MAGNETIC RESONANCE IMAGING AND SPECTROSCOPY
FOR THE STUDY OF TRANSLATIONAL DIFFUSION:
APPLICATIONS TO NERVOUS TISSUE
By
ELIZABETH L. BOSSART
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
For Juan
ACKNOWLEDGMENTS
The work in this thesis could not have been accomplished without the help of a
great many people it is truly a collaborative effort. First and foremost, I would like to
thank Dr. Tom Mareci for his help, time and patience over the last few years. Working
with him, I learned a lot about research, and a lot about myself. I would also like to thank
Dr. Steve Blackband for all his counsel and support. Among other things both scientific
and personal, through him I learned when to relax. I would also like to thank Dr.
Raymond Andrew for his encouraging words and advise throughout my graduate career.
His kind words always made me smile.
I would like to thank several people in the lab and around the Brain Institute for
their help in my endeavors. I would like to acknowledge Ron Smith in the Neurosurgery
department of the UF Brain Institute for providing the human brain samples. I would like
to thank Drs. Ed Wirth, Paul Reier and Doug Anderson for their input on and expertise
with the neuroanatomy. With their help I managed to learn more biology than I ever
thought I would know. Also want to acknowledge Ed's help in reading over a couple of
chapters of my thesis and giving me his opinion. I would also like to thank Drs. Ben
Inglis and Dave Buckley. My discussions with them helped make clear many of my
ideas. I would also like to thank Ben Inglis, Jim Rocca and Dan Plant for helping me
learn many of the more practical things about NMR, like how to run the equipment. My
appreciation goes to Xeve Silver for his patience with the "stupid biology questions" I
iii
was constantly asking, for his aid in mixing up solutions, and for doing a lot of the animal
handling. My appreciation also goes to Emma Mercer for all her help with animal
handling. Without her, many of the normal studies and injury studies could not have
been completed. Thanks also go to Emma for providing the artwork in Appendix B and
for proof reading a couple of thesis chapters. I would also like to acknowledge Dr. Abbas
Zaman in the University of Florida Engineering Research Center for not only allowing
me to use his viscometer, but showing me how to use it as well. For their suggestions,
encouragement, and friendship I would like to thank Cecile Mohr, Alan Freeman and Jon
Bui.
I would also like to acknowledge five people without whom I might never have
made it through graduate school. First I would like to thank my parents, Marilynn and
William Bossart, for not setting limits on what I could do. Their support has been
invaluable. Next, I want to recognize my sister Cathy. Through her I learned what it
means to be a survivor. I would also like to acknowledge Koren Okuma. Through the
years her friendship has meant more to me than I know how to say. She has been there
for the ups and the downs and consistently reminded me how to laugh. Last, but not
least, I want to thank Juan Villar for standing by my side, loving me and encouraging me.
His love and support have helped to make the time go by more smoothly. I know this has
not been easy for him, but it has made things easier for me.
TABLE OF CONTENTS
page
A C K N O W L E D G M E N T S ..................................................................................................iii
LIST OF TABLES .................................... ............... viii
LIST OF FIGURES.......................................... ............... ix
LIST O F A BBREV IA TION S ................................................................... ................ xi
A B S T R A C T ..................................................................................................................... x iii
CHAPTERS
1 IN TR O D U CTIO N ......... .. ..................................... ........................... .. ............ 1
Introduction to R elaxation.. .................................................................... .............. 3
D description of T R elaxation.................................. ....................... .............. 4
D description of T2 R elaxation.................................. ....................... .............. 5
Relaxation Processes ....................... ............ ..............................6
E change P rocesses ..... .. ........................................ ........................ .. . ......... 6
D ip olar Interaction s ..... .. ........................................ ......................... . ...... ..... 8
Q uadrupolar Interactions......................................... ......................... .............. 8
Chem ical Shielding A nisotropy ....................... .............................................. 8
S calar R elax action ............................................................................................. 9
Introduction to D iffu sion ............................................ ............................ ............. 9
The Solution to the BlochTorrey Equation...................................... .............. 10
D ata C o llectio n ..................................................................................................... 1 1
C alculating b V alues ... .. ........................................ ....................... . . ..... ..... 12
M multiple L inear R egression ..................................... ....................... .............. 15
Problem with the MonoExponential Form .............. ................................... 17
M ultiexponential D iffu sion ......................................................................................... 2 1
P orous M edia T heory ................................................................ ..... ...... ......... .. 2 1
An Analytical Model of Restricted Diffusion................................................. 23
M ultiexponential M odel ........................................ ......................... .............. 25
Study D directions .......................................................................................................... 27
2 FIX A TION EFFECTS ..... ................................................................. .............. 30
Materials and Methods ..... ............ ............................................ 32
R elaxation M easurem ents in Solution .............................................. .............. 32
v
Im aging M easurem ents in Tissue...................................................... .............. 33
R e su lts ................. .... .................................................. ..................................... 3 4
Fixative Solution M easurem ents....................................................... .............. 34
Tissue M easurem ents ..... .. ............................... ...................................... 35
D iscu ssio n ............................................................................................................... . .. 3 7
3 RELAXATION AND DIFFUSION MEASUREMENTS ON FIXED HUMAN
B R A IN SA M P L E S ................................................................... ................. ... 44
M materials and M ethods ... ...................................................................... .............. 46
Sam ple Preparation ..... .. ................................ .......................... ......... ..... 46
N M R M easurem ents .. .................................................................... .............. 47
P o stP ro cessin g ..................................................................................................... 4 8
R e su lts .............. ........................................................................................ . .......... 4 9
D iffusion Tensor M easurem ents ....................................................... .............. 49
T2 R elaxation M easurem ents ....................... ................................................ 52
D iscu ssio n ............................................................................................................... . .. 5 3
4 MULTIEXPONENTIAL DIFFUSION TENSOR IMAGING OF NORMAL RAT
SPINAL CORD ....................... .. .......... .............................. 56
M materials and M ethods ...... .. ................................. .......................... .............. 57
Sam ple Preparation .............................................. .................. .. .. .. .......... 57
Diffusion Tensor Measurements and Post Processing .................................... 57
R e su lts ......................................................................................................... ............ 5 8
D iscu ssio n ............................................................................................................... . .. 6 7
5 MULTIEXPONENTIAL DIFFUSION TENSOR IMAGING OF NORMAL AND 1
MONTH POST INJURY RAT SPINAL CORDS ............... ................................... 70
M materials and M ethods .... ... ......................................... ....................... ......... ..... 7 1
Sam ple Preparation ..... .. ................................ ........................................ 71
N M R Experim ents ...... .. ..................................... ........................................ 72
R esu lts ....................... ................................................................................ . . . 73
N orm al R at Spin al C ord ........................................................................................ 77
R at Spinal Cord 1M onth Post Injury ............................................... .............. 85
D iscu ssio n ............................................................................................................... . .. 9 0
vi
6 SUMMARY AND CONCLUSIONS................................................................. 94
G L O S S A R Y .................................................................................................................... 1 0 0
APPENDICES
A C O M PU TER C O D E ..... .................................................................. .............. 104
B P U L SE SE Q U E N C E S ............................................................................................... 125
C RAT SPINAL CORD ANATOMY AT VERTEBRAL LEVEL LI ..................... 127
L IST O F R E F E R E N C E S ................................................................................................ 129
BIO GRAPH ICA L SK ETCH ................................................................ .............. 137
vii
LIST OF TABLES
Table page
21. Measured SNR in gray and white matter regions of the spinal
cord for various tissue preparations ...................... ............... 31
22. Relaxation rates in GM/WM for different tissue preparations .............. 35
31. Diffusion rates found for various types of fitting routines ....................... 50
32. T2 relaxation rates and volume fractions for GM and WM samples ............... 52
33. Diffusion volume fractions with and without the T2 contribution ................ 53
41. Diffusion rates for the regions if rat spinal cord shown in Figure 41 d .......... 61
LIST OF FIGURES
Figure page
11. Sim ple pulse sequence exam ple ....................................................... 13
12. Natural log of the signal intensity vs. b value graphed for b to 1500 s/mm2 ..... 18
13. Natural log of the signal intensity vs. b value for simulated data ................. 20
14. Natural log of the signal intensity vs. b value graphed for b to 10000 s/mm2 .... 20
15. Natural log of the signal intensity vs. b value for a region of interest
in the rat spinal cord ......................................................... 29
21. Graph of 1/T1 vs. concentration ......................................... ........... 34
22. Graph of 1/T2 vs. concentration .......................... ........... 36
23. Graph of dynamic viscosity vs. concentration .......................... ............ 36
24. Fast diffusion rate component vs. slice position graphs for regions of
interest in fixed rat spinal cords measured in PBS ............................. 38
25. Slow diffusion rate component vs. slice position graphs for regions of
interest in fixed rat spinal cords imaged in PBS ............................ 39
31. The natural log of the signal intensity vs. b value graph for brain white matter .. 45
32. The natural log of the signal intensity vs. b value graph for brain gray matter ... 45
41. Zero diffusionweighted images .......................... ........... 58
42. The full biexponential diffusion tensor ............... .................... 59
43. The color diffusion tensor trace .......................... ........... 64
44. The grayscale diffusion tensor trace .......... ....................... 65
45. The anisotropy/isotropy images .......................... ........... 66
51. Diffusion trace images 7 mm rostral to the epicenter of injury ................... 74
52. Diffusion trace images at the epicenter of injury ....................... .............75
53. Diffusion trace images 7 mm caudal to the epicenter of injury ................... 76
54. Anisotropy/isotropy images 7 mm rostral to the epicenter of injury ............. 78
55. Anisotropy/isotropy images at the epicenter of injury ............................ 79
56. Anisotropy/isotropy images 7 mm caudal to the epicenter of injury ............. 80
57. Graphs showing the fast and slow diffusion rate trace vs. slice position for
normal and injured rat spinal cords ............... ..................... 82
58. Graphs showing the fast and slow fractional anisotropy vs. slice position for
normal and injured rat spinal cords ......................... ...... .......... 83
59. Graphs showing the fast and slow volume ratio vs. slice position for
normal and injured rat spinal cords .......... ......................... 84
LIST OF ABBREVIATIONS
ADC : apparent diffusion coefficient
ADT : apparent diffusion tensor
Bo : the magitude of the main magnetic field
B1 : the magnitude of the applied magnetic field
by : b value in the i, j direction with respect to the gradient directions, where i, j = x, y or
z
cTR : color scale diffusion tensor trace
Dij : diffusion in the i, j direction with respect the gradient directions, where i, j = x, y, or
z
I [D D (TR(D))I2
3 i=x,y,zj=x,y,z
FA : fractional anisotropy, FA = 
1= x,y,zJ= x,y,z
7: gyromagnetic ratio
GM : gray matter
gTR : grayscale diffusion tensor trace, (Dx + Dyy + Dzz) /3
M (r, t) : the nuclear magnetization transverse to the static magnetic field; M+ = Mx+ iMy
Mo : equilibrium magnetization
Mi(t) : the net magnetization in the transverse plane, perpendicular to the magnetic field
B0, at time t
MR : magnetic resonance
MRI: magnetic resonance imaging
Mz(t) : magnetization along the longitudinal magnetic field at time
NA : number of averages
NMR : nuclear magnetic resonance
PBS : phosphate buffered saline solution
RF : radio frequency
SNR : signal:to:noise ratio
Tc : correlation time; the order of time it takes a molecule to turn through 1 radian, or the
time for a molecule to move through a distance comparable to it's dimensions
T : tortuosity
TE: echo time
TR: repetition time
Deter min ant(D)
VR : volume ratio, VR =
[TR(D) / 3]3
WM : white matter
co : Larmor frequency; coo = 7 Bo
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
MAGNETIC RESONANCE IMAGING AND SPECTROSCOPY
FOR THE STUDY OF TRANSLATIONAL DIFFUSION:
APPLICATIONS TO NERVOUS TISSUE
By
Elizabeth L. Bossart
August 1999
Chairman: T.H. Mareci
Major Department: Physics
In all stages of trauma and disease in the brain and spinal cord, it is important to
know the amount of the physical damage, how far the damage will extend, and how the
structural changes relate to the final amount of functionality. Though it is fairly
straightforward to measure this damage ex vivo through histological sectioning,
assessment of internal physical damage in vivo has been difficult to do. The innovation
of magnetic resonance (MR) imaging, in particular the measurement of water diffusion,
has been an important step towards quantifying structural changes in living systems.
Water diffusion, once considered a single diffusion rate (or single diffusion rate
tensor) process, appears to be a multiple diffusion rate process. To the limits of the
gradients available (300 mT/m), two unique diffusion regimes have been seen in fixed
CNS tissue: a fast diffusing component and a slow diffusing component. Others have
speculated that the fast and slow diffusing components in tissue represent diffusion in the
extracellular and intracellular spaces, respectively.
The aim of these studies was to find the best tissue preparation for ex vivo
measurements, determine the best model with which to fit diffusion studies done on CNS
tissue, and to provide a basic understanding of the information contained in the fast and
slow diffusion compartments. These measurements should provide a solid background
for future in vivo experiments, allowing a better understanding of the information
provided by the fast and slow diffusion components. This will be useful for
understanding the role of diffusion in normal tissue and for quantifying changes due to
trauma, which could lead to better diagnostic techniques in the future.
CHAPTER 1
INTRODUCTION
Observations of soft tissues in vivo were difficult prior to the advent of 1H
magnetic resonance imaging (MRI; Bottomley, 1982; Callaghan, 1991; Hinshaw & Lent,
1983; Lauterbur, 1973). However MRI techniques were needed to quantify the
differences between normal and abnormal tissue. Among the techniques that have been
used to show contrast in biological tissues are the more conventional T, relaxation and T2
relaxation methods (Akber, 1996; Becerra et al., 1995; Bottomley et al., 1984; Moseley et
al., 1984; Willcott, 1984), and more recent translational water diffusion (Callaghan, 1991;
Le Bihan et al., 1993; Moseley et al., 1990; Stejskal, 1965; Stejskal & Tanner, 1965;
Torrey, 1956). For the measurement of changes due to diffuse injury (i.e. stroke or
edema that causes cell swelling), Ti and T2weighted images show very few changes
from normal tissue (Becerra et al., 1995; Moseley et al., 1990; Pierpaoli et al., 1996). In
these cases, translational water diffusion imaging better characterizes the changes from
normal tissue (Ford et al., 1994; Kirsch et al., 1991; Moseley et al., 1990; Pattany et al.,
1997; Pierpaoli et al., 1996; van Gelderen et al., 1994).
Translational water diffusion has been used to define solid porous media samples
(Borgin et al., 1996; Ek et al., 1994; Helmer et al., 1995; Latour et al., 1993; Latour et al.
1994; Mitra et al., 1993; Mitra et al., 1992) and to look at biological tissues (Basser et al.,
1994a & 1994b; Basser et al., 1993; Inglis et al., 1997; Kirsch et al., 1991; Moseley et al.,
1990; Norris et al., 1994; Pattany et al., 1997; Pierpaoli et al., 1996; Szafer et al., 1995a
and 1995b). Measurements of diffusion coefficients both in vivo and in vitro have aided
in elucidating structure in tissues such as brain and spinal cord (Basser et al., 1993;
Basser et al., 1994b; Chenevert et al., 1990; Ford et al., 1994; Gulani et al., 1997; Inglis
et al., 1997; Kirsch et al., 1991; Le Bihan et al., 1993; Moseley et al., 1990; Ono et al.,
1995; Pattany et al., 1997; Pierpaoli & Basser, 1996; Pierpaoli et al., 1996; Szafer, et al.
1995a & 1995b; Thompson et al., 1987; van Gelderen et al., 1994). By taking a series of
diffusionweighted images, it is possible to calculate the apparent diffusion coefficient
(ADC) for water molecules moving in the direction of the applied diffusionweighted
gradients. ADC images have been measured for many tissues (Norris et al., 1994; Szafer
et al., 1995a & 1995b; Pattany et al., 1997). However, if the tissue being looked at is
anisotropic, the ADC values are dependent on the orientation of the structure with respect
to the gradient axes. Because imaging gradients will cause some diffusion crossterms,
an ADC is only an approximation of a complete apparent diffusion tensor (ADT). A full
ADT map of a tissue gives an indication of the fiber tract orientations within the tissue
(Basser et al., 1993; Basser et al., 1994b; Inglis et al., 1997; Kirsch et al., 1991; Moseley
et al., 1990; Pierpaoli et al., 1996). ADC and ADT mapping have been used extensively
to characterize both brain and spinal cord tissue (Basser et al., 1993; Basser et al., 1994b;
Inglis et al., 1997; Kirsch et al., 1991; LeBihan et al., 1993; Moseley et al., 1990;
Pierpaoli et al., 1996).
Diffusionweighted imaging and spectroscopy are two of the methods thought to
give an indication of function in both pre and post injury spinal cords. ADC and ADT
maps of these tissues both in vivo, in vitro and ex vivo have been studied to this end. In
order to compare ex vivo and in vivo measurements of diffusion, more must be
understood about the effects that fixation has on tissue samples. To understand the
fixation effects better, relaxation measurements should be made both on fixative solutions
and on fixed tissue. The measurements on fixative and fixed tissue should aid in the
understanding of how fixation changes the tissue, leading to a better understanding of the
diffusion in fixed tissue. The diffusion maps may be able to give some understanding of
the underlying tissue structure and processes, but first more must be understood about the
diffusion of water through a tissue. In particular, a better understanding is needed of how
relaxation, compartmentalization, exchange and anisotropy effects the diffusion within
tissues.
This introduction begins by outlining different relaxation processes (including
spinlattice relaxation T1, and spinspin relaxation T2), and explaining some of the
mechanisms that affect those relaxation processes. Next, the steps necessary to create an
apparent diffusion tensor will be described, followed by a sketch of the inherent problems
in the monoexponential model created by Stejkal and Tanner. The current models that
are used to fit diffusion data are presented and analyzed. From this discussion comes the
presentation of what must be elucidated in order to have a complete model of diffusion in
biological tissue. Finally, a brief outline of the studies done will be presented.
Introduction to Relaxation
Before getting too far into the descriptions of relaxation, one thing should be
noted. In this work a symbol B (e.g. Bo) will be called the magnetic field. Although
calling B the magnetic field is not unusual, it is actually a misnomer. In fact, B is the
magnetic flux density, or magnetic inductance, through a material, and is given by the
equation B = p(H + M) where p. is the permeability of the material, M is the
magnetization of the material, and H is the magnetic field strength. Getting back to
relaxation, a resonant radio frequency (RF) pulse effects a spin system by disturbing it
from its thermal equilibrium state. Equilibrium can be restored via several types of
relaxation processes. The following pages will contain a summary of two relaxation
processes, T1 relaxation and T2 relaxation, then describing some of the mechanisms that
cause changes in these relaxation rates (i.e. T2* relaxation, dipoledipole interactions,
exchange).
Description of Ti Relaxation
T1, "spinlattice," or longitudinal relaxation is characterized by the return of the
net magnetization to the ground state (i.e. the state where the net magnetization is along
the main magnetic field) from the highenergy state induced by a RF pulse. For a set of
mutually independent nuclei coupled to a thermal bath, T1 relaxation can be defined as
dMz (Mz Mo)
_ [11]
dt Ti
with the solution
Mz(t)= Mz(0)et/Tl + MO(1et/T ) [12]
where Mo is the equilibrium magnetization along the longitudinal magnetic field Bo
(which is assumed to be along the zaxis), and Mz(t) is the magnetization along the
longitudinal magnetic field at time t. This means that at time t = TI, approximately 63%
of the magnetization has returned to the ground state. The spin system is considered to
be fully relaxed in the longitudinal direction after 3 to 5 T1 periods has passed.
T1 relaxation is magnetic field strength dependent and is fastest when the nuclear
motion, or tumbling rate, matches the Larmor frequency (coo). At this tumbling rate, T1
reaches its characteristic minimum. The Larmor frequency is the precessional frequency
of a nucleus in a magnetic field. It is governed by the equation coo = y Bo where y is the
gyromagnetic ratio for the nucleus and Bo is the strength of the magnetic field. As the
strength of the magnetic field increases, so does the Larmor frequency. When the Larmor
frequency increases, the characteristic T1 minimum gets longer. Therefore, the higher the
magnetic field, the longer it takes for a nucleus to relax in Ti.
Description of T2 Relaxation
For a perfectly homogeneous magnetic field, Bo, the decay of magnetization in the
xy plane is governed by T2, "spinspin," or transverse relaxation. T2 relaxation is
characterized by adjacent spins in high and low energy states exchanging energy without
losing that energy to the surrounding lattice. For a set of mutually independent nuclei
coupled to a thermal bath, T2 relaxation can be defined as
dMi Mi
[13]
dt T2
with solution
Mi(t) = Mi(0)et/T2 [14]
where i = x or y and Mi(t) is the net magnetization in the transverse plane (along the i
axis), perpendicular to the magnetic field Bo, at time t. This means that after time t = T2,
the net magnetization in the transverse plane has been reduced by approximately 63%.
As with longitudinal relaxation, the spin system is considered to be fully relaxed in the
transverse plane after 3 to 5 T2 periods has passed.
In practice, however, the Bo field is inhomogeneous. This means that in different
parts of the sample the nuclei experience disparate magnetic fields, causing the nuclei to
process at slightly different frequencies. The result is a signal loss due to the dephasing
of individual magnetizations. The consequence of the dephasing is a loss in transverse
magnetization at a rate that is greater than that due to T2 relaxation alone (i.e. the free
induction decay, FID, disappears more rapidly than it would due to T2 relaxation alone).
This loss is known as T2* relaxation and is characterized by the following equation
1 +(yAB) [15],
T2 T2
where AB is the inhomogeneous variation in the magnetic field. T2* relaxation results in
an inhomogeneous broadening in the MR spectrum, or severe distortions in MR images
when the data is taken at a narrow bandwidth (Callaghan, 1993). Unlike the irreversible,
homogeneous broadening due to T, and T2 relaxation processes, this type of broadening
is ordered and can be "undone" by the use of an appropriate pulse sequence.
Like TI, T2 is magnetic field dependent. Transverse relaxation time is always less
than or approximately equal to the longitudinal relaxation time. For pure water, the two
relaxation times for 'H NMR are approximately equal, but for biological tissues T2 is
almost always less than Ti.
Relaxation Processes
Exchange Processes
Protons in water molecules experience two types of interactions: (1) dipolar
interaction with protons on the same molecule and (2) intermolecular interaction from
protons in neighboring molecules (Kaplan & Fraenkel, 1980; Andrew, 1958; Callaghan,
1993). These interactions fluctuate as the water molecule diffuses via rotational and
translational motions. For free water, the rotational correlation time (the time it takes for
the molecule to turn through a radian) is much shorter than the Larmor period (1/coo
where coo = 7Bo), so the line width is extremely narrow and T1i T2. An impurity in the
water, such as oxygen, will act as a relaxation center for the water. The dipolar
interaction between the proton and the impurity ionic moment is modulated by the
relative motion between the water and the ion. This interaction causes a shortening of the
relaxation times T1 and T2.
Water molecules closely associated with larger molecules or solid surfaces (i.e.
bound water) will tumble more slowly. The slower rotational motion leads to a reduction
in T1 and T2 relaxation times. This shortening continues until the correlation time (i.e.
the time it takes a molecule to turn through 1 radian, or the time for a molecule to move
through a distance comparable to it's dimensions; Andrew, 1958) for the dipolar
fluctuation is approximately equal to the Larmor period (Tco 1/coo). When the correlation
time for the dipolar fluctuation equals the Larmor period, T1 relaxation reaches its
characteristic minimum.
Water molecules in close proximity to solid surfaces and slowly moving
macromolecules will have their proton relaxation rates thoroughly affected. Slowing of
reorientational motion in water will inevitably lead to an altered proton relaxation for this
phase. Translational diffusion induces an exchange of molecules between the bound and
free phases. Rotational motion in the bound phase can be quite slow (i.e. correlation
times are much longer than the Larmor period), so divergence of T1 and T2 relaxation
occurs with T1 becoming significantly longer than T2.
Dipolar Interactions
The intrinsic magnetic moment associated with each nuclear spin dipole exerts a
large influence on its neighbors via the magnetic field produced by this dipole acting on
the dipole moments of remote spins (Callaghan, 1993; Tycko, 1994; Kaplan & Fraenkel,
1980; Wasylichen, 1987). For solids, where internuclear distances are fixed, this process
dominates the line shape. In all liquids except the most viscous, the tumbling motion of
the molecules is rapid, and the dipolar interaction strength is comparatively weak.
Therefore, the dipolar interactions do not contribute to the broadening of the line shape.
However, for viscous liquids, where tumbling is slower, dipolar interactions begin to take
effect, shortening T2.
Quadrupolar Interactions
A nucleus with spin > 1/2 possesses an electric quadrupole moment (Wasylichen,
1987; Tycko, 1994). The quadrupole moment is collinear with the magnetic dipole
moment for the nucleus. For these nuclei, the interaction between the nuclear quadrupole
moment and fluctuating electric field gradients provide a source for nuclear relaxation.
In fact, for these nuclei, quadrupolar interactions are the principal contributor to spin
relaxation.
Chemical Shielding Anisotropy
Atomic or molecular electron clouds interact with nuclear spin angular
momentum (Callaghan, 1993; Wasylichen, 1987). These interactions characterize the
local electronic environment for an atom or molecule. The principle influence of the
surrounding electron cloud is magnetic shielding which results when electronic orbitals
are perturbed by an applied magnetic field. This phenomenon results in chemical
screening or shielding. The local field of a nucleus with less than tetrahedral symmetry
will be dependent on the orientation of the molecule in the applied magnetic field.
Reorientation of the molecule results in a fluctuation of the field at the nucleus, providing
a source for relaxation.
Scalar Relaxation
The finest structural details observed in the liquid state NMR spectroscopy are
from scalar spinspin coupling or J coupling (Wasylichen, 1987). Indirect interaction
between two nuclei, I and S, is mediated by the electrons present in the molecular orbital.
Nuclear spin causes a slight polarization in the electron cloud. This electron
delocalization is transmitted to neighboring molecular nuclei, leading to the spinspin
interaction. Fluctuations in the magnetic field at some nucleus I arise due to one of two
interactions. Either the coupling between the nuclei is time dependent due to rapid
chemical exchange (scalar relaxation of the first kind), or one nucleus has a T1 relaxation
time that is short compared to the inverse of the scalar coupling between the nuclei
(scalar relaxation of the second kind). Scalar coupling of the second kind is often a
reason for T2 being much less than T1.
Introduction to Diffusion
Now that relaxation mechanisms have been treated, it is time to treat translational
water diffusion. In diffusion, particles move from one location to another as a result of
random motion due to thermal or equilibrium processes. Using magnetic resonance, this
random process can be tracked. In the following discussion, a background of diffusion
will be presented, followed by the methods used to collect and process MR diffusion
weighted data. Afterwards, the shortcomings of the monoexponential diffusion
formulation are presented, and some competing descriptions are explored.
The Solution to the BlochTorrey Equation
The BlochTorrey equation gives a generalized treatment of diffusion and flow
through a sample due to magnetic fields (Torrey, 1956; Callaghan, 1991). In the case of
isotropic diffusion and spatially independent velocities, this equation reduces to
aM, (r, t) M+(r, t) \\
M(,t) iyrgM+(r,t) + DV 2M(r,t) V vM(r,t) [16]
dt T2
where D is the diffusion, v is the velocity of spins due to flow, and M+(r, t) is the nuclear
magnetization transverse to the static magnetic field. This is written in complex notation
as M+ = Mx+ iMy. The solution for this equation has the form
M+ (r, t) = S(t) exp iyr .f g(t')dt']exp [17].
Putting this solution back into the BlochTorrey equation, it is found that
(t)= [ Dy2 ( g(t')dt')2 + iyv. (Jg(t')dt')]S(t) [18].
This differential equation has the solution
S(t)= S(0)exp Dy2 'g(t )2dt exp i v g(t")dt)dt'] [19].
In order to look exclusively at diffusion, consider the case where there is no flow
through the sample, i.e. when v = 0. That is, the following equation needs to be solved
S(t) = S(0) exp Dy2 f g(t")dt" dt' [110].
This is generally rewritten as
S(t) = S(0)ebD [111]
where the diffusionweighting coefficient, b, is defined as
b =y 2 '(f "g(t")dt" )2dt' [112].
The BlochTorrey equation changes when the diffusion is considered to be
anisotropic. If diffusion is anisotropic, the BlochTorrey equation becomes
aM+ M+
iyr gM+ +V. DVM+ VvM+ [113]
at T2
and the solution to the equation when there is no flow becomes (Callaghan, 1991)
S(t)k =S(0)exp I bijkDij [114]
i=x,y,zj=x,y,z
where the k subscript indicates the particular gradient strength used to get the signal. The
i and j subscripts on the b and D indicated the direction of the gradients, and the
diffusionweighting coefficient, by, is now defined as
b j= y gi(t)dt" gj(t)dt" t' [115].
Data Collection
In solving the anisotropic form of the BlochTorrey equation, a solution was
found that was dependent on b and D, both of which are matrices. If an instantaneous
picture of Di and Dji (where i i j) could be taken, then it is possible that Dj would not
equal Dji. However, an instantaneous picture cannot be taken, and through averaging Di
can be assumed to be equal to Dji. With this assumption, the solution to the BlochTorrey
equation becomes
bxxkDxx + byykDyy +b zzkDzz
S(t)k = S(0)exp + 2(bxykDxy + bxzkDxz + byzkDyz [116]
There are seven unknowns in this equation: S(0) and the six diffusion coefficients, Dij. In
order to solve this equation the data should be taken with the gradients on in seven
different directions. The directions normally chosen are x, y, z, x = y, x = z, y = z, and x
= y = z (or x = y, x = y, x = z, x = z, y = z, y = z, and x = y = z). The direction of
diffusion can be calculated by looking at diffusion in each of these seven directions.
Theoretically these seven unknowns could be found using only seven data points
(one point in each of the seven directions) and the corresponding seven calculated b
values. This is not the best way however since every minor variation in the data would
cause problems in fitting the data to the equation above. To aid in getting a better fit to
the value of the diffusion rate D, three or more data points are taken in each direction.
Each signal intensity, Sk, is taken at different gradient strengths such that every point has
a different b value. This allows the value of diffusion to be quantified more accurately.
In many of the data sets five different gradient strengths were used in each direction.
This gives a total of thirtyfive data points and thirtyfive times six b values.
Calculating b Values
After the data has been collected, a fit must be made to find the diffusion
constants in each direction. Before a fit can be made, the b values for each direction must
be found. As was stated previously, the b value is given by Eq. [115]. A simple
example will be utilized to illustrate the method used to calculate the b values. For the
sequence where an RF pulse is followed by one positive and one negative gradient
(Figure 11) the following is defined:
( IGk fortk t tk+ 8k
k) 0 otherwise
F(t, k) = ftk k g(t', k)dt' [118],
tk
(bj) = TE F(t,k)F(t, )t [119],
(bij = (bijk [120],
so bij = bji [121]
where k, i indicate the timing of the sequence (in this case k, = 1, 2 or first gradient
pulse in the sequence, second gradient pulse in the sequence) and i, j indicate the
RF
g(t,k)t G i t2 G
A J< A "
TE
Figure 11. Simple pulse sequence example.
direction. Each term must be added together to get the total b value
bii = (bii)bI + (bii)22 + (bii)12 + (bii)21 = (bii) + (bii)22 + 2(bii)12 [122].
Solving all the above equations for the simple example provided gives a final b value of
b = 72G282(A8/3) [123].
This answer is the same as others have found using the Heaviside functions (Hinshaw &
Lent, 1983; Torrey, 1956), however the method described here is simpler for more
complex pulse sequences than the method described previously.
What has been described so far may be easily applied to simple pulse sequences.
In order to apply this method to more complex pulse sequences, the effect of RF pulses
within the time sequences must be taken into account (Bodenhausen et al. 1984; Mareci,
1988). One way to do this is to look at RF pulses in terms of coherence transfer
pathways. The coherence order is the difference in the magnetic quantum number of two
eigenstates. Coherence transfer pathways are defined by the value of the coherence
order, pi, after the RF pulse. In a system of isolated spins with spin quantum number 1/2,
the coherence order can take on the values 1 < pi < 1. An echo will form during an
interval, n, due to precession in the main field only if
n
Xpiti =0 [124]
i=l
where Ti is the duration of the ith interval. In the same way, an echo is formed due to an
applied gradient field when the following condition is satisfied for Eq. [118],
npkF(8k,k)= 0 [125].
i=l
The coherence order can be included by modifying Eq. [117] (Yang et al., 1994) so that
(tk)JpkG(t) fortk
g(t, k) = k k k k [126].
0 otherwise
With the inclusion of the coherence order, b values can be calculated from any pulse
sequence easily since the specification of the coherence order at each point in the
sequence accounts for the effect of the RF excitation sequence.
Multiple Linear Regression
In order to quantify the rate of diffusion in each of six different directions, the
following set of equations must be solved
S(t=S(O)exp bkDi
1 xx
2 yy [127]
3 = zz
i =< k = 1,...,n
4 =xy
5 = xz
6= yz
once the data has been taken and the b values have been calculated (as before, k indicates
a particular gradient strength) (Stejskal, 1965). Keep in mind that the values bij where i
#j are actually multiplied by 2 in order to appear like Eq. [116] (e.g. b'4 = bxy + byx =
2bxy). The first thing that must be done to make this problem simpler is to create a linear
equation by taking the natural log of both sides of Equation [127]. That means
6
y(t)k =0 bikDi + k [128]
i=l
where y(t)k = ln(S(t)k), yo = ln(S(0)) and k is the uncorrelated random uncertainty for
measured data points (Montgomery, 1976). The intercept should be redefined as y'o in
order to take into account the average value of b
6
y =yo XbDi [129]
i=l
where
b bik [130].
n k=l
Using this definition for the intercept, the linear equation becomes
6
Yk =YO (bikbi i+) k [131].
i=1
This can be written in matrix form as
y =b'a+ Z [132]
where y and e are vectors of size 1 x n, a is a vector of size 1 x 7, and b' is a matrix of
size 7 x n. The matrix b' is
1.0 ( b'I I) ( b' 1) ... (b b' I1
1= .0 (b'b'2) (b b2) (b b2 [133]
1.0 (b' b'n) (b b'2n) (b' b'n)
and the vector a is
Transpose() = [y' D1 D2 D3 D4 D5 D6] [134].
A least squares fit is performed to find the values for a. For the least squares fit
n 2
L= ek =CTI =(yb'a) (yb') [135].
k=1
The derivative of L with respect to a is set equal to zero in order to find the optimum
values of a. This means
L = 0 = 2bTy + 2bYTbo' [136].
b Tb', = bTy [137].
Solving for cx gives the equation in its final form
a b= (Tb bTy [138].
The solution to this equation is analytical, and a routine written in IDL
(Interactive Data Language Research Systems Incorporated) will rapidly process the
entire diffusion tensor (64 x 64 matrix, 5 b values in 7 directions) in about half a minute
on an SGI Onyx computer.
Problem with the MonoExponential Form
To date, most diffusion weighted images have been taken with b values in the
range of 0 to 1500 s/mm2 (Basser et al., 1994a and 1994b; Basser et al., 1992; Kirsch et
al., 1991; Moseley et al., 1990; Norris et al., 1994; Pattany et al. 1997; Pierpaoli et al.,
1996; Szafer et al., 1995a and 1995b). This is due to the fact that most clinical MR
systems are limited in their gradient strengths. These diffusion measurements seemed to
indicate that diffusion is a monoexponential phenomenon, as can be seen in Figure 12.
This monoexponential model was described for liquids by Stejskal and Tanner (Stejskal,
1965; Stejskal & Tanner, 1965). The curve in Figure 12 was generated from actual data
taken on a sample of human corpus callosum. With the use of better and stronger
gradient systems in both clinical and nonclinical MR systems, diffusion measurements
are being taken with b values up to 40000 s/mm2 (Assaf & Cohen 1998; Bossart et al.,
1999a and 1999b; Borgin et al., 1996; Buckley et al., 1999; Bui et al., 1999; Ek et al.,
1994; Helmer et al., 1995 and 1999; Inglis et al., 1997; Kraemaer et al., 1999; Latour et
18
al., 1993; Latour et al., 1994; Mitra et al., 1993; Mitra et al., 1992; Mulkern et al., 1999;
Niendorf et al., 1996; Pfeuffer et al., 1999; Stanisz et al., 1997; van Zijl, et al., 1991).
In (Signal Intensity) vs b value
For Gradients in the z direciton
4.8
4.7
451I
0.0 300.0 600.0 900.0 1200.0 1500.0
b value (s/mm^2)
Figure 12. Natural log of the signal intensity vs. b value graphed for b to 1500 s/mm2.
Inglis et al. (1997) made some diffusion measurement in spinal cord with b values
up to ~ 10000 s/mm2. The diffusion values found from these measurements were much
smaller than those previously reported. A closer look showed that the data curved much
more than would be expected for monoexponential character. This led to the speculation
that the system might more closely approximate a biexponential curve of the form
S(t)k =S(0)(f) exp I I bijkDi(f)
= i=x,y,zj=x,y,z
+S(0)(s)exp bijkDij(s)
Si=x,y,zj=x,y,z
[139].
' Peak height data
 y = 128.7 exp(0.000194 b)
'.,
**ss *
. .
In this equation the f and s subscripts stand for fast and slow diffusion rates, respectively.
In order to get an idea about what happens when a monoexponential curve fit is
used on biexponential data, biexponential data was generated using a simplification of
Eq. [139], namely
S(t)k = S(0)(f) exp( bkD(f)) + S(0)(s) exp( bkD(s)) [140]
with all values of b, D and S(0) set to given values. The data generated a curve of the
form in Figure 13. This curve looks like actual measured data from a human corpus
callosum sample (shown in Figure 14). A monoexponential fit to the whole data set
gives very inaccurate results. Instead the data was fit by breaking it up into three regions:
a region of small b values (0 to 1000 s/mm2) where the natural log of the data appears to
be a straight line, a region of large b values (6000 to 10000 s/mm2) where the natural log
of the data also appears to be a straight line, and a region between those two where the
data is obviously better fit to a curved line. The two regions where the data appears to be
straight were fit using a monoexponential curve of the form
S(t)k = S(0)exp(bkD) [141]
where the b values were calculated and the S(t)k values were the generated data point.
Next, the calculated fits to the straight regions were used as a starting point for a non
linear regression analysis of the curve, this time fitting the entire curve using Eq. [140].
The monoexponential fit to the region of high b values gave a diffusion value very close
to that found by the biexponential fit, therefore fitting a monoexponential curve to this
region is a fairly accurate estimate. The monoexponential fit for the region of low b
values, however, is at least a factor of two smaller than answer found by using the
biexponential fit. This suggests that the diffusion values given by investigators working
In(Signal Intensity) vs. b value
Simulated data
2000.0
4000.0
6000.0
8000.0
10000.0
b value (s/mm2)
Figure 13. Natural log of the signal intensity vs. b value for simulated data.
In (Signal Intensity) vs b value
For Gradients in the z direction
Peak height data (human brain white matter)
 y = 128.7 exp(0.000194 b)
Iy y = 81.9 exp(0.000032 b)
 y = 119.8 exp(0.000095 b)
* y = 50.1 exp(0.000552 b) + 79.3 exp(0.000028 b)
2000.0
4000.0
6000.0
8000.0
10000.0
b value (s/mm^2)
Figure 14. Natural log of the signal intensity vs. b value graphed for b to 10000 s/mm2.
simulated data
 y=92.55exp(0.000039b)
 y=99.76exp(0.000099b)
 y=82.35exp(0.000023b)
y=20.01exp(0.00005b)+80.00exp(0.00002b)
.. S.
.'
.. ..
'S .h~ 
S
4.4 1
4.0 L
0.
0
'4.
e
S4.5
g) 4.4
.

4.2
4.1
4.0
0.
0
in the b value range of 0 to 1000 s/mm2 may not be accurate if the actual diffusion rate is
a biexponetial. The diffusion values should be higher than one would anticipate from
looking at such a limited range of data.
Exploring the biexponential behavior of the data is the next task. In the following
three sections, different models will be discussed that propose to fit the curvilinear data.
Each model has its positive and negative aspects, but each is incomplete. These aspects
of the models will be described and explored briefly.
Multiexponential Diffusion
Porous Media Theory
Solidboundary porousmedia theory uses the timedependent diffusion
coefficient of the fluid in the interstitial spaces of the medium, D(t), to give structural
information for the medium (Borgin et al., 1996; Ek et al., 1994; Helmer et al., 1995;
Latour et al., 1993; Latour et al., 1994; Mitra et al., 1992; Mitra et al., 1993). At short
times (i.e. (Dot)1/2 << pore size, where Do is the free diffusion coefficient) the diffusion is
given by (Helmer et al., 1995)
D(t) 4 A
Do 9I V DOt +O(Dot) [142]
where A and V are the pore surface area and pore volume respectively. In the long time
limit
im D(t) [143]
t>oDo T
where T is the tortuosity, or connectivity of the medium. In the intermediate region a
Pade approximate is used to interpolate between the two regions (Latour et al., 1993).
This model has been demonstrated to work very well for solid systems that have
liquids or gases filling their interstitial spaces. Many of the experiments performed to fit
this model varied diffusion time instead of gradient strength. Usually the experiments
run on tissue vary gradient strength and leave the time constant. This makes the
experiments a bit difficult to compare with the tissue measurements. Latour et al. (1993)
used this model to fit data from water in the "pores" of a monosized sphere pack (spheres
a diameter of 96 jpm), and water in packed human red blood cells (1994). Helmer et al.
(1995) extol the virtues of this model in its suggestion that the curvilinear appearance of
the diffusion rate vs. b value data is due to geometrical structure within the sample rather
than the number of distinct compartments within the sample. However, they do not
attempt to fit their biological tissue data (sampling from nonnecrotic and necrotic
regions of a tumor) to this model. That suggests some difficulty in applying this model to
tissues. Tissues are not just porous, but have semipermeable membranes that allow the
passage of some substances through but not others. This implies that the cell membrane
is impermeable to all solutes except for very small, uncharged molecules. So diffusional
motion through biological tissue is not as simple as in porous media.
Inside the cell, the diffusion rate is slowed due to water molecules running into
proteins, organelles and other substances contained within the cell. The diffusion rate
outside of the cell will be slowed by the closeness of the cells to one another, similar to
the manner in which the diffusion rate was changed by moving around the packed beads.
Both rates should be modified by exchange between the cells and the surrounding
extracellular space, and the exchange rate will be effected by the permeability of the
membrane in question. The porous media theory does not take into account the
cytoskeletal structures or other substances within the cells, nor does it include exchange.
As well as this theory works for solid substances, it is not clear that it will work for
biological systems because of these aforementioned facts.
An Analytical Model of Restricted Diffusion
The second model was created by Stanisz et al. (1997 and 1998) to look at
restricted diffusion in bovine optic nerve using a three pool model. The model has two
intracellular compartments (spherical and ellipsoidal cells) and one extracellular
compartment. One of the assumptions made in this model is that the diffusion rates for
the two intracellular compartments are equal in magnitude, and the intracellular diffusion
rate is different than the extracellular diffusion rate. This model has three analytical
parts: (a) water motion in the extracellular spaces; (b) restricted diffusion inside cells;
and (c) exchange between the intracellular and extracellular compartments.
The diffusion rate in the extracellular compartment is found by
DEAPP D [144]
T
which is the same equation as was given for the longtime behavior in porousmedia
theory (Helmer et al., 1995). The tortuosity is orientation dependent. This accounts for
anisotropy in the observed data.
The intracellular diffusion rate is modeled using a onedimensional
approximation. The model is for restricted diffusion within an infinite parallel
membrane. The equation for the diffusion rate is (Stanisz et al., 1997)
1 cos(yG5 j)
1 (yGS J)2
APP 1
b + 4(,G J ) exp 2 7 2DI 2 l(l) cos ) [1451
n 2 ((G 2()2 n)2
where t is the average restricted distance the diffusing species experiences within a cell
of type J = S (spheres), T ellipsoidss).
The exchange rate, K, between the intracellular compartment and the extracellular
compartment is found by multiplying the membrane permeability, P, by the surfaceto
volume ratio A/V. This exchange rate is used by the modified Bloch equations which
were extended according to Karger et al. (1988) to be
dMT 2G22DTAPPMT KTMT +KEME [146]
dt
dt y2G22D Ms KsMs +KEME [147]
d tM
dME = 2G22DEAPPME KEME + KMS +KTMT [148]
dt
with the appropriate initial conditions. Here, E represents the extracellular compartment,
S represents the spherical intracellular compartment and T represents the ellipsoidal
intracellular compartment. Using a Monte Carlo simulation, nine parameters are fit into
the three parts of this model. These nine parameters are the intracellular diffusion rate
(DI), the long and short axes of the ellipsoidal cells (aT(L) and aT()), the diameter of the
spherical cells (as), the volumes of each cell type (VT and Vs), the membrane
permeability of each cell type (PT and Ps), and the extracellular diffusion rate (DE).
There are a few problems with this model. Stanisz et al. (1997) fit the parameters
to within only 15% using this model. The model seems to fit well at smaller b values, but
the larger the b value, the worse the fit seems to get. That means that the model does not
fit the data quite well enough. Their errors probably stem from two major sources: (a)
the longtime diffusion model used for the extracellular space and (b) the one
dimensional model used to characterize the diffusion in the intracellular space. The long
time diffusion model is the same as is used in porous media theory, and the drawbacks of
that model were explained in the previous section. As before, exchange between the
intracellular and extracellular spaces will change how well this model fits the data. The
onedimensional model used to define diffuison in the intracellular space is inaccurate
because of the infinite parallel barrier approximation that was used. This may be a
reasonable approximation for water travelling down the long axis of an ellipsoid, but it
will not fit as well for water travelling in the small spherical cells, or along one of the
minor axes of the ellipsoid. Also, this model takes into account only very regular shaping
of the cells. Cells are very irregularly shaped, and filed with structures and substances
with which water interacts. True biological systems do not have such regular shapes.
Multiexponential Model
The final model was introduced by Karger et al. (1988) and was reiterated by
Niendorf et al. (1996). Of late, this model has gained popularity, and has been used in
presentations at a few meetings (Bossart et al., 1999a and 1999b; Bui et al., 1999; Helmer
et al., 1999; Kraemer et al., 1999; Mulkern et al., 1999; Pfeuffer et al., 1999). When the
system under consideration is composed of different subregions, then the observed signal
attenuation can be given by a superposition of the contributions from individual
subregions
S(b) = fi exp( bDi) [149].
i
For tissues it can be assumed that the two compartments are an intracellular
compartment and an extracellular compartment. Exchange between the two
compartments occurs on a time scale related to the mean lifetime (Tin(ex)) of water
molecules in a compartment. In the short diffusion time limit (t << tin(ex)) the signal is
just a linear superposition of two monoexponentials (Neindorf et al., 1996)
S(b)
S(b) = fin exp( bDin)+ fex exp( bDex) [150].
At the long diffusion time limit (t >> Tin(ex)), complete exchange occurs between
the two compartments and the signal attenuation will show a monoexponential
dependence
S(b)
S(b) = exp( b(finDin +fexDex)) [151].
In the intermediate range, the signal attenuation appears as the sum of two
monoexponentials as in the short time limit
S(b)
= fin exp( bDin)+ fex exp( bDex) [152];
however, the volume fractions (f'in(ex)) and the diffusion rates (D'in(ex)) in the intermediate
time periods become mixtures of the rates found in the short time limit due to the
exchange of water between the two compartments (Karger et al., 1988). The only thing
lacking in this model is an accounting of the anisotropy. Tensor data taken by our group
and by other groups shows that there is anisotropy in some biological samples, otherwise
there would be no difference between the diffusion images along the diagonal of the
tensor. That is to say, the x, y and z images would be exactly the same and the off
diagonal elements would be zero if the tissue were isotropic. Perhaps the anisotropy
comes from the geometry of the sample, or perhaps it comes from the variety of
substances impeding the diffusion through the tissue sample.
Study Directions
The purpose of this study is to verify that the biexponential model proposed by
Karger et al. (1988) is an accurate one, to adjust the model to take into account anisotropy
in tissues, and to find the roles that compartmentalization, exchange and relaxation play
in diffusion behavior. Although the model proposed by Stanisz et al. (1997 and 1998)
retains some of the characteristics of the Karger model (e.g. assuming distinctly different
diffusion rates in the extracellular and intracellular spaces), it does not seem to fit the data
as closely as the model proposed by Karger et al. (1988). The biexponential model is a
very good twocompartment model with exchange. This model fits well to mono
directional data taken in rat spinal cord and human corpus callosum (Figures 14 and 1
5). However, as stated before, the model does not adequately explain anisotropic
diffusion in tissues. Also, given large enough diffusion gradients, three or more diffusion
coefficients have been seen. Assaf and Cohen (1998) describe observing a system that
seems to fit the sum of three exponential terms when the gradients are large enough. The
question to answer is whether those different diffusion rates are due to different
compartments or if it is due to other factors. Therefore, the following series of studies are
proposed to aid in determining if this really is a good multicompartment model, to
determine how exchange effects the apparent diffusion rate, and to determine how the
anisotropy fits into the model.
1. TI and T2 relaxation studies of fixative solutions and the effect fixative solutions have
on relaxation in biological tissues were done. These studies will aid in determining
how tissue preparation affects the ex vivo tissue results. It will also help to determine
the best tissue preparation method for comparison with in vivo studies, as well as for
doing histological comparisons.
2. T2 relaxation and diffusion spectroscopy studies were done on human white and gray
matter brain samples. Human brain samples were chosen because nearly
homogeneous tissue samples can be cored from the human brain. The data taken from
brain white and gray matter samples could compare qualitatively with the white and
gray matter in the spinal cord. Pulsed gradient spinecho spectroscopy was done
because complete data can be taken much more rapidly than with imaging. Also, T2
relaxation imaging experiments are inherently diffusion weighted due to the imaging
gradients used, making the deconvolution of diffusion and relaxation much more
difficult than with spectroscopy studies.
3. Diffusion imaging studies in normal rat spinal cord were done. These studies could
confirm or deny some of the ideas presented in the previous studies on the brain
samples, as well as give the first images of the two different diffusion compartments.
4. Comparison studies of diffusion in normal and injured rat spinal cord were done.
These studies further explore the meaning of the two diffusion compartments, as well
as give some further confirmation of ideas presented in the previous studies.
It is the hope that this process gives a more complete model of diffusion in biological
tissues. This will be useful for understanding the role of diffusion in normal tissue and
for quantifying changes due to trauma, and maybe lead to better diagnostic techniques.
In(Signal Intensity) vs b value
Rat Spinal Cord Gray Matter
5.4 Data from Region of Interest
5.2  y=1.04exp(0.000128b)
5.2  y=1.13exp(0.000612b) + 0.399exp(0.000037b)
5.0 , y=1.53exp(0.000439b)
5.0  y=0.404exp(0.000037b)
4.8 
S4.6 ',
. 4.4
S 4.2 ,
4.0 ,.
3.8 
3.6 .
0.0 3000.0 6000.0 9000.0 12000.0 15000.0
b value (s/mm2)
Figure 15. Natural log of the signal intensity vs. b value for a region of interest in the rat
spinal cord.
CHAPTER 2
FIXATION EFFECTS
Relaxation measurements have been made on both fixed and "fresh" (or unfixed)
tissues ex vivo in order to determine the effects that death and fixation have on tissue as
observed using 'H NMR (Akber, 1996; Baba et al., 1994; Bottomley, et al. 1984; Carvlin
et al., 1989; Fischer et al., 1989; GyorffyWagner et al., 1986; Kamman et al., 1985;
Moseley et al., 1984; Nagara et al., 1987; Pattany et al., 1997; Thickman et al., 1983;
Tovi & Ericsson, 1992; Willcott, 1984). Such measurements provide a greater
understanding of how in vitro and ex vivo tissue studies relate to in vivo tissue studies.
These relaxation experiments on tissue samples in vitro and ex vivo have shown that
tissue death and fixation have a very large effect on the T2 relaxation times and a less
pronounced effect on the T, relaxation time. However, these early studies did not explore
the reasons for these relaxation changes or the signaltonoise ratio (SNR) differences
observed between fixed and unfixed tissue samples.
In this study, experiments were performed to find the best tissue preparation for
imaging in vitro. The SNR measurements presented in Table 21 show that tissue
preparation has a huge effect on the SNR. Each measurement was performed on a single
sample of this type. The SNR for each tissue preparation was calculated from images
taken at 600 MHz with the following parameters: TR = 3 s, TE = 36 ms, matrix = 128 x
128, and NA = 2. The SNR was determined to be very poor for fixed samples that were
imaged in the fixative solution. It was also determined to be poor in fixed samples that
were poorly perfused (i.e. lightly fixed and containing blood). However, unfixed samples
and fixed samples that were washed with phosphate buffered saline (PBS) had an
improved SNR when imaged in PBS. It has been shown that the T2 relaxation shortens
rapidly in the first few hours after excision (Thickman et al., 1983; Nagara et al., 1987;
Carvlin et al., 1989; Fischer et al., 1989; Tovi & Ericsson, 1992; Baba et al., 1994).
Therefore, it is not possible to use unfixed samples for long experiments because the
tissue constantly degrades and measurements will be unstable over the course of an
image acquisition. To observe tissue samples ex vivo or in vitro, the best sample
preparation will be one that prevents degradation during the time of the measurements,
maintains sufficient SNR for the relevant calculations to be performed on the data, and
most closely resembles the in vivo tissue.
Table 21. Measured SNR in gray and white matter regions of the spinal cord for
various tissue preparations.
Tissue preparation GM WM
SNR SNR
unfixed rat spinal cord imaged in PBS 49.1 45.8
4% formaldehyde fixed rat spinal cord 3.7 3.0
imaged in 4% formaldehyde
4% formaldehyde fixed rat spinal cord 150.1 102.3
imaged in PBS
4% formaldehyde fixed injured rat 9.3 6.4
spinal cord imaged in 4% formaldehyde
4% formaldehyde fixed injured rat 148.2 79.6
spinal cord imaged in PBS
4% formaldehyde fixed, poorly 9.8 4.4
perfused rat spinal cord imaged in 4%
formaldehyde
4% formaldehyde fixed, poorly 105.4 63.7
perfused rat spinal cord imaged in PBS
For multiexponential T2 and multicompartment diffusion experiments, it is
important to have a high SNR (> 60). If the SNR is too low, fitting the data to nonlinear
curves is difficult and inconclusive. It is important to understand why these SNR
changes occur in order to avoid these difficulties. The relaxation rates of varying
concentrations of fixative solutions were measured in order to interpret the differences
seen between imaging tissue in fixative solution and PBS. Afterwards, the relaxation rate
for a spinal cord tissue sample fixed with 4% paraformaldehyde solution was measured
both in fixative and in PBS.
Materials and Methods
Relaxation Measurements in Solution
Relaxation times were measured for solutions of formaldehyde (CH20) and
gluteraldehyde (OCH(CH2)3CHO) at concentrations of 1, 2, 4 and 8 percent. They were
also performed on full, half and quarter Karnowsky's solution (full Karnowsky's solution
= 5% gluteraldehyde + 4% formaldehyde). All measurements were performed using a
600 MHz Varian spectrometer with the samples at physiological pH (7.2) and at a
temperature of 250 C. Ti measurements were made using an inversion recovery
sequence, and T2 measurements were made using the CarrPurcellMeibloomGill
(CPMG) sequence (see Appendix B for the pulse sequences). Each sample spectrum was
performed only once with eight averages.
Imaging Measurements in Tissue
The water relaxation measurements in tissue were acquired using an imaging
spinecho sequence. Repetition time (TR) was arrayed for the T1 measurements, and
echo time (TE) was arrayed for the T2 measurements. These water relaxation
measurements were performed on a single tissue sample with four averages. To provide
the tissue sample, a rat was transcardially exsanguinated with saline solution and heparin,
then perfused with a 4% formaldehyde solution. Water relaxation measurements were
performed on the water within the fixed cord in immersed in fixative solution. The tissue
was then washed with PBS 4 times over a 36 hour period, placed in fresh PBS, and the
water relaxation measurements were repeated.
Finally, rat spinal cord tissue was prepared by exsanguinating in vivo with saline
solution, followed by perfusion with one of three different fixative solutions prior to
removal from the animal: 4% formaldehyde (n = 3), half Karnowsky's (n = 3) and full
Karnowsky's solution (n = 3). A total of 9 diffusion experiments were performed. Each
tissue sample was taken from the same spinal cord location, at T13. Following fixation,
these cords were washed with PBS three times over a 36 hour period, and placed in fresh
PBS. A full biexponential diffusion experiment, as described in the Materials and
Methods section of Chapter 4 and Chapter 5, was then performed on these cords at 600
MHz. Numerical data was calculated at specific regions of interest in the cord to find out
if the fixative preparation changes the observed diffusion rates. After the experiments, all
tissue samples were replaced in the fixative solution.
Results
Fixative Solution Measurements
The presence of fixative in water slightly decreases its Ti relaxation time (Figure
21) and greatly reduces its T2 relaxation time (Figure 22). The Ti and T2 relaxation
rates increase linearly with increasing concentration. The graph of Ti relaxation rate vs.
concentration (Fig. 21) shows that while Ti relaxation rates change very little with
fixative concentration, there is a definite relationship. This increase is greater for
1/T, vs. Concentration
0.360 ....*
Formaldehyde s
0.350 Gluteraldehyde ,
A Karnowsky's solution ,
0.340 ,
0.330 .
0.320 .
0.310
0.300 '*' *
0.0 2.0 4.0 6.0 8.0 10.0
% Concentration
Figure 21. Graph of 1/T1 vs. concentration.
Graphs shown for formaldehyde, gluteraldehyde and Karnowsky's solution.
Karnowsky's solution than it is for the formaldehyde and gluteraldehyde solutions. The
relationship between Ti and fixative concentration may be due to an increase in the
viscosity of the solution (Andrew, 1958) as polymerizing agents such as a fixative is
added. To test this hypothesis, the viscosity of each fixative solution was measured at
250 C using a capillary viscometer. The results of the viscosity measurements are shown
on the graph of viscosity vs. fixative concentration in Figure 23. The trends seen in the
slopes of the lines on Figure 23 are the same as the trends seen in the lines on Figure 21
(i.e. Karnowsky's solution vs. concentration has the steepest slope and formaldehyde
solution vs. concentration has the shallowest slope). The large effect that fixative has on
the T2 relaxation rate of water is most likely due to chemical exchange processes going
on in the solution.
Tissue Measurements
The relaxation rates observed in both the gray matter (GM) and the white matter
(WM) of the rat spinal cord samples show that washing the sample does not significantly
change T1 (Table 22). However, washing increases the T2 by about 40% in both gray
and white matter. Since this is the case, it would appear that the short T2 measured in
fixed tissue imaged in the fixative may be due to free fixative exchanging with free water
in the tissue. Therefore, the relaxation rates of the fixed spinal cord tissue are dependent,
in part, on the solution surrounding the tissue.
Table 22. Relaxation rates in GM/WM for different tissue preparations.
GMTi WMTi GMT2 WM T2
Tissue preparation (ms) (ms) (ms) (ms)
4% formaldehyde fixed rat spinal 159 245 22 18
cord imaged in 4% formaldehyde
4% formaldehyde fixed rat spinal 178 253 32 28
cord imaged in PBS
1/T2 vs. Concentration
150.0
100.0
'()
50.0
oB
0.0
0.0 2.0
4.0 6.0 8.0
% Concentration
Figure 22. Graph of l/T2 vs. concentration.
Graphs shown for formaldehyde, gluteraldehyde and Karnowsky's solution.
Dynamic Viscosity vs. Concentration
2.0 4.0 6.0 8.0
10.0
% Concentration
Figure 23. Graph of dynamic viscosity vs. concentration.
Graphs shown for formaldehyde, gluteraldehyde and Karnowsky's solution.
10.0
A
Figures 24 and 25 show graphs of the fast and slow diffusion rates vs. slice
position for rat spinal cords fixed in three different solutions. The data on each graph
was averaged from measurements on three different spinal cords, for a total of nine rat
spinal cords used for these measurements. The regions of interest used are indicated in
Figure 41d, with the exception of the lateral funiculus. The lateral funiculus would be
located between the dorsal lateral funiculus and ventral lateral funiculus on that figure.
These graphs show that the fast and slow apparent diffusion rates observed appear
dependent on the tissue preparation. For the fast diffusion rate graphs, the diffusion rates
are similar in all parts of the cord for fixation with 4% formaldehyde. When fixed with
half Karnowsky's solution, the fast diffusion rates much more spread out from one
another. With full Karnowsky's solution, the diffusion rates are close to one another
again, but the order of fastest to slowest is different than before. Although there are
changes in the fast diffusion rate graphs, the changes are larger in the slow diffusion rate
graphs, where the split between GM and WM changes radically when different fixation
processes are used. A GM/WM split is obvious in both the 4% formaldehyde fixed cords
and the half Karnowsky's solution fixed cords, with the split being larger in the cords
fixed with half Karnowsky's solution. This split disappears when full Karnowsky's
solution is used to fix the spinal cord.
Discussion
It has been shown that changes in the relaxation rate are directly proportional to
changes in viscosity (Andrew, 1958). The changes in T1 relaxation rate appear to be
consistent with changes in viscosity for the various solutions measured, as seen in the
trends of the slopes in Figures 21 and 23. As stated in Chapter 1, the spins will be fully
a) Fixed with 4% Formaldehyde b) Fixed with Half Karnowsky's Solution
800 800
750 750
I 700 700 ,
650 650
600 , 600  ......
) 550 5 550  .
." '
50 1 2 3 4 5 6 7 50 1 2 3 4 5 6 7
Slice Position Slice Position
c) Fixed with Full Karnowsky's Solution
800
W 750
S'\ ** Dorsal Funiculus (WM)
u700 Dorsal Horn (GM)
+^ **Lateral Funiculus (WM)
S650 oVentral Funiculus (WM)
o 60 .  ..... ,Ventral Horn (GM)
C 550
500
0 1 2 3 4 5 6 7
Slice Position
Figure 24. Fast diffusion rate component vs. slice position graphs for regions of interest in fixed rat spinal cordsmeasured in PBS.
a) Fixed with 4% formaldehyde. b) Fixed with half Karnowsky's solution. c) Fixed with full Karnowsky's solution.
a) Fixed with 4% Formaldehyde
b) Fixed with Half Karnowsky's Solution
0 1 2 3 4
Slice Position
5 6 7
c) Fixed with Full Karnowsky's Solution
140
120 "
E 100
3  .
80 "..
Ca   .
0 1 2 3 4
Slice Position
1 2 3 4
Slice Position
*e Dorsal
H Dorsal
+ Lateral
o Ventral
A Ventral
Funiculus (WM)
Horn (GM)
Funiculus (WM)
Funiculus (WM)
Horn (GM)
5 6 7
Figure 25. Slow diffusion rate component vs. slice position graphs for regions of interest in fixed rat spinal cords imaged in PBS.
a) Fixed with 4% formaldehyde. b) Fixed with half Karnowsky's solution. c) Fixed with full Karnowsky's solution.
*4
S4~ .
k 
*
 i J. A.


~
5 6 7
I "i" J
relaxed at approximately 5 times the T1 relaxation time. The relaxation rate for pure
water is approximately 3 s at 600 VMHz, but the fixed tissue samples presented here have
much shorter relaxation rates of approximately 0.5 s. Therefore, full T1 relaxation of the
spins occurs with a TR of approximately 2.5 s. This shortening of T1 relaxation rate
makes it possible to measure the observed signal intensities without T1 dependence.
For the fixative solutions, the shortened T2 relaxation rate is probably due to
chemical exchange. Both formaldehyde and gluteraldehyde are molecules that
polymerize in water, crosslinking the water into a more solid matrix (Hahnenstein et al.,
1994 and 1995; Hopwood 1972; Kawahara et al., 1992; Jayakrishnan and Jameela, 1996;
Prento 1995). The chain length and reaction kinetics are dependent on the pH,
concentration and temperature of the solution. In solution, formaldehyde exists as an
equilibrium mixture of free formaldehyde, a small amount of methylene glycol and the
polymer poly[oxymethylene] glycol (Hahnenstein et al., 1994 and 1995). Each species
of this solution is highly reactive, though the new bonds formed can be easily broken and
reformed. In solution, gluteraldehyde exists as small amounts of free aldehyde, mono
and dihydrated monomeric gluteraldehyde, in equilibrium with monomeric and polymeric
cyclical hemiacetals, and various ca, j3unsaturated polymeric aldehydes (Hopwood, 1972;
Jayakrishnan and Jameela, 1996). The prominent reactive species for gluteraldehyde in
solution are the ca, j3unsaturated polymeric aldehydes. The other polymeric species
formed have a lower reactivity. In both of these solutions, the exchange of water
molecules in and out of these polymer chains, as well as the hydrogen bonds forming
between the water and the polymers, acts as a relaxation sink, as discussed in Chapter 1
(in the section called Exchange Processes). Therefore, the T2 relaxation time is
significantly reduced from that of water alone. This is similar to the observation made by
Kennan et al. (1996) in polyacrylamide gels of varying crosslink density. The observed
response of the T1 and T2 relaxation curves for the polyacrylamide gels were similar to
those seen here for various concentrations of formaldehyde and gluteraldehyde. That is,
the greater the number of polymers in solution and the stronger their crosslinking
capability, the greater the drop in T2 relaxation rate.
Formaldehyde and gluteraldehyde fix tissues by crosslinking proteins,
glycoproteins, nucleic acids and polysaccharides, thereby "fixing" their structure and
preventing them from breaking down (Helander, 1994; Hopwood, 1972). Both
formaldehyde and gluteraldehyde fix tissue rapidly. Formaldehyde fixation bonds in
tissue are reversible; they have been shown to break down over time when placed in
solutions not containing the fixative (Helander, 1994; Hopwood, 1972). Gluteraldehyde
solutions react with a wider variety of chemical substances in tissue than gluteraldehyde.
This means that tissue fixed in gluteraldehyde is more solidly bound than tissue fixed in
formaldehyde. Gluteraldehyde creates bonds that are unlikely to break down over time
when the sample is placed in solutions which do not contain the fixative. However,
because formaldehyde is a small molecule and its solution has fewer nonreactive
species, it penetrates and fixes the tissue more rapidly than gluteraldehyde solutions
(Hopwood, 1972). This slow penetration of gluteraldehyde into tissues is why
gluteraldehyde, even though it gives a much stronger fix to the tissue, is rarely used on its
own. Instead it is used with a faster fixative, like formaldehyde, in combinations such as
Karnowsky's solution. In these fixative solutions, the formaldehyde rapidly penetrates
the tissue and fixes it to prevent further degradation. Over time, the gluteraldehyde then
replaces the weaker formaldehyde bonds, creating a much more stable tissue sample.
As observed in Figures 24 and 25, when the tissue is perfused with a different
fixative, the observed diffusion rates changes, even if these tissue samples have been
washed. This is probably due to the different bonds that formaldehyde and
gluteraldehyde create in the tissue. Since formaldehyde bonds are weaker than
gluteraldehyde bonds, the tissue perfused with formaldehyde alone probably more closely
approximates the in vivo case than solutions containing higher amounts of
gluteraldehyde. The relationship between fixative and diffusion rate is probably due to
the structural changes that occur inside the cell due to fixation.
In the experiments to determine the best tissue preparation method for imaging,
the extremely shortened T2 relaxation rate resulted in reduced SNR. The excessive
shortening ofT2 observed in fixed tissue measured in fixative solution can be partially
attributed to free fixative solution in the interstitial spaces of the tissue. T2 shortening
effects of fixative solution can be counteracted, to some extent, by washing the tissue
with PBS (minimizing the amount of free fixative in the tissue) and imaging the tissue in
fresh PBS.
There seemed to be no tissue degradation or fixation reversal during relatively
short time period that the tissue sample was immersed in PBS. Even after being washed,
the T2 relaxation rate is still fairly fast compared to T2 rates seen in vivo, which are
generally twice as long. In order to avoid large amounts of signal loss, the TE must be
chosen as short as possible. Unfortunately, due to the limits of the gradients used here,
43
the TE chosen for diffusion measurements generally must be on the order of the T2
relaxation rate, making T2 a large contributor to signal loss in these measurements.
CHAPTER 3
RELAXATION AND DIFFUSION MEASUREMENTS
ON FIXED HUMAN BRAIN SAMPLES
As seen in Chapter 1, at low diffusion gradient weightings (b < 1000 s/mm2), the
diffusion appears to be a single diffusion rate or a single diffusion rate tensor (Fig. 13).
At higher gradient weightings, the data points can no longer be fitted to a single diffusion
rate or single diffusion rate tensor (Figs. 31 and 32). Instead, the data seems to be
comprised of multiple diffusion rates or a multicomponent diffusion rate tensor. Towards
the gradient limits (b z 15000 s/mm2), two unique diffusion components were seen in
brain and spinal cord tissue: a fast and a slow compartment. These compartments have
been ascribed to the extracellular and intracellular spaces, respectively (Karger et al.,
1988; Neindorf et al., 1996; van Zijl et al., 1991).
Data acquired by nonNMR methods have shown that the volume
fractions for the extracellular space and intracellular space is approximately 0.20 and
0.80, respectively (Nicholson & Sykova, 1998; Sykova, 1997). To test the hypothesis
that the fast component represents extracellular water and the slow component represents
intracellular water, this ratio was taken to be the standard for comparison. In the
biexponential fits (Figure 31 and 32), the volume fractions differ from the expected
volume fractions. This could be because the two compartments being visualized by
diffusion are not the extracellular and intracellular compartments, or that there are other
In(Signal Intensity) vs. b value
Human Corpus Callosum
4.7 ,I
peak height data
4.. y = 37.5exp(0.000589b) + 62.5exp(0.0000172b)
 y = 95.2exp(0.000232b)
4.5 y = 55.0exp(0.0000162b)
y = 81.3exp(0.0000941b)
4.4
4.2 \ .
S4.0 1  % % %,
3.9 \ ,
3.8 I
3.7 .. ^ "
0.0 5000.0 10000.0 15000.0
b value (s/mm2)
Figure 31. The natural log of the signal intensity vs. b value graph for brain white matter.
In(Signal Intensity) vs. b value
Human Cortical Gray Matter
4.64 s peak height data
4 y = 43.3exp(0.0004315b) + 56.7exp(0.0000290b)
4.4 y = 93.8exp(0.0001486b)
4.2 y = 18.2exp(0.0000282b)
 y = 39.4exp(0.000098b)
4.0 
c 3.8 
c 3.6
L) 3.4 
E 32 '
0.0 5000.0 10000.0 15000.0
b values (s/mm^2)
Figure 32. The natural log of the signal intensity vs. b value graph for brain gray matter.
factors that effect the volume fractions calculated. Some of the parameters which may
change the observed volume fractions are relaxation weighting, in particular T2 (Chapter
2), water exchange between compartments, restriction, and proton density differences
between each compartment.
To investigate the origin of these two compartments, diffusion tensor and T2
relaxation measurements were performed with NMR spectroscopic methods on small
homogeneous samples of postmortem, excised, fixed human corpus callosum (white
matter) and human cortical gray matter. These two samples were chosen for later
comparison with white matter and gray matter in the spinal cord. Spinal cord tissue itself
was not used because it is difficult to core a homogeneous sample from the spinal cord.
Diffusion tensor experiments were performed on these samples using echo times and
diffusion times similar to what would be used in Chapters 4 and 5 for the diffusion tensor
measurements. Also, a multiexponential measurement of T2 relaxation rates allowed the
relaxation to be taken out of the volume fractions found by the diffusion tensor
measurements.
Materials and Methods
Sample Preparation
The cadaver was flushed using a 10% formaldehyde solution, and then perfused
with a 4% solution of formaldehyde mixed with glycerin. The brain was then excised
and stored in 70% ethanol. Cylindrical sections approximately 3 mm in length and 4 mm
in diameter were cored from the corpus callosum (WM) and the cortical gray matter
(GM) of the brain. The samples were soaked in PBS for 6 hours, then sealed in a 5 mm
NMR tube to prevent dehydration. There was no buffer solution surrounding the tissue
samples.
NMR Measurements
'H spectra for each sample of human brain tissue were acquired on a Bruker
Avance 600 MHz spectrometer (14.1 T magnet, 54 mm bore) using a commercially
available 5 mm, threeaxis, activelyshielded gradient probe capable of generating 50
G/cm per axis. The samples were maintained at 200 C throughout the NMR
measurements.
T2 relaxation measurements
T2 relaxation measurements were obtained using the CarrPurcellMeibloomGill
sequence (CPMG; see Appendix B for the sequence), with TR = 8 s and NA = 8. To
enable calculation of more than one relaxation rate, eighteen time points were taken
between TE = 2 ms and TE = 1000 ms.
Diffusion tensor measurements
ADT spectra were obtained using the method described by Basser, et al. (1992;
1994a; 1994b; Basser, 1995; Basser & Pierpaoli, 1996). For the diffusion experiments,
diffusionweighting gradients were applied along the gradientwinding directions x, y, z,
x = y, x = z, y = z, and x = y = z. To obtain the diffusionweighted (DW) spectra of
human brain tissue, forty different gradient weightings were applied in each direction.
Gradient strengths were chosen so that the b values would be linearly spaced between 0
and 15000 s/mm2. To acquire the spectra, the acquisition parameters were TR = 3 s, TE
= 36 ms, 8 = 9 ms, A = 20 ms, and NA = 4.
PostProcessing
T2 relaxation spectra
The peak heights of each spectrum from the T2 relaxation data were fit to a
biexponential curve of the form
Sij = Soa exp( TE / T2a)+ Sob exp( TE/T2b) [31],
where TE is the echo time, T2a(b) the long (short) relaxation times, and SOa(b) is the signal
fraction for T2a(b). As with the diffusion data, this fit was performed using a gradient
expansion algorithm (IDL, Research Systems Incorporated).
Diffusion tensor spectra
The matrix of b values, bij (i, j = x, y, z) (Mattiello et al., 1994), was determined
for each DW spectrum in IDL. All gradient auto and crossterms were taken into
account with the exception of terms involving the imaging phaseencoding gradient,
which was assumed to be zero, i.e. its central value. To obtain a robust biexponential fit,
estimates of the fast and slow diffusion rates were made by separately fitting the peak
heights for the low and high ranges of b. Accordingly, the low (b = 0 to 1000 s/mm2) and
high (b = 6000 to 10000 s/mm2) diffusionweighted series were fit to the
monoexponential curve in Eq. [116] using a multivariate linear regression (Montgomery,
1976). These estimated diffusion rates and the calculated relaxation rates were used as
starting points for fitting the whole data set to the biexponential curve of the form given
in Eq. [139], and to a biexponential curve that includes the T2 relaxation weightings:
S= So(f) exp(TE/T2a )exp bijDij(f)
1 J [32]
+ So(s) exp( TE/ T2b)exp i bijDij(s)
where the subscripts f and s denote fast and slow components of diffusion, respectively,
and i, j = x, y, z. This fit was performed on the peak heights using a gradientexpansion
algorithm (IDL, Research Systems Incorporated, See Appendix A for examples of the
computer code).
Results
Diffusion Tensor Measurements
Figures 31 and 32 show the natural logarithm of signal intensity vs. b value for
gray matter (GM) and white matter (WM), respectively, with gradients applied along the
z direction. Table 31 lists the So and the diffusion rate values for a monoexponential fit
to the low diffusionweighting regime (b = 0 to 1000 s/mm2), the high diffusion
weighting regime (b = 8000 to 15000 s/mm2), the whole data set, and a biexponential fit
to the whole data set. In the high b regime, most of the signal from the fast diffusing
fraction is lost, so the diffusion rates calculated are similar to the slow component values
found by the biexponential fit. In the low b regime, however, there is a significant
contribution from both diffusion rates, so values found by monoexponential fits are
generally a factor of 2 to 3 smaller than the fast component diffusion values found by the
biexponential fit. Nevertheless, these values provide a good starting point for use in a
nonlinear curve fitting routine.
A monoexponential curve was fit to the entire b range (0 to 15000 s/mm2). This
monoexponential fit is shown as a dotdashed line on the graphs in Figures 31 and 32.
These lines clearly do not fit the data, though the fits have fairly low X2 values (0.012).
Using this fit, the mean WM diffusion value was found to be 87.8 Pm2/s, and the mean
GM diffusion value was found to be 98.4 pm2/s. The differences between the dashed and
the dotdashed line in Figures 31 and 32 give an indication of how the monoexponential
fit changes as the maximum b value increases. Fitting to a monoexponential curve gives
smaller diffusion rates for increasing b value because the slow component of diffusion
contributes an increasing amount, making the diffusion rate appear to be smaller.
Although a monoexponential curve fits the data taken for b values of approximately 0 to
1000 s/mm2, it will not properly fit data when higher b values are used. At higher b
values, the X2 may indicate a good fit even when a visual inspection of the data at a single
pixel shows that to be untrue. Therefore, a different model is needed.
Table 31. Diffusion rates found for various types of fitting routines.
Monoexponential Fit Biexponential Fit
B =0 to 1000 b = 6000 to b = 0 to 10000 GM WM
s/mm2 10000 s/mm2 s/mm2
GM WM GM WM GM WM Fast Slow Fast Slow
So 93.8 95.2 22.2 55.0 39.1 81.3 43.3 56.7 37.5 62.5
Dxx 226.7 148.9 34.9 8.1 97.9 82.4 478.9 27.6 361.7 9.1
Dw 167.7 167.2 39.4 11.9 109.2 86.9 496.5 41.0 406.6 12.5
Dzz 148.6 231.9 28.2 16.2 98.0 94.1 431.5 29.0 589.0 17.2
D, 86.6 17.0 2.0 1.4 16.4 6.7 87.2 5.6 29.6 1.7
Dxz 140.7 32.7 1.7 2.4 16.6 8.4 82.3 6.3 51.0 2.8
Dz 134.4 16.4 2.7 4.9 17.5 5.5 80.7 6.8 11.9 4.5
TR 181.0 182.7 31.7 12.1 98.4 87.8 468.9 32.5 452.4 13.0
The solid line in each graph is derived from the zerodiffusionweighting values
(Sof and Sos) with the fast and slow Dzz values from the full tensor fit to Eq. [137]. The
diffusion values and zerodiffusionweighting values found using the biexponential fit are
given in Table 31. The data points (Figures 31 and 32) follow the solid curves very
closely (in the x and y directions as well, not shown), indicating the biexponential fitting
routine gives an accurate fit to the tensor. For white matter, the fast component average
diffusivity (gTR, where gTR = (Dxx + Dyy + Dzz) / 3) is 452.4 pm2/s and the slow
component average diffusivity value is 13.0 pm2/s. For gray matter, the fast component
average diffusivity is 468.9 pm2/s and the slow component average diffusivity is 32.5
[Pm2/s. The diffusion rates found by a monoexponential fit to the entire curve are
approximately a factor of 5 smaller than the fast diffusion component, and approximately
a factor of 2.5 larger than the diffusion values found by the monoexponential fit to the
low b value regime.
The fractional contribution of fast and slow diffusion regimes to the total signal
intensity can be found by dividing each of the zerodiffusion weighting values (Sof and
Sos) by the sum of the two values. In the brain white matter, the fast component
contributes a volume fraction of approximately 0.375 and the slow component
contributes a volume fraction of approximately 0.625 to the total signal intensity. In the
brain gray matter, the fast component contributes a volume fraction of approximately
0.433 and the slow component contributes a volume fraction of approximately 0.567 to
the total signal intensity. As stated in the introduction, these differ from the expected
values of approximately 0.20 for the fast and 0.80 for the slow component. To eliminate
the contribution of T2 relaxation to these diffusion volume fractions, T2 relaxation
measurements were performed and taken into account.
T2 Relaxation Measurements
The T2 relaxation data was best fit by two relaxation rates: a long and a short
relaxation rate. The T2 relaxation rates found for both the GM and the WM samples, and
the So values for each, are found in Table 32. Both T2 relaxation rates are longer for the
GM than for the WM. Also, the difference between the two relaxation rates is larger for
the GM than for the WM.
Table 32. T2 relaxation rates and volume fractions for GM and WM samples.
Sample Soa SOb T2a T2b
GM 26.7 73.3 663.2 48.4
WM 14.8 85.2 255.3 31.9
In both GM and WM, the volume fractions for the long and the short T2 relaxation rates
are similar to the values expected to be found in the extracellular and intracellular spaces,
respectively (Sykova, 1997). This implies that the T2 relaxation values could represent
the water species present in these spaces. Assuming that the species with the long (short)
relaxation time is the same species as the fast (slow) diffusing species, the relaxation
rates can be removed from the diffusion via Eq. 32. After doing this deconvolution, the
volume fractions given in the diffusion experiment are approximately the volume
fractions seen in the T2 relaxation measurements (Table 33).
Table 33. Diffusion volume fractions with and without the T2 contribution.
Diffusion volume fraction Diffusion volume fraction
with T2 contribution without T2 contribution
Sample Sof Sos Sof Sos
GM 43.3 56.7 27.7 72.3
WM 37.5 62.5 18.3 82.7
Discussion
Single exponential diffusion rates and diffusion rate tensors have been measured
in central nervous system (CNS) tissues for a variety of human and animal subjects
(Chenevert et al., 1990; Ford et al., 1994; Guliani et al., 1997; Inglis, et al., 1997; Le
Bihan et al., 1993; Moseley et al., 1990; Pierpaoli & Basser, 1996; Pattany et al., 1997;
Pierpaoli et al., 1996; Thompson et al., 1987; van Gelderen et al., 1994). Diffusion rates
in normal GM and WM are, on average, similar in humans and animals. In most of these
studies, the maximum b value used was 1000 s/mm2 (Chenevert et al., 1990; Ford et al.,
1994; Le Bihan et al., 1993; Moseley et al., 1990; Pierpaoli & Basser, 1996; Pattany et
al., 1997; Pierpaoli et al., 1996; Thompson et al., 1987; van Gelderen et al., 1994). The
diffusion rate trace values in these studies approximate 300 Pm2/s in GM and 650 [Pm2/s
in WM. These values are both larger than the diffusion rates found by doing a
monoexponential fit to the brain data taken at b = 0 to 1000 s/mm2. The WM values are
similar to the fast diffusion rate values found by doing a biexponential fits to the entire
data set, and the GM rates are about a factor of two smaller. Because of this unexpected
result, it was necessary to look closer at the previous studies. Two major differences
stand out between this study and ones done previously: i) the diffusion times (TD) are
longer than those in this study, and ii) the echo times (TE) are longer (by a factor of two
or more) than those used in this study.
Although diffusion time may not play a significant role in the observed
differences, echo time most likely does. The T2 relaxation rate measurements done on
the brain tissue samples showed that T2 relaxation is biexponential. From the volume
fractions found here, and by observations made by others (van Dusschoten et al., 1995;
Stanisz & Henkelman, 1998), the two rates can be ascribed to the extracellular space and
the intracellular space, with the longer T2 relaxation rate being extracellular and the
shorter T2 relaxation rate being intracellular. By revisiting the assumption that the fast
diffusing species has a long T2 and the slow diffusing species have a short T2, then at
long TE the fast diffusing species will contribute more to the overall signal intensity than
the slow diffusing species. The diffusion trace rates found previously appear to support
this supposition (Chenevert et al., 1990; Ford et al., 1994; Le Bihan et al., 1993; Moseley
et al., 1990; Pierpaoli & Basser, 1996; Pattany et al., 1997; Pierpaoli et al., 1996;
Thompson et al., 1987; van Gelderen et al., 1994). The data in these experiments was
taken at echo times ranging from 70 to 110 ms. The echo times were a minimum of 2
times the short T2 relaxation rate in WM and a minimum of 1.5 times the short T2
relaxation rate in GM., So the slowly diffusing component taken at these long TE values
makes little, if any, contribution to the WM, and a slightly greater contribution to the GM
By taking the T2 relaxation time contribution out of diffusion volume fractions,
the diffusion volume fractions come close to the expected 0.20 and 0.80 for extracellular
and intracellular spaces, respectively. Since the two T2 components appear to represent
the extracellular and the intracellular spaces, and the spaces that the T2 components
represent seem to match the spaces the diffusion components represent, the diffusion
components may also represent the extracellular and intracellular compartments. Also,
because taking the T2 relaxation contribution away from the diffusion volume fractions
gives volume fractions close to the expected values, other mechanisms may have less of a
contribution to the changes in volume fractionation than the relaxation, at least for fixed
tissue. For example, exchange is very slow, if there is any exchange at all. So, for fixed
tissue, it may be possible to completely separate the extracellular spaces from the
intracellular spaces, making it possible to image intracellular and extracellular
compartments. In this case, the Sof and Sos images will be the relaxationweighted images
of the extracellular and intracellular compartments, respectively, and the fast and slow
diffusion rate tensor images will show, specifically, diffusion in the extracellular and the
intracellular spaces, respectively.
CHAPTER 4
MULTIEXPONENTIAL DIFFUSION TENSOR
IMAGING OF NORMAL RAT SPINAL CORD
The ability to clearly visualize normal tissue and tissue structure so that clear
comparisons can be made with injury and disease states is important. Although
histological sampling of tissue has given clear insight to both normal and pathological
tissues, such methods cannot give images of the tissue in vivo serially and over the long
term. Though not in the detail of histology, magnetic resonance imaging can give an
overall picture of the tissue and its structure through TI, T2, proton density, and
diffusionweighted imaging. Such experiments can be done in vivo, allowing serial
measurements to be done.
The apparent diffusion tensor (ADT) can show an overall picture of tissue
structure making it useful for imaging central nervous system tissues (Basser et al.,
1994a; Basser, 1995; Basser & Pierpaoli, 1996; Chenevert et al., 1990). As we saw in
Chapter 3, however, the diffusion values found by making a single ADT map will depend
on the experimental parameters used to take the data (i.e. TE and diffusion time). This is
due to the fact that there are distinct diffusion pools within the tissue that have different
T2 relaxation rates.
Also in Chapter 3, two unique diffusion and T2 relaxation components were found
in homogeneous samples of excised, fixed human brain tissue, perhaps explaining why a
single ADT map depends on the experimental parameters. Further, evidence was seen
that the two components within the tissue could represent the extracellular and
intracellular compartments. Since the two diffusion components were different enough to
be separable, imaging experiments were done to visualize the two components.
Materials and Methods
Sample Preparation
A SpragueDawley rat was euthanized under deep general anesthesia (sodium
pentobarbital, 100 mg/kg, i.p.), exsanguinated, and perfused transcardially with
physiological (0.9%) saline followed by 4% formaldehyde solution. A 3 cm section of
spinal cord was removed from the rat (from the midthoracic region through the lumbar
enlargement) following euthanasia and stored in 4% formaldehyde solution.
Approximately 24 hours prior to the NMR measurements the spinal cord was rinsed in a
bath of phosphate buffered saline (PBS). Three times over the 24 hours the cord was
removed from the PBS and rinsed in a fresh bath of PBS. This procedure removed as
much of the free formaldehyde as possible from the cord since, as seen in chapter 2,
formaldehyde dramatically shortens the T2 of the sample.
Diffusion Tensor Measurements and Post Processing
Diffusion tensor measurements and post processing were done almost as
described in the previous chapter. A PGSE imaging pulse sequence was used for ADT
imaging (LeBihan and Breton, 1985). For both spectroscopy and imaging, diffusion
weighting gradients were aligned along the gradientwinding directions x, y, z, x = y, x =
z, y = z, and x = y = z. DWimages were created used ten different gradient weightings
per direction, and were chosen so that the b values would be logarithmically spaced
between 0 and 10000 s/mm For imaging, acquisition parameters were TR = 3 s, TE =
36 ms, 8 = 9 ms, A = 20 ms, matrix = 128 x 128, and NA = 2. The images were fit
pixelwise using the method described in the previous chapter.
Results
The fast and slow equilibrium images, Sof and Sos, are shown in Figures 41a and
4lb, respectively. In gray matter, the volume fractions for the fast and slow components
are 0.55 and 0.45 respectively. In the white matter, the volume fractions for the fast and
slow components are 0.35 and 0.65, respectively. The sum of these two images gives the
pure relaxationweighted image (Figure 41c). The calculated image in Figure 4ic looks
like an image taken using the same parameters as the diffusion images but without active
diffusion gradients (Figure 41d). Any differences between the two images are due to the
diffusion weighting from the imaging gradients used to make the image in Fig. 41d.
h h9
Figure 41. Zero diffusionweighted images.
Calculated images a) Sof image, b) Sos image, and c) sum of the Sof and Sos images gives
the pure relaxationweighted image. d) An image taken with no diffusion weighting. The
labels a through h are described in table 41.
Figures 42a and 42b show the full diffusion tensors for the fast and slow
diffusion components, respectively, in a rat spinal cord. These were obtained using a
biexponential fit to Eq. 139. Note that Dyx = Dxy, D = Dx, and Dzy = Dyz. The image
b
Figure 42. The full biexponential diffusion tensor.
a) Fast component of the diffusion tensor. Thresholds for these images are set between 0
and 2000 s/mm2. b) Slow component of the diffusion tensor. Thresholds for these images
are set between 0 and 200 s/mm2
Dzz describes diffusional water motion along the length of the spinal cord (in and out of
the image plane). The image Dyy describes diffusional water motion left and right in the
image plane. The image Dxx describes diffusional water motion up and down in the
image plane. Table 41 contains the diffusion values for several anatomical regions of
the spinal cord. The regions chosen for Table 41 are numbered 1 to 8 and overlaid on
Figure 41d. A more complete picture of spinal cord anatomy can be found in Appendix
C.
The first noticeable difference between the fast and slow regime tensors is that the
slow regime image does not show any buffer surrounding the cord. This is expected
since free water diffusion, unlike the diffusion of water in tissue, fits a monoexponential
and does not have a slow component. The ring artifacts around the sample, noticeable in
the offdiagonal elements of the fastdiffusion tensor and in all the elements of the slow
diffusion tensor, arise from a poor fit to regions at boundaries due to partial volume
effects.
The fast diffusion tensor (42a) appears as expected from the spinal cord
structure, with the white matter being oriented along the z direction (like long, thin tubes)
and the gray matter being more isotropic. The threshold in the fast diffusion trace images
is set so that the darkest points are 0 pmm2/s and the brightest points are at 2000 Pm2/s.
Though there is some diffusion in these elements, the offdiagonal elements appear
almost uniformly dark. This is because the diffusion rates in the offdiagonal elements
are much slower than the than their corresponding diagonal elements. The uniform
threshold values chosen to show the diagonal elements to their best advantage causes the
offdiagonal elements to disappear.
Table 41. Diffusion rates for the regions of rat spinal cord shown in Figure 41d.
The biexponential fit was done to b values in the range of 0 to 10000 s/mm2. Then anatomical regions are shown in Figure 41d.
Fast Component Diffusion Rates (Pm2/s) Slow Component Diffusion Rates (Pm2/s)
Region Dx, 'Std Dyy Std Dzz Std gTR Std Dx Std Dyy Std Dzz Std gTR Std
'DF 166.1 34.4 79.9 24.8 1590.7 93.3 618.4 45.6 7.5 6.0 16.5 12.5 77.8 9.7 33.4 6.3
2VLF 131.6 35.6 125.2 26.5 1497.1 153.7 586.2 52.2 7.1 5.4 5.8 4.2 47.5 5.8 19.8 4.2
'VF 201.0 38.8 98.2 23.9 1477.0 154.1 594.2 57.2 3.7 2.8 4.1 3.5 39.5 5.0 15.2 2.5
4DLF 119.0 42.5 120.5 24.4 1505.7 121.5 582.2 46.9 9.4 13.2 8.6 4.7 55.3 5.5 23.9 3.8
'SG 181.6 80.8 115.6 47.2 632.8 322.4 309.5 147.9 5.2 8.2 8.1 12.3 23.6 35.4 12.2 18.2
'DH 493.5 53.9 268.2 32.9 1165.7 106.2 642.1 58.4 60.5 14.8 45.3 10.3 140.6 11.3 83.0 12.3
SVH 561.4 55.5 443.0 54.1 1074.2 122.9 692.4 60.7 77.5 12.4 65.7 12.2 132.5 15.4 92.0 11.1
307.3
69.1
389.1
79.0 870.2 264.3 521.7
128.8 25.5 27.6
42.9 45.2 73.9
75.4 49.7 51.8
1 Dorsal Funiculus
2 Ventral Lateral Funiculus
3 Ventral Funiculus
4 Dorsal Lateral Funiculus
5 Substantia Gelatinosa
6 Dorsal Horn
7 Ventral Horn
8 Gray Commissure
i Standard deviation over the pixels in the region of interest
GC
The slow diffusion tensor (Fig. 42b) has the threshold set such that the darkest
points are at 0 pmm2/s and the brightest points are at 200 Pm2/s, a factor of 10 less than
used in the fast diffusion tensor. The diffusion elements Dx and Dyy have a similar
contrast to the fast tensor. The Dzz element, however, has contrast opposite to that seen in
the fast tensor (i.e. the GM is brighter than the WM). Table 41 shows explicitly that in
the fast regime, the white matter Dzz is approximately 1.4 times larger than the gray
matter Dzz, whereas in the slow regime, the gray matter Dzz is approximately 2.5 times
larger than the white matter Dzz. The relative decrease of Dzz from fast to slow is much
larger for WM than for GM. The offdiagonal elements are more obvious in the slow
diffusion tensor, but the high amount of noise in these images makes in doubtful that they
are any more significant.
Color diffusion trace (cTR) images are one method to quickly see the structural
information contained within the diffusion tensor. cTR images are not independent of the
relative orientation of the gradients to the sample, so care must be taken in orienting the
spinal cord tissue samples similarly in the magnet. The cord is placed in the gradients
such that the length of the cord (the rostralcaudal extent) is aligned with the zaxis
gradients, the lateral extent of the cord is aligned with the yaxis gradients, and the
anteriorposterior extent is aligned with the xaxis gradients. Therefore, water diffusing
in the rostralcaudal extent (i.e. in and out of the imaged plane) is defined by Dzz, water
The slow diffusion tensor (Fig. 42b) has the threshold set such that the darkest
points are at 0 pmm2/s and the brightest points are at 200 Pm2/s, a factor of 10 less than
used in the fast diffusion tensor. The diffusion elements Dx and Dyy have a similar
contrast to the fast tensor. The Dzz element, however, has contrast opposite to that seen in
the fast tensor (i.e. the GM is brighter than the WM). Table 41 shows explicitly that in
the fast regime, the white matter Dzz is approximately 1.4 times larger than the gray
matter Dzz, whereas in the slow regime, the gray matter Dzz is approximately 2.5 times
larger than the white matter Dzz. The relative decrease of Dzz from fast to slow is much
larger for WM than for GM. The offdiagonal elements are more obvious in the slow
diffusion tensor due to the difference in the threshold. The signal intensity in these
images is approximately equivalent to the signal intensity seen in the diagonal elements
of the fast diffusion tensor.
Color diffusion trace (cTR) images are one method to quickly see the structural
information contained within the diffusion tensor. cTR images are not independent of the
relative orientation of the gradients to the sample, so care must be taken in orienting the
spinal cord tissue samples similarly in the magnet. The cord is placed in the gradients
such that the length of the cord (the rostralcaudal extent) is aligned with the zaxis
gradients, the lateral extent of the cord is aligned with the yaxis gradients, and the
anteriorposterior extent is aligned with the xaxis gradients. Therefore, water diffusing
in the rostralcaudal extent (i.e. in and out of the imaged plane) is defined by Dzz, water
diffusing laterally (i.e. left and right in the image plane) is defined by Dyy, and water
diffusing anteriorposterior (i.e. up and down in the image plane) is defined by Dxx.
Figures 43a and 43b show the fast and slow cTR images, respectively. Although these
images do not give different information that the full diffusion tensor, the information is
presented in a way that can be understood much more quickly. For example, in the cTR
images it is immediately obvious that the WM has a bias towards the z direction from the
red coloring present. It is also clear that the GM is much more isotropic. Variations in
color can be seen throughout the gray matter, especially in the slow diffusion cTR.
Fast Slow
Figure 43. The color diffusion tensor trace.
a) The fast component of the cTR. b) the slow component of the cTR..
Three parameters, invariant to the relative orientation of the gradient and sample
axes (PJ Basser, 1995), are available from the full tensor: i) the grayscale trace of the
tensor, where gTR = (Dxx + Dyy + Dzz)/3; ii) the fractional anisotropy index (FA) given by
I I [Dij (TR(D))Iij]2
FA = 3 1i=x,y,zj=x,y,z
SiA
1i=x,y,z j=x,y,z
where Iij is the identity matrix; and iii) the volume ratio (VR) given by
Deter min ant(D)
VR=
[TR(D) / 3]3
The fast and slow grayscale diffusion trace (gTR) images are shown in Figs. 44a and 4
4b at display scales identical to the ones used for the fast and slow diffusion tensors in
Fig. 42. The fast diffusion trace (gTRf) appears almost uniform across the spinal cord,
indicating that, on average, there is very little differentiation between GM and WM for
the fast diffusing species. In the slow diffusion trace (gTRs), the GM and WM are
completely differentiated, showing there are innate structural differences between the
GM and WM. The ratio of the diffusion rate in the gray matter to the diffusion rate in the
white matter is 1.10 and 3.80 for gTRf and gTRs, respectively.
Fast Slow
Figure 44. The grayscale diffusion tensor trace.
a) The fast component of the gTR. b) the slow component of the gTR.
The fractional anisotropy and volume ratio give measures of the anisotropy or
isotropy of the tensors, respectively. Although the FA and VR are complementary, at
times more subtle variations appear more clearly in one than the other. One example of
this is the brightened outline around the gray matter in the slow FA image appears more
clearly than in the slow VR image. Also, the variations of isotropy across the gray matter
from dorsal to ventral stand out more readily in the VR than in the FA images. The fast
and slow FA images are displayed in Figures 45a and 45b. Overall, the average FA for
the fast component (FAf) is 0.44 for the gray matter and 0.74 for the white matter, i.e. the
white matter is more anisotropic than the gray matter. The FA for the slow component
(FAs) is 0.40 for the gray matter and 0.71 for the white matter. Overall, the fast
component of diffusion is marginally more anisotropic than the slow component.
Fractional Anisotropy
Fast
Volume Ratio
Fast
Slow
Slow
Figure 45. The anisotropy/isotropy images.
a) The fast and b) the slow FA images. c) The fast and d) the slow VR images.
The fast and slow VR images are displayed in Figures 45c and 45d. These
images are complimentary to the FA images displayed in Figures 45a and 45b. On
average, the VR for the fast component (VRf) is 0.68 for the gray matter and 0.11 for the
white matter. The VR for the slow component (VRs) is 0.74 in the gray matter and 0.15
in the white matter. The VR results imply that the gray matter is more isotropic (or less
anisotropic) than the white matter, consistent with the previous result for the FA. Also,
the slow component of diffusion is marginally more isotropic than the fast component,
which is also constant with the previous FA result.
Discussion
As stated in Chapter 3, if the fast and slow diffusion rates are the extracellular and
intracellular components, the fractional contributions to the signal should be compared to
the physiological values for the extracellular and intracellular volume fractions (fex ~ 20%
and fin 80%) (Nicholson & Sykova, 1998; Sykova, 1997). Although the volume
fractions detected were not the same as the physiological values, they were much closer
to the physiological values than those observed by Neindorf, et al. (1996). They found
the relative magnitudes of the volume fractions were reversed compared to physiological
values, though these volume fractions are effected by other NMR considerations, such as
the T2 and exchange between components discussed in the previous chapter.
As shown in the previous chapter, differences between these values and the
physiological values may be largely due to the effects of multicompartment T2 (Buckley
et al., 1999; Li et al., 1998; Stanisz et al., 1998; Stanisz & Henkelman, 1998; van
Dusschoten et al., 1995). As observed with the brain tissue samples in Chapter 3, the
shorter T2 intracellular relaxation rate would mean more signal loss for that volume
fraction. Other factors could contribute to the differences observed. Exchange of water
between the compartments could also contribute to the observed differences in fractional
contribution due to mixing between compartments. Schoeninger, et al. (1994) showed
that in Aplysia neurons the nucleus has a much larger diffusion rate than the surrounding
cytoplasm. The spinal cord GM has more cell bodies in it than the WM, perhaps causing
the volume fractions to be significantly changed by the nuclei or other subcellular
organelles having a faster diffusion rate than the surrounding material. Also, Sykova
(1997) observed that white matter is approximately 18% extracellular space, whereas
gray matter is approximately 22% extracellular space which could also explain part of the
difference between the fractional contributions.
The fast component of the diffusion tensor appears as expected from what is
known about spinal cord structure. That is to say, the axons in white matter are oriented
predominantly along z whereas gray matter parenchyma is more isotropic. The slow
component of the diffusion tensor has some unexpected features. The slow Dzz
component exhibits a very slow diffusion rate for WM compared to GM. This implies
that the apparent barriers to diffusion are further apart for the gray matter, or there are
perhaps other restrictions to free diffusion, e.g. macromolecular interactions, giving rise
to the differences between the gray and white matter at this level. Structures such as
myelin, microtubules or neurofilaments, which are much more prevalent in white matter,
could account for the slower white matter diffusion rates.
So the differences between the fast and the slow component of the diffusion
tensor could be largely due to the difference in the type of restriction seen by intracellular
and extracellular water. The fast and slow diffusion coefficients appear to be the average
components of the extracellular (fast diffusing) water and the intracellular (slow
diffusing) water, respectively (Norris & Neindorf, 1995; Neindorf et al., 1996; van Zijl et
al., 1991). The homogeneity of gTRf indicates that the average diffusivity of the
extracellular water fraction is similar in WM and GM, in spite of very different diffusion
anisotropies. gTRs shows that the intracellular water fraction is much more restricted for
WM than for GM, although the anisotropies are very similar to the fast diffusion regime.
These results seem due to similarities in external architecture (at least for tissue fixed
with 4% formaldehyde) for cells in each tissue type, and differences in internal cellular
architecture in each tissue type. It must be kept in mind, however, that the data was taken
at fairly long diffusion times (> 10 ms) and the two components likely cannot be
completely separated into intracellular and extracellular water due to exchange between
the compartments (Karger et al., 1988; Norris & Neindorf, 1995; Neindorf et al., 1996;
van Zijl et al., 1991).
CHAPTER 5
MULTIEXPONENTIAL DIFFUSION TENSOR IMAGING
OF NORMAL AND 1MONTH POST INJURY
RAT SPINAL CORDS
In all stages of trauma and disease in the brain and spinal cord, it is important to
know the current amount of the physical damage, how far the damage will extend, and
how the structural changes relate to the final amount of functionality. Though it is fairly
straightforward to measure the extent of internal damage ex vivo through histological
sectioning, assessment of internal physical damage in vivo has been difficult. The
innovation of magnetic resonance (MR) imaging has been an important step towards
quantifying structural changes in living systems. One MR contrast mechanism that has
proved useful for the study of nervous tissue is water translational selfdiffusion
(Chenevert et al., 1990; Ford et al., 1994; Gulani et al., 1997; Inglis et al., 1997; Kirsch et
al., 1991; Le Bihan et al., 1993; Moseley et al., 1990; Ono et al., 1995; Pattany et al.,
1997; Pierpaoli & Basser, 1996; Thompson et al., 1987; van Gelderen et al., 1994). MR
images of water diffusion have been used extensively to aid in the elucidation of
structure. In cases of trauma or ischemic stroke, diffusion studies have been used to
demonstrate the magnitude of the damage to the tissue (Becerra et al., 1995; Ford et al.,
1994; Kirsch et al., 1991; Moseley et al., 1990; van Gelderen et al., 1994).
Water diffusion is usually considered to occur as a simple, single diffusion rate
(or single diffusion rate tensor) process (Stejkal & Tanner 1965; Basser et al. 1994a &
1994b). Only recently, with the availability of much stronger gradients, has it been
discovered that water diffusion in tissue is a multiple diffusion rate processes (Assaf &
Cohan 1998; Karger et al. 1988; Neindorf et al. 1996; Stanisz et al. 1997; van Zijl et al.
1991). To the limits of the gradients available on our instumentation, two unique
diffusion regimes have been measured for fixed CNS tissue: a fast diffusing component
and a slow diffusing component. It has been speculated that the fast and slow diffusing
components in tissue represent diffusion in the extracellular and intracellular spaces,
respectively (Karger et al., 1988; Neindorf et al., 1996; van Zijl et al., 1991).
Using the multiple component diffusion tensor measurement protocol seen in
Chapter 4, we examined normal and injured rat spinal cord ex vivo. The aim of these
studies was to provide a basic understanding of the information contained in the fast and
slow (i.e. proposed extracellular and intracellular) diffusion compartments. The
information found in these ex vivo measurements will provide a background for future in
vivo experiments, allowing a better understanding of the information provided by the fast
and slow diffusion components.
Materials and Methods
Sample Preparation
Five adult female Sprague Dawley rats were used for this study. Under general
anesthesia, three rats were given a moderate contusion injury to the low thoracic region
(T13) of the spinal cord. These rats were allowed to recover for one month before the
spinal cord was excised.
After the one month recovery period, all rats (normal and injured) were
euthanized under deep general anesthesia by exsanguination and transcardial perfusion
with physiological saline solution followed by a 4% paraformaldehyde solution, the same
procedure that was done in Chapter 4. A 3 cm section of spinal cord, centering on the
region of injury, was removed from each rat and stored in a 4% paraformaldehyde
solution. Approximately 24 hours prior to NMR measurement, the spinal cords were
placed in a bath of phosphate buffered saline (PBS). These cords were placed in fresh
baths of PBS three more times before being inserted into the magnet. This procedure was
done to remove as much free paraformaldehyde as possible from the cord since free
paraformaldehyde in the solution surrounding the tissue will dramatically shorten the T2
relaxation rate of the sample as seen in Chapter 2.
NMR Experiments
Magnetic resonance microscopy and diffusion measurements were performed on
all rat spinal cords using a Varian Unity 600 spectrometer (14.1 T magnet, 51 mm bore).
The gradient probe used was a 5 mm radiofrequency coil, threeaxis activelyshielded
Varian gradient probe capable of 300 mT m1 per axis. Sample temperature was
maintained at 200 C through the entire measurement protocol. The microimaging
measurements were performed using a pulsed gradient spin echo (PGSE) sequence.
Images of the spinal cords were taken in contiguous 250 jpm thick transverse slices.
Other acquisition parameters included TR = 8 s, TE = 14 ms, and a matrix of 256 x 256.
Overall measurement time was 1.5 hours.
The ADT imaging procedure used for the spinal cord was adapted from that
described by Basser, et al. (1994a & 1994b), the same procedure that was utilized in
Chapter 4. Diffusionweighted (DW) images were acquired using a PGSE imaging
sequence with diffusion gradients aligned along the directions x, y, z, x = y, x = z, y = z
and x = y = z. Ten gradient weightings were applied in each direction to give b values
that were approximately logarithmically spaced between 0 and 10000 s/mm2. Other
acquisition parameters included TR = 3 s, TE = 36 ms, 8 = 9 ms, A = 20 ms, matrix = 64
x 64 and NA = 4. The inplane resolution was 78 jam x 78 jam and the slice thickness
was 1.0 mm. The overall measurement time was 16 hours.
For each cord, three sets of diffusion and microimaging experiments were done.
One set (6 diffusion slices) was at the epicenter of injury. One set was rostral to the
epicenter (with slice 6 of this set being the same as slice 1 of the epicenter set). The third
set was caudal to the epicenter of injury (with slice 1 of this set being the same as slice 6
of the epicenter set). This procedure allowed for 16 diffusion slices to be made through
the spinal cord, allowing the visualization of damage to the cord as much as 78 mm
rostrally and caudally.
Results
Figures 51 to 53 show diffusion trace images for both normal and 1month post
injury rat spinal cords. Selfdiffusion in the spinal cord is given in the coordinate system
of the gradient axes. The gTR images (b, d, f and h in Figs. 51, 52 and 53) give an
average diffusion rate across the cord that is independent of the cord orientation. As in
Chapter 4, the cTR images (a, c, e and g in Figs. 51, 52 and 53), in which red indicates
Dzz, green indicates Dyy, and blue indicates Dxx are not independent of cord orientation
inside the gradients. Although the cTR is not independent of cord orientation, all the
tissue samples were equivalently aligned within the gradients so that the cTR images
from different cords could be compared. The cTR images show some structural
differences more readily than the gTR images.
Normal Injured
Fast
Color Diffusion Trace
Grayscale Diffusion Trace
L 4 1 4I P
Slow
Color Diffusion Trace
Grayscale Diffusion Trace
Figure 51. Diffusion trace images 7 mm rostral to the epicenter of injury.
Fast component of the a) cTR and b) gTR; slow component of the c) cTR and d) gTR for
normal rat spinal cord. Fast component of the e) cTR and f) gTR; slow component of the
g) cTR and h) gTR for a rat spinal cord 1month post injury.
Normal Injured
Fast
Color Diffusion Trace
Grayscale Diffusion Trace
lrM&I Pri
I
U
I
S
uI
I
Slow
Color Diffusion Trace
Grayscale Diffusion Trace
Grayscale Diffusion Trace
%W; 
Figure 52. Diffusion trace images at the epicenter of injury.
Fast component of the a) cTR and b) gTR; slow component of the c) cTR and d) gTR for
normal rat spinal cord. Fast component of the e) cTR and f) gTR; slow component of the
g) cTR and h) gTR for a rat spinal cord 1month post injury.
Normal Injured
Fast
Color Diffusion Trace
Grayscale Diffusion Trace
Slow
Color Diffusion Trace
Grayscale Diffusion Trace
Figure 53. Diffusion trace images 7 mm caudal to the epicenter of injury.
Fast component of the a) cTR and b) gTR; slow component of the c) cTR and d) gTR for
normal rat spinal cord. Fast component of the e) cTR and f) gTR; slow component of the
g) cTR and h) gTR for a rat spinal cord 1month post injury.
Figures 54 to 56 give comparative views of the anisotropy for each of the
images shown in Figures 51 to 53. The fractional anisotropy (Basser, 1995; FA; a, b, e
and f in Figs. 54, 55, and 56) is a direct view of the anisotropy of the diffusion images
where the white represents highly anisotropic regions and black represents completely
isotropic regions. The volume ratio (Basser, 1995; VR; c, d, g, h in Figs. 54, 55, and 5
6) is a compliment to the FA. For the VR, white represents complete isotropy, and black
represents complete anisotropy.
Normal Rat Spinal Cord
In order to discern what is occurring in the injury case, the normal images at
various positions within the spinal cord should be interpreted first. Figures 5la, c; 52a,
c; and 53a, c (the lefthand column in each figure) are representations of the cTR for the
normal spinal cord. The dark red/black region immediately surrounding the gray matter
in the slow cTR images (Figs. 51c, 52c and 53c) is a feature that is not apparent in the
fast cTR images (Figs. 5la, 52a and 53a). This region of the cord, known as the
fasciculus proprius (or propriospinal) system, consists of short, thin ascending and
descending fibers. Shorter path lengths are available for water in this region as compared
to water in other white matter regions, as the dark red/black color indicates. This means
that the average diffusion rate is slower in this region than in other white matter regions,
a feature that is visible in the grayscale trace (gTR) images (Figs. 51d, 52d, and 53d).
This shortened path along the length of the cord, just outside the gray matter, is not seen
in the fast (or extracellular) diffusing component, indicating that the extracellular
structure is similar to other regions of white matter. This is one piece of evidence that
Normal
Fractional Anisotropy
Fast
Injured
Slow
Volume Ratio
Fast
Slow
Figure 54. Anisotropy/isotropy images 7 mm rostral to the epicenter of injury.
FA a) fast and b) slow; and VR c) fast and d) slow for normal rat spinal cord. FA e) fast
and f) slow; and VR g) fast and h) slow for rat spinal cord 1month post injury.
Normal
Fractional Anisotropy
Fast
Slow
~W)
Sw 
Volume Ratio
Fast
pF.,
Slow
Slow
Figure 55. Anisotropy/isotropy images at the epicenter of injury.
FA a) fast and b) slow; and VR c) fast and d) slow for normal rat spinal cord. FA e) fast
and f) slow; and VR g) fast and h) slow for rat spinal cord 1month post injury.
Injured
i
Normal
Fractional Anisotropy
Fast
Slow
Volume Ratio
Fast
Slow
Figure 56. Anisotropy/isotropy images 7 mm caudal to the epicenter of injury.
FA a) fast and b) slow; and VR c) fast and d) slow for normal rat spinal cord. FA e) fast
and f) slow; and VR g) fast and h) slow for rat spinal cord 1month post injury.
Injured
seems to confirm the proposition that the two components of diffusion represent the
extracellular (fast) and intracellular (slow) compartments.
For the fast component, the graph of the diffusion trace values vs. slice position
(Fig. 57a) shows that the average diffusion rate in the gray matter regions is slightly
higher than the average diffusion rate in the white matter regions. This information leads
to the conclusion that, on average, if the fast component corresponds to water diffusion in
the extracellular space, the rate of diffusion is similar for both gray and white matter.
This would be expected since the extracellular matrix should be similar. For the slow
component, the graph of the diffusion trace values vs. slice position (Fig. 57b) shows
that the average diffusion rate in the gray matter regions is approximately two times
faster than the average diffusion rate in the white matter regions. This difference may be
reflective of the difference in internal structure of the gray and white matter, where gray
matter is composed of mostly cell bodies and white matter is composed mostly of axons.
In the fast cTR images (Figs. 5la, 52a and 53a), the majority of the gray matter
regions appear to be varying shades of blue/green to yellow/gray, reflecting the more
isotropic nature of the extracellular spaces in the gray matter. Variations in the color
appear to coincide with the variation in the differing cell types occurring in each gray
matter lamina. These variations are enhanced in the slow cTR images (Figs. 51c, 52c
and 53c) where the gray matter appears in more distinct shades of green, blue and red,
reflecting the orientation of cell bodies and fibers, as well as different internal cell
structure, for each lamina.
From the FA and VR images (a, b, c and d in Figs. 54, 55, and 56), as well as
the graphical representations of each (a and b in Figs. 58 and 59), gray matter is much
a. Fast
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Diffusion Trace vs Slice position
Normal Rat Spinal Cord
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low Diffusion Trace vs Slice Position
Normal Rat Spinal Cord
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c. Fast Diffusion Trace vs Slice Position
.. Rat Spinal Cord One Month Post Injury
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d. Slow Diffusion Trace vs Slice Position
Rat Spinal Cord One Month Post Injury
1 2 3 4 5 6 7 8 9 1011 1213141516
< Rostral Caudal >
** Dorsal Funiculus
.. Lateral Funiculus
,, Ventral Funiculus
*. Dorsal Horn
A Ventral Horn
Figure 57. Graphs showing the fast and slow diffusion rate trace vs. slice position for normal and injured rat spinal cord.
 *
Fractional Anisotropy vs Slice Position
Normal Rat Spinal Cord
2 3 4 5 6 7 8 9 10111213141516
< Rostral Caudal >
Fractional Anisotropy vs Slice Position
Normal Rat Spinal Cord
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c. Fast Fractional Anisotropy vs Slice Position
1.0 Rat Spinal Cord One Month Post Injury
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d. Slow Fractional Anisotropy vs Slice Position
Rat Spinal Cord One Month Post Injury
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* Dorsal Funiculus
H Dorsal Horn
** Lateral Funiculus
> Ventral Funiculus
A. Ventral Horn
Figure 58. Graphs showing the fast and slow fractional anisotropy vs. slice position for normal and injured rat spinal cord.
a. Fast
1.0.
2 3 4 5 6 7 8 9 1011 1213141516
< Rostral Caudal >
a. Fast Volume Ratio vs Slice Position
Normal Rat Spinal Cord
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b. Slow Volume Ratio vs Slice Position
Normal Rat Spinal Cord
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7
c. Fast Volume Ratio vs Slice Position
1 2 3 4 5 6 7 8 9 10111213141516
< Rostral Caudal >
d. Slow Volume Ratio vs Slice Position
, Rat Spinal Cord One Month Post Injury
** Dorsal Funiculus
u Dorsal Horn
* Lateral Funiculus
> Ventral Funiculus
Ak Ventral Horn
1 2 3 4 5 6 7 8 9 10111213141516
< Rostral Caudal >
< Rostral Caudal >
Figure 59. Graphs showing the fast and slow volume ratios vs. slice position for normal and injured rat spinal cord.
more isotropic than white matter in all regions of the cord for both the fast and the slow
diffusing components. Rostral to caudal in the sixteen slices of the rat spinal cord, the
fast diffusing species in the gray matter becomes slightly more isotropic in all regions.
Rostral to caudal in the sixteen slices of the rat spinal cord, the slow diffusing species in
the dorsal gray matter becomes slightly more isotropic down the length of the cord, but
ventral gray matter becomes slightly more anisotropic. These variations probably reflect
the changes in structure of the gray matter in various parts of the spinal cord. However,
the white matter has almost no variation in its diffusion isotropy across these sixteen
slices. This is seen in the graphs of FA and VR, as well as in the images in the three
regions: for FA the white matter always appears almost white, and in the VR it appears
the same shade of dark gray. Moving rostral to caudal, the major differences in white
matter are the relative proportions of ascending to descending fibers, but there are few, if
any, differences between the structure of ascending and descending fibers beyond the
location of the main cell body. Therefore, it seems reasonable that the diffusion isotropy
in white matter would not change significantly, but that the diffusion isotropy in gray
matter might change along the length of a normal rat spinal cord.
Rat Spinal Cord 1Month Post Injury
With a basic understanding of the structures and structural differences seen with
diffusion tensor imaging of the normal cord, the injured cord data can be interpreted.
First, the fasciculus proprius may no longer be easily discerned in the cTR and gTR
images of the slowly diffusing component (Figs. 51g, 51h, 52g, 52h, 53g and 53h).
It has become less differentiated from the surrounding white matter because the
surrounding white matter has, in general, become more isotropic and exhibits slower
diffusion rates. There should be some amount of disruptions inside the axons due to
degeneration and apoptosis, where internal cellular structures are breaking apart; or cell
survival strategies, where cells produce more organelles in an effort to survive. These
changes would cause slower diffusion rates to occur in the long white matter axons,
making them indistinguishable from the propriospinal regions of the cord.
Epicenter of injury
At the epicenter of injury, it is clear that there are major changes (Figs. 52eh and
55eh). A fluid filled cavity has appeared in the cord. Diffusion in the remaining white
matter tissue is much more isotropic, particularly in the dorsal and lateral sections of the
cord. This is seen in the FA (Figs. 55e, f) and VR images (Figs. 55g, h) as well as in
the FA or VR vs. slice position graphs (Figs 58 and 59). In the most ventral portion of
the cord, the fast component appears to be more anisotropic, and thus the tissue appears
more structured than the rest of the cord. This is indicative of spared white matter in this
region. The slow component of diffusion in that same region appears more isotropic,
which is more consistent with the rest of the cord. Gray matter regions are, on average,
slightly more anisotropic than normal for the fast diffusing species and slightly more
isotropic than normal for the slow diffusing species. Slices 58 and 59 (epicenter of the
injury) in the graphs of the diffusion trace vs. slice position (Fig. 57) show that the fast
diffusion rate is faster than in normal tissue, approaching that of free water, for all
regions of the cord. The slow diffusion rate, however, is slower than in normal tissue for
the gray matter regions, but appears almost normal for the white matter regions. The
quicker than normal rate of diffusion for the fast compartment could indicate a rising
amount of free space due to surrounding cellular damage and death, dissolution of the
