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Special issue: Original article

Studying connections in the living human brain

with diffusion MRI

Derek K. Jones*

CUBRIC, Cardiff University Brain Research Imaging Centre, School of Psychology, Cardiff University, Cardiff, Wales, UK

a r t i c l e i n f o

Article history:

Received 7 June 2007

Reviewed 6 September 2007

Revised 24 October 2007

Accepted 9 December 2007

Published online 23 May 2008

Keywords:

Diffusion

MRI

White matter

Tractography

Connectivity

a b s t r a c t

The purpose of this article is to explain how the random walks of water molecules under-

going diffusion in living tissue may be exploited to garner information on the white matter

of the human brain and its connections. We discuss the concepts underlying diffusion-

weighted (DW) imaging, and diffusion tensor imaging before exploring fibre tracking, or

tractography, which aims to reconstruct the three-dimensional trajectories of white matter

fibres non-invasively. The two main classes of algorithm – deterministic and probabilistic

tracking – are compared and example results are presented. We then discuss methods to

resolve the ‘crossing fibre’ issue which presents a problem when using the tensor model

to characterize diffusion behaviour in complex tissue. Finally, we detail some of the issues

that remain to be resolved before we can reliably characterize connections of the living

human brain in vivo.

ª 2008 Published by Elsevier Masson Srl.

1. Introduction

In 1827, Robert Brown – a botanist from Montrose in Scotland,

returned from a triptoAustraliawith variousflorasamples. In-

trigued by the mechanisms of fertilization in plants, he took

a sample of pollen grains from Clarkia pulchella and suspended

them in water beneath his microscope (Fig. 1). On close inspec-

tion, Brown found the grains to be in constant motion – as if

having a life of their own. Indeed, with investigations into the

‘essence of life’ being very much in vogue at the time, Brown

wondered whether he was observing life force itself. However,

as a true scientist, Brown looked for the phenomenon in inor-

ganic material including rock and coal samples, and even

‘rock from the Sphinx’. He observed the same phenomenon, i.e.,

the particles were constantly in motion (Brown, 1828).

We now know that what Brown observed was not the pol-

len grains moving of their own accord, but rather the water

molecules that they were suspended in (invisible under the

light microscope) undergoing constant thermal agitation

known as ‘Brownian motion’, or ‘diffusion’. Diffusion is an

essential physical process for the normal functioning of living

systems.Forexample,the transport of metabolites intocells is

facilitatedby diffusion.Thisphenomenon, omnipresent in the

water in living tissue, has the potential, through diffusion-

weighted (DW) magnetic resonance imaging, to provide in-

sights into cell physiology, cell structure and potentially the

connections of the living human brain.

2.

microstructure?

The random walk as a probe of tissue

Imagine a cube-shaped volume of 2.5?2.5? 2.5 mm in di-

mension. There are approximately 1020water molecules

* CUBRIC, School of Psychology, Cardiff University, Park Place, Cardiff CF10 3AT, UK.

E-mail address: jonesd27@cf.ac.uk

0010-9452/$ – see front matter ª 2008 Published by Elsevier Masson Srl.

doi:10.1016/j.cortex.2008.05.002

available at www.sciencedirect.com

journal homepage: www.elsevier.com/locate/cortex

cortex 44 (2008) 936–952

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contained within this volume, each undergoing a random

walk as part of the diffusion process. Here, by random walk,

we mean that each molecule stays in a particular place for

a fixed time, T, before moving to a random, new location in

space. This process continues for each molecule so that a ran-

dom path is drawn out in three-dimensional space. It would

perhaps seem impossible to characterize the behaviour of

this set of molecules undergoing such random behaviour. It

is certainly impossible to predict the pathway that any one

of these molecules will take. Fig. 1b shows a simulation of

a randomwalk for five single particles in an isotropicmedium,

such as our cube of water. It is clear that we cannot accurately

predict any single molecule’s position at a given time. How-

ever,in 1905Einsteinprovedthat, providedthenumberof par-

ticles is sufficiently large, at least one aspect of the behaviour

could be characterized, namely the squared displacement of

molecules from their starting point over a time, t, averaged

over all the molecules in the sample, Cr2D is directly proportional

to the observation time (Einstein, 1905). The constant of pro-

portionality is the self-diffusion coefficient. In equation form

(the ‘‘Einstein’s equation’’):

Cr2D ¼ 6Dt

The distribution of squared displacements takes a Gaussian

form, with the peak being at zero displacement and the

probability of displacing a given distance from the origin is

thesame–nomatterinwhichdirectionitismeasured(Fig.1d).

For the cube of water at body temperature (37?C), the diffu-

sion coefficient of water is 3? 10?3mm2s?1. Thus, if we

observe water molecules for, say, 30 ms, they will have

displaced, on average, 25 mm in all directions.

As stated in Section 1, DW magnetic resonance imaging

(MRI) utilizes this motion to probe tissue microstructure.

The astute reader, new to the topic, may then pose the ques-

tion ‘‘If we are looking at the diffusion of water, and we know

what the diffusion coefficient of water at body temperature is a

(1)

Fig. 1 – (a) Robert Brown, 1773–1858. (b) Brown’s original microscope with which he reportedly observed the pollen of Clarkia

pulchella in constant motion. (c) Simulation of trajectories of five molecules undergoing a random walk in an isotropic

medium, after 10, 20, 40 and 80 steps. (d) A histogram of displacement from the origin of 1 million molecules, after 100, 400,

900 and 1600 steps. The full-width at half-maximum (FWHM) scales in the ratio of 1:2:3:4, which mirrors the ratio of

O100O400:O900:O1600, demonstrating the consistency of Einstein’s equation for Gaussian diffusion.

cortex 44 (2008) 936–952

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constant – then how can we possibly use that to garner information

abouttissue microstructure?’’ The answerlies in Einstein’sequa-

tion (Eq. (1)), which says that the mean-squared displacement

is directly proportional to the observation time. It is important

to note that in DW MRI, we do not measure the diffusion coef-

ficientdirectly,butthemeandisplacement ofwatermolecules

within each three-dimensional volume element, or voxel, that

forms the image (typically, these are cube-shaped and about

2.5?2.5? 2.5 mm in dimension). The presence of cell mem-

branes, inclusions, macromolecules and so forth present in

tissue serve to hinder the pathway of the molecules undergo-

ing their random walks. As a result, their overall displacement

from their starting point in a fixed period of time is reduced

and compared to their mean displacement when they were

in ‘free’ water. Thus, it appears to us that the diffusion coeffi-

cient is lower than it is – which is partly why the term ‘appar-

ent diffusion coefficient’ (or ADC ) was coined to reflect

the fact that we realize that we are subject to the effects

of hindrances, etc. The average ADC in tissue is around

0.7?10?3mm2s?1, about 4 times smaller than in free water.

ADWmagneticresonance(MR)sequencesensitizestheMR

signaltodiffusionbyimposingagivenphasetoamoleculethat

is dependent on its overall displacement (Stejskal and Tanner,

1965). Under the random walk process that is diffusion, we get

a distribution of displacements and thus a distribution of

phases.Thisspreadofphasesmeansalossofsignalcoherence

and therefore a reduction in signal amplitude, which in an im-

age, means that the image appears darker. The greater the

spread of displacements (the higher the ADC), the greater the

spread of phases – and thus the greater the loss of signal –

and the image appears dark. Conversely, the lower the rate of

diffusion, the lower the spread of phases – and thus the lower

the loss of signal and the image appears bright.

It is clear that tissue microstructure fundamentally affects

the apparent diffusion properties of water and diffusion

should therefore act as a sensitive probe to any changes in

cellular structures that alter the displacement per unit time.

Thus, the introduction of diffusion imaging (Le Bihan and

Breton, 1985) was met with enthusiasm as a non-invasive

method of gaining new contrast within the brain. The most

useful clinical application to date is the use of the DW scan

in acute ischaemia in which there is a reduction in the

voxel-averaged displacement of water molecules per unit

time, hence a reduction in the ADC, therefore less signal

attenuation – and the lesion appears hyper-intense (Moseley

et al., 1990a).

About the same time as Moseley’s initial observations of

the reduced ADC in ischaemia, it was noted that the ADC in

certain regions of the mammalian brain appeared to depend

on the direction of the applied diffusion-encoding gradient

(Moseley et al., 1990b). In other words, the ADC was direction-

ally dependent. This effect had been known for some time in

ex vivo samples of muscle and brain tissue dating back to the

pioneering work of Hansen (1971) and Cleveland et al. (1976).

Shortly after Moseley’s observation in the cat brain, the

directional dependence of the ADC was reported in human

white matter by Doran et al. (1990) and Chenevert et al.

(1990). This is illustrated in Fig. 2, which shows the same

(human) brain imaged 3 times, each time with the diffusion-

encoding gradient applied along one of three orthogonal axes.

In certain regionsof the brain, the DW intensityis the same

in all three images suggesting that the ADC is the same in all

directions. Diffusion in these cases is described as isotropic.

However, in the regions highlighted by arrows this is not the

case and diffusion in these regions is referred to as aniso-

tropic. From just these three DW images, we can infer a sub-

stantial amount of information about the structure indicated

by the arrows. First, the large differences in DW intensities

that are observed as the direction of the diffusion-encoding

gradient is changed, suggest that the tissue here is highly

ordered on the voxel scale. Second, as there is high signal

attenuation in 2a (in which the diffusion-encoding gradients

were applied in a left–right orientation), we can infer that

diffusion is relatively unhindered along this direction. Con-

versely, in the two perpendicular orientations (Fig. 2b and c),

the signal attenuation is much less, indicating that the

mean-squared displacement per unit time is reduced and

that something is therefore hindering the displacement of

water molecules along these orthogonal axes. Therefore,

from just these three images, we are able to infer an ordered

structure which has predominantly a left–right orientation.

These inferences are entirely consistent with the fibres of

the corpus callosum, a white matter (WM) structure which

passes through this region (Dejerine, 1895; Crosby et al., 1962).

3.

anisotropy?

What is the source of diffusion

Initial suggestions for the mechanisms underlying diffusion

anisotropy in WM included the myelin sheath (Thomsen

Fig. 2 – Effect of changing the direction of the diffusion-

sensitizing gradients on the DW intensity (top row) and

computed ADC (bottom row). The figure shows the same

brain slice, with gradients applied in the left–right

direction (a and d), anterior–posterior direction (b and e),

and superior–inferior direction (c and f). The amount of

diffusion weighting (b[1000 s/mm2) was the same in all

three cases.

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et al., 1987), local susceptibility gradients (Hong and Dixon,

1992; Lian et al., 1994), axonal cytoskeleton and fast-axonal

transport. Myelin itself, however, does not appear to be neces-

sary in order for diffusion to be anisotropic in the brain. This

conclusion was first suggested by the demonstration of aniso-

tropic diffusion in the immature rat brain where there was no

histological evidence of myelin (Wimberger et al., 1995; Prayer

et al., 1997). Furthermore, Gulani et al. (2001) reported aniso-

tropic diffusion in the spinal cord of a myelin-deficient rat.

In the mid-1990s, Christian Beaulieu and Peter Allen con-

ducted a series of experiments to try to elucidate the origin

of anisotropy in WM (Beaulieu and Allen, 1994a, 1994b, 1996)

and were able to rule out the effects of susceptibility induced

gradients, axonal cytoskeleton and fast-axonal transport.

They concluded that the main determinant of anisotropy in

nervous tissue is the presence of intact cell membranes and

that myelination serves to modulate anisotropy. For a thor-

ough and excellent review, see Beaulieu (2002).

4. The diffusion tensor

It should be clear that, in contrast to our cube of free water,

when we have ordered tissue in our sample, we can no longer

characterize the behaviour of the water molecules with a sin-

gle ADC. The ADC we measure will depend on the direction in

which we measure it. The more ordered the tissue is within

the sample, the more the ADC will depend on the measure-

ment direction. The next most complex model to characterize

Gaussian diffusion in which the displacements per unit time

are not the same in all directions is the diffusion tensor.

This is a 3 ?3 symmetric matrix of numbers that character-

izes three-dimensional displacements. The diagonal elements

correspond to diffusivities along three orthogonal axes, while

the off-diagonal elements correspond to the correlation be-

tween displacements along those orthogonal axes. For those

less mathematically inclined, the tensor can be thought of in

another way. Consider a gedanken experiment in which we

place a drop of ink in the centre of our cube of water. As the

ink particles displace over time, the outer profile of the dis-

placements would resemble a spherical envelope, since diffu-

sion in isotropic media is isotropic. However, in an anisotropic

medium, the ink particles would diffuse further along the

principal axis of the anisotropic medium than in a perpendic-

ular orientation. Clearly, the displacement profile can no lon-

ger be described by a sphere and is more correctly described

by an ellipsoidal envelope with the long axis parallel to the

long axis of the anisotropic medium (Fig. 3). The diffusion ten-

sor is often thought of in terms of this ellipsoid – a surface rep-

resenting the distance that a molecule will diffuse to with

equal probability from the origin. The diffusion tensor formal-

ism provides an ‘internal reference frame’, namely the eigen-

system. The principal axes of the ellipsoid are given by the

eigenvectors, and the lengths are given by the diffusion dis-

tance in a given time, t. Eq. (1) shows that the displacement

in a given time is proportional to the square root of the diffu-

sivity. Hence, the ellipsoid axes are scaled according to the

square root of the eigenvalues. The tensor is estimated by col-

lecting a number of samples of the DW signal, the direction in

which the diffusion sensitization is applied being varied for

eachsample.Thetensorissubsequently estimated fromthese

signals using multivariate regression. Full details are provided

in Beaulieu and Allen (1994a).

5. Trace

The traceofthediffusiontensorisequalto thesum ofitsthree

eigenvalues and provides a rotationally invariant index of the

overall amount of diffusivity within each image voxel. It is

equivalent to 3 times the average diffusivity and so a more

commonly reported index is the ‘mean diffusivity’ which is

the trace divided by three. The trace can also be computed

by taking the average of three ADCs measured along any three

orthogonal axes.

Fig. 3 – Schematic of the diffusion ellipsoid. The ellipsoid is the envelope where a spin – placed at its centre – will diffuse to

with equal probability. The axes are scaled according to the square root of the eigenvalues, l1l2, and l3, and the principal

axes are given by the corresponding eigenvectors, 31, 32, and 33. The eigenvalues are sorted according to their magnitude

such that l1‡l2‡l3. The tensor in (a) is prolate, where l1>l2[l3. The principal eigenvector is designated as 31. In (b), the

tensor is oblate, i.e., l1[l2>l3and the principal eigenvector is therefore poorly defined.

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6. Diffusion anisotropy

The eigenvector/eigenvalue system provides a framework

that rotates with the diffusion tensor, and thus any index of

anisotropy that is defined within this framework, will be inde-

pendent of the orientation of the tensor with respect to the

laboratory frame of reference. By far, the two most commonly

used indices are the relativeanisotropy (RA) and fractional an-

isotropy (FA) (Basser and Pierpaoli, 1996).

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l

RA ¼

ffiffiffi

1

3

r

ðl1? lÞ2þðl2? lÞ2þðl3? lÞ2

q

(2)

FA ¼

ffiffiffi

3

2

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

l2

ðl1? lÞ2þðl2? lÞ2þðl3? lÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where l ¼ ðl1þ l2þ l3Þ=3.

q

1þ l2

2þ l2

3

q

(3)

The numerator for both terms is the same and is related to

the variance of the three eigenvalues about their mean. The

FA index normalizes this measure by the magnitude of the

tensor as a whole. Just as the magnitude of a vector can be

found from the sum of the squares of its individual compo-

nents, the magnitude of the tensor is found from the sum of

the squares of its eigenvalues. Thus, FA measures the fraction

of the tensor that can be assigned to anisotropic diffusion. The

FA index is appropriately normalized so that it takes values

from 0 (when diffusionis isotropic) to 1 (whendiffusioniscon-

strained along one axis only). The denominator of the RA

index is simply the mean diffusivity. This index is mathemat-

ically identical to a coefficient of variation, i.e., standard devi-

ation divided by the mean. Fig. 4 shows an example FA map in

axial and coronal formats. The higher the pixel intensity, the

higher the FA, thus gray matter and cerebro-spinal fluid

(CSF) appear dark, while white matter appears, on the whole,

bright. Voxels that contain fibres that run in highly parallel

Fig. 4 – Example FA images from the whole brain in (a) coronal and (b) axial orientations. The pixel intensity is directly

proportional to the FA which ranges from zero (diffusion is isotropic) to the theoretical limit of 1 (where water molecules are

only free to move along one axis).

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