High Angular Resolutioo Diffusion MR for the Determination of Fibre Structure

Elisabeth von dem Hagen

A thesis submitted in conformity with the requirements for the Degree of Master of Science

Graduate Department of Medical Biophysics University of Toronto

O Copyright Elisabeth von dem Hagen 2001

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High Angular Resolution Diffusion MR for the Determination of Fibre Structure

Elisabeth von dem Hagen Master of Science, 2001

Department of Medical Biophy sics University of Toronto

Abstract

Diffision MR has recently been developed as a technique for tracking neuronal pathways

in the brain. Current methods, however, are unable to resolve fibre orientation in voxels

with multiple fibre tract directions. In this thesis, a physical mode1 of restricted difision

analogous to that in nerve fibres is presented. Difision measmments at high angular

resolution in sarnples with different fibre orientations are compared with theoretical

calculations for restricted diffision in a cylindrical geornetry. Orientational diffusion

measurements are shown to reflect fibre geometry and theoretical predictions to within

10%. Theoretical calculations of signal decay for a single fibre cm be used to predict the

angular dependence of diffision coefficients for any distribution of fibre orientations, and

simulations of fibre crossings at high gradient strengths refiect the presence of the

crossed fibres. Theoretical predictions rnay therefore allow experimental measurements

of similar in vivo fibre geometry to be resolved.

Acknowledgements

1 would like to express my gratitude and thanks to everyone who has helped me

throughout the past two years. At Sunnybrook, 1 want to thank Greg Stanisz for being an

indispensable source of 'diffbion' and SMIS information. Thanks also to Carneron

Chiarot for providing me with the EM images of the plastic tubing. 1 am especially

grateful to al1 those who took the time to read through this thesis, and provided me with

helpful comments and suggestions. 1 would like tu thank the membea of my supervisory

cornmittee, Simon Graham and Peter Burns, for their help and support. i would especially

like to thank my supervisor Mark Henkelman for his guidance and for teaching me to

think like a scientist. The past two years would not have been the same if it Iiadn't been

for al1 those who dragged me out and made me laugh: Mark, Michelle, Warren, Gal,

Nick, Nom, Bart, my roommates and many others. Thanks to Josette, my training

partner, for several triathlons, a marathon, and countless conversations. 1 also had the

pleasure of being a part of the U of T Nordic ski team for which 1 am very grateful.

Finally, and most importantly, 1 would like to thank my fmily for their continued

support.

iii

Contents . . Abstract ............................................................................................. 11 . . Acknowledgements ............................................................................ 111

Contents ............................................................................................ iv .................................................................................... List of Figures v

........................................................................... List of Abbreviations vi

Chapter 1 ........................................................................................................................ 1 Introduction

1 . 1 Motivation ............................................................................................................... 1 1.2 Outline ..................................................................................................................... 2 1.3 Diffision in MR ...................................................................................................... 3

.......................................................................... 1.3.1 Measurement o f Difision 4 1.3.2 Restriction ....................................................................................................... 7 1.3.3 Anisotropy .................................................................................................. 10

.................................................................................................. 1.3.4 Applications 1 1 1.4 Thesis Staternent ................................................................................................... 17

Chapter 2 .................................................................................................................... 18 High Angular Resolution Diffusion MR for the Determination of Fibre Structure

Submitted as a paper to Magnetic Resonance in Medicine Sept 2001 2.1 Introduction ........................................................................................................ 18 2.2 Experimental Methods ......................................................................................... 20 2.3 Theoretical Methods ........................................................................................... 24 2.4 Results and Discussion ......................................................................................... 26 2.5 Conclusions ........................................................................................................... 36

Chapter 3 .................................... .... ......................................................................... 38 Future Work

........................................................................................................... 3.1 Introduction 38 3.2 Orientational Diffision Measuements in White Matter ...................................... 39

3.2.1 A Modei of White Matter .......................................................................... 40 ........................................................................... 3.2.2 Determining fibre structure 45

3.3 Other MR techniques for the rneasurement of difision ...................................... 47 ............................................................................. 3.3.1 Weighting of Eigenvalues 47

......................................................................... 3.3.2 Diffision Spectrum Imaging 48 ........................................................................... 3.3.3 Multiple Diffision Tenson 49

3.4 Conclusions ........................................................................................................... 50

List of Figures

Fig . 1 : Stejskal-Tanner pulse sequence ................................................. 4

............................................................. Fig . 2: Diffusion decay curve 7

Fig . 3: Diffision tensor and ellipse ...................................................... 1 5

Fig . 4: Electron micrograph ............................................................... 21

Fig . 5: Fibre sample orientations .......................................................... 22

Fig . 6: PFG MSE pulse sequence ........................................................ 23

Fig . 7: Coiled fibre approximation ...................................................... 26

Fig . 8: Decay curves for X. Y. Z directions .......................................... 27

Fig . 9: Experimental and theoretical ADC contours ............................... 29

................................. Fig . 10: Theoretical decay curves to hi& b values 31

Fig . 1 1: Simulated ADC contours for crossed fibres .............................. 33

Fig . 12: In vivo nerve fibre mode1 ...................................................... 44

Fig . 13: Crossing and kissing fibres .................................................... 46

List of Abbreviations

ADC

DT-MRI

F m

MR

MRI

PFG MSE

RF

SNR

Apparent Diffision Coefficient

Diffusion-Tensor Magnetic Resonance Irnaging

Full Width at Half Maximum

Magnetic Resonance

Magnetic Resonance Imaging

Pulsed Field Gradient Multi Spin Echo

Radiofiequency

Signal-to-noise Ratio

Chaptea 1

Introduction

1.1 Motivation

Neuronal connections mediate complex behavioural and cognitive processes by

facilitating signalling between different areas of the brain. The neural networks involved

in these higher order cerebral functions are poorly understood, due in part to the difficulty

in elucidating the elaborate structures underlying integrated nervous system activity.

Determining nerve fibre tract connectivity wouid enable the establishment of a neural

network map to link regions of the brain. The pathways or fibre tracts associated with

specific functions could be used to correlate functional regions, which may provide

insight into human behavioural processes. In addition, the detection of changes in the

organisation and structure of nerve axons or fibres rnay be early indicators of pathologie

degradation of the nervous system.

Until recently, establishing fibre connectivity in humans has been bas-d on invasive

tracer studies in animals and subsequent human-primate conelations (1 -3). These

methods provide limited information on complcx human cognitive functions or disorders

associated with higher level cerebral processes, since these cannot always be inferred

fiom animal studies.

The sensitivity of magnetic resonance (MR) to molecular diffusion, specifically of water,

has been the focus of much research since it was established as an imaging signal contrast

method in the early 1980s with numerous clinical applications (4-6). In brain white

matter, water preferentially difises along nerve fibres. This preferred direction of

molecular motion can be detected by MRI and used to establish fibre tract orientation and

connectivity non-invasively (7-10). This technique, however, is still at the pre-clinical

stage since several difficulties must be resolved before accurate and continuous tracking

is possible.

Multiple fibre bundle orientations within a single voxel are dificult to resolve with

current techniques. While several new methods have been proposed to address this

problem (1 1 -14), in some cases their usefulness in establishing fibre orientation has yet to

be determined. This thesis will adàress one such technique, high angutar resolution

diffision measurements, in an in vitro mode1 malogous to nerve fibres, in order to

determine its adequacy in i n f ' n g fibre orientation.

The background for both the principles behind diffision MRI and its applications are

discussed in the rest ofchapter 1. At the end of the chapter, the specific aims of the thesis

2

are revealed. Chapter 2 describes the development and application of the experimental

and theoretical models to examine diffusion in single voxels with different fibre

orientations. Finally, in Chapter 3 the implications of the thesis for possible fuRue work

are discussed.

1.3 Diffusion in MR

In MR, the measured signal originates from protons of hydrogen nuclei. When placed in a

magnetic field, the magnetic moment vectors of the protons, or spins, tend to align in the

direction of the magnetic field, and resonate at the Larmor frequency au, which is

proportional to the magnetic tield Bo:

0 0 = yB0 [il

where y is the gymmagnetic ratio equal to 42.58 M H f l . By applying a radiofrequency

(RF) pulse tuned to the Larmor frequency a,, the spins can be excited out of equilibriurn

into the transverse plane, i.e. orthogonal to the main magnetic field Bo. The rotating

magnetisation vectors of the spins induce an electromotive force in a receiver coil, which

constitutes the MR signal.

Water molecules, which account for most of the hydrogen in the body and are therefore

the basis of the MR signal, are not stationaiy and exhibit random Brownian motion. As a

result, spins may move during the time of the experiment. In the presence of magnetic

field inhomogeneities, spins will move and precess at different fiequencies depending on

the gradient strength at their location. This loss of phase or coherence of spins results in a

loss in detected signal. Although undesirable in most MR applications, it is this change in

signal due to molecular diffision which forms the basis for MR diffision studies.

1 Al Measurement of Diffusion

Although the measurement of diffision using spin echoes was first described by Hahn

(1 5) in 1950, it was Stejskal and Tanner's seminal experiments in 1965 which enabled

very precise measurements of difision through the use of pulsed tield gradients (16).

Their expenmental pulse sequence is depicted in Fig. 1 and is based on the simple spin

echo sequence. In spin echo, a RF pulse tips the magnetisation by 90' (n/2 radians), such

that al1 the magnetisation lies in the transverse plane. The application of a second n RF

pulse reverses the loss of phase experienced by spins precessing at different frequencies.

By applying time-dependent gradients, or magnetic fields of a certain duration that

change strength linearly with position, the spin echo sequence can be used to mesure

diffision by deliberately inducing phase shifts in the spin population as follows.

Fig. 1. StejskaCTanner Pulse Sequence with gradient strength G, gradient duration 6. and diffusion time A. The applied RF pulses are denoted by the

their tip angles in radians.

When a gradient G is applied along a certain axis, e.g. the z-axis, the spins along that axis

are given a phase shift that depends on their position:

4, = y 1: Gzi dt = y06zi [2]

where zi is the spin position, G is the gradient strength, 6 its duration, and y is the

gyromagnetic ratio. Following the n RF pulse, the phase shift due to the first gradient

pulse 4l will become -41. AAer a tirne A fiom the onset of the first gradient pulse, the

application of a second gradient pulse will impart an additional phase shift to the spins:

+2 = y / ~ A + S Oz2 dt = yGGzt [3]

where z2 is the spin position during the second gradient pulse, and G, 6, its strength and

duration. The net phase shifi following the application of the second gradient pulse

becomes:

A+ = $2 - 41 = yGw1-z~) [4 1

if the spins have remained stationary, zl=q and the net phase shift is zero. However, if

the spins have moved, the phase shift will be non-zero and will cause a loss in measured

signal. In MR, the detected signal is proportional to the magnetisation. The attenuated

signal due to diffusion, SB,, is then simply the surn of the individual magnetisations of

the moving spins (Zexp (iyG6(zl - zr))). This surn can be evaluated if the probability

distribution P(z11z2, A) of a spin beginning at 21 and moving to z2 in a time A is known.

The attenuated signal equation becomes:

Siso = p(z) P(zi 1 22, A) exp (iyG6(z1 - 22)) dzi dz2 F I

where p(z) is the spin density, and P(zI Jz2, A) is the spin displacement probability .

For free diffusion in one dimension, the probability that a spin will move fiom position zi

to 22 in a time A is Gauuian (17):

P(zi 1 22, A) = I I(~IIDA)'~ exp (-(zi - z ~ ) ~ / ( ~ D A ) ) [61

where D is the diffision coefficient. Integmtion of the signal equation by substituting Eq.

[6] into Eq. [5] yields:

SIS, = exp ( - ( y ~ ~ ) 2 AD) [7]

or altematively:

ln (S/S,) = - (?~6)~ AD 181

which relates difisional signal loss to the difision coefficient. Equation [8] assumes

that 6 is short with respect to A. However, this assumption is rarely mie in MR difision

experiments, where gradient strength limitations result in the application of longer

gradient durations to produce observable diffusion effects. In these cases, Eq. [8]

becomes, to a first approximation in 6, (1 7):

ln (SB,) = - ( y ~ ~ ) 2 (A-613)D

This equation can be rewritten as:

In (SB,) = - b~ El01

where b = ( @ ~ ) ~ (A-W3). In order to determine experimentally the diffision coefficient,

the signal attenuation is measured at different b values. The b value is changed by

increasing either the gradient strength, duration or diffision time. The logarithm of the

measured signal attenuation is plotted as a function of b, and the dope of this line is the

diffision coeficient (see Fig. 2 for example).

Fig. 2. Diffusion decay curve or 'b-plot' where b=(y~6)2 (6513). The dope of the line is the diffusion coefficient D.

For fieely diffising spins, the measurement of the difision coefticient is stniightforward

since the slope is constant, Le. the relationship between the signal attenuation and the

diffision coefficient is monoexponential. In tissues, however, the translational motion of

spins may be restricted due to cellular structures, which affect the signal decay curve in a

number of ways.

1.3.2 Restriction

Diffision is restricted when boundaries obstnict the movement of molecules. In tissues,

the presence of cellular structures, such as cell membranes, create barriers to the diffising

spins. The measurement of restriction is dependent on experimental parameters. If the

experimental difision tirne is short, spins may not have time to reach barriers and will

appear to be diffising fieely. In this case, the displacement probability remains Gaussian.

As the diffision time is increased, a greater number of spins will encounter restrictive

boundaries, and their displacement distribution deviates fiom the case of unbounded

diffusion. This deviation is due to the relationship between diffision time and the

distance spins will have moved in that time. Einstein showed that the root mean square

displacement for fiee diffision in one dimension is related to the diffusion coefficient and

diffision thne A by (1 8):

= DA)'^ [ l u

In a restrictive environment, the root mean square displacement or diffision distance will

no longer Vary linearly with difision tirne. Instead, when the difision distance

approaches the length of the restrictive cornpartment, the distance spins have travelled

will no longer increase in direct proportion to the diffision time. As a result, the effects

of restriction in MR experiments become apparent when the diffision distances are

comparable to the restrictive length.

These effects are dependent on the type of restriction and the shape of the restrictive

volumes. For simple geometnes, such as spins diffising between planes, in spheres or in

cylindrical geometry, the deviation of the spin displacement distribution fiom the

Gaussian fiee diffision distribution has been calculated (1 9-2 1 ), and the exact behaviour

of the signal decay curve is known.

In tissues, however, restricted diffision is much more cornplex. Spins may be diffbsing in

many different geometries due to the presence of water in multiple tissue compartments.

In addition, barriers may be permeable, ailowing exchange between compartments. The

presence of macromolecules may also affect diffising spins by acting as additional

obstacles to their fkee movement. As a result, the presence of restricted diffusion in

tissues is evidenced experimentally by qualitative changes in the signal decay curve.

One of the primary charact~ristics of restricted diffusion is the detection of non-linearity

in the semilog plot of signal attenuation vs b. Although the origins of this upward

cwature - implying slower diffision - remain somewhat controversial, several

explanations have been put forward. Since water is present in multiple tissue

compartments and the voxels in diffision experiments may incorporate several

compartments, the non-monoexponential behaviour of the decay curve has been ascribed

to the presence of multiple diffision coefficients representing diffision in the different

compartments (22-27). Sirnilar changes in the signal decay curve, however. are seen in

expenments with a single restricted cornpartment (28). In such cases, the upward

curvature has been ascribed to complete dephasing of spins with high mobility near the

centre of the restrictive compartqent since higher gradient strengths cause greater

dephasing, such that the detected signal is originating solely fiom spins close to the

restrictive boundary. Thus, diffusion will appear more restricted at higher b and hence

less signal loss will occur, which is depicted by the upward curvature of the signal decay

c w e .

An upward curvature of the signal decay curve alone, however, is not suficient to

determine whether diffision is restricted Le. confined to one or more compartments. It

has been shown that diffusion in certain systems, such as water diffising in the presence

of glass beads, will also demonstrate non-monoexponential signal decay (29). In this

case, spins are said to be obsûucted rather than restricted. When spin movement is

limited by boundaries however, diffision decay curves will aiso exhibit dependence on

diffusion time, by shifting upwards as the latter is increased. As mentioned earlier, this

relationship results fiom a greater number of spins encountering restrictive barriers as the

experimental difision time is increased, which causes a decrease in the measured

difision coefficient. Since the signal equation, as solved in Eq. [9], constrains spins to

have a Gaussian displacement distribution, the presence of restriction will manifest itself

as a decrease in the measured diffision coefficient, i.e. a difference in signai attenuation

for the same b value.

Although the dependence on diffision time and the upward curvature of the decay curve

are signs of restricted diffusion, the b values, or gradient strengths, used in current

clinical applications are rarely high enough for the detection of non-monoexponential

signal decay. The presence and degree of restriction are therefore measured by changes in

tfie initial slope of the diffision decay curve. At these low b values, the relationship

between signal attenuation and diffision coefficient is monoexponential, and the

measured slope on a semilog plot is called the apparent diffision coefficient or ADC.

1.3.3 Aniso'tropy

Due to the presence of restrictive barriers, diffision may not be the sarne in al1 directions.

This dependence of diffision on direction, or diffision anisotropy, can also occur in the

absence of restriction (30,3 l), but in tissues it is ascribed to spins encountering differing

levels of restriction as a function of direction. In such cases, a single scalar diffision

coefficient is insufficient to describe diffision within a voxel. Basser et al. described the

use of the diffusion tensor to replace the diffision coefficient in anisotropic systems (32).

The signal attenuation equation then becomes:

S/S,=~X~(- b g T ~ g )

where D is the diffusion tensor

D=

g is the diffision encoding unit vector, and b = ( y ~ 6 ) 2 ( ~ - ~ 3 ) . The diffusion tensor

diagonal components, Du, D,, and D,, reflect correlations between molecular

displacements dong x, y, and z, whereas the off-diagonal components, Dxy, Dxz, and Dy,,

reflect conelations in difision in orthogonal directions. Since D is symmetric, six

noncolinear diffision measurements are required to estimate the diffision tensor, in

addition to a measurement in the absence of diffision encoding gradients to determine S,.

The diffision tensor is used to determine the degree of anisotropy present in a voxel, and

to detemine rotationally invariant diffision quantities. These measures allow cornparison

of diffisivities in voxels so that relative changes due to pathology may be detected

despite anisotropy (33). Howevcr, its most recent application is in fibre tractography, as

will be discussed below.

1.3.4 Applications

The availability of MR as a tool to measure molecular displacements has been the basis

for extensive research into possible applications for the study of normal and diseased

tissues. One of the primary clinical applications of diffusion MR is in the detection of

changes in water diffision during cerebral ischemia or stroke. Moseley et al.

demonstrated a dramatic reduction in ADC in the region of the inf'arct (5). This regional

ADC decrease, and the resulting change in signal contrast in diffusion-weighted MR

images, is apparent long before changes in conventional Tz-weighted images. As such, it

provides a means of earlier detection of stroke. Changes in ADC have also been obsewed

in a number of other pathologic processes which affect the brain, such as Alzheimer's

disease (34), multiple sclerosis (6), and brain tumours (35). In emphysematous lungs,

measurement of the increased diffision coefficient using hyperpolarized gas techniques

reflects the destruction of lung tissue structure (36).

Changes in tissue structure, and the subsequent change in the measured diffision

coefficient within a voxel, are often indicators of pathology, such as in the detection of

stroke and emphysema. However, the detection of restricted or anisutropic diffision cm

also be used to examine healthy tissue structure. As mentioned above, the diffision

tensor is used to characterize anisotropy in highly structured tissue. To date, the

dependence of diffision on direction has been reported in a nurnber of tissues with a high

degree of structural organisation, such as kidney (37), skeletal muscle (38), cardiac

muscle (39) and most notably, brain white matter (40,41).

Diffusion in White Matter

The dependence of difision with direction in white matter has been attnbuted to the

presence of nerve fibres. Whereas grey matter consists largely of neurons and suppon

cells, white matter is made up prYnarily of bundles of nerve fibres. Axons, or nerve

fibres, are the projection amis of the neuron ce11 bodies found in grey matter. They range

between 1 and 2 0 p in diameter, but may be over a metre in length. In white matter,

their role is to provide communication between different regions of the brain. The mode

of signalling dong the fibre is through electric impulses, and nerve fibres are highly

stnictl;isd xi* ri surrounding layer of insulation known as the myelin sheath.

MR difhsion measurements in white matter have found that diffision along the fibres is

much greater than difision across the fibres. In studies of healthy hurnan volunteers, the

diffision coefficient measured with fibres paralle1 to the gradient is on the order of

1 .OX 1 0%m2/s (4 1). and has been reported as high as 1 . 5 ~ 1 0"cm2/s parallel to the corpus

cailosum (33). Across the fibres, however, the diffision coefficient is much lower,

ranging fiom 0.3 to 0.6~10'~crn~/s (33,41). In addition, diffision measurements at higher

b values (> 1 500 s/mm2) demonstrate non-monoexponential signal decay (22,23 &),

which may suggest that diffision in neural tissue is restricted as well as anisotropic.

The observation that diffision is greatest along the length of nerve fibres has provided a

non-invasive method for establishing fibre directionality. The direction of highest

difision can be used to infer fibre tract orientation within successive voxels and thus

infer the anatomic path of neuronal connectivity.

Fibre Tracking

To date, fibre tracking has been performed using diffision tensor MM (DT-MW). As

mentioned eariier, the diffision tensor for a voxel incorporates more information than a

single scalar diffusion coefficient. ADC measurements along each of the three orthogonai

axes alone are insufficient to properly detemine anisotropy and fibre orientation, since

the gradient or laboratory x, y and z axes rarely coincide with the local fibre axes. The

offdiagond elements of the diffision tensor therefore provide additional information on

fibre structure and correlate displacements along x, y and z when the mobilities in these

directions differ (43).

In order to establish fibre tract orientation within a voxel, six non-collinear diffision

measurements are made to determine the elements of the diffision tensor. Since D is

positive definite and symmetric, its three eigenvectors and associated eigenvalues are

orthogonal:

D E = E A 1141

where A is the diagonal matrix of eigenvalues Li, k2, k3, and E contains the

corresponding orthonormal eigenvectors, arranged in columns. The eigenvalues and

eigenvectors represent the effective diffusivities and local fibre CO-ordinates i i yactively.

They therefore eliminate the need for the laboratory or gradient axes to coincide with the

voxel's fibre axes. The eigenvalues and eigenvectors are used to describe the diffision

ellipsoid, the surface of which represents the mean displacement of spins within the

diffusion time. The eccentricity of the ellipse describes the degree of anisotropy present

within a voxel and provides a pictorial representation of the information contained in the

tensor. Figure 3 depicts the relationship between difîüsing environment (upper row),

difision ellipsoids (middle row), and diffision tensor (bonom row).

Fig. 3. Relationship between diffusing environment (upper row), diffusion ellipsoids (middle row), and diffusion tensor (bottom row). In isotropic diffusion (a), ellipse is spherical and diffusion represented by one diffusion coefficient D. In anisotropic diffusion with coincident fibre and laboratory axes (b) ellipse is elongated and the tensor's diagonal elements are the eigenvalues of the ellipse. In (c) the fibre and laboratory axes do not coincide and the nine tensor elements depend on the relative orientation of these axes.

In Fig. 3% diffision is free and isotropic, represented by the spherical ellipsoid and only

one diffision coefficient. Fig. 3b, on the other hand, depicts a case of restricted diffision

with coincident laboratory and fibre axes. There are three tensor elements representing

diffision along the three principal axes. Findly, in Fig. 3c, the fibre frame has been

rotated with respect to the laboratory frame. Al1 nine elements of the tensor are required

to correlate diffision along the different axes. It is apparent fiom these images that the

eigenvalue with the greatest difisivity represents diffision along the direction of the

fibre and its eigenvector provides the fibre tract orientation. Based on this observation, a

number of tracking algorithm have been developed to perfom continuous tracking fiom

one voxel to the next (7-10).

Although some of the larger fibre bundles in the brain have been traced using DT-MN,

many dificulties must be resolved before fibre backing can become a useful clinical tu01

(10). DT-MN studies are plagued with low signal-to-noise-ratio (SNR), due in part to

fast imaging techniques. In addition, validation of tracking is problematic since the

current gold standard for fibre tractography is histology, which is inappropkite for in

vivo studies.

Currently, one of the largest challenges for fibre tracking is resol-{ing fibres lacking

directional coherence within a single voxel. Although some fibre bundles are relatively

large and highly directional, consisting of 800,000 or more individuai nerve fibres / mm2,

many fibre bundles are smaller, have collateral branches, or are in close proximity to

other fibre tracts. As a result, multiple fibre tracts may be present in an imaging voxel,

each with independent orientations. An example of this situation occurs where fibre

bundles cross. Due to the lack of a predominant direction of diffision, the diffision

tensot and its corresponding ellipsoid are unable to resolve fibre tract orientations in

these voxels. In such cases, six independent difision measurements are insufficient to

characterize the complex molecular motion.

An alternative method to resolving the presence of multiple fibre tracts was introduced by

Tuch et al. (12) and Frank (14). They described the use of high angular resolution

diffusion measurements to provide additional information on difision in regions of low

fibre directionai coherence. This technique consists of applying diffusion-encoding

gradients at a high number of angles to adequately represent diffision in al1 possible

directions, instead of limiting these measurements to the six axes required for the

difision tensor. The resulting ADC measurernents as a fùnction of gradient angle have a

higher degree of angular variability than the diffision ellipsoid. Whereas the ellipsoid

represents the spin displacements in the 'fibre' m e , the measured ADC contour

represents the projection of the spin displacements ont0 the gradient or laboratory axes.

In the case of two crossed fibres, the ADC contour has a cloverleaf shape, in contrast to

the spherical diffusion surface defined by the tensor formalism for the same fibre

structure. Although these diffusion coefficient contours reflect a higher angular

variability, the degree to which they reflect fibre bundle orientation has yet to bc

deterrnined.

1.4 Thesis Statement

The goal of this thesis was to determine the role of high angular resolution diffision

measurements in the determination of fibre tract orientation. An in vitro mode1 of

anisotropic diffision analogous to that produced by nerve fibres was developed, so that

experiments could be performed in a controlled sample with known fibre orientations.

Experimental results for different fibre orientations were compared with theoretical

calculations for diffision in a cylinder to detennine the accuracy and feasibility of

difision meesumnents at multiple angles in determining complex fibre structure.

Chapter 2

High Angular Resolution Diffusion MR for the Determination of Fibre Structure

This Chapter was submitted as a paper to Magnetic Resonance in Medicine in September 200 1,

2.1 Introduction

The signal contrast in MRI is due to tissue molecular content and structure, which affect

such MR parameters as Ti, T2, magnetisation transfer and difision. In diffision imaging

studies, the contrast is based on spin dephasing and the rate of diffision can be measured

by the application of diffusion-encoding gradients (1 6). The presence of ordered

biolopicd membranes, however, can provide obstacles to the difising spins, restricting

their movement in certain directions.

The degree of restriction encountered by spins is thus a reflection of cellular structure.

Echo attenuation curves due to diffision have been denved for spins diffising between

planes (1 9,2O), in spheres (19,20), and in cylindncal geometry (20,2 1). Measurements of

the differences in these echo attenuation plots, however, require extremely strong

diffision gradients or high b values. In clinical imaging, b values are generally

sufficiently small ( < lx1 0' s/cm2) that the diffision decay curve can be approximated as

monoexponential, and estimates of the diffision coefficient can be made by measuring

the slope, or apparent diffision coefficient (ADC). The level of restriction encountered

by spins, or the presence of cellular structures, is then determined by relative changes in

the ADC with diffision gradient orientation.

The dependence of diffision on direction, or diffisional anisotropy, due to restrictive

barriers has been reported in a variety of tissues exhibiting a high degree of structural

organisation. Cleveland et al. first observed anisotropy of difision in skeletal muscle

(38), Reese et al. in cardiac muscle (39), and, more recently, diffision anisotropy has

been linked to the highly structureci nerve fibre tracts in brain white matter (40,41).

In white matter, nerve fibres have a high degree of restriction across their width and

relatively free diffision dong their length. Thus, the measured diffision coefficients

reflect fibre tract orientation (7- 10). By following the direction of highest difision fiom

one voxel to the next, fibre bundles can be tracked and neuraI connections established.

Fibre tracking algorithrns are king actively developed and have successfully traced

many of the larger known fibre bundles in the brain (7-1 0). Preliminary studies have

already shown that establishing in vivo neural connectivity may provide a means of early

disease detection for pathologie processes afTecting the integrity of neuronal connections

(44,45), as well as providing a greater understanding of fùnctional relationships between

different regions of the brain.

There are, however, outstanding issues which must be resolved before fibre tracking can

successfully be implemented as a clinical tool. Current techniques based on diffision

tensors are inappropriate for assessing multiple fibre tract orientations within a voxel,

which may be attributed to the fact that diffision coefficients dong six noncolinear axes

are insufficient to describe the complex underlying fibre structure (1 0). Wedeen et al.

recently proposed imaging diffision spectra (1 1) as an alternative, in which the

displacement probabilities of spins are used to detemine fibre orientation. High angular

resolution difision measurements have also been explored to characterize diverse fibre

structure within a single voxe1(12,14), however, the ability to resolve fibre orientation

using this technique bas not been addressed. Although many new techniques are being

explored, to date there has been no report of evaiuating diffision acquisition schemes

using a model of restricted diffision similar to nerve fibres.

In this paper, we present a physical model andogous to diffusion in nerve fibres. By

performing single voxel diffision experiments, we examine the effects of changing fibre

orientation and multiple fibre orientations on diffision measurements at high angular

resolution. The experimental results are compared with a known theoretical model for

diffusion in a restricted cylindrical geomeuy, and these theoretical calculations are used

to assess the use of difision rneasurements at multiple angles and high b values in

determining fibre bundle orientation.

2.2 Experimental Methods

Hollow plastic fibres (PTFE ultramicrobore tubing P-06417-70, Cole-Parmer Instrument

Company) were used to create an experimental model of white matter fibre bundles. Each

sample was prepared with a length of 2.5 m of tubing. The tubing, or fibres, had an inner

diameter of approximately 50 Pm (roughly twice the size of larger scale nerve fibres in

the brain) and an outer diarneter of 325 p, as show in the eiectron micrograph in

Figure 4.

Fig. 4. Electron micrograph of plastic tubing in cross-section with outer diameter 325 Fm and inner diameter 50 Fm.

Water was inserted into the fibre by placing the tip inside a 22 '/1 gauge needle and gluing

the fibre to the needle. Water was injected manually into the iength of fibre and was

sealed by melting both ends after removing the needle. The fibres were then arranged to

mate three different fibre bundle orientations: aligned, coiled, and oriented at random

(Fig.5).

Fig. 5. Orientation of fibre samples A) aligned fibres 8) wound fibres C) randomly oriented fibres. x, y, and z represent the CO-ordinate axes, G the gradient axis, 0 the angle between x and G, and Ba the direction of the main magnetic field.

The samples were 1 cm in height, 0.5 cm in width, and were placed inside borosilicate

g las tubes and oriented such that the long axis of the sample was perpendicular to the Bo

direction (Fig. Sa) inside the superconducting 1.5T magnet (Naiorac, Martinez, CA)

controlled by a programmable console (SMIS, Surrey, England). Diffision measurements

were acquired using a pulsed field gradient multi spin echo (PFG MSE) pulse sequence

(Fig. 6). This sequence uses a series of x refocusing pulses between the difhsion gradient

pulses to diminish the effects of susceptibility and diffision caused by susceptibility

gradients (46) (any additional gradients may cause unwanted diffision).

Fig. 6. Pulsed Fiekl Gradient Muhi Spin Echo Sequence (PFG MSE) with 85 x pulses between gradients and TR=3s.

The diffusion-encoding gradient used was a three mis gradient (XL Resonance, London,

Ontario) with a 5 cm inner diameter and novel multilayer coi1 design to achieve high

gradient strengths (47). Eddy currents in this gradient were < 0.5% afler 100 ps, and the

gradient was unifonn over 4 cm. Diffision-encoding gradients were rotated in both the

x-y and x-z planes in increments of 15". At each orientation, the gradient strength was

varied fiom O to 1500 mT/m in 10 unifonn steps. The gradient pulse duration was 300 ps

with 65 n pulses between the diffision-encoding gradients for a total diffision time of

130 ms. This diffision time corresponds to a 23 pn fiee diffision length for water at

20°C with Dc2.02 x IO-' cm2/s (48). Following the last gradient pulse, 10 echoes were

obtained to measure signal strength. Data were averaged 80 times (TR=3s) to improve

signal-to-noise-ratio.

The echo attenuation for each angular gradient orientation was plotted against b to obtain

the initial dope (ADC), as described by the Stejskal-Tanner formula (16). ADC values

were plotted as r function of gradient angle in polar co-ordinates to display ADC

contours for the three sample orientations.

2.3 Theoretical Methods

The signai attenuation SIS, as a function of gradient strength in restricted cylindncal

geometry has been derived by S6deman et al. (21) by solving Fick's second law of

diffision with appropriate boundary conditions. The relationship is given by:

where ak., are given by the roots of the Bessel equation J',(a)=O (with the convention

that aio=O), q=ygW2x and Kn, are integer constants, where Knm=l if n=m=O, Knm=2 if

n#O and m=O, or m t O and n=O, K,,,=4 if n, mtO. The angle between the gradient

direction and the symmetry axis of the cylinder is given by 0. Actual expenmental values

were used for difhision time (A= 130 ms), and gradient strength (g=O to 1500 m T h ) and

duration (6=300 ps), with a difision coefficient of water ~ = 2 . 0 2 ~ 1 0 ~ ~ c r n ~ / s . The length

of the cylinder, L, and its radius, R, were set equal to 1 cm and 50 pm respectively, in

order to represent the previously described sarnple fibre geometry.

Theoretical results for echo attenuation as a fùnction of gradient strength were computed

using Matlab (Mathworks Inc., Natick, MA) according to Eq. [IS]. Summation was

perfonned to k, m=10 and n=1000, since higher order terms did not contribute

significantly to signal attenuation. Calculations were repeated for vatious gradient angles

and the ADC obtained f ~ r cach gradient orientation.

Calculations of diffision decay curves at high b values were obtained by convolving the

theoretical results for a single fibre with the fibre orientation probability distribution:

S(0') = S(û) P(û'-û)dû 161

where 0 represents the direction of diffision in three dimensions, S(0) is the signal decay

in any direction for a single cylinder, P(9) the probability distribution of fibre

orientations, and S(8') the resulting signal decay for the given fibre orientation

distribution. By using Eq. [16], signal decay curves for any set of known fibre

orientations can be obtained. For the case of the wound fibres in the plane of the winding

(y-z plane) and the randomiy onented fibres, the fibre orientation probability densities

were assumed to be equivalent to a star-shaped distribution of fibres (see Fig. 7). This

simplification was deemed appropriate since the mean fiee path of the spins is small

during the time of the experiment. The estimated diffision coefficient at different b

values for simulated fibre orientations was calcuiated by taking the dope of a straight line

joining b=O and the chosen b value on the diffusion decay cuves.

Fig. 7. Coiled Fibre Approximation

2.4 Results and Discussion

In Figure 8, the signal attenuation as a function of gradient strength is plotted for the X

@=O0), Y (e=90°) and Z directions in the aligned fibre bundle, where 0 is the gradient

rotation angle in the x-y plane. Diffision dong the fibres resembles fkee diffision of

water, ADC= 1.90k0.03~ 1 O-' cm2/s, whereas across the fibres, the ADC is decreased by

almost one half, ADC= 1.05k0.03~ 10" cm2/s. This is an anisotropy factor of 1.8 : 1,

which is comparable to that reported experimentaily in brain white rnatter (1.7-2.3 : 1 for

gradients parallel to and perpendicular to nerve fibres) (4 1). The lines represent the

theoretical calcuiations with no fiee parameters for diffision at 8=0° and 0=90° in a

cylinder, and agree well with experimental measurements. In ail cases, experimental and

theoretical calculations agree to within 10%.

Gradients along

I Theoretical calculations - 8=û0

- 1 1 0=90° 1

Fig. 8. Experimental and theoretical diffusion decay curves for gradients along the X, Y, Z directions in the aligned fibre bundle

A slight upward deviation with increasing b value is apparent in the experimental results

for the aligned fibres with the gradient oriented along X. This deviation h m theoreticai

calculations cm be explained by the small signal strength, which is no longer

distinguishable fiom the standard deviation of the noise. We know from theoretical

predictions that an upward curvature, or bi-exponentiality of the signal decay curve, as

has previously been observed in restricted diffûsion (22,23,42), would appear at much

higher b values than those obtained in our experiments.

Although not readily apparent in Fig. 8, there was a tendency for the first data point, the

measmd signal in the absence of diffusion encoding gradients @=O), to be slightly lower

than the signal attenuation following the application of the h t gradient increment. It is

possible that this discrepancy was caused by the presence of a slight gradient dong die

sample, which would cause a loss of signal in the absence of the diffusion-encoding

gradients. Calculations of the ADC, however, were unaffected whether or not this low

first data point was taken into account. As a result, al1 ADC calculations were perforrned

with the aberrant fint data point for consistency.

The ADC is plotted in Figure 9 as a function of angle between the diffusion gradient and

the axis of the fibres. Difision gradients were rotated in both the x-y and x-z planes for

al1 fibre samples, but only the x-y data are shown here. At 8=0° and 1 80°, diffision is

essentially the fiee diffision of water at 20°C for the aligned fibre bundle (Fig. 9a) and

was determined experimentally to be 1.90*0.03x 1 O-' cm2/s and 2.1 0k0.03~ 1 o - ~ cm2/s

respectively. However, as the gradient angle increases to 90' and 270°, the degree of

restriction is increased, which is reflected in the decreased ADC values,

ADC= 1 .OS&O.O~X 10" cm2/s and l.13k0.03~ 105 cm2/s respectively. The experimentally

obtained peanut-Iike ADC contour for the a l i p d fibre bundle is compared with

theoretical caîculations for a single cylinder (solid line, Fig. 9a). The calculation has the

same shape, with a somewhat higher degree of restriction across the fibres. The peanut-

like shape of the ADC contour for a single fibre can be expleinecl by the nature of the

ADC measurement. Whereas the difision ellipse represents the actual spin

displacements, the ADC contour is the projection of these spin displacements onto the

diffising gradient axis.

When the fibres were wound in a coil-like structure, the highly restricted dimension, the

diameter of the fibre, has now been rotated by 90° and lies dong the x-axis (Fig. 9b). The

ADCs measured across the diameter of the fibres dong X at 0=0° and 1 80°,

ADC= 1.06k0.03~ 1 O" cm2/s and 1.1 1 k0.03~ 10" cm2/s, are consistent w ith ADCs across

Fig. 9. Experimental results (dots) and theoretical calculations (solid line) of AOC vs gradient angle 9 for samples shown in Figure 5. A) aligned fibres 6) wound fibres C) randomly oriented fibres. All AOC contours are of the same scale with concentric circles 0 . 5 ~ 1 ~~crn* /s apart. Errors on AOCs t 0.03x105cm2is.

the fibres in the case of the aligned fibre bundle. As the gradient is rotated into the y-z

plane, the ADC increases but remains significantly less than fiee diffusion. The reduction

in the ADC in the y-z plane occurs due to the equal representation of al1 fibre directions

and is equal to the average of the free difision dong the fibres and the restricted

diffision across the fibres. Theoretical calculations for the wound fibres were obtained by

convolving the individually derived cylindrical diffision ADC contour with a circular

distribution of possible fibre orientations, as described in Eq. [16], and agree well with

experimental measurements.

Figure 9c depicts the ADC contour for a sample of randomly oriented fibres. The ADCs

are similar regardless of direction since spins will experience on average the same degree

of restriction in dl orientations. Whereas, previously in the wound sample, the

probability of fibre orientations was equally distributed in the y-z plane only, in the

randomly oriented sample, al1 fibre orientations are represented equally in three

dimensions. This is reflected in the theoretical ADC contour, which is spherical, and

results frorn an equal probability of fibre orientations in al1 directions. Some discrepancy

between the experimental results and theoretical calculations in the case of the randomly

oriented fibres can be attributed to the difficulty in assuring a truly random orientation for

scrambled plastic tubing (there is a slight tendency for fibres to track dong the wall of the

g las tube).

To explore the behaviour of the signal decay curves as a îùnction of gradient angle at

high b values, Fig. 10 is a plot of the theoretically calculated diffision decay for a single

fibre or cylinder, but calculated to a much higher b value (bmm=7.5~10S s/cm2). The

highest decay curve represents signal loss due to diffusion with the gradient onented

along the transverse axis of the fibre (8=90°). Successive decay curves are plotted in S0

increments until the gradient is oriented along the long axis of the fibre (8=0°).

Fig. 10. Theoretical diffusion decay curves to high b value (7.5x10~s/crn~) for a single fibre with gradient angles from O0 to 90° in 5" increments. The dashed line

indicates the estirnated Ob at b=4.2x10~s/cm~ for 8=90°.

The upward curvature at high b values in these calculations cannot be explained by the

presence of water in multiple compartments, since the theoretical calculations are based

on diffision inside an impermeable cylinder. It may, however, be the result of complete

spin dephasing near the centre of the cylinder at high gradient strengths so that the

measured signal attenuation is pnmarily due to spins close to the cylinder walls. As a

result, diffision will appear more restricted, manifested by an upward cwature in the

decay curve. This is further supported by the fact that the cwature is higher as the

gradient is aligned with the symmeûy axis of the cylinder (8=0°). Since the cylinder is a

closed structure and has lids at its top and bottom, unlike nerve fibres within a voxel, the

signal will appear to be the result of even more restricted diffusion at higher gradient

strengths, since it will originate primarily with spins close to the lids, as well as the

swrounding walls (28).

Fig. 1 la is a simulation of the change in shape of the estimated difision coefficient

contour for a single fibre as the b value is increased. The b values were chosen at

b=0.5xl0~ dcrn2, 4.2~10' s/cm2, 13.4~10' s/cm2. The dashed line on the signal decay

curve in Fig. 10 represents the estimation method for the difhsion coefficieiii. While this

method is a gross approximation, since the diffusion is clearly non-monoexponential, it is

the method rnost ofken reported in the literature for clinical difision measurements. At

low b values, this technique is equivalent to measuring the ADC, but as the b value

increases, it is apparent that the estimated diffision coefficient Db will be biased

depending on the b value. The decrease in the estimated diffusion coefficient at high b

might be misinterpreted as a higher level of restriction or a change in the structural

geometry within the voxel, but is simply reflective of the upward curvature of the signal

decay curve. As a result, as show in Fig. 1 1, the peanut-like shape becomes more

constrained dong the transverse axis of the fibre with increasing b and, at very high b,

becomes constrained dong its long axis as well.

This is fiuther illustrated in the case of the wound fibres in the plane of the winding and

for the randomly oriented fibres (Fig. 1 1 b). As b value is increased, the estimated

diffision coefficient Db for the randomly oriented fibres decreases due to the bend in the

decay cuves, and the contour becomes progressively smaller. However, it is not possible

to resolve the orientational distribution of the fibres.

Fig. 11. Estimated diffusion coefficient Ob vs gradient angle at b=0.5xi ~ ~ s l c r n ~ (outermost contour), 4 .2~1 ~ ~ s / c r n ~ , 13.4~1 ~ ~ s l c m * for A) single fibre 6) wound fibres in plane of winding and randomly oriented fibres C) two perpendicular fibres oriented along O0 and 90° D three fibres oriented along 0°, 45O, and 90°. 4 Concentric circles are 0.5x10~cm 1s apart.

In order to address the ability of high angular resolution diffision measurements coupled

with high b values to resolve fibre structure, Fig. 1 lc is a simulation of two identical

fibres oriented perpendicular to each other. n ie fibres are oriented along 0=0° and 0=90°.

At low b value (b=0.5xl0~slcm~), which is the approximate regirne for clinical ADC

measurements, the estimated difision coefficient is equal in al1 directions and provides

no information on fibre structure. As the b value is increased, however, the shape of the

resultant difision coefficient contour suggcsts the presence of two fibres, but the lobes,

which becorne more pronounced as b increases, are oriented at 4S0 to the fibre directions.

In the direction of the fibres, the estimated diffision coefficient decreases at high b

values due to the logarithmic relationship between signal decay and the diffision

coefficient. In the presence of multiple cornp&ments as seen in fibre crossings with

equivalent fibres, the estirnated diffision coeficient will be governed by the most

restricted cornpartment. In the case of two perpendicular fibres, the lowest diffision

coefficient appears dong ïhe gradient angle where the long axis of the first fibre is

parallel to the short axis of the second fibre. In these directions, the spins moving across

the highly restricted transverse axis of the second fibre dominate the signal decay,

because the rapidly decaying signal fiom the fieely diffising spins moving dong the long *

axis of the other fibre contribute negligibly. As a result, the measured diffision

coefficient contour is counter-intuitive and appears to have more 'restricted' diffision

dong the long axis of the fibres.

As the number of crossing fibres increases, the shape of the diffision coefficient contour

becomes more complex. In Figure I Id, the estimated diffision coefficient Db at three b

values is plotted as a function of gradient angle for three fibres oriented at 0°, 4S0 and 90"

respectively. As seen previously, the most restricted dimension of the three fibres appears

to have the highest estimated Db. The mon complex shape makes the detection of fibre

directionality much more difficult.

Current clinical b values are best represented by the lowest b value used in Our

simulations (b=0.5xl oSs/cm2). Our calculated diffision coefficient contours at these b

values do not demonstrate a high degree of variability with respect to the difision-

encoding gradient orientation. The upward cwvature in signal decay, however, has been

observed at much lower b values in white matter (22-27). Whereas our simulations do not

reflect non-monoexponentiality until b < 3 . 0 ~ 1 ~~s/cm*, in vivo studies of white matter

have shown this behaviour occurs at much lower gradient strengths b < 1.5xl0~s/cm~. We

would therefore expect difision coefficient contours measured in vivo to display much

greater variation with fibre orientation at lower b values than were observed in our

simulations at similar b values. The advent of stronger and faster clinical gradients will

also see the use of higher gradient strengths for diffision measurements, making the

detection of diffision coefficient contours with greater angular variability clinically

feasible.

Deteminhg the contours done, without a priori knowledge of fibre structure, is

insufficient to resolve multiple fibre orientations within a voxel. We have shown however

that knowledge of signal decay as a function of gradient angle for a single cylinder allows

us to detennine diffusion coefficient contours for any arrangement of fibres. A

comparable mode1 of diffision for white matter could be established by incorporating

multiple water compartments, permeable membranes, and possible exchange between

compartments. Similar models have already been suggested in the literanire (49-52).

Knowledge of the signal decay in such an environment would enable the determination of

diffusion coefficient contours for any probability of fibre orientations. These contours

could then be fit to experimental diffusion measurements at high angular resolution to

resolve complex fibre structure.

2.5 Conclusions

We have described a physical mode1 of restricted difision which demonstrates similar

anisotropy to that observed in nerve fibres. Our orientational diffision measurements

reflect the fibre geomeûy of the sample, and theoretical calculations for diffusion in

cylindrical geometry are in good agreement with the experimental results. Although the

magnitude of the measured ADCs are much larger than those reported in white matter,

the model's di ffisional anisotropy and structural similarity to nerve fibres may have

potential applications in the evaluation and cornparison of difision tensor imaging

sequences, as well as other methodologies for the study of restricted diffision.

Simulations of signal decay at high b values show diffision coefficient estimates which

assume monoexponentiality of the diffision decay curve to be very dependent on the b

value of the expenment. While the shape of the subsequent diaision coefficient contours

may provide information on the orientations of fibre bundles present within the voxel, the

apparent directionality of the contour alone does not indicate fibre orientation.

Our results suggest that high angular resolution diffision measurements provide

information on the presence of restriction and reveal dinusional anisotropy in fibre

bundles, which rnay not be detected using other analysis methodologies such as the

diffision ellipse. Further work is needed to fully characterize changes to the diffision

coefficient contour as the nurnber and orientation of fibres within a voxel change before

difision measurements at high angular resolution can become a clinically feasible

method for resolving fibre structure.

Chapter 3

Future Work

3.1 Introduction

In the previous chapter, experimental measurements and theoretical calculations were

used to examine the capability of difision measurements at multiple angles to resolve

complex fibre structure within a voxel. The problem of fibre crossings and branchings

must be addressed before fibre tractography and its applications to the detection of

diseases afl'ecting white matter can be successiùlly used in a clinical setting. A better

understanding of diffision in regions with low fibre directional coherence is essential for

continuous fibre tracking and understanding the organisation of complex neural networks

in the brain. The previous chapter discussed orientational diffision measurements at

multiple angles as a proposed solution to this problem. The experimental and theoretical

results were in good agreement, and m e r theoretical calculations demonstrated high

angular variability of the predicted diffision coefficient for simulated fibre orientations.

38

The experimentd model presented, however, is a simplification of the restricted and

anisotropic diffision in nerve fibre bundles, and may not accurately represent difision in

brain white matter. The theoretical calculations were based on diffusion in a simple

cylindtical, restricted geomeuy, which agreed well with our experimental model of

cylindrical fibres, but does not account for many of the more complex structural and

organisational characteristics of white matter.

In this chapter, possible implications for translating this study h m an in vitro model to

in vivo applications are discussed. The limitations of the current model in describing

diffision in highly structured tissues like brain white matter are described, as well as

suggested improvements to the model to better incorporate white matter tissue

characteristics. Finally, the feasibility of resolving difision and fibre orientation l using

diffision measurements at high angular resolution with an appropriate in vivo model are

discussed.

Orientational diffision measurements, however, are not the only proposed solution for

the stdy of molecular motion in reg ions of varied nerve fibre bundl e structure. Several

other techniques have also k e n suggested as alternatives to the diffision tensor

formalism. These methods will be discussed in the latter half of this chapter.

3.2 Orientational Diffusion Measurements in White Matter

In order to translate this study to an in vivo model, several additionai factors must be

incorporated into the model. These factors will be discussed below in detail. Previously

reported models of white matter and their limitations for this study will also be described.

3.2.1 A Model of White Matter

As mentioned in Chapter 1, brain white matter consists primarily of bundles of nerve

fibres, which facilitate communication between different regions of the brain, as well as

providing signalling to and fiom the rest of the body. The structure of individual nerve

fibres is cylindrical and their inner diameter ranges between 1-20 Fm. The experimental

and theoretical models presented in this thesis have the same approximate geornetry, and

demonstrate a similar degree of anisotropy to that reported in white matter. However, the

models assume diffision occurs in a single cornpartment, and is solely due to

contributions fiom spins inside the fibres. This is a simplification, since water is present

in al1 tissue compartments and the upward curvature of the signal decay curve has often

been attributed to different spin compartments (22-27).

Nevertheless, since the theoretical mode1 displays many of the characteristics associated

with diffision in white matter such as anisotropy and non-mononexponential signal

decay, the first step in translating the technique presented in this thesis to an in vivo

setting is performing a set of difision measurernents at high angular resolution in a white

matter tissue sample. These results can then be compared with theoretical calculations for

diffision in a cylinder to determine their validity in approximating white matter. Certain

discrepancies, which are intrinsically apparent, include the appearance of a rnuch delayed

upward curvanire in signal decay for the theoretical calculations. In addition, the top and

bottom wdls or lids of the theoretical cylinder create boundaries to the spins, which are

not present in white matter fibre geometry.

In order to address these issues, the theoretical model for diffusion in a cylinder could be

expanded to better represent white matter. Although difision in white matter is

undoubtedly anisotropic and restricted, controversy remains as to the origins of this

anisotropy and restriction. The axonal membrane may provide barriers to both

intraaxonal and extracellular diffision of water. In addition, the intnnsic structure within

the nerve fibre itself, consisting of neurofilarnents and microtubules associated with

axonal transport, may be contributing factors to the observed diffusion anisotropy. Some

studies are attempting to explain the behaviour of the signal decay curve as the result of

many water compartments with di fferent structurai characteristics (22-27).

Compartrnentalizing the signal, however, is difficult due to the presence of exchange

between the various compartments.

It is apparent from these previous snüiies that many factors must be incorporated into a

model of diffision in white matter. Several theoretical models of diffision in tissues have

been previously described and combine many of these elements (49-52). Szafer et al. (5 1)

presented a two-pool model for diffision in tissues with ceIl membrane permeability

between intria- and extracellular compartments. They showed that, considering the

typically low biological ceIl membrane pemeability, intra- and extracellular

compartments c m be regarded as independent and only weakly linked through membrane

pemeability. Their model, however, does not address the geometry specific to white

matter. Stanisz et al. (52) descnbed an analytical model of restricted diffision in bovine

optic nerve, incorporating both axons and spherical glial cells. They found that omitting a

third difising cornpartment consisting of sphekai glial support cells resulted in a poor

fit to their expenmental measurements of bovine optic nerve tissue sample. However,

their model did not take into account the possibility of multiple axonal orientations within

a voxel and the distribution of these restricted dimensions.

As demonstrated by the above models, an appropriate description of diffision in white

matter would have to consist of several components. Based on previous work, at least two

compartments, intraaxonai and extracellular, should be modelled, and contributions fiom

both must be accounted for in diffision signal decay. Axonal water signal contributions

can be represented by the theoretical model of cylindrically restricted diffusion used in

Chapter 2. Extracellular water in tissue models has been show to be dependent on the

degree of tortuosity encountered by the spins (51). In white matter, this tortuosity would

primarily depend on the distribution of nerve fibres, but rnay also be influenced by other

cells, such as glial cells, which provide obstacles to diffising spins. The degree of

tortuosity due to axons is dependent on direction in a similar manner to the anisotropy of

diffision of intraaxonal water.

In addition to the presence of two diffising water compartments, another factor which

must be taken into account is the presence of exchange between these compartments.

This is determined by the permeability of the axonal membrane. Permeability and

exchange rates are related to each 0 t h via the surface-to-volume ratio. The degree to

which the permeability will af5ect signal contributions, however, is very dependent on

experimental pammetea. For short diffision times, few water molecules will have time

to diffuse across the membrane, whereas longer diffision times may see complete mixing

of the compartments.

Another factor to be considered when examining diaision in multiple compartments is

the relative volume fractions of these compartments. Studies in the literature place the

extracellular volume fraction at roughly 0.2 and the intracellular volume fraction at 0.8.

However, these may not be easily distinguishable, depending on experimental

parameters.

Using these approximations to diffusion in white matter, the measured signal will be a

combination of water in two compartments:

sm(e, b) =X si@, b, p) +f, Se@, b, p) 1171

where Si, Se represent the signal fiom spins difising intraaxonally and extracellularly

respectively,J and5 the relative spin Fractions, and p the pemieability. Si may be

detemined fiom Eq. [15] for signal attenuation in a cylinder, with the 'lids' of the

cylinder moved to infinity, and with appropriate axon dimensions. In addition, the

intraaxonal signal will be afFected by exchange, spins leaving the axonal cornpartment

and spins moving into the axon from the extracellular space. Diffusion in the extracellular

space can be described by:

De@) = Df / ~ ( 0 ) ~ [18]

where h is the tortuosity, which is dependent on the gradient angle 0. The signal in the

extracellular space will also be govemed by exchange in a similar manner to the

intraaxonal space. The angular dependence of both the axond and extracellular signal

contributions will be govemed by the orientational distribution of fibre bundles, P(8).

The measuced signal Sm will therefore be a hction of P(B), Di, De, p,fi,fr, and b. Fig. 12

depicts the tissue mode1 with its pararneten.

Fig. 12. In vivo nerve fibre rnodet with intraaxonal diffusion coefficient Di, extracellular diffusion coefficient De, and permeability P.

This model of diffision in white matter fibre bundles is still grossly simplified, but may

incorporate some aspects of tissue diffision, which are neglected when diffision is

approximated by a single compartment inside a closed, impermeable cylinder, as

presented in this thesis. The addition of more parameters, however, increases the

complexity of the model and thus experimental parameters must be modified accordingly.

The presence of exchange assumes knowledge of the scale of this exchange, whether it

occurs rapidly (complete mixing of the compartments), slowly (no mixing of the

compartments) or at some time M e between those extremes. Several studies have been

aimed at associating diffision fractions and exchange regimes to T2 components in an

effort to isolate compartments according to their transverse relaxation parameter

(26,37,42,53). It is therefore important to be aware of the experimental parameters such

as diffision tirne, echo t h e , and gradient strength, to properly assess whether signal

contributions are originating predominantly from one compartment, two individual

compartments, or complete mixing of both spins groups (54). Assuming the presence of

only two compartments may be somewhat oversimplified (52), however the adequacy of

two compartments should be determined before unnecessarily complicating the model.

3.2.2 Determining fibre structure

Once the signal decay behaviour due to diffision in white rnatter has been modelled, and

is defined for any diffusion-encoding gradient orientation with any distribution of fibre

bundles, difision coefficient contours can be obtained for any fibre orientation. If the

single cylinder model used in Chapter 2 is sufficient to adequately describe in vivo white

matter diffision, the signal decay for any orientation of fibre bundles can be obtained

using Eq. [16]. If a more complex model of white rnatter is required, the theoretical signal

decay can be detemined according to Eq. [17], taking care to account for experimental

parameters which affect the permeabilities, exchange rates and relative volume fractions

of the diffising compartments.

Experimental measurements can then be performed in regions of known fibre

arrangement in the brain to assess the capability of the theoreticai model in replicating

signal decay due to diffusion. With an adequate model, the orientation of fibres can be

determined by finding a theoretical fit to the experimentally obtained diffision

coefficient contours, using the fibre orientation distribution as a fiee parameter.

There are limitations, however, to this technique. As demonstrated in the experiments and

simulations presented in this thesis, a completely random orientatiop of fibres is

indistinguishable fiom isotropie diffision, aside fiom an overall decrease in the measured

diffision coefficient. Using a relative decrease in the measured difision coefficient to

infer fibre structure rnay be difficult due to inter-subject vanability and the additional

dependence of the measured diffusion coefficient on b value. There is, therefore, a limit

to the number of fibre bundle crossings which can be resolved using this method. This

limit has not been determined. However for the detection of simple fibre crossings, or a

limited number of multi-directional bundles, diffusion measurements at high angular

resolution cm reveal a great deal of detail in the variability of the diffision coefficient

contours at high b values. Combined with a priori knowledge of diffision contours for

any orientation of fibres as determined by theoretical calculations, complex fibre bundle

distributions might be resolved.

Another limiting factor will be the difficulty in distinguishing crossing fibres from

kissing fibres (see Fig. 13).

Fig. 13. Crossing and kissing fibres

The ability for diffision measurements at multiple angles to resolve these similar fibre

structures is unclear since the diffision contours will likely appear very similar. Some a

priori anatomical information about fibre tract directions may be required since the

models of white matter suggested here do not take curvatwe of the fibres into account.

3.3 Other MR techniques for the measurement of diffusion

Although this thesis suggests that diffision measurements at multiple angles may be a

promising solution to the problem of diffision in multiple tibre bundles within a voxel,

several other techniques have been deveioped in an attempt to address the complexity of

difision in these regions. Three of these techniques and their possible limitations are

discussed below. Although, diffision spectrurn imaging shows the greatest promise in

resolving fibre structure, both the weighting of eigenvalues and the use of multiple

tenson are also discussed below since they provide insight into the presence of several

fiber tract orientations.

3.3.1 Weighting of Eigenvalues

Current fibre tractography makes use of the direction and magnitude of the largest

eigenvalue/eigenvector combination of the diffision ellipse to determine fibre orientation

within a voxel, and to track the fibre bundle in successive voxels. Wiegeil et al. (13)

suggested using the information in the second and third eigenvalues and vectors to further

characterize molecular motion in regions of higher fibre structure complexity. This

technique, however, does not provide a direct solution to the resolution of fibre

orientation. Instead, it serves simply as an indicator of the degree of anisotropy and

suggests the presence of multiple fibre bundles orientations without the capability to

resoive them.

3.3.2 Diffusion Spectrum Imaging

Difhion Spectnun Imaging is a terni coined by Tuch et al. (12) to describe a novel

technique for the examination of diffision in complex structurai regions. It is based on q-

space imaging, which was first described by Callaghan (1 7), and uses spin displacement

distributions to observe molecular difision. This technique stems fiom the origin of the

signal equation due to diffision. The signal attenuation Eq. 151, which was presented in

Chapter 1, can be rewritten as:

where P(zi( 22, A) is the displacement probability distribution, p(z) is the spin density, and

q=yg6/2n. Q-space is therefore simply an alternative representation of the same data. 1 t is

apparent fiom Eq. [19] that the signal attenuation is simply the Fourier transform of the

displacement probability distribution, where q is the reciprocal space vector:

By taking the inverse Fourier transform of the measured signal attenuation, the spin

displacement distribution can be obtained. In regions of complex fibre structure, such as

crossing fibres, the displacement distribution functions deviate strongly fkom a Gaussian

probability distribution and demonstrate highly irregular shapes, which reflect the

convoluted molecdar motion in these voxels. Whereas conventional MR difision

measwments are based on solving the integral in Eq. [5] assuming a Gaussian

probability distribution of spin displacements, q-space representation of dimision

examines the shapes of these displacement distributions, and therefore does not constrain

the movement of spins to a Gaussian probability distribution. Q-space diffision

measurements are often used to determine the restricted dimension of the diffusing

cornpartment since the full-width-at-ha1 f-maximum (F WHM) of a Gaussian represents

the mean displacement of the spins during the time of the experiment (55-59).

In the case of diffision in white matter, the displacement distributions obtained using

difision spectnun imaging have been show to deviate strongly from the 'free' Gaussian

distribution. The highly lobulated displacement distributions have been used to

characterize diffision in regions of complex fibre structure, and show great promise in

resolving fibre orientations within single voxels.

3.3.3 Multiple Diffusion Tensors

Another method that attempts to describe diffision in regions of complex tibre structure

involves the application of more than one diffision tcnsor (60,61). The rationale for this

technique stems fiom the observation that diffision signal decay in highly structured,

restrictive tissues, such as white matter, is non-rnonoexponential, and is often described

as bi-exponential. A bi-exponential fit to signal decay curves results in two diffision

coefficients to represent diffision along a certain axis within the voxel of interest.

Most often, the measwment of the diffision coefficient is the initial slope or ADC of the

decay curve, and bi-exponentiality is neglected. The presence of this bi-exponentiality,

however, has prompted some researchers into iworporating multiple diffision

coefficients into the description of tissue diffision. Specificaily, severai attempts have

been made to link the two decaying components along an axis with two contributing

water compartments (22-27). The application of two diffision tenson is attempting to

describe diffision decay behaviour based on spins in two compartments within a voxel.

The signal equation becomes:

SE, + (1 [21]

where Di and D2 represent the two diffision tenson, and f represents the weighting

fraction of the two diffising components.

This work was first applied to observe cardiac muscle fibre orientation, and preliminary

results used the anisotropy of the two tensors to compartmentalize the signal (62).

Although this technique seems promising in distinguishing difising water

compartments, it is dificult to separate compartments completely due to the movement

of water across membranes. As such, the use of multip!~ tensors will require fiuther

development and experimental verification before it may be applied to distinguishing

contributions from spins in noncolinear fibre bundles.

3.4 Conclusions

In vivo fibre tractography has widespread potential clinical applications. When combined

with fùnctional imaging for exarnple, it may provide insight into the complex neural

networks associated with higher order cerebral processes by linking functional regions of

the brain. Other highîy structured tissues, such as the myocardium, can also benefit fiom

tractography methods to detect changes in heart muscle structure. Difision MRI is

idealIy suited for these tasks due to its sensitivity to molecular motion in the micron

range. As a result, diffision MRI is an excellent probe of tissue structure and is able to

establish fibre connectivity. The inability to resotve a distribution of fibre tracts within a

voxel, however, is a severe limitation in current tractography using diffusion tensor MRI,

particularly in the brain which is known to have complex branching patterns.

In this thesis, orientational diffusion measurements were used to study fibre orientation.

Diffision measurements at high angular resolution in plastic fibre samples with different

fibre anangements were shown to be reflective of fibre structure and in good agreement

with theoreticai predictions for diffision in a restricted cylindrical geometry. Simulations

of diffision coeficient contours for fibre crossing patterns showed that the direction of

greatest diffision in these measurements does not correspond to fibre orientation.

However, the variability of the measured diffision coefficient with gradient angle does

provide detail about the presence of multiple fibres. Moreover, with the theoretical

calculation for diffision in a single fibre, the difision contours for any distribution of

fibre orientations may be obtained.

Knowledge of the contours for any arrangement of fibre bundles within a voxel would

allow the resolution of fibre orientation by finding a theoretical fit to diffision

measurements at multiple angles. The theoretical model presented in this paper must be

compared with in vivo fibre bundle measurements to determine its adequacy in

representing diffision in tissue. Although some modifications of the model to incorporate

certain white matter diffision characteristics may be required, once these have been

determined, high angular resolution diffusion rneasurements show great promise in

resolving the crossing of fibre tracts within a single voxel.

In vivo fibre tractography with diffision MR is a technique still in its infancy. As such,

many issues remain to be addressed before its establishment as a mutine clinical tool.

This thesis has approached only one of these issues. The plethora of applications for

tractography, however, from the study of white matter disease to the understanding of

basic neuronal connectivity, will ensure the rapid development of this technique as a

powerfùl research and diagnostic tool.

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