficient. Under equilibrium conditions, the diffusion of the molecules is called Brownian motion (Callaghan, 1991), and p s has the initial condition: P s (r\r ,t) = S(rr ) . [1.2] For unrestricted diffusion, p s Â» 0 as ?' Â»Â°Â°. This combined with Eq. [1] yields Ps(f'f ,t) = (47iDt)3/2 exp[(f'f ) 2 /4Dt] . [1.3] This equation means the probability of finding a molecule originally at a position r' at a position f at time t , is a Gaussian distribution.
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5 The diffusion distance (RMS, root mean square displacement) ^< (F F ') 2 > is given by: This, socalled Einstein equation, gives direct interpretation of the diffusion coefficient. In one dimensional case: Diffusion Effects On NMR Molecular selfdiffusion was first observed to have an effect on NMR in 1 954 (Hahn, 1954). In Hahn's spin echo experiment, measured T 2 values were found to be abnormally low for liquids, such as water, an effect exacerbated by the poor homogeneity of the magnet at that time. The diffusion effect on magnetization in a magnetic field can be solved using the Bloch equation (Torrey, 1956, Stejskal and Tanner, 1965). In a single quantum coherence spin echo experiment, one gets: <(FF') 2 >=jdr(rr') 2 e (??' flADt [1.4] V< (rf'f > =4lDt [1.5] t M=M Q *e Tl e bD [1.6] te t' [1.7] 0 0 in which G(t) is the magnetic field gradient. In the pulsed gradient spin echo sequence (Table 1.2), [1.8]
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6 This is called the Stejskal and Tanner equation. Here 8 is the gradient duration, and A is the separation between two pulsed gradients. Methods for Apparent Diffusion Coefficient (ADC) Measurement In biological systems, the diffusion of water molecules is not only hindered by other molecules, but also restricted by some obstacles such as cell membranes, which cause the mean square displacement to be shortened. The measured diffusion coefficient, which is based on the signal attenuation of NMR signals ( Eq. 1.6), is called the Apparent Diffusion Coefficient (ADC). Based on the basic principles discussed in the previous section, several NMR techniques have been proposed to study the diffusion. The most popular techniques are the bipolar gradient pulse spin echo sequence and the stimulated echo sequence. Bipolar gradient pulse spinecho technique The effect of diffusion on spin echo signals in the presence of a pair of gradients ( Table 1 .2) is given by: ~t M=A%*er 2 e 1 ' , [1.9] where 5 is the duration of each pulse, and A is the pulse interval. This is usually called Spin Echo ADC Method (Table 1.2). Compared with the technique using a constant gradient pulse, the advantages of using bipolar gradients are (1) It is better suited for imaging, and (2) It has a welldefined diffusion time A.
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7 Bipolar gradient pulse stimulatedecho technique A stimulated echo is generated from a sequence comprised of three RF pulses separated by time intervals t, and x 2 between the pulses (Table 1.2). After the end of the second RF pulse, part of the transverse magnetization (exactly half in the case where 90Â° pulses are used) is stored as longitudinal magnetization, which becomes insensitive to field inhomogeneities. The third RF pulse returns the stored magnetization to the transverse plane at time x, after the third pulse. The amplitude of the stimulated echo is, in the case where the three RF pulse are 90Â° pulses ( Hahn, 1950): r2 M M=I^*e T ' e Tl e 1 ' [1.10] This is called the stimulated echo ADC method (Table 1.2). The advantages of using bipolar gradients are: This technique is particularly useful for qspace diffusion experiments where a long diffusion time is required. Because T[ is usually much larger than T 2 in tissues, longer diffusion times can be achieved with a stimulated echo sequence than with a spin echo sequence without the usual signal loss due to T 2 decay. When the tissue has very short T 2 , the spin echo cannot be used, while the stimulated echo is useful due to the signaltonoise consideration. When the gradient system has large eddy current, the stimulated echo will eliminate most of the effect due to the eddy current, while spin echo cannot. Problems for these two techniques These two methods have been used (Stejskal, 1965, Cleveland, 1976, Cooper, 1974, Ford, 1994) to calculated ADC in spectroscopy and imaging experiments.
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8 However, the Stejskal and Tanner equation does not take into account the diffusionweighting factor contributed by imaging gradients, which includes bvalue contributed by each imaging gradient and bvalue contributed by the interaction between imaging and diffusion gradients, which are called cross terms. Thus the ADC methods only measure approximate diffusion coefficients ( Mattiello, 1994) and need to be improved. Diffusion Coefficient as an Important Parameter in Biomedical Systems MRI and spectroscopy are the only ways to study the molecular process in vivo noninvasively. The Brownian motion of molecules in liquids is restricted by the obstruction inside the biological systems, such as fibers, intracellular organelles (Nicholson, 1981) and membranes. So, water diffusion coefficients vary according to tissue microstructure, physiological or pathological state. This is a potential source of tissue contrast for characterization of functional studies. Using Eq. 1 .6, one will be able to measure the diffusion coefficient by varying the b values and keeping diffusion time the same. If the measured diffusion coefficient decreases with diffusion time t, this would indicate that the molecules are experiencing an impermeable or semipermeable barrier and can not diffuse beyond this barrier freely. In normal diffusion experiments, typical diffusion times lie in the range 10200ms. For water in room temperature (20 Â°C) with D~2*10" 3 mnr/s, this range of diffusion time yields RMS path length of 630 um, which is of the same order as nerve cell diameter( 1030 um) and larger than nerve fiber diameter (1pm). So water diffusion
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9 allows microdynamic studies on a scale that is much smaller than the resolution of macroscopic NMR images. Clinical application of diffusion MR imaging It is found that water diffusion in different tissues vary (about 2 to 10 times less than that of pure water (table 1.1)). Also, even in the same tissue like white or gray matter, diffusion coefficient is not the same in different directions (LeBihan, 1990, Inglis et al., 1996), in different pathological conditions (Ford, 1994). Diffusion thus appears as a new source of contrast for MR imaging. It has been shown that there is no correlation between the diffusion coefficient and the relaxation times T,,T 2 . It has been found that T,,T 2 may be normal in the diseased state, while diffusion is changed such as in early brain ischemia (Moseley, 1990). Unsolved Problem: bValue Calculation for Any Order of Quantum Coherence bValue calculation problem in different sequence As we have discussed before, bvalue calculation is the key to diffusion coefficient measurement, but it is also very complex because it consists of: 1 . bvalue contributed by each gradient. 2. bvalue contributed by the interaction between two gradients, which are called cross terms. The Stejskal and Tanner equation ( Eq. 1.8) for a pair of rectangular gradients is unable to calculate the diffusion weighting factor of the following cases: 1 . In normal experiments there are always background gradients. 2. In imaging experiment, there are many imaging gradients. 3. There are different shapes of gradients, such as trapezoidal gradient, sinusoidal gradient.
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10 4. In multiple quantum coherence (MQ) experiments, there are gradients in different coherence pathways (Table 1.2). Some scientists try to use different pulse sequences to get crossterm free images. Noeman (1990) ran the experiment twice with the diffusion gradient in opposite directions. The geometric average of the signal values has no cross term effect from imaging and diffusion gradients, but the cross terms between imaging and imaging gradients are still existing. The multiecho pulse method proposed by Van (1991) and Williams (1978) reduces medium and longrange gradients. The combination of alternating field plus multiple RF pulse method proposed by Karlicek and Lowe (1980), and Hong (1992) for spin echo and Cotts (1989) for stimulated echo eliminate medium and longrange localgradient completely. All these methods are unable to derive offdiagonal terms of the diffusion tensor. On the other hand, Mattiello ( 1994) successfully calculated all the crossterms and self terms of the gradients by a pairwised method in single quantum coherence in 2D FT spin echo sequence, but it is not clear how to solve this crossterm problem in gradient echo, stimulated echo, and in MQ conditions. According to Torrey (1956), the work by Stejskal and Tanner's (1965) and Tanner (1970), the bvalue expression for GE, SE, and STE are quite different. In a review paper (LeBihan, 1991), one can find that, when a pair of rectangular gradients apply in STE, one between first and second 90 Â°, one after third 90Â°, the StejskalTanner equation still applies. This might seem to imply the bvalue calculation for spin echo and stimulated echo are actually the same, but this conclusion is not supported by either experiment or theoretical analysis.
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11 bValue calculation in multiple quantum coherence The bvalue calculation of multiple quantum coherence has been studied (Martin, 1978; D.Zax, 1983) in a special case, when a pair of rectangular gradients is applied in the double quantum evolution time (Table 1 .2), where b = y 2 2 2 G 2 S 2 (AS/3), [1.11] which is 2 2 times larger than that of the single quantum case. This relation was supported by a set of experiments. However, it is not clear what the crossterm is between gradients in single quantum evolution time and gradients in multiple quantum revolution time, for example, in double quantum COSY experiments (Fig 1.1) Conclusion In summary, diffusion spectroscopy and imaging have been very powerful methods for studying tissue structure in different biological states. However, these methods need to be improved in order to show more accurate diffusion sensitivity. In this dissertation, a new theory and methods and their applications to the study of the spinal cord and brain will be presented.
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12 Fig 1 . 1 Double Quantum COSY experiments
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13
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14
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15
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16 .1 2 Â£ J= Â£ C3 .9 5b 55 s= C O c is
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CHAPTER 2 A COHERENCE TRANSFER APPROACH TO GRADIENT WEIGHTING FACTOR (bVALUE) CALCULATION As we have discussed in Chapter 1 , calculating the diffusion weighting factor (revalue) is the key to diffusion coefficient measurements, but calculating these factors is different for different sequences under different orders of quantum coherence. Calculation is also very difficult when there are many gradients. Here the application of the concept of coherence transfer pathway to diffusion weighting factor calculation will be introduced, which will give a general expression for the diffusion weighting factor calculation for any sequence with an arbitrary order of quantum coherence. Theory Scientists have been trying to derive bvalue expressions for different pulse sequences. In 1956, Torrey obtained a general expression (Torrey, 1956) for the Gradient Echo sequence ( Fig 1.1) using the Bloch equation. Nine years later, Stejskal and Tanner derived a general expression (Stejskal, 1965) for the Spin Echo sequence using Bloch equation and a step function. In 1970, Tanner successfully used the treatment of Carr and Purcell (Carr, 1954) to get a general expression for the stimulated echo sequence (Tanner, 1970). The bvalue calculation of multiple quantum coherence sequences has been so difficult that nobody has been able to use the old methods mentioned above to solve it. 17
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18 The bvalue calculation of multiple quantum coherence has been studied (Martin, 1982) in a special case when a pair of rectangular gradients applied in the double quantum evolution time (Fig. 1.1), where b = y 2 2 2 G 2 S 2 {k5l'S) [2.1] which is 2 2 times larger than that of the single quantum case. This relation was supported by a set of experiments. Spin Evolution in the Presence of Anisotropic Diffusion By measuring the signal attenuation caused by diffusion, scientists will be able to measure the diffusion coefficient. The diffusion effect can be described by the BlochTorrey equation of motion for transverse magnetization (Callaghan, 1991): Â— Â— = iyrGM + +V T DVM\ [2.2] at where M + = M x +iM y , D is the diffusion tensor, G is the gradient, and V r is the transpose of V . The solution to this equation has the general form: M + (r,t) = A(t)e^ r ^ [2.3] where F(t) = jdt' G(t') . A(t) is the timedependent amplitude of the magnetization and 0 F(t) can be visualized as the contribution of an applied gradient to the precessional phase, cp(t) = yrF(t) , of the magnetization at time t . Information on the diffusion process is contained in the timedependent amplitude coefficient, A(t) . Expanding Eq.[2.3] in terms of this general solution, given in Eq.[2.3],
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19 results in the following definition of this coefficient, ' Â— T = Â— y 2 \dt'F (t')DF(t') A(t) = M 0 e 0 [2.4] Expanding the diffusion dependent portion of the exponential factor in terms of Cartesian coordinates results in, F T ( t ).D.F(t)= x x mpjm i = x,y,zj =x,y,z Then the timedependent coefficient becomes, I I M<)^ A{t) = M 0 e '=W=w [2.5] where M 0 is the equilibrium magnetization and the diffusion weighting factor, by , can be written as the following, t bijtf^y^dt'F^Fjit') [2.6] o In general, a pulse sequence can be considered as a series of RF pulses and timedependent gradient pulses. If each gradient pulse is described by a timedependent function, g(k,t) for the kth gradient pulse, the general form of all gradients can be written as, G(/) = Â£g(M [2.7] Jt=l where,
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20 g(M = \g(t) ,t k t k +5 k [2.8] Using this timedependent form of the gradient, n t F{t)=Y J \dt'z{kJ'), k=\ 0 [2.9] the diffusion weighting factor becomes, b ij {t) = r 2 f j f j \dt'F i {k,t')F j {l,t') k=\l=\ 0 n n t ft' \ff = r 2 X Â£ J*' \dt" gj (i,t") k=n=i o vo Ao Jfc=l/=1 where [2.10] [2.11] Here k, 1 are series number of gradients, and i, j are the directions that the gradient applies. Eq.[l 1] gives the diffusion effect of a pair of gradients in the pulse sequence, and the sum of such pairwise combination of all gradients gives the total diffusion effect of all gradients in the pulse sequence (Eq. [10]), so Eq. [1 1] is the start point of bvalue calculation for any specific pulse sequence. Recently, Mattiello (1994) introduced analytical expressions for the evaluation of the diffusion weighting matrix. Since they
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21 explicitly include the effect of a spin echo RF pulse sequence in their analysis, it is only applicable to spin echo imaging sequences. By contrast, our analysis is completely general in terms of RF, for known but arbitrary shaped gradients. Explicit Effect of RF Pulses The method of calculating diffusion weighting factors developed above has been done by Torrey (Torrey, 1956) in the case of isotropic diffusion. It can be applied directly to relatively simple sequences, such as the gradient echo sequence which has only a single RF excitation pulse followed by a series of gradient pulses. However, in the more complicated sequence with more than one RF pulse, Eq. [2.10] can not be applied directly. Using the step function to deal with the effect of second 1 80Â° RF pulse (Fig. 1.1), Stejskal and Tanner derived a general expression (Stejskal, 1965) for Spin Echo sequence using the Bloch equation. Later, Tanner successfully used the treatment of Carr and Purcell (Carr, 1954) to deal with the second and third 90Â° RF pulse (Fig 1.1) and obtained a general expression for Stimulated Echo sequence (Tanner, 1970) . The bvalue calculation of multiple quantum coherence sequences has been so difficult that nobody can use the old methods mentioned above to solve it. The effect of any sequence of RF pulses can be understood in the context of coherence transfer pathways (Bodenhausen). The coherence transfer pathway is defined by the value of the coherence order, p t , during the interval following the z'th RF pulse. For systems of isolated spin1/2, p x can be 1, 0, or 1. The magnetization will form an echo during interval n, due to precession in the main field, whenever
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22 n T d p i r i =o. [2.12] Here r,is the duration of the z'th interval. Using the same formulation, an echo is formed, due to precession in the applied gradient field, when the following condition is satisfied, Since the effect of the RF pulse sequence on the applied gradients is contained in the coherenceorder pathway (through Eq. [2.13]), this can be incorporated into the calculation of the diffusion weighting factors in the following fashion. The RF pulse sequence can be included through the definition of an effective gradient, which includes the coherence order, by modifying Eqs. [2.7] and [2.8] to be, n [2.13] ti+Si where F(i, Si ) = \dt' g(i,t'). ti G*(t)= Â£g\k,t) [2.14] where ) ,t k t k +S k [2.15]
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23 This modified form of the gradient can be substituted directly into Eqs [2.91 1], resulting in the final diffusionweighting factor for any sequence of any order of quantum coherence. Application to Different Pulse Sequences Now we apply the concept of coherence pathway to the bvalue calculation for some specific sequences, consider a general sequence (Fig 2.1), the general equation of btensor becomes bijit)= r X t\dt\ \dt"g t (k /) * Sdt'glQ /) k=\k=\Q [2.16] The self and cross terms of gradient G k , G, are: (bij(t) ) =r 2 p k P l G k Gi(T l + T 2 t l S l /2 s 12) [2.17] (b ij (t)) vv = y 2 *p 2 v Gl(S 2 v (tt v 2*S v /3Â£/2)S v * Â£ 2 /l2 + Â£ l/60) [2.18] where pÂ„ p k is the coherence order, and v = k or 1 , 8 is the raising time of the gradient. The bvalue of any sequence (including GE, SE, STE for any order of quantum coherence) are the sum of different combinations of these two buildingblock terms. So this provides people a straightforward way to analyze and create crosstermfree sequences or doing diffusiontensor calculation. The above formulas are tested by applying them to some wellknown sequences (such as single quantum coherence spinecho and stimulatedecho experiments) and getting same results as what has been done by other scientists using other methods, also they give predictions about multiquantum coherence gradient diffusion weighting which has not been done before.
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24 Applying the Theory to Single Quantum Coherence Sequences in Isotropic Condition Gradient echo sequence For the general sequence (Fig 2.1), with a =90, ft = 0,rl = t2, the sequence reduces to Gradient echo sequence, where p k = P m = Pi = 1 Â• Consider only two gradients in isotropic diffusion: Gk Â— Â— G/, Sk Â— Si b (0= [b (t) ) kk + (b (0 ) +2* [b (t) ) = r 2 *Gl(Sl((tit k )S k /3)S k *Â£ 2 /6 + Â£ 3 /30) [2.19] which is consistent with the results in Callaghan's book (Callaghan, 1991). Spinecho sequence For the general sequence (Fig 2.1), with a = 90, j5 = 180, rl = xl = tel 2 , the sequence reduces to Spinecho sequence, where p k =l, p,=1 . Using the condition that the echo is formed: t 'jdig(k>t')=i ]dtgik.i) t 2 2 Â°] 1=1 0 1=1 rl The general equation of bvalue becomes b (0= r 2 tt\dt(idt"(g (k ,t"< r,)g (k ,t"> rii) =i {F(t)2*t*f(t))*{F(t)2*Z*f(t))dt [221]
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25 where J(t) = ^G\(t)dt = 0,when t < rl, <; = 1 when f > rl, which yields the Stejskal Tanner's equation (Stejskal, 1965). Stimulatedecho sequence For the general sequence (Fig 2.1), with a = 90, ft = 90, rl * t2, another RF pulse 0= 90 at t = rl, the sequence reduces to stimulatedecho sequence, where p k =L P m =0, Pr1. Using the condition that the echo is formed: n rl n rl+r2 Z Idi'g (k ,t")= X sdt'g (k ,/") Â«=i o /=1 t2 The general equation of bvalue becomes n n I t\ rl rl+r2\ ^(rl+r2)= y 2 II J+J +  U' *=l*=l\ 0 rl t2 I !dt"g*(k /) n rl ft Y (rl+r2 Y rl +f r2 f^l + r2 =/ 2 (Z(JA' g (k ,t) +(r2rl) I dt"g{k,t") + ) dt"\ J di'g (k ,t") J v r 2 J t2 *=l o [2.22] which yields J.E Tanner's equation (Tanner, 1970). Double quantum coherence In doublequantum COSY experiment (Fig 2.2): (V> ) =r 2 G*G/(n + r 2 t,5,l 2 s I 2 ) (tyO) = 7 2 * 4^ (/ t m 2 * S m I 3 e 1 2) S m * s 2 1 12 + s 3 / 60) etc.
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26 These formulas are still waiting for experimental verification. If G k , G m are both between 0 and p, and G k = G m = G, s=0: b = r 2 rG 2 s\ L tk s/3) Which yields to Martin's result (Martin, 1982). It will be seen that the general formula developed in this chapter not only includes special cases previously used, but is also valuable for the experimental work documented in this thesis and for other new sequences.
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27 P A N x2 coherence pathway Fig 2.1 General pulse sequence t + 8  + s a \ / \ / w \ / \ 1/ t k t k+ 5 k+ ^ /t + 8, + p k =1 p. = 1 P =2 m xl i2 coherence pathway for doublequantum coherence COSY experiment. Fig 2.2 Double quantum coherence sequence and coherence pathway
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CHAPTER 3 DIFFUSION TENSOR METHODS As we have discussed in Ch. 2, the signal attenuation due to the diffusion of molecules is described by Eq. [2.5], and diffusion weighting due to the gradient is expressed by Eq. [2.16]. In this chapter, we will apply these equations on the spin echo and the stimulated echo sequence and build up methods for diffusion tensor imaging. Data Acquisition and Processing Formulas for Data Acquisition Taking the logarithm of Eq. [2.5], we obtain: In 5 =T.bij*DijS(0) [3.1] ij where S is the NMR signal intensity with diffusion gradient, and S(0) is the signal intensity without diffusion gradient. This is a 6variable linear function where bij=bji. We use multivariable linear regression (Rao, 1965) of Eq. [3.1] to calculate Dij in a voxel. All echo intensities are measured for a series of gradient pulse sequence in which gradients are applied in seven nonlinear directions. In each direction, m measurements of S are made at different gradient strength. These 7*m observations are stored as an (7*m)*l column vector. 28
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29 Y=lnS J \nS : InS 2 UnS 7 * m y [3.2] We also define the column of vector of parameters to be determined: D = Du D\2 D22 D23 D33 In 5(0) J [3.3] Then Eq. [3.1] can be written as: X*D=Y [3.4] where X is the bvalue matrix: X= 2*b l 21 ; 21 1 b 2\ I b 3x 2*bA, 2*b 31 2*b Â°22 b\ 2 31 Â°22 1 * l1 1 b 32 2*b 32 2*b\ 2 Â°33 *33 1 1 bÂ» n 2* b Â« 2l 2*bÂ» n bÂ» 22 2* b Â» 32 bÂ» 33 1 [3.5] Then the diffusion tensor D is (Roa, 1965): D = (X'X)~ l X'Y [3.6]
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30 Define Y and P as: Y = PY [3.7] P = X(X'X)~ l X [3.8] and the square root of standard error is : SSE = (YY) = TPY [3.9] Data Processing Eq. [3.5] is the start point to design the experiment and data processing of the diffusion tensor imaging. In Fig. 3. 1 and 3.2, flow chart of data processing are shown. The value of each by element was calculated using Mathematica (Wolfram Research, Inc.) (Fig. 3.1) and matrix linear regression was performed using Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign) (Fig. 3.2). Spin Echo and Stimulated Echo Diffusion Tensor Imaging bMatrix of Spin Echo Sequence For the diffusionweighted spin echo ("Sems.c") sequence (Fig 3.3), the bMatrix can be derived from Eq. [2.16]. Here r = read, p = phase, s = slice, and the number i denotes the time interval of the pulse sequence. For example, gdro * gror */w231 gives the cross terms between gradients gdro and gror (Fig 3.3).
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31 b rr = y {gdro * m22 + 2 * gdro * gror * m23 1 + gror * 77133 1 + 2 2* gror* gro*m37\ +gro^*mT]} b n r = 7" {gdro* gdpe *m22 + gdpe* gror *m23\ + gdro*gpe *m232 + gpe* gror *w3321 +gpe*gro * m 312 } b pp = 7 [gdpe~ * m22 + 2 * g d P e *gpe*m232 + gpe~ * 771332} 2 b sr = Y {gss*gdro *(ml2/w233 /2 + w24) + ^ 5 ' s *^ ro *(wl7'Â«373 12) + gss*gror *(w 131 m333i/2 + 771341) + gdss * gdro * m22 + gdss*gror *m23\) 2 b S p = r {gss*gdpe*(m\2m233 /2 + m24) + gw*gpÂ£?*(m 132 7713332/ 2 + m342) + gdpe* gdss *m22 + SP e *gdss*m232 } 2 2 ^55 = / ( 8 SS * (wnwi 33 + 2 *wi4 + /w333/ 4/77343 + 77244) + gss * gdss(2 * ml 2 7W233 + 2 * Â™24) + gdss 2 * m22} here : 'mil ' ml 2 w22 wil3 w23 /Â«33 ml 4 m24 m34 m44 K m\l mil mhl m41 mil,
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32 1/4*(Â£,*(7ESi/2)S,*s 2 /6 + Â£/30) [l/2*S l *S 2 (t 2 2tn ^(A 2 ^Mzf 6 12 120 i! i; 7 2 2 bMatrix of Stimulated Echo For the Diffusionweighted Stimulated Echo ("Stems.c") sequence (Fig 3.4), the bMatrix can be derived from Eq. [2.16]. Here r = read, p = phase, s = slice, and the number i denotes the time interval of the pulse sequence. For example, gdro*gror *m23\ gives the cross terms between gradients gdro and gror (Fig 3.4). b, , = grol (kj m22\ + m77) + gro*gdro(2m67 + 2Â£,m231) + gdro 2 (m33 + m66) b 22 = gpe 2 (w222) + gpe * gdpe * (2m232) + gdpe 2 (m33 + w66) Â£33 = gss 2 1 + 2ml 4 + m44 + k]m223 2k 2 ml23 2k 3 m243 + m55) + gss * gdss(2ml3 + 2m3A 2k 3 m233 + 2m56) + gdss 2 (m33 + w66) b 2i =gpe* gss(m\22 + w242 Â£ 3 m2232) + gdpe * gss(m\3 + m34 k 3 m233 + m56) + gpe * gdss(m232) + gdpe * gdss(m33 + m66)
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33 b n = gro * gpe(k ] m222 1) + gro * gdpe(m23 1 + m67) + gpe * gdro * m232 + gdpe * gdro * (m33 + m66) b u = gro* gss{k x m\2\ + k x m24\k y ki m223\ + m51) + gdro * gss(m\3 + m34 k 3 m233 + m56) + gro * gdss(k 1 m23 1 + m67) + gdro * gdss(m33 + m66) Here mij= mji, 1 is gro direction, 2 is gpe, 3 is gss. 'will ml2 m22 m\3 m23 m33 m\4 m24 m34 m44 f 5 /480 + r ,j;/4 1 / 2*Â£, S 2i * (r, ~t 2 2/3*S' 2 ,Â£ 2 5 2i 'i2 + \/2*S l S i (rrt i Â£2 &/122/3*Â£ + Â£,/16*(Â£ + Â£ 2 /3) ^ 2 ,/8*(^ + f 2 /3) Â£,/8*(Â£ + f J /3) ^/24 + Â£ 3 /120 Â»22y52L^!^ + Z +W . (r _ /2) _M._MÂ£ 6 12 120 2 ' 2 ' VT ; 2 2 m55 Vm57 w67 w77y
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34 r $/24 + Â£3 /480 +
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35 Small amount of CuS0 4 (about 0.01 ml) was added to water to decrease T,. A pulsed field gradient spin echo imaging sequence was implemented on a SISCO imaging spectrometer (Varian NMR Instruments, Palo Alto, California) equipped with a 4.7 T, 33 cm bore magnet (Oxford Instruments Ltd., Oxford, England) and an actively shielded gradient/shim system (Nalorac Cryogenics Corp.) capable of 5 G/cm along x, y and z. Ten accurately calibrated gradient amplitudes from 0.5 to 5 G/cm in 0.5 G/cm increments were applied along the seven directions, yielding maximum bmatrix values of 1 200 s/mm 2 . Each image was acquired as a 128 x 128 matrix and the overall experiment time was 5 hours. On a pixelbypixel basis, a sevenparameter matrix linear regression (Rao, 1965) was used to perform a fit for each element of the diffusion tensor (D z y). This was accomplished by determining the natural logarithm of the ratio of measured signal to the signal with no diffusionweighting gradients, then determining the diffusion tensor using the known values of by according to Eq. [3.1]. The value of each by element was calculated using Mathematica (Wolfram Research, Inc.) and matrix linear regression was performed using Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign). The calculated apparent diffusion tensor is shown below along with its standard error matrix (Rao, 1967): D = r 1.54 0.04 0.05^ 0.04 1.54 0.03 v0.05 0.03 1.52 j + ^ Â±0.01 Â±0.01 Â±0.0 1 A Â±0.01 Â±0.01 Â±0.01 U0.01 Â±0.01 Â±0.01/ xlO" [3.10] The correlation coefficient, r 2 = 0.9993. As expected for a sample exhibiting isotropic translational diffusion the diagonal terms of the diffusion tensor are equivalent within experimental error, while the offdiagonal terms are nearly zero. The offdiagonal terms
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36 are a little bit out of error value, this might be due to the nonuniform magnetic field. In this case the diffusion tensor correctly reflects the scalar diffusion of the water. The accuracy of diffusion weighting factors deduced using our formalism was also tested by applying the diffusion tensor imaging method to pure water at 20.5Â±1.0Â°C using the multiple stimulated echo sequence (Mareci, 1986), two gradient amplitudes (2.5G/cm and 4.5 G/cm) were applied along 7 directions (x, y, z, xy, xz, yz, and xyz), 2 yielding maximum bmatrix values of 1300 s/mm (please refer to Ch. 2 about the bvalue calculation). 6x7 diffusionweighted images were acquired in a total acquisition time of 56 minutes. The calculated apparent diffusion tensor is shown below along with its standard error matrix (Rao, 1967): D = f 2.12 0.12 0.19^ 0.12 2.10 0.25 v0.19 0.25 2.20 J + Â±0.08 +0.0 Â±0.10^ Â±0.10 Â±0.12 Â±0.10 U0.10 Â±0.10 Â±0.1 0) xlO 3 '"'Â»>/ [3.11] The correlation coefficient, r 2 = 0.9788. Again as expected for a sample exhibiting isotropic translational diffusion the diagonal terms of the diffusion tensor are equivalent, within experimental error, while the offdiagonal terms are nearly zero. Errors are ten times larger than those in spin echo experiments, that might be because of the less number of points used in linear square fit. The values of D are significantly higher in the second experiment compared with the first experiment; this is attributed to the difference of temperature in the two experiments (12.5Â°C and 20.5Â°C respectively)"
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37 Conclusion In this chapter, methods for diffusion tensor imaging using spin echo and stimulated echo sequence are built and successfully tested by the experiments. Now can apply these method to study biological structures. Start processing Input the parameter from the image file Calculate the time for each gradient Calculate bvalue for all gradient combination Calculate total bvalue for all bmatrix Multiple variable linear regression Output regression result Fig 3.1 Flow chart of data processing for multiple linear regression
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38 I Input multiple linear regression result Input images, do Fourier Add weight for each image from multiple linear regression results Add all images together including weighting Fig 3.2 Flow chart for image processing.
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39 90Â° 180o & gss gdss gss gdss slice ^ A /"A gror gdro gdro gro read /"AM gpe gdpe gdpe Phase q/~A /~\ time period III I I  I 1 2 3 4 5 6 7 Fig 3.3 Spin echo sequence
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40 90Â° 90" 90Â° G gss gdss g SS gdss gss gdss slice ^ MA m AA gror gdro gdro gdro gro read */~\ /~\ Am 8P e g d P e gdpe gdpe Phase O AA time period _JJ __  I 1 2 3 4 5 6 7 Fig 3.4 Stimulated echo sequence
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CHAPTER 4 FAST DIFFUSION TENSOR IMAGING USING A MULTIPLE STIMULATED ECHO SEQUENCE Diffusionweighted imaging is an effective technique to identify brain tissue pathology since it permits discrimination of grey and white matter structures (Moseley, 1990) and normal and edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (Ti, T2 ). Diffusionweighted imaging, however, is only an approximation to the actual diffusion sensitivity of an anisotropic sample. A much improved estimation can be achieved by measuring an apparent diffusion tensor. In this way it should be possible to identify regions with varying degrees of edema. Diffusion tensor imaging has been previously performed using a spinecho sequence (Basser, 1994). However, the total acquisition time is typically more than 3 hours, which is prohibitive for in vivo experiments. Here, we introduce a faster diffusiontensor imaging method using stimulated echoes which reduces the acquisition time to less than an hour and which is successfully used to obtain in vivo images from the rat brain. Methods bmatrix Calculation for Stimulated Echo Sequence The bvalue expression (Eq. 1.7) for a gradient echo is given by (Torrey, 1956): Introduction [4.1] 41
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42 Here we substitute G(t") by including the coherence order P : : (Bodenhausen, 1984, Bain, 1984, Mareci, 1988) G(0 =!*(*.') k=] where, [4.2] g (*,0= \PkSif) ,t k t k +8 k Then we get b expression for stimulated echo sequence: b(T\+z2) =y 2 Y(]dt"G*(t"))dt' Â° V 0 J [4.3] This expression is equivalent to Tanner's expression (Tanner, 1970 ), this is shown as follows: For stimulated echo: 1 ,0<;
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43 General Library for bMatrix Calculation for Stimulated Echo From eq [4.6], rewrite it in matrix form: rl+r2 f V Z>,(rl + r2) = rÂ£ Â£ \dt\ \dt" gl *(k,t")\\\dt" gj *{l,t") =ii(M'))Â„ [4.7] where rl+r2 ft' M'))Â„=r 2 Ja' Ja" & Â•(*,/") J>' g ,*(/,r) [4.8] This formulation of the diffusion weighting is central to the calculation of diffusionweighting factors and forms the starting point for the explicit application to specific pulse sequences. The indices k and / refer to a pair of gradient pulses applied in the i and / directions, respectively. Therefore, the calculation involves the explicit evaluation of pairwise autoterms (k = l) and crossterms (k * I ) in the diffusionweighting factors, where the total diffusion weighting is the sum of these pairwise terms. Here we give an example for bij term for a pair of trapezoidal pulse shape gradient ( other shape gradients can be calculated in the same way): =r 2 plG ik G Jk (S 2 k {tt k 2S k /3Â£ /2)S k *s 2 /U + e* /60) and the cross term as, (by {t)) u = Y 2 p k p, G ik G, 8 k S,(t hSjlel 2) [4.9] [4.10] The advantage of using Eq 4.8 rather than Tanner's expression (Eq. 4.6) is, we have the library for b value calculation for stimulated echo sequence, which is the same
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44 as the library for gradient echo and spin echo sequence, for any order of quantum coherence. Multiple Stimulated Echo Sequence Stimulated echo and multiple stimulated echoes method have been used to measure T, relaxation time (Haase, 1986) (Mareci, 1986)and diffusion cofficient (Tanner, 1970, Merboldt, 1985), diffusion coefficient in inhomogeneous fields (Latour, 1993) and in Heterogenous System(Cotts, 1989). In a system with long T, and short T 2 (such as in injured part of spinal cord, T 2 ~ 25ms(Ford, 1994)), the advantage of using stimulated echo (MacFall, 1988, Lebihan, 1991) is that, the echo time TE can be decreased, and TL can be increased so as to have enough signalto noise ratio, while with long diffusion time(A) , the diffusion weighting factor b will still be large enough to ensure a good fit for diffusion tensor calculation. Stimulated echo is also very useful in studying the timedependant restricted diffusion (Gates, 1994)). The multiplestimulated echo sequence (Fig. 4.1) (Frahm, 1985, Mareci, 1986) is used to acquire three diffusionweighted echoes (one spinecho and two stimulatedecho) per transient. Crush gradients gcrush (Fig. 4.1) are used to get rid of unused spinecho. The signal intensity of the three echoes is given by(Mareci, 1986): f nl \ "I I M')4, An (t) = M 0 sin a n * [Tl cosa/J * e~ TE ' h e TLnl ^ e [4. 1 1] Here n is the number of echoes in the sequence, if we treat first spin echo as a special stimulated echo where TL=0, then TL0=0, a 0 =90Â° for the first spin echo, TL1, a,=45Â° for
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45 the first stimulated echo, and TL2, a 2 =90Â° for the second stimulated echo. The diffusionweighting factor(bij) caused by the diffusion gradient is given by: b = r 2 G 2 di(r S 2 (A n S /3) [4.12] Here A 0 = Ase, A,= A slel , A 2 = A ste2 , G diff and 8 is the strength and time duration of the diffusion gradient. The diffusion weighting factor increases as the time (A) between the first (dephasing) diffusion gradient and each rephasing diffusion gradient increases, so the three echoes in the pulse sequence have different by values. When the data are processed, each diffusion weighted image is divided by its corresponding nondiffusionweighted image, thus z I w\ A n (G dlff , t) I An (0, t) = e [4.13] So each echo is used as one single point to perform linear regression for each element of tensor according to eq (3), having taken natural logarithms. For six elements of diffusion tensor, seven different gradient weighting directions are required. We typically use six by values (two different diffusion gradient amplitudes) per direction to ensure a good fit to the data. One set of nondiffusionweighted images were also collected to normalize the data by removing the differential T\ effects produced by the different diffusion weighting times (A).Therefore, a total of eight experiments are required: one nondiffusionweighted and seven differently weighted, Gdiff oriented experiments.
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46 Using the generalized library for calculating the diffusion weighting factor (bfactor) used for all sequences (Yang, 1995), the bjj of stimulatedecho sequence were calculated, including the crossterms between the diffusion and imaging gradients. On a pixelbypixel basis,a sevenparameter matrix linear regression(Rao, 1967 ) was used to perform a fit for each element of tensor according to eq. [3.6], having taken natural logarithms. Viewit (National Center for Supercomputing Applications, University of Illinois at UrbanaChampaign) was used to perform matrix linear regression and to process the 42 diffusionweighted images to obtain the diffusion tensor images. Results The fast diffusion tensor imaging method using multiple stimulated echo sequence was first tested using water and celery samples, then applied to the rat brain in vivo experiments. Water and Celery Experiments Using the multiple stimulated echo sequence described above, 42 diffusionweighted images were acquired in a total acquisition time of 56 minutes. Diffusion gradients were applied along x, y, z, xy, xz, yz, and xyz, while varying by between 40 and 1300 s/mm 2 . The accuracy of bij was measured using the general formalism Eq. [4.7] and this was tested by applying the fast diffusion tensor imaging method to a sample of pure water and a piece of celery at 20.5Â± 1.0Â°C. The water is contained in a cylindered glass vessel (2 cm in diameter and 3 cm in length). Small amount of CuS0 4 (about 0.01 ml) was put into the water inorder to decrease T,. A piece of fresh celery (2.5 cm in length, 2 cm in width) was
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47 put into the vessel, with its fiber orienting along vessel. The vessel was put into the 4.7 T magnet, with its axis along the magnetic field. The calculated apparent diffusion tensor is shown below along with its standard error matrix. The diffusion coefficient of water surround the celery is (Fig 4.2): D 2.12 0.12 0.19 0.11 2.10 0.24 0.19 0.24 2.20 J + 'Â±0.08 Â±0.11 Â±0.10^ Â±0.10 Â±0.12 Â±0.10 Â±0.10 Â±0.10 Â±0.10; xlO" [4.14] The diffusion coefficient of the celery fiber is (Fig 4.2): D = 0.45 0.27 0.14 0.27 0.43 0.06 0.14 0.06 1.37 ; + 'Â±0.21 Â±0.05 Â±0.06^ Â±0.05 Â±0.15 Â±0.05 Â±0.06 Â±0.05 Â±0.147 xlO" [4.15] The diffusion coefficient of the celery (nonfiber region) is (Fig 4.2): D = ( 0.98 0.06 0.11 0.06 1.01 0.16 I0.11 0.16 0.90; + 'Â±0.10 Â±0.10 Â±0.l0 Â±0.05 Â±0.12 Â±0.07 U0.11 Â±0.07 Â±0.13; xlO" [4.16] Each value of is obtained from a plot of ln(S/S 0 ) against b(t). The correlation coefficient of this process is r 2 = 0.9788. As expected for a sample exhibiting isotropic translational diffusion, the diagonal terms of the diffusion tensor in [4.14] are equivalent, within experimental error, while the offdiagonal terms are nearly zero. In this case the diffusion tensor correctly reflects the scalar diffusion of the water. In the region of the fiber inside celery, Dzz (1.37 Â± 0.14 *10 " 3 mm 2 /s ) is much larger than Dxx (0.45 Â± 0.21*10 3 mm 2 /s) and Dyy (0.43 Â± 0. 1 5 * 1 0 3 mnr/s), which shows diffusion anisotropy inside the celery fibers. In the nonfiber region inside the celery, the diagonal terms of the diffusion tensor are equivalent (Eq. 4.16), but less than
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48 those of pure water (Eq. 4.14), while the offdiagonal terms are nearly zero, which shows isotropic but restricted diffusion inside celery (nonfiber region). Rat Brain in vivo Experiments The fast diffusion tensor imaging method was used to acquire in vivo brain images from 21 dayold rats. A diagram of rat brain anatomical structure is shown in Fig. 4.3. The rat cerebral cortex at 21days after birth has a long Ti(550 ms) and a short T2 (60 ms) (Harris, 1992) and therefore TE and TL were optimized to 33 ms and 210 ms, respectively. This enabled us to attain b values up to 1300 s/mm^ and to acquire a good signaltonoise ratio, both of which are critical for the tensor calculation. The in vivo diffusion tensor data is shown in the table below (Table 4.1, Fig 4.3). As expected the corpus collosum exhibits anisotropic diffusion; greater in the x plane (parallel to the white matter fiber) than in the perpendicular y or z directions while the diffusion in the cortex is almost isotropic. The diffusion tensor values are in reasonable agreement to measurements made using localized spectroscopy in human brain (Gates, 1994). These experiments on in vivo rat brain were carried out as an example of the method. More deatailed experiments on rat brain are reported in Chapter 6.
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49 Table 4. 1 Diffusion tensor values of rat brain.*unit: 1 0 mm /s Rat Human (Gates, 1994) corpus cerebral corpus collosum cortex collosum temp. ( C) 37.0 37.0 ^37.3 Dxx 1.07(0.29) 0.64(0.26) 1.15 Dyy 0.43(0.14) 0.74(0.16) Dzz 0.46(0.23) 0.65(0.23) 0.71
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50 TL2 TL1 90Â° f 90Â° 46Â° 90Â° 4 Â— Â£ gss gssr gdss gss gdss gcrush gss gdss gcrush gss gdss slice read phase gror gdro gdro gro gcrush gdro gro gcrush gdro gro gpe gdpe gdpe gcrush gdpe gcrush gdpe Ase I 1 AstelAste2 0 I first spin echo first stimulated echo second stimulated echo Fig 4. 1 Multiple stimulated echo sequence.
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Fig 4.2 Water and Celery diffusion tensor maps.
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52 measurement region 2 Fig 4.3 Basic anatomical structure of the rat brain illustrating the regions used for the diffusion tensor measurements in the table 4. 1 .
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CHAPTER 5 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF RAT SPINAL CORD IN VITRO Introduction Diffusion Tensor Imaging Diffusionweighted imaging (DWI) is an effective technique to identify nerve tissue pathology, since it permits discrimination of grey and white matter structures (Moseley, 1990) and between normal and injured regions (Ebisu, 1993) with higher specificity than conventional contrast (Ti, T2 ). Apparent diffusion coefficient (ADC) images have also been obtained for spinal cord by using a series of diffusion weighted images. However, when the rootmeansquared (RMS) path length of the diffusion of the molecules is so long that the molecules hit the boundary, diffusion becomes anisotropic and is a tensor rather than a scalar quantity. So the ADC value measured for a specific structure depends on the directions of the applied gradients and is only an approximation to the accurate anisotropic diffusion coefficient measurement. Diffusiontensor images (Basser, 1994) and color trace imaging has been used successfully to show neuronal orientation and pathological changes of rat spinal cord using microimaging (Inglis, 1996). Here we use the same method to study grey and white matter organization and tissue injury in the spinal cord of young rats. Anisotropic apparent diffusion tensors are computed for data on neuronal orientation, and the 'trace' is computed as a measure of 53
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54 orientationindependent tissue space available for water diffusion, which shows neuronal damage in rat spinal cord. Anisotropy Index Diffusion approaches have been used to measure white matter tissue anisotropy. The most widely used is the ratio of ADCs measured with diffusion gradients presumed to be parallel and perpendicular to the fiber direction ( ADC(90)/(0) ) (Douek, 1991, van Geldren, 1994, Basser, 1994). Also another scalar index of anisotropy was proposed using ADCs measured with diffusion gradients applied in three orthoganal directions [ Standard Deviation Index]. On theoretical ground, ADCs are just approximation of the tissue diffusion nature. Pierpaoli et al (Pierpaoli, 1995) proposed another index: ratio of the principal diffusivities (eigenvalues A.,, X 2 , A 3 ) of diffusion tensor X.,/X 3 , which are invariant to rotations of tissue within the NMR magnet . However, due to the complexity of brain and spinal cord tissue structure, diffusion tensor of each image pixel has different principal axis and need to be normalized seperately. Practically this requires huge amount of data processing and still unable to give a general overview of whole brain's tissue structure. Here we use color trace and anisotropy index: Anisotropyindex = 2*D 2Z /(D xx +D yy ) Which clearly show tissue structure and detail informations about tissue damage of brain and spinal cord. Here zz is parallel, yy and xx is perpendicular to the fiber direction.
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55 Anatomic Structure of Spinal Cord Gray matter of spinal cord The central cellular region of the spinal cord is referred to as the gray matter. Afferent fibers of peripheral nerves form the dorsal root and enter the spinal cord at the dorsal margin (Fig 5.1). The Dorsal horn and the intermediate zone contain the posterior marginal nucleus, the substania gelatinosa, the nucleus proprius, Clarkes's nucleus and the intermediolateral nucleus. Substania gelatinosa locates on the top of dorsal horn. Fibers from peripheral nervous system ( medial division and lateral division of dorsal root) enter the spinal cord here, and send branches that ascend and descend in the cord and some of them arborize in the spinal gray matter, some of them terminate on the top of dorsal horn, or pass through the superficial dorsal horn to terminate at the bottom of dorsal horn. Largediameter fibers have been observed to pass through intermediate zone and terminate directly in motor nuclei which are at bottom of ventral horn. Also inside gray matter there are neuronal processes that interconnect sensory neurons in the dorsal horns with motor neurons in ventral horns. In summary, at Substania gelatinosa fibers travel in the anterposteral direction and lateral direction, while in lower part of dorsal horn, intermediate zone and ventral horn, the predominant fiber orientation is lateral. Ventral horn contain largest nerve cells (motor nuclei), their axons form ventral root which go across white matter into muscles and directly innervate them. Inside white matter ventral root travels mainly in lateral direction.
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56 In ventral and dorsal gray commissures there are a large number of primary afferent axons that are transverse to the colateral gray matter, so in this region the fibers travel in dorsoventral direction. White matter of spinal cord The area surrounding the gray matter is called the white matter of the spinal cord. Located in the white matter are the major ascending and descending nerve fiber tracts as well as pathways made up of the short axons that connect local regions of the cord. Methods Fixation of Rat Spinal Cord Injuries are performed at the T13 vertebral level 90 minutes after a single level laminectomy. An impounder is dropped from 25mm to generate a moderate to severe injury model. After injury the tissue is closed in layers. At the desired time intervals the injured animals are euthanized and perfused with a fixative. After a deep level of anesthesia is obtained the thoracic cavity is opened and the animal perfused via transcardial exsangination with 250ml of saline followed by 250ml of 4% paraformaldahyde. The animal is kept in a refrigerator for 24 Hrs for complete tissue fixation. The spinal cord is removed by a full laminectomy from the level of the cauda equina to the cervical cord or other levels depending on the length of cord desired. Diffusion Tensor Imaging Experiments Three normal and two injured rat spinal cords were imaged on a 600 MHz Varian Unity microimaging system using a diffusionweighted spin echo sequence. Sample temperature was maintained at 20Â°C throughout. Acquisition parameters were:
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57 TR/TE=3000/44 ms, slice thickness = 0.75 mm, 4 averages, 0.5x0.5 cm FOV ( 39x39 urn in plane resolution), diffusion weighting duration 5 = 10 ms, diffusion weighting gradient pair separation A = 26 ms, data matrix = 128*128. At first, routine imaging was performed, which included sagittal imaging and multislice imaging over region of interest. Second, diffusionweighted images were acquired with diffusion weighting gradients aligned at polar angles, (0,
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58 was observed for free water at diffusionweighting gradient strength above 3G/cm for readout direction and above 9 G/cm for all other directions, which lead to poor linear square fit for water. From the series of images, on a pixelbypixel basis, multivariate matrix linear regression was used to find the six unique elements of the diffusion tensor (Rao, 1965), each tensor element being an image displaying anisotropic diffusion coefficients (Moseley, 1990). The complete ADT were calculated using Viewit software. By adding the diagonal tensor elements as images with separate monochromatic (8bit) color scales, a color image was constructed that depicts tissue organization, using red, green and blue for D xx , D^, and D H respectively (Ebisu, 1993). An anisotropic index map was also made by dividing D s by the average of D xx and D yy as images. Data Analysis The color trace images greatly enhances the recognition of anatomical features, for which the diffusion tensors can then be extracted ( Table I). Two color trace images are presented. Fig. 5.2 shows a color trace image of a normal rat spinal cord, Fig. 5.3 shows four color trace images of a injured spinal cord, Fig. 5.4 shows the full tensor of a normal spinal cord. Comparisons are made in Fig 5.6, 5.7, 5.8. Regions of interest were marked using Dispunc software and a mean ADT calculated. Results Diffusion Tensor Values of Normal Rat Spinal The diffusion tensor values of three normal rat spinal cord are pretty consistent, within experiment error. For example, D of Dorsal funiculus are 0.22+/0.08, 0.27+/
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59 0.10, 0.32+/0.09. The variation of diffusion values may also due to the biological difference between the rat spinal cord samples. Diffusion Tensor Values of Injured Spinal Cord The diffusion tensor values of two injured rat spinal cord are different. For example, D yy of Ventrolateral funiculus are 0.48+/0.13, 0.77+/0.13. The variation of diffusion values may due to the pathological difference between the rat spinal cord samples, i.e. more serious injury cause more white and gray matter damage, which also increase the diffusion tensor value (Table 5.15). Diffusion Tensor Values of Rat Spinal Cord All the diffusion values of rat spinal cord are much less than that of free water at 20Â°C (2.5 *10" 3 mmVs), which are as predicted. The restrictions and obstacles inside the tissue slow down the diffusion. Color Trace Images Color trace images of rat spinal cord clearly indicate the predominant diffusion direction of the spinal cord, which is also the predominant nerve fiber tracts orientation. Colors in Gray Matter Region of Normal Rat Spinal Cord As it has been discussed in anatomy of spinal cord (Fig. 5.1), at substania gelationosa fibers travel in the anterposteral direction and in plane, this region is expected to show orange color ( red+blue or red+green) in color trace images (Fig. 5.2, 5.1). In lower part of dorsal horn , intermediate zone and ventral horn, the predominent fiber orientation is lateral, so we expect to see mainly green in these area. Ventral root goes across the white matter laterally, then it is supposed to show green among the red
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60 backgrounds ( white matter) (Fig. 5.2, 5.1). At last, fibers travel dorsoventrally in gray commissures, so blue color would be the right color (Fig. 5.2, 5.1). Color of the White Matter of Normal Rat Spinal Cord In white matter fibers mainly ascend and descend ( anterposteral), so it should all be red (Fig. 5.2,5.1). Therefore, the observed diffusion anisotropy is consistent with the prevailing orientation of fibers in the spinal cord. Color Trace Images of Injured Rat Spinal Cord The red color of white matter turns into gray in injured area, and the color of gray matter changes to gray or white (Fig 5.3), which shows different degree of injury. Trace is an Important Parameter for Different Nerve Tissues In normal rat spinal cord, the average trace among different white matter regions (DF, VLF, VF, DLF) is 0.82 Â± 0.12 xlO" 4 mm 2 /s (table 5.15.5, fig 5.6), while the average trace among different gray matter regions (SG, DH, VH, GC) is 1.64 Â± 0.14 xlO 4 mm 2 /s (table 5.15.5, fig 5.6). These data suggest that trace value can be used to differentiate white matter and gray matter tissues. We also note that all the diffusion values of the spinal cord are much smaller than that of pure water (25* 10" 4 mm 2 /s) due to the restriction, tortuosity, reflection, etc. Anisotropy Index of Normal Rat Spinal Cord The anisotropy indexes of all normal samples in different white matter regions (DF, VLF, VF, DLF) are ranging between 3.2 to 7.1 (table 5.15.5, fig 5.7), which shows high diffusion anisotropy among white matter tissues. The anisotropy indexes of all
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61 normal samples in different gray matter regions regions (SG, DH, VH, GC) are close to 1 (table 5.15.5, fig 5.7), which shows isotropic diffusion in gray matter. In Substantia gelatinosa(SG), anisotropy indexes of all samples are around 2, which suggests that both white matter and gray matter are coexisting in this area. This is is consistent with the prevailing anatomic structure of the spinal cord. Anisotropy Index of Normal and Injured White Matter Tissues In all white matter regions of injured sample, the anisotropy indexes are all decreased (table 5.15.5, fig 5.8), which suggests that anisotropy index is related to white matter damage. However, it is still unknown about how the degree of injury is related to anisotropy index. Diffusion Tensor Imaging of Human Spinal Cord Diffusion tensor imaging was also applied in human spinal cord, the result was published in SMRM conference in 1997 (Inglis, 1997)(Fig. 5.5). Conclusion Diffusion tensor imaging and color trace are very powerful methods in studying gray and white matter organization and tissue injury, and become an important reference in future spinal cord study.
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62 Table 5.1 Diffusion coefficient of normal spinal cord (sample #1). Anatomical Region Dvv Dxx Dzz trace Dzz/(0.5*(D xx+Dyy)) Dorsal funiculus(DF) 0.22(0.08) 0.30(0.09) 1.46(0.02) 0.66 5.6 Ventrolateral funiculus(VLF) 0.30(0.12) 0.33(0.12) 1.44(0.30) 0.69 4.5 Ventral funiculus(VF) 0.23(0.08) 0.33(0.09) 1.23 (0.24) 0.64 4.9 Dorsolateral funiculus(DLF) 0.26(0.08) 0.29(0.11) 1.30(0.21) 0.62 4.7 Substantia gelatinosa(SG) 1.01(0.17) 1.22(0.18) 2.27(0.39) 1.50 2.0 Dorsal horn(DH) 1.40(0.18) 1.51(0.17) 1.83(0.19) 1.58 1.25 Ventral horn(VH) 1.44(0.18) 1.42(0.17) 1.54(0.24) 1.47 1.08 Gray commissure(GC) 1.92(0.15) 1.66(0.24) 2.29(0.32) 1.96 1.28 Table 5.1 5.5. and Fig. 5.1 5.2. Apparent diffusion coefficients (xlO 4 mm 2 /s) recorded at 20 Â°C in 8 regions from fixed rat cervical spinal cord. Dyy is the lateral, Dxx is the dorsoventral, and Dzz is the anteroposteral direction. Different values between the three directions indicate tissue anisotropy. Values were ADT calculated in the region of interested.
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63 Table 5.2 Diffusion coefficient of normal spinal cord (sample #2). Anatomical Region Dyy Dxx Dzz trace Dzz/(0.5*(D xx+Dyy)) Dorsal funiculus(DF) 0.27(0.10) 0.30(0.07) 2.00(0.22) 0.86 7.1 Ventrolateral funiculus(VLF) 0.40(0.09) 0.50(0.07) 1.91(0.18) 0.94 4.2 Ventral funiculus(VF) 0.34(0.11) 0.42(0.08) 1.90(0.25) 0.88 5.0 Dorsolateral funiculus(DLF) 0.53(0.09) 0.47(0.06) 1.92(0.21) 0.97 3.88 Substantia gelatinosa(SG) 0.87(0.06) 1.35(0.08) 2.63(0.24) 1.62 2.36 Dorsal horn(DH) 1.35(0.21) 1.60(0.14) 1.89(0.20) 1.61 1.1 Ventral horn(VH) 1.80(0.12) 1.68(0.12) 1.45(0.16) 1.64 0.8 Gray commissure(GC) 2.26(0.20) 1.28(0.16) 2.06(0.23) 1.86 1.2
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64 Table 5.3 Diffusion coefficient of normal spinal cord (sample #3). Anatomical R peri on Dyy Dxx Dzz trace Dzz/(0.5*(D xx+Dyy)) Dorsal fiiniculusfDF) 0.32(0.09) 0.49(0.07) 1.80(0.46) 0.87 4.5 Ventrolateral funiculus(VLF) 0.46(0.09) 0.53(0.13) 1.58 (0.38) 0.86 3.2 Ventral funiculus(VF) 0.39(0.07) 0.67(0.18) 1.89(0.37) 0.98 3.6 Dorsolateral funiculus(DLF) 0.46(0.09) 0.38(0.09) 1.58(0.43) 0.81 3.8 Substantia gelatinosa(SG) 0.93(0.16) 1.26(0.10) 2.19(0.36) 1.46 2.0 Dorsal horn(DH) 1.53 (0.22) 1.90(0.29) 1.84(0.29) 1.76 1.1 Ventral horn(VH) 1.51(0.26) 1.67(0.21) 1.72(0.32) 1.63 1.1 Gray commissure(GC) 1.73(0.38) 1.48(0.20) 1.84(0.29) 1.68 1.2
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65 Table 5.4 Diffusion coefficient of injured spinal cord (sample #4538). Anatomical Region Dyy Dxx Dzz trace Dzz/(0.5*(D xx+Dyy)) Dorsal funiculus(DF) 1.60(0.63) 1.54(0.65) 1.91(0.60) 1.7 1.2 Ventrolateral funiculus(VLF) 0.48(0.13) 0.57(0.13) 1.30(0.33) 0.8 2.5 Ventral funiculus(VF) 0.58(0.17) 0.66 (0.11) 1.05(0.20) 0.8 1.7 Dorsolateral funiculus(DLF) 0.46(0.12) 0.35(0.13) 1.28(0.24) 0.7 3.2 Substantia gelatinosa(SG) 0.54(0.10) 0.75(0.14) 1.25(0.13) 0.9 1.9 Dorsal horn(DH) 2.23(0.40) 2.21(0.43) 2.21(0.44) 2.2 1.0 Ventral horn(VH) 1.54(0. 63) 1.45(0.64) 1.74(0.31) 1.6 1.2 Gray commissure(GC) Unable to locate Unable to locate Unable to locate
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66 Table 5.5 Diffusion coefficient of injured spinal cord (4529). Anatomical Region Dyy Dxx Dzz trace Dzz/(0.5*(D xx+Dyy)) Dorsal funiculus(DF) 1.49(0.54) 1.64(0.55) 1.77(0.52) 1.6 1.1 Ventrolateral funiculus(VLF) 0.77(0.13) 0.82(0.11) 1.05(0.15) 0.9 1.3 Ventral funiculus(VF) 0.48(0.14) 0.50(0.14) 1.24(0.21) 0.7 2.5 Dorsolateral funiculus(DLF) Unable to locate Unable to locate Unable to locate Substantia gelatinosa(SG) Unable to locate Unable to locate Unable to locate Dorsal horn(DH) Unable to locate Unable to locate Unable to locate Ventral horn(VH) Unable to locate Unable to locate Unable to locate Gray commissure(GC) Unable to locate Unable to locate Unable to locate
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67 Lateral direction Anterposteral direction Fig 5.1 Anatomic structure of spinal cord. Color legend in the color trace images: Dxx (Dorsoventral, blue), Dyy (Lateral, green), Dzz (Anterposteral, red).
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Fig 5.2 Diffusion tensor (color trace) of normal rat spinal cord
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Fig 5.3 Diffusion tensor (color trace) of injured rat spinal cord
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70
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71 Fig 5.5 Diffusion tensor (color trace) of Normal human spinal cord
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72
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73 opui (dOJJOSllUT
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in 10 g. a c = $ s a s s H ft 5 tapu ] ^dojjosiuv
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CHAPTER 6 HIGHRESOLUTION DIFFUSION TENSOR AND COLORED TRACE IMAGING OF NORMAL, HYDROCEPHALUS, AND SHUNTED RAT BRAIN IN VITRO Introduction Infantile Hydrocephalus Hydrocephalus is a serious brain disease encountered in some infants at birth or shortly after. It is caused by the accumulation of fluid in the brain, especially in young children, which makes the head enlarge and cause mental handicap. It can be treated by inserting a shunt tube which drains the excess fluid. One hypothesis is that the primary cause of neuronal injury in hydrocephalus is the damage in the periventricular white matter and the corpus callosum. Abnormal transependymal absorption of cerebrospinal fluid (CSF) results in increased white matter water content, causing axonal damage which leads to subsequent dendritic and synaptic degeneration in grey matter. Pathological changes in the periventricular white matter have been reported for hydrocephalus in human infants (Weller & Shulman, 1972) in rabbits (Del Bigio & Bruni, 1988) and in the kitten model of infantile hydrocephalus (Del Bigio et al, 1994). In the kitten, the changes such as delayed myelination and gliosis persisted despite shunt treatment (Del Bigio et al ,1994, Chumas et al, 1994). Hence a comparison between hydrocephalic and agematched control rats will provide information on the timing and nature of the pathological changes due to the hydrocephalus. The HTx rat is 75
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76 an ideal model for this study. It has inherited hydrocephalus due to aqueduct stenosis with an onset in late gestation, a stage which is equivalent to pre midgestation in the human. Ventricular enlargement starts prenatally and progresses rapidly after birth, with death occurring at 46 weeks (Jones and Bucknall, 1988). Ventriculomegaly is associated with severe thinning of the cerebral cortex, disruption of the grey matter laminar structure by 10 days after birth (Jones et al, 1991) and abnormal pyramidal cell morphology in layer V by 21 days (Boillat et al, 1993, Harris et al, in preparation). White matter pathology has not been studied in this model. Shunt Treatment of Hydrocephalus In Dr. Jones' recent studies, the hydrocephalus has been treated by inserting a ventriculosubcutaneous shunt. Conventional T,weighted spinecho MR imaging was then used to measure the extent of ventriculomegaly in both untreated and treated rats. The effect of the shunt was to reduce the ventriculomegaly, prevent further dilation and to restore cortical thickness (Harris et al, 1994, Jones et al, 1995). The effect of shunt treatment at 36 days after birth was indistinguishable from treatment at 812 days after birth when the rats were examined at 21 days, suggesting that shunt timing may not be important. However, using other techniques we have shown that, although early treatment is more effective than later treatment, neither cortical ultrastructure, neuronal dendritic organization or cortical metabolism are completely normalized by shunting (Boillat et al 1993, Harris et al 1995, and in preparation). White Matter Pathology in Hydrocephalus Hitherto, Dr. Jones' studies have concentrated on cortical grey matter changes, when the primary cause of cortical damage may be pathology in the periventricular white
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77 matter. Dr. Jones et al. have observed edematous white matter in the periventricular regions of HTx rats with advanced hydrocephalus by conventional histology and also thinning of the corpus callosum and an absence of the septum pellucidum, which can be seen on T,weighted MR images. A recent finding by us is that the concentration of many metabolites in the cerebral cortex is reduced in hydrocephalic HTx rats (Harris et al, 1995 and in preparation) and that tissue water content is increased. These changes may indicate mild cytotoxic edema, when intracellular water increases and the tissue would be expected to become less anisotropic. In kaolininduced hydrocephalus, however, there was a large increase in anisotropy, similar to that seen in vasogenic edema, due to an increase in extracellular water content (Ebisu et al, 1993). This discrepancy may be due to differences between infant and adult hydrocephalus and is one that can be resolved with the proposed tensor mapping experiments. MR Imaging Techniques Diffusionweighted imaging (DWI) is an effective technique to identify brain tissue pathology, since it permits discrimination of gray and white matter structures (Moseley, 1990) and between normal and edematous brain regions (Ebisu, 1993) with higher specificity than conventional contrast (Ti, T2 ). Diffusiontensor images (Basser, 1994) and color trace imaging has been used successfully to show neuronal orientation and pathological changes of rat spinal cord using microimaging (Inglis, 1997). The apparent diffusion coefficient (ADC) can be obtained using DWI but it provides a quantitative measure of diffusion along the diffusion weighted axis only. Conversely, DTI provides quantitative diffusion coefficients (in mm 2 /s) in all directions. Another parameter which can be extracted is the trace, which is invariant to the orientation of the
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78 brain with respect to the scanner axes. This information can be relayed in the form of pseudothree dimensional images, two spatial and one color, thus illustrating fiber orientation within the image plane (Inglis, 1997). Here we use the same method to study grey and white matter organization and tissue edema in young rats with infantile hydrocephalus and in shunttreated hydrocephalic rats. Anisotropic apparent diffusion tensors are computed for data on neuronal orientation, and the 'trace' is computed as a measure of orientationindependent tissue structure, which shows neuronal damage in infantile hydrocephalus. Anatomy Structure of Rat Brain Organization of the cerebral cortex: cerebral cortex contains three type of neurons: pyramidal, stellate and fusiform cells (Fig. 6.1). Pyramidal cell are the most prominent of three types of cells. They have two types of dendrites, basilar dendrites that arborize near the layer of origin of the cell, and apical dendrites that ascend perpendicularly toward the surface of the cortex. Stellate cells are starshaped, some of their axons project horizontally through a local cortical region. Fusiform cells are spinal shaped and their axon often project out of the cortex (Fig 6.1). Methods Fixation of the Rat Brain HTx rats have inherited infantile hydrocephalus which develops in late gestation. 21 day old rats were perfused intravascularly with a fixative and the brain excised from the skull. The fixative procedure is as follows: rat brains were anesthetized with 60mg/kg
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79 pentobarbitone. The rat chests were opened and a needle was put into the left cardiac ventricle. We perfuse 5ml saline and 1020 ml fixative: Fixative = 4% paraformaldehyde in 0.1 M phosphate buffer, PH 7.3 Shunt Surgery The shunts were prepared from Teflon tubing( OD 0.76 mm, ID 0.3 mm) which was preshaped in a 90Â° bend to form the proximal end to insert into the lateral ventricle. The distal end was attached to an openended 1to 1.5 cm silicone tube ( OD 1.5mm, ID 0.5mm) and the shunts were stored in Cidex solution. Prior to use, shunts were rinsed in sterile water and filled with sterile artificial cerebrospinal fluid containing 0.1 mg/ml Gentamicin antibiotic. Microimaging Brains from 3 control rats, 3 hydrocephalic rats, and 3 rats shunttreated at 1 0 days after birth were imaged on a 300 MHz Varian Unit microimaging system using a diffusionweighted spin echo sequence. A threeaxis gradient set was constructed in our laboratory to fit onto a broadband X{'H} Nalorac probe in place of the variable temperature dewar. The X nucleus coil was removed to leave the 'H decoupler coil of internal diameter 18 mm. This coil served as the transmit/receive coil and was retuned for optimal signaltonoise. Gradient strengths of 10.7, 1 1.4 and 25.6 G/cm along x, y and z, respectively, were attained with 20 A Highland power supplies. The brains were approximately 15 mm in cross section. Acquisition parameters were: TR/TE=3000/70 ms, slice thickness = 1.0 mm, 4 averages, 1.5x1.5 cm FOV (107x107 um in plane resolution). 35 diffusionweighted images were acquired in a total acquisition time of 15
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80 hours. Five different values of diffusion gradient were applied along x, y, z, xy, xz, yz, 2 and xyz, varying bij (diffusionweighting factor) between 20 and 1 846 s/mm . Conventional T2 weighted images were also acquired for comparison. Data Processing From the series of images, the apparent diffusion tensor was computed, each tensor element being an image displaying anisotropic diffusion coefficients (Basser, 1994). Matrices of diffusionweighting factors were computed using Mathematica software (Fig 3.1) and the complete ADT by a sevenparameter matrix linear regression using Viewit software (Fig 3.2). By adding the diagonal tensor elements as images with separate monochromatic (8bit) color scales (Figs 6.3, 6.4, 6.5), a color image was constructed that depicts tissue organization, using red, green and blue for D yy , D xx , and respectively (Inglis, 1997). The color trace image greatly enhances the recognition of anatomical features, for which the diffusion tensors can then be extracted ( Table I). Color trace images are presented (Fig 6.2, 6.3, 6.4). Regions of interest were outlined using Dispunc software (Fig 3.1) and a mean ADT calculated. Results Diffusion Tensor Values of Normal Rat Brain The diffusion tensor values of three normal rat brains are pretty consistent, within experiment error. For example, D yy of white matter are 0.92+/0.25, 1.06+/0.15, 1.07+/0.10. The variation of diffusion values may also due to the biological difference between the rat brain samples.
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81 Diffusion Tensor Values of Hydrocephalic Rat Brain The diffusion tensor values of three hydrocephalus rat brain are different. For example, D xx of white matter are 0.50+/0.08, 0.77+/0.12, 1.05+/0.28. The variation of diffusion values may due to the pathological difference between the rat brain samples, i.e. more serious hydrocephalus cause more white and gray matter damage, which also increase the diffusion tensor value (Table 6.1, 6.2, 6.3). Diffusion Tensor Values of Rat Brain All the diffusion values of rat brain are much less than that of free water at 20Â°C (2.5 *10" 3 mm 2 /s), which are as predicted. The restrictions and obstacles inside the tissue slow down the diffusion. Color Trace Images The color trace images of the rat brains clearly indicate the predominant diffusion direction in each pixel of the brain, which also corresponds to the predominant orientation of the nerve fiber tracts (Fig 6.3, 6.4, 6.5, 6.7). Color in Cerebral Cortex(CX) As it has been discussed in 'anatomy of rat brain' (Ch6), in cerebral cortex the predominant fiber orientation is perpendicular to the pia surface, i.e., in lateral cortex, fibers travel along dorsoventral direction, so green color is expected in this region; in dorsal cortex, fibers travel along lateral direction, so red color is expected in this region (Fig. 6.3).
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S2 Color in Corpus Callosum Near lateral cortex, the fibers of corpus callosum is along lateral direction, so red color should been seen, while near dorsal cortex, the fibers of corpus callosum is along lateral direction, so green color should been seen. All these color expectations match the diffusion tensor color images of rat brain. (Fig 6.3). Anisotropy Index of White Matter in Normal and Hydrocephalic Rat Brain The anisotropy indexes of the white matter in normal rat brain are above 2 (table 6.16.4, fig 6.10), in the hydrocephalic cases, they drop to near 1, in the shunted case, it increases back to normal. All these show that white matter is damaged in the hydrocephalic conditions and may be recovered after shunted treatment. Diffusion Tensor Trace of Lateral and Dorsal Cortex The diffusion trace of lateral cortex in the hydrocephalic cases are higher (0.83 Â± 0.04) than that of normal cases (0.80 Â± 0.01 xlO 3 mm 2 /s) (table 6.16.4, fig 6.10). The diffusion trace of dorsal cortex in the hydrocephalic cases are also higher (0.85 Â± 0.03 xlO" 3 mm 2 /s) than that of normal cases (0.79 Â± 0.02) (table 6.16.5, fig 6.9). The changes in hydrocephalus and in shunted case are much less than were found in chapter 5 for spinal cord injury, which suggest a much less dramatic change in anatomic structure. It is concluded that hydrocephalus results in increased water content in the brain surrounding the ventricles. This has been confirmed by independent measurements (Jones, 1996). It is found that, at 21 days, cortical H 2 0 (ml/kg wet weight) increased by
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83 1.5%, and H,0 (ml/kg dry weight) increased by 9%. Shunt treatment normalized the gray matter but there remained an increase of water content in the white matter. Stimulated Echo Diffusion Tensor Imaging Stimulated echo diffusion tensor imaging was also applied in rat brain (Fig.6.6), which seem to show more anatomical structure of rat brain. Conclusion Diffusion tensor imaging and color trace are very powerful methods in studying gray and white matter organization and tissue edema of rat brain, also in studying the effect of shunttreatment, and become an important reference in future rat brain study.
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84 Table 6.1 : Diffusion coefficient of rat brain (normal). Control 1 D YY (green) u zz (ume) X 1 uV ^color in UUlUl 111 color map direction Dxx(red) white matter 0 40 CO 1 I s ) 0.55 green dorsal CX n qq 7i\ 0 70 fO 1 0^ 0 63 CO 16^ 0.77 red lateral CX A "7C /A 1 /1\ (J. O (U.14J 0.91 (0.19) 0.70 (0.17) 0.79 green striatum n 7Â£ /A 1 c\ 0.64 (0.13) 0.64 (0.14) 0.67 red uontroiz white matter 1.06(0.15) 0.42(0.10) 0.62 Green dorsal CX 1 i e/ft 1 i\ 0.68(0.11) 0.59(0.09) 0.79 red lateral CX U. /'f^U.U / ) 0.99(0.13) 0.65(0.08) 0.81 green striatum A OJ/A 1 1 \ U.oi(U.l 1 ) 0.86(0.12) 0.65(0.11) 0.78 red comroij wnite matter U. JU^U.UO j 1.07(0.10) 0.57(0.11) 0.71 green dorsal CX 1.17(0.19) 0.66(0.11) 0.60(0.09) 0.80 red lateral CX 0.77(0.07) 0.94(0.09) 0.70(0.07) 0.81 green striatum 0.78(0.11) 0.89(0.13) 0.65(0.10) 0.77 red Table 6.2: Diffusion coefficient of rat brain(hydrocephalic3).
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85 Hydrocephalic 1 UYYlg reen / L'ZZvlJlUC ) Trace color in color map direction D xx (red) white matter 0.60 (0.08) ft 11 (C\ flT> U.JU ^u.uo ) ft 61 dorsal CX l.zU (U.UZ) 0.88 red lateral La A 0,1 / A 1 ^\ 0 84 CO 1 3"> 0.87 green striatum ft O/l /T> 1 A'V 0.72 (0.10) 0.78(0.10) 0.81 red Hydrocephalic2 white matter U. / 1 (U.U /) 0.75 (0.12) 0.77 (0.12) 0.74 green dorsal la 1 fts /"n 1 ^ 0.72 (0.08) 0.70 (0.07) 0.83 red lateral LA U. /U yJ. in) 1.00 (0.18) 0.72 (0.15) 0.80 green striatum ft Oft /A 1 1 ^ U.7U [V. L L) 0.85 (0.11) 0.68 (0.10) 0.81 red Hydrocephalics white mailer 0.90(0.23) 1.05(0.28) 0.86 light green dorsal la 0.79(0.09) 0.74(0.16) 0.84 red lateral CX 0 11(0 08*> 1.01(0.14) 0.74(0.09) 0.83 green striatum 0.86(0.11) 0.85(0.11) 0.66(0.09) 0.79 red Table 6.3: Diffusion coefficient of rat brain (shunted).
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86 shunted r*r\lAr in UUlUi in pnlnr man white matter 0.47(0.02) 1.21(0.32) 0.46(0.07) n 7 1 U. / 1 dorsal CX 1.10(0.18) n 7 ico in 0 55C0 10) n 7Q U. / " til CC/11 lateral Vsj\. 0 10(0 09) 0.95(0.12) 0.66(0.12) 0.77 red striatum 0.81(0.12) 0.68(0.10) 0.65(0.09) 0.71 green Table 6.16.3. Apparent diffusion coefficients (xlO 3 mm 2 /s) recorded at 20 Â°C in 4 regions (periventricular white matter, dorsal and lateral cortex and striatum) from fixed 21 day rat brains. Dxx is the lateral, Dyy is the dorsoventral, and Dzz is the anteroposteral direction. Different values between the three directions indicate tissue anisotropy. Values were ADT calculated in the region of interest. The figures in bold type are the largest of the components. Table 6.4: Degree of anisotropy of white matter.
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87 degree of anisotropy ( Dyy/0.5(Dxx+Dzz)) Control 1 2.43 Control2 2.65 ControB 2.00 Hydrocephalic 1 1.32 Hydrocephalic2 1.01 Hydrocephalic3 1.07 Shunted 2.60
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88 pia surface of cerebral cortex Fig 6.1 Neurons' shapes and their dendritic and axonal branching patterns of the cerebral cortex. 1, Pyramidal cell 2, Stellate cell 3, fusiform cell.
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Fig 6.2 Rat brain T2weighted image with region of interest Labelled.(0.5cm * 0.5cm)
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Fig 6.3 Diffusion tensor (color trace) of normal rat brain
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Fig 6.4 Diffusion tensor (color trace) of hydrocephalus rat brain
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Fig 6.5 Diffusion tensor (color trace) of shunted rat brain
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Fig 6.6 Diffusion tensor (color trace) of normal rat brain using Stimulated echo Diffusion Tensor Imaging.
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94 Dxx(Recl) Dyy (Green) Dzz(Blue) Fig 6.7 Color legend of Color diffusion tensor of rat brain (compare to fig 6.2). Dxx is Lateral, Dyy is dorsoventral, and Dzz is anteroposteral.
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95 Z v rain/s (HvOI* ) 33B J I
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96 ;Z v umi/s (HvOT*)33KJ
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97
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CHAPTER 7 SUMMARY AND CONCLUSION Introduction In this thesis, the theory of diffusion weighting factor of gradient for any sequence was revisited and a new comprehensive formula was found which not only unified all former results, but also predicted new results for unsolved pulse sequences. Then, on the basis of this new formula, two diffusion tensor imaging techniques were built, tested and applied to rat spinal cord and brain, which shows biological and clinical significations of diffusion tensor imaging. Achievements and Signification A New Formula for Diffusion Weighting Factor There have been many attempts to get an expression of diffusion weighting factor of gradients in different pulse sequence since 1956 (Torrey, 1956, Stjeskal, 1965, Tanner, 1970, Martin, 1982). Up to now, three different expressions for single quantum gradient echo, spin echo and stimulated echo sequence have been found, but nobody has been able to derive a formula for any double quantum and multiple quantum coherence sequences, only under special case (bipolar gradients, a pair of rectangular gradients) had this problem been solved experimentally. In this thesis, a new method was introduced, which not only unifies the all different expressions for single quantum coherence into one, but 98
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99 also predicts expressions for double quantum and multiple quantum coherence sequence, which is consistent with experiment result (Martin, 1982) under bipolar gradient condition. This new method will be very useful for both multiple quantum spectroscopy and imaging, where complicated gradients are applied and diffusion weighting can not be ignored. Two Diffusion Tensor Imaging Techniques Using the new formula for diffusion weighting factor, two diffusion tensor imaging techniques were built: spin echo diffusion tensor imaging and multistimulated echo diffusion tensor imaging. The EPI spin echo diffusion tensor imaging has been built in 1994 (Basser, 1994), but the multiecho stimulated echo diffusion tensor imaging technique, along with Basser' s EPI stimulated echo diffusion tensor imaging technique (Basser, 1996), were the first two techniques in the world that used stimulated echo sequence in diffusion tensor imaging. These two techniques described in this thesis are also tested using water and celery phantoms, which shows isotropic diffusion in free water and anisotropy diffusion in celery fibers. Diffusion Tensor Imaging of Normal and Injured Spinal Cord In this thesis, diffusion tensor imaging were first applied to normal and injured rat spinal cord, color trace images were presented which clearly shows fiber orientations of the spinal cord that is not accomplished by any other MRI techniques (for example, grayscale trace images).
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100 It is also found that the injury of the spinal cord causes the loss of anisotropy of white matter, which is consistent with other results using ADC (Ford et al, 1994). It is interesting that there were substantial changes in diffusion characteristics in the injured area which appeared normal by conventional imaging (Ford et al, 1994). These results imply that there are consequences of spinal cord injury which dramatically alter axon structure but do not change water content, so are not detected by conventional imaging. Diffusion Tensor Imaging of Normal, Hydrocephalic and Shunted Rat Brain In this thesis, diffusion tensor imaging were applied for the first time to the normal and hydrocephalus and shunted rat brain, color trace images were presented which shows fiber orientations of rat brain that is not accomplished by any other MRI techniques (for example, grayscale trace images). It is also found that the hydrocephalus of the rat brain causes the loss of anisotropy of white matter, and hydrocephalus results in increased water content in the brain surrounding the ventricles. This has been confirmed by independent measurements (Jones, 1996). This is a new direction in the noninvasive investigation of normal and abnormal brain development by using a new imaging technique to study tissue organization, pathological changes, and recovery with treatment. The technique could be ultimately be applied to clinical situations (in the same way as ADC is now being applied to measure the progress of clinical stroke). Diffusion Model at the Molecule Level Although there is no general ab initio theory of selfdiffusion in liquids which calculates the selfdiffusion coefficient D in terms of fundamental atomic quantities,
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101 progress can be made on a semiempirical basis. Molecules in a liquid diffuse or migrate in a random manner into interstices between neighbors which open up in the course of thermal motions. One approach is to consider a molecule as a sphere of radius a moving in a liquid which is regarded as a continuum of viscosity r\. This approach leads to the classical StokesEinstein equation: D=kT/(67ira) [7.1] Since the water molecule is not spherical and the surrounding fluid is not a perfect continuum on the molecular scale, the coefficient 671 should not be taken too seriously. Nevertheless if we insert in equation (7.1) the value of D = 2.5* 10" 5 cm 2 /s for water at 25Â°C, T=298K) and the value of r=0.89*10" 2 poise at the same temperature and the known value of Boltzmann's constant( k=1.38*10" 16 erg/K), we find a value of a = 1.0 A which is a not unreasonable value for the effective radius of the water molecule (The OH bond length is 1.1 A and the interproton distance is 1.8 A in the water molecule). A feature of both D and r\ is their very strong dependence on temperature. However equation [7.1] suggests that the product Drj/T should be constant for a given fluid over a range of temperature and this is found to be a reasonable approximation (Franks, 1972). The strong dependence of D and r\ on temperature provide good illustrations of the empirical Arrhenius activation equation: ti=Tio exp(E An /kT) [7.2] D=D 0 exp(E AD /kT) [7.3] Where E A are activation energies. For a given liquid E Atl and E AD are in fact found to be similar. For water the Arrhenius equation fits the temperaturedependence of D well
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102 at higher temperatures with E A =6.7 KJ/mole (Franks, 1972). At 25Â°C the value of E A is around 20 KJ/mole. Arrhenius type of behavior has frequently been interpreted in terms of the theory of rate processes of Glasstone, Laidler and Eyring (1941). When considering a chemical reaction, or the rotation of a molecule in a solid, or a molecule diffusion or moving in a viscous medium, the molecule is visualized as requiring sufficient energy to surmount a barrier in order to proceed. If this energy is E A above the ground state, the proportion of molecules able to move over the barrier is given by the Boltzmann factor e" EA /kT, thus providing the basis of understanding the Arrhenius behavior in terms of statistical mechanics. We may expect E A to be of the order of the energy required to create a molecular vacancy in the fluid into which another molecule may move. A good approximation to the energy we needed to create a molecule vacancy is provided by the latent heat of evaporation of the liquid, which for water is 41 KJ/mole. Since a complete vacancy is not needed expect this to be an upper bound for the activation energy for diffusion. The observed value for water at 25Â°C is about 20KJ/mole, which is consistent with the hypothesis. Values of D for water encountered in tissues may be expected to be smaller that those in free water at the same temperature, whether one is considering the trace of the diffusion tensor or any of its components. This expectation arises from three factors, discussed by Lebihan(1996). First the free path of a diffusion water molecule is not only hindered by other surrounding water molecules, but also by solute molecules all much larger than the water molecules, for example amino acids, peptides, proteins, nuclei acids
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103 and by organelles, cellular components such as ribosomes, mitochrondia and cell nuclei. Secondly whether a molecule of interest is inside a cell or outside, its flight is interrupted by collision with membranes which are only partially permeable. Thirdly the path of a molecule between any two points is abnormally extended by the need to navigate around somewhat impermeable obstacles ( the tortuosity factor). For these three reasons the values of D, as defined by Fick's law can be expected to be reduced below those for free water at the same temperature. If we examine all the values D;j in the tables and charts of chapter 5, we see that they fall in the range between 0.26 and 0.02 10" 5 cm 2 /s. The highest values are an order of magnitude lower than in free water, and the lowest values are two orders of magnitude lower than in free water. It is clear that the water molecules are severely hindered in rat spinal cord. It is not surprising that the molecules should be more hindered perpendicular to the nerve fibers than parallel to them as is found, showing greater freedom parallel to the axons. For each component of the diffusion tensor, or its trace, we could write: Dexpt/Dpw = exp(AE/kT) [7.4] Where D exp t is the experimental value, and D pw is the diffusion constant for pure water. The values of AE so obtained would then provide a measure of the additional hindrance suffered by the water molecules in that anatomical region at that temperature. We note too that in chapter 5 the values of anisotropy index range from 7.1 in white matter to 1 .0 in gray matter. In a similar way we may write anisotropy index I: I = exp(AE'/kT) [7.5] AE' would be a measure of the additional hindrance to diffusion perpendicularly to the fiber directions relative the diffusion parallel to the fiber directions. Table 7. 1
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104 shows AE' in different area of spinal cord. It is interesting to see the additional hindrance energies in whiter matter are 10 times larger than those in gray matter. Table 7.2 shows AE' of white matter in normal, hydrocephalus cases, the additional hindrance energies in normal whiter matter are 10100 times larger than those of hydrocephalus. It is not clear however that such values of AE from equation 7.4 and 7.5 would give great insight into the diffusion processes and their hindrance since the structures of the tissues are too complex to be characterized by two parameters such as AE and AE'. Another approach to the analysis of the data is to devise a simplified model of the biological tissue and to compare the data with properties of the model. Such a procedure was attempted by Szafer, Zhong and Gore (Szafer, 1995). The tissues were regarded as a periodic array of boxes surrounded by partially permeable membranes (these were models of the cells), embedded in an extracellular medium. The intracellular and extracellular diffusion constants differ, Di and D E respectively. The model generated anisotropic values of the measured diffusion constants which were less than for pure water and were dependent on the dimensions lx, ly, lz of the boxes and their periodic spacing a, and on the permeability P of the cell walls. By making lz Â» lx, ly we may imitate long nerve cells and find expressions of the anisotropy index. For example for an array of long parallel fibers Safer et al find under certain assumptions: Dp^fDi+afpE [7.6] D ^rf/ Sw + ^ )D Â° /Dr /D,
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105 [7.7] With 2 1 =4 a [7.8] Two steps were adopted by Szafer et al (Szafer, 1997) to test their theoretical model. One of these consisted of a Monte Carlo simulation of water diffusing in their model of the tissue, in which the motions of 10000 molecules were followed and the decay of the NMR signal was compared with the predictions of the theoretical analysis. In the second step calculations were made for the model and compared with some experimental results in the literature. In both cases fair agreement was obtained. The work has been extended by Stanisz, Szafer et al (Stanisz, 1997) in an application to bovine optic nerve. Here the authors were able to make use of electron micrographs of the tissue to assess the best size and shapes of their model cells. It will be seen that equations that equations (7.6) to (7.8) contain the following undetermined adjustable quantities f, a, P, D E , Di. The authors proceed by assuming values for these parameters and seeing what values of D they lead to. The parameters may then be adjusted by trial and error. The theory has not yet reached the stage where one may take measured values of diffusion constant and directly discover the values of the adjustable parameters; indeed there is insufficient data to do this. It was found that the value of P was not critical. We can see that equation (7.6) for D para has a simple interpretation of diffusion in two parallel compartments. For tightly packed fibers the packing factor f is nearly unity and D para = Di; for very low packing D para = D E . The expression in (7.7) for D per p is
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106 somewhat more complex because of the transverse geometry. Nevertheless in the limit of f= 1 , D P erp=Di and for f=0, D para =D E again. These results are readily comprehensible. So it is of interest to examine a situation between these limits, for example taking f=0.5. Since the value of P is not critical we take it large enough that Dc is approximately equal to Di. With this simplification we now try some values of D E and Di to see if we can account for measured values of D in the tables and charts of chapters 5 and 6. With D E =0.28 * 10 e" 5 cm 2 /sec and Di= 0.004 * 10" 5 cm 2 /sec, we obtain D para = 0.142 10" 5 cm 2 /sec and D pe rp = 0.032* 10" 5 cm 2 /sec. Now we look at the first entry in Table 1 for the measured diffusion tensor of normal rat spinal cord (sample #1), dorsal funiculus, where the values recorded are 0.146 * 10" cm /sec for D zz and 0.030 * 10" cm /sec for D xx , and we see that the theoretical an experimental values are in close agreement. This is not a unique solution; by starting with a different value of f and adjusting for P and Dc somewhat different values may be obtained for D E and Di. The same process may be applied to all the results of Chapters 5and 6, and it is seen that the general approach Szafer et al can indeed account for the measurements. Although the theory does not allow definitive values D E and Di to be determined for each tissue we can draw several conclusions. First the intracellular diffusion constant Di is substantially less than the extrecellular diffusion constant DE in anisotropic situations. As we discussed earlier, this is no doubt on account of the many additional hindrances to molecular motion in these nerve cells. Secondly we notice as mentioned in Chapter 5 that the extracellular diffusion constant is always lower than in pure water in bulk. Here
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107 again there are many additional hindrances not lest the narrowness of the passages through which the molecules must migrate in the fine structure of the rat spinal anatomy. One other unknown factor is the effect of the fixative which is likely to reduce further the diffusion constant for all measurements in vitro. Future studies Double Quantum Diffusion Tensor Imaging and Spectroscopy The general formula for diffusion weighting factor (bvalue) (Chapter 2) has solved the problem of bvalue calculation of double quantum(DQ) sequence. Now we try to find a good application for DQ diffusion tensor imaging or spectroscopy. Spinal Cord Injury Model We are trying to build up relationship between diffusion tensor values and degree of injury of spinal cord, which can provide important information for doctors to diagnose and treat the spinal cord injury. In Vivo Diffusion Tensor Imaging In order to get diffusion tensor (ADT) images of human being, we need faster ADT technique. One option is EPI diffusion tensor imaging. Summary The examples of rat spinal cord and hydrocephalus in rat brain demonstrate the great value of diffusion imaging in providing local anatomic information concerning the nerve fibers, their integrity and damage. The diffusion anisotropy tensor provides a unique local directional parameter at the cellular level in the tissues which is not provided by any other NMR parameter such as relaxation time or chemical shift. Measurements of this kind have significant clinical potential.
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108 Table 7.1 Rat spinal cord anisotropy index and addition hinderance energy to diffusion perpendicular to the fiber directions relative the diffusion parallel to the fiber directions. Tissue type Anatomic region Anisotropy Index Additional hinHprance enertrv (AE'*kJ/mole) White matter Dorsal funiculus(DF) J.O 4? ft? White matter Ventrolateral funiculus^ v Lr ) White matter v entrai funiculus(VF) 4 Q 39 31 WIllLC IIlaLLCI T^nrcn 1 a fpra 1 L/Ul > v ' 1 cl L ^ 1 ui funiculus(DLF) 4.7 38.28 White matter Substantia gelatinosa(SG) 2 17.09 Gray matter Dorsal horn(DH) 1.25 5.53 Gray matter Ventral horn(VH) 1.08 1.92 Gray matter Gray commissure(GC) 1.28 6.02 Table 7.2 White matter of rat brain: anisotropy index and addition hinderance energy to diffusion perpendicular to the fiber directions relative the diffusion parallel to the fiber directions. Anisotropy Index Additional hinderance energy (AE'*kJ/mole) Control 1 2.43 21.91 Control2 2.65 24.14 Control3 2 17.09 Hydrocephalic 1 1.32 6.86 Hydrocephalic2 1.01 0.24 Hydrocephalic3 1.07 1.68 Shunted 2.6 23.59
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REFERENCES Bain A.D. J. Magn. Reson . 56: 418 (1984). Barker G., and Mareci T. H. J. Magn. Reson : 83, 1 1 (1989). Basser P. J., Mattiello J., and LeBihan D. J. Magn. Reson. B 103 : 247 (1994). Basser P. J., Mattiello J., and LeBihan D. Biophys. J . 66: 259 (1994). Bax A., deJong P.G., Mehlkopf A.F., and Smidt J. Chem. Phys. Lett . 69: 567 (1980). Bloch F., Hansen W.W., Packard M.E. Phys. Rev . 69: 127 (1946). Bodenhausen G., Kogler H., and Ernst R.R. J. Magn. Reson . 58: 370 (1984). Callaghan P.L . Principles of Nuclear Magnetic Resonance Microscopy , pp. 331 Oxford: Oxford Science Publication (1991). Carr H.Y. and Purcell E.M. Phys. Rev . 94: 630 (1954). Crank, J. The Mathematics of Diffusion Oxford: Oxford University Press (1975). Copper, R.L., Chang D.B., Young A.C. Biophys. J. 14:161177 (1974). Cotts R.M ,Hoch M.J.R, Sun T., Markert J.T. J. Magn. Reson . 83: 252266 (1989). Douek P., Turner R, Pekar J., LeBihan D. J. Comput. Asst. Tom L5: 923 (1991). Dusschoten D.V., As H.V., Jager A. D. SMRM Abstract 10: 785 (1991). Ebisu T., Naruse S., Horikawa Y., Ueda S., Tanaka C, Umeda M., Higuchi T. JMRI 3: 863868(1993). Ford J.C., Hackney B.H., Alsop D.C., Jara H., Joseph P.M., Hand CM., Black P. MRM 31: 488494(1994). Frahm J., Merboldt K.D., Hanicke W., Haase A. J. Magn. Reson . 64: 8193 (1985). 109
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110 Franks F. Water, A Comprehensive Treatise New York: Wiley (1972). Gates L., Cameron J. SMRM abstract 1037 (1994). Gelderen P.V., Olson A., Moonen C.T.W. J. Magn. Reson .(A) 103: 04 (1993). Gillies R.J. NMR in Physiology and Biomedicine . Phanix: Academic Press (1994). Haase A., Frahm J., matthaei D., Hanicke W., Bomsdorf H., Kunz D., Tischler R. Radiology 160 : 787790 (1886). Harris N.G., Jones H.C., Williams S.C. Exp. Neurol . 118 : 16 (1992). Hong X. and Dixon W.X. J. Magn. Reson . 99: 561 (1992). Inglis B.A., Yang L., Wirth III E.D., Plant D., and Mareci T.H. Magnetic Resonance Imaging . Vol. 15, No.4, pp.441450 (1997). Inglis B.A., Yang L., Wirth ED, Plant D., and Mareci T.H. SMRM abstract (1997). Inglis B.A., Wirth III E.D., Plant D., Yang L., and Mareci T.H., Histology of Excised Human Spinal Cord using HighResolution MR Imaging and Diffusion Tensor MR Images. Submitted to Magnetic Resonance Imaging, 1997. Jones H.C., Anderson R.W. Journal of Neurotrauma 13:590 (1996). Latour L., Li M., Sotak C.H. JMR 101(B): 7277(1993). Latour L., Mitra P.P., Kleinberg R.L., Sotak C.H. JMR 101(A): 342346(1993). Lauterbur P.C. Nature(London ) 242 : 190(1973). LeBihanD. Magnetic Resonance Quarterly , 7: 130(1991). LeBihan D. SMRM abstract #1324 (1996). LeBihan D., and Breton E. C. R. Acad. Sci .(Paris) 301 : 1 109 (1985). MacFall JR. Radiology 169 (1988). Mareci T. H. Pulse Methods in ID and 2D Liquid Phase NMR " (W. S. Brey, Ed.), 259 San Diego: Academic Press (1988). Mareci T.H., Sattin W., Scott K.N. JMR 67: 5565,1986.
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Ill Mareci T.H., Yang L, Inglis B.A., "A formalism for generalized gradient weighting with application to diffusion tensor imaging", submit to JMR , Feb 1995. Martin J.F., Selwyn L.S., Void R.R., Void R.L. J. Chem. Phys. 76: 26322634 (1982). Mattiello J., BasserP. J., and LeBihan P., J. Magn. Reson. A 108 : 131 (1994). Merboldt K. D., Hanicke W., and Frahm J., J. Magn. Reson . 64: 479 (1985). Moseley M.E., Cohen Y., Kucharczyk J., Mintorovitch J., Asgari H.S., Wenland M.F., Tsuruda J. Radiology 176 : 439445 (1990). Neeman M., Freyer J.P., and Sillerud L.O. J. Magn. Reson . 90: 303(1990). Purcell E.M., Torrey H.C., Pound R.V. Phys. Rev . 69: 37(1946). Pierpaoli C, Linfante I. SMRM abstract #902 (1996). Rao C. R. Linear Statistical Inference and Its Applications New York: Wiley (1965). Schrimsher, G.W., Reier, P.J. Exp. Neurol. 120 : 264 (1993). Stanisz G.J., Szafer A., Wright G.A., Henkelman R.M. Magnetic Resonance in Medicine 37: 103111 (1997). Stejskal, E.O., and Tanner J.E. J. Chem. Phys. 42: 288292 (1965). Stepisnik J. Physics 104B : 350 (1981). Stepisnik J. Prog. NMR Spectro . 17: 187 (1985). Szafer A., Zhong J., Gore J.C. MRM 33: 697712 (1995). Tanner J. E. J. Chem. Phys . 52: 2523 (1970). Tanner J.E. J. Chem. Phys . 69: 1748 (1978). Taylor D.G., and Bushell M. C. Phys. Med. Biol. 30: 345 (1985). Torrey H.C. Phys. Rev. 104 : 563565 (1956).
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112 van Geldren P., Marloes H.M., Vlesschouwer D., DesPres D., Pekar J., Zijl P., Moonen C. MRM 31:15463(1994). Williams W.D., Seymour E.F.W, Cotts R.M. J. Magn. Reson. 31: 271(1978). Yang L., Mareci T.H., Harris N.G., Wirth ED, Inglis B.A. SMRM abstract #902 (1995).
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BIOGRAPHICAL SKETCH Lei Yang was bora in Hoyang, Fujian, China in January 1967. He graduated from Fuzhou First Middle School in 1984 and four years later received a B.S. degree in physics from Zhejiang University. While at Zhejiang University he received prizes in 19841985 and 19861987 for excellent study. Then he worked as an electrical engineer in a hotel for two years In 1991 he entered the Ph.D. program at the University of Florida, passing the written qualifying exam two months after he arrived in Gainesville. He was lucky to meet Dr. Andrew and Dr. Mareci and conducted his research under their guidance. While at the University of Florida he received the International Student Academic award in 1995 and 1996. His publications include the following: Inglis B.A., Yang L., Wirth III E.D., Plant D., and Mareci T.H. (1997) Diffusion Anisotropy in Excised Normal Rat Spinal Cord Measured by NMR Microscopy. Magnetic Resonance Imaging . 4: 441450. Inglis, B.A., Yang,L., Wirth, E.D., Plant, D., Mareci, T.H. (1997) Diffusion tensor of human spinal cord. Vancouver: Society of Magnetic Resonance in Medicine Conference Abstract 3: 267 (Poster and Oral presentation). Mareci, T.H., Yang,L., Inglis, B.A., Wirth, E.D., Plant, D.(1995) Translational diffusion as a structure parameter in magnetic resonance imaging. Talahasse: 27th Annual Southeastern Magnetic Resonance Conference (Oral presentation). Yang, L., Mareci, T.H, Harris, N.G., Wirth, E.D., Inglis, B.A. (1995). Diffusion tensor imaging of rat brain in vivo using a multiple stimulated echo sequence. Paris: Proceedings of Society of Magnetic Resonance in Medicine 2: 902. 113
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114 Wirth, E.D., Mareci, T.H., Yang, L., Inglis, B.A., and Anderson D.K.(1995). In vivo diffusionweighted MR imaging of subacute and chronic spinal cord injury. Paris: Proceedings of Society of Magnetic Resonance in Medicine 1:344 (Oral presentation). Mareci, T.H., Inglis, B.A., Yang,L., Wirth, E.D., Plant, D. (1995). NMR microscopy at 300 and 600 Mhz: Images and diffusion tensor maps of excises rat, cat, and human spinal cord. Boston: 36th Experimental Nuclear Magnetic Resonance Conference 265. Yang, L., Mareci, T.H, Xu, W., Inglis, B.A. (1994). A buildingblock approach to the analysis of generalized gradient weighting: application to diffusion tensor measurements. Pacific Grove: 35th Experimental Nuclear Magnetic Resonance Conference 143.
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. E. Raymond Andrew, Chairman Graduate Research Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy, Thomas H. Mareci, Cochairman Associate Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ;iTS\ Sullivan Neir Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Phjio^ophy. 1^ ^ CvW Pradeep Kumar Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Longan Qiu Associate Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard W. Briggs U(/ Associate Professor of Biochemistry and Molecular Biology I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John M. Yelton Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wallace S. Brey VJ Professor of Chemistry This dissertation was submitted to the Graduate Faculty of the Department of Physics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May, 1999 Dean, Graduate School

