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INSTITUTE OF PHYSICS PUBLISHING INVERSE PROBLEMS

Inverse Problems 19 (2003) 1031–1046 PII: S0266-5611(03)56256-6

Persistent angular structure: new insights fromdiffusion magnetic resonance imaging data

Kalvis M Jansons1 and Daniel C Alexander2

1 Department of Mathematics, University College London, Gower Street,London WC1E 6BT, UK2 Department of Computer Science, University College London, Gower Street,London WC1E 6BT, UK

E-mail: [email protected]

Received 14 November 2002, in final form 13 June 2003Published 22 August 2003

Online at stacks.iop.org/IP/19/1031

Abstract

We determine a statistic called the (radially) persistent angular structure (PAS)

from samples of the Fourier transform of a three-dimensional function. The

methodhasapplications in diffusionmagnetic resonanceimaging (MRI),which

samples the Fourier transform of the probability density function of particle

displacements. The PAS is then a representation of the relative mobility of

particles in each direction. In PAS-MRI, we compute the PAS in each voxel

of an image. This technique has biomedical applications, where it reveals the

orientations of microstructural fibres, such as white-matter fibres in the brain.

Scanner time is a significant factor in determining the amount of dataavailable in clinical brain scans. Here, we use measurements acquired for

diffusion-tensor MRI, which is a routine diffusion imaging technique, but

extract richer information. In particular, PAS-MRI can resolve the orientations

of crossing fibres.

We test PAS-MRI on human brain data and on synthetic data. The human

brain data set comes from a standard acquisition scheme for diffusion-tensor

MRI in which the samples in each voxel lie on a sphere in Fourier space.

1. Introduction

Diffusion magnetic resonance imaging (MRI) measures the displacements of particles that are

subject to Brownian motion within a sample of material. The microstructure of the material

determines the mobility of these particles, which is normally directionally dependent. The

anisotropy of particle displacements gives information about the anisotropy, on a microscopic

scale,of thematerial in which theparticlesmove. Theprobabilitydensity function p of particle

displacements (in three-dimensional space) reflects the anisotropy of particle displacements.

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1032 K M Jansons and D C Alexander

Diffusion MRI and, in particular, diffusion tensor magnetic resonance imaging (DT-

MRI) [1] is popular in biomedical imaging because of the insight it provides into the

microstructure of biological tissue [2, 3]. In biomedical applications, the particles are usually

water molecules. Water is a major constituent of many types of living tissue and water

molecules in tissue are subject to Brownian motion, i.e. the random motion driven by thermalfluctuations. The microstructure of the tissue determines the mobility of water molecules

within the tissue. Our primary interest is in using diffusion MRI to probe the microstructureof

brain tissue. However, the new approach we introduce here could be used much more widely.

For the following discussion, it is useful to have in mind the important lengthscales of the

brain tissue and the measuring process:

(i) The voxel volume is of order 10−9 m3.

(ii) The diffusion time is of order 10−2 s and, over this time, the root-mean-squared

displacement of water molecules is in the micrometre range.

(iii) The diameters of axon fibres in human white matter can reach 2 .50 × 10−5 m, but most

fibres have diameter less than 10−6 m [3–5].

(iv) The packing density of axon fibres in white matter is of order 1011

m−2

[3–5].

White matter in the brain contains bundles of parallel axon fibres. At the diffusion

lengthscale applicable in MRI, the average displacement of water molecules along the axis of

the fibres is larger than in other directions. The function p has ridges in the fibre directions in

a material consisting solely of parallel fibres on a microscopic scale.

In diffusion MRI, an inverse problem is solved to determine from MRI measurements

features of the material’s microstructure that affect the distribution of particle displacements.

One useful statistic is the root-mean-squared particle displacement

R =

R3

|x|2 p(x) dx

12

. (1)

Statistics based on the root-mean-squared particle displacement (see section 2) highlight, for

example, regions of brain tissuedamaged by stroke[6],head trauma[7],and neurodegenerative

diseases such as multiple sclerosis [8], in which the value of R often increases. This increase

is thought to occur because of cell loss, which reduces the hindrance to the movement of water

molecules.

Statistics that depend on the anisotropy of p provide information about the anisotropy

of the material’s microstructure. Scalar statistics of this kind also highlight tissue that has

been affected by neurodegenerative diseases [8, 9]. The reduction in anisotropy observed in

affected areas is thought to result from microstructural changes, such as loss of axons and

gliosis (scarring due to axonal damage).

Diffusion-tensor MRI, which we discuss further in section 2, is a standard technique in

diffusion MRI. In DT-MRI, a Gaussian profile for p is assumed. However, this simple model

is often a poor approximation in material with complex microstructure, for example in areas

of the brain where white-matter fibres cross. Here we introduce a technique called PAS-MRI.In PAS-MRI, we determine a feature of p which we call the (radially) persistent angular

structure (PAS). The PAS represents the relative mobility of particles in each direction, which

can have many peaks. In brain imaging, these peaks can be used to determine the orientation

of white-matter fibres. The new technique can resolve the directions of crossing fibres, which

DT-MRI cannot. We demonstrate this in simulation and show results from white-matter fibre-

crossings in the human brain. The human brain data were acquired originally for DT-MRI

using a clinical acquisition sequence.



Persistent angular structure 1033

A common application of diffusion MRI is fibre tracking, or ‘tractography’ [10, 11]. The

goal of tractography is to map the connectivity of a piece of fibrous material, such as a brain,

using MRI data that depends on the local fibre orientations. Most tractography algorithms

use DT-MRI and are unreliable at fibre-crossings. We expect similar algorithms based on

PAS-MRI to produce more detailed and reliable results.In section 2, we provide some background on diffusion MRI. In section 3, we list the

parameters used to acquire the human brain data we use to test PAS-MRI and comment on

its properties. In section 4, we use an algorithm based on the maximum entropy method to

define the PAS of p. We do not attempt a full maximum entropy reconstruction of p, but

focus instead on extracting just the PAS. In PAS-MRI, we determine the PAS by fitting its

parameters to the MRI measurements in each image voxel. We discuss the implementation of

PAS-MRI in section 5. In section 6, we test PAS-MRI on synthetic data and then show results

from the human brain. We conclude in section 7 with some comparisons of PAS-MRI with

other diffusion MRI techniques and a discussion of areas for further work.

2. Background

In diffusionMRI,measurements arecommonlymadeusing theStejskal–Tanner pulsed gradient

spin echo (PGSE) method [12], which samples the Fourier transformof p. Two magneticfield

gradient pulsesof duration δ and separation are introducedto thesimplespin-echosequence.

Assuming rectangular pulse profiles, the associated wavenumber is q = γ δg, where g is the

component of the gradient in the direction of the fixed field B0 and γ is the gyromagnetic

ratio [13, 14].

TheMRI measurement A(q;X , ) is definedat each locationX of a finite regular image

grid in three-dimensional space and depends on the wavenumber,q, and the pulse separation,

. In PAS-MRI, we treat each voxel separately and use a fixed . We therefore drop the

dependence of A on and X from the notation.

The normalized MRI measurement A(q) is given by A(q) = ( A(0))−1 A(q). When

−1

δ is negligible [13, 14],

A(q) =

R3

p(x) exp(iq · x) dx. (2)

We ignore non-zero −1δ effects [15], as even when −1δ is not negligible, these effects will

not significantly change the principal directions extracted by PAS-MRI.

The evolution of p is due to the combination of advection and diffusion. The main

contribution to advection is likely to come from movements of the sample with respect to the

MRI scanner, but there will be other contributions. In this study, however, we are interested

in the directional structure due to diffusion alone, and not the effects of advection. Assuming

the local advection velocity is zero, we expect that p(x) = p(−x) is a good approximation,

so that (2) becomes

A(q) = R3

p(x) cos(q · x) dx. (3)

One way to proceed is to measure A at each location on a regular grid of wavenumbers,

q, and then to use a fast Fourier transform (FFT) algorithm to obtain values of p on a discrete

set of displacements, x. This approach is called ‘q-space imaging’ and is commonly used to

probe p in one dimension,as in [13, 16, 17]. Wedeen et al [18] succeed in acquiring data from

healthy volunteers for a direct FFT inversion in three dimensions. They measure A with 500

different wavenumbers in each scan. To acquire all the measurements in scanner time that is




tolerable for conscious subjects, they increase the voxel size from the usual size in MRI (of

order 10−9 m3) to 64 × 10−9 m3.

We shall refer to the motion of particles by diffusion alone in a medium for which the

diffusion tensor is constant and homogeneous as simple diffusion. In simple diffusion, the

probability density function of particle displacements has a Gaussian profile [19] so that p(x) = G(x;D, t ), where

G(x;D, t ) = ((4π t )3 det(D))− 12 exp

−xT

D−1x

4t

, (4)

D is the diffusion tensor and t is the diffusion time.

In DT-MRI, the apparent diffusion tensor is determined on the assumption of simple

diffusion. With the simple diffusion model, (2) gives

A(q) = exp(−t qTDq). (5)

Taking the logarithm of (5), we see that each A(q) provides a linear constraint on the elements

of D. To fit the six free parameters of D, we need a minimum of six measurements of A(q)

with independent q. Because of the effects of noise, acquiring more than six A(q) provides

a better estimate of D. A common approach [20] is to acquire M measurements of A(0)

together with measurements A(q j ), 1 j N , for which δ, , and |q| are fixed, but each q̂ j

is unique. Then A(q j ) = ( ¯ A(0))−1 A(q j ), where ¯ A(0) is the geometric mean of the A(0)

measurements. In the literature, 1 M 20 and N < 100. Fewer measurements of A(q)

are required for DT-MRI than for the three-dimensional q-space technique of Wedeen et al

[18]. Thus, in DT-MRI, smaller voxel sizes can be used for images acquired in the same

scanner time. The q̂ j are chosen so that they are well separated on the sphere and are not

directionally biased [21].

With the simple diffusion model, R = (2t Tr(D))12 . Medical researchers often work with

Tr(D) directly rather than R to highlight damaged brain tissue, as in [6–8]. Scalar indices of

diffusion anisotropy can be derived from the diffusion tensor. A popular anisotropy index is

the fractional anisotropy ν [2], which is given by

ν =

32

3i=1

(µi − 13

Tr(D))2

12 3

i=1

µ2i

− 12

, (6)

where µi , i = 1, 2, 3, are theeigenvalues of D. Manytractographyalgorithmsuse the principal

eigenvector of D as an estimate of fibre orientation [10, 11].

A drawback of DT-MRI is that the simple diffusion model is poor in many regions,

in particular those in which fibres cross within a voxel. With more than six A(q) with

independent wavenumbers, some departures from the simple diffusion model may be detected

and observations of such departures are in the literature, e.g. [22–24].

One alternative to the simple diffusion model has p as a mixture of Gaussian

densities [22, 25]:

p(x) = Mn(x;D1, . . . ,Dn; a1, . . . , an; t ), (7)

where

Mn(x;D1, . . . , Dn; a1, . . . , an; t ) =n

i=1

ai G(x;Di , t ), (8)

each ai ∈ [0, 1] and

i ai = 1.

Equation (7) could, for example, model a macroscopically homogeneous system, which

consists of n interlocking subsystems on a microscopic scale, where we suppose that the rate




of exchange of particles between the different subsystems is negligible. In this case, the ai

are the probabilities that a randomly chosen test particle is in subsystem i of the partitioned

medium, and the Di are the corresponding diffusion tensors. A simple extension to this model

would allow exchange of particles between subsystems.

3. Data

In this section, we give details of the data we use for testing PAS-MRI, acquired using a

standard DT-MRI sequence for human brain imaging.

The University College London Institute of Neurology provided data sets from healthy

volunteers together with details of their acquisition sequence [26]. All subjects were scanned

with the approval of the joint National Hospital and Institute of Neurology ethics committee

and gave informed written consent.

For these data sets, M = 6 and N = 54. Also = 0.04 s, δ = 0.034 s and

|g| = 0.022 T m−1, which gives |q| = 2.0 × 105 m−1. The 54 values of q̂ j come from

an electrostatic spherical point set with equal and opposite pairs (see section 5.1) from which

we take a point from each equal and opposite pair.Thedata sets contain 60 axial slices, with separation 2.3×10−3 m. Each row of each slice

of the acquisition undergoes a linear phase correction step. In each slice, the 62 × 96 array of

measurements is padded with zeros to a 128 × 128 grid from which an image is created via a

FFT. After the Fourier transform, the phase is discarded by taking the modulus of the complex

value in each voxel, so phase errors are unimportant. In each slice, thevoxels are 1.7 ×10−3 m

squares.

The total scan time is around 20 min. During this time subject motion can cause slight

misalignments among the 60 MRI volumes. Image distortion can also arise from magnetic

field inhomogeneity. An imageregistration scheme basedon automatedimageregistration [27]

improves the alignment of each set of corresponding slices in the measurement volumes. One

of the measurement volumes for which q = 0 provides a reference set of slices. Automated

image registration computes two-dimensional affine transformations that minimize the sum of

square differences between each slice of the other volumes and the corresponding reference

slice.

4. Method

We do not know the probability density function, p, directly, but rather have a finite set of

values for the Fourier transform of p. In the test data that we use, all of the wavenumbers have

the same modulus and the same diffusion time, i.e. all the points lie on a sphere in Fourier

space.

We use a method based on the principle of maximum entropy [28] to obtain an expression

for the function that contains the least information subject to the constraints from the data. We

use −S[ p], where S is the entropy [28], for the information content of a probability density

function p defined over a set . Thus the mathematical expression for the information contentof p is

I [ p] =

p(x) ln p(x) dx. (9)

Since we have only Fourier components of p for a small set of points, which may lie on

a sphere in Fourier space, we are unable to extract any useful information about the radial

structure of p and, in any case, the greatest interest is in the angular structure. To extract




useful information about the angular structure in a computationally efficient way, we restrict

attention to determining a probability density function of the form

p(x) = ˜ p(x̂)r −2δ(|x| − r ), (10)

where δ is the standard one-dimensional δ distribution, r is a constant and x̂ is a unit vectorin the direction of x. In a sense, we are projecting the angular structure from all radii onto the

sphereof radius r , and ignoring any information about the radial structure in the data, which is

often very limited. The final result is weakly dependent on the choice of r (as discussed later)

but the important features of the angular structure are not, provided we are inside the range of

r for which the method is reliable.

We determine a function of the above form that has minimum relative information with

respect to p0 = (4πr 2)−1δ(|x| − r ) and subject to the constraints of the data. We call ˜ p

the (radially) PAS. The domain of ˜ p is the unit sphere, as it represents only orientational

information.

Therelativeinformationof theprobabilitydensityfunction pwith respect to theprobability

density function p0 is given by

I [ p; p0] =

p(x) ln

p(x)

p0(x)

dx. (11)

The constraints on p from the data, given by equation (2), can be incorporated into the

expression above using the method of Lagrange multipliers to yield

I [ ˜ p] =

˜ p(x̂) ln ˜ p(x̂) − ˜ p(x̂)

N j=1

(λ j exp(iq j · r x̂)) − ˜ p(x̂)µ

dx̂, (12)

where q j , 1 j N , are the non-zero wavenumbers for the MRI measurements, the λ j are

Lagrange multipliers for the constraints from the data and the Lagrange multiplier µ controls

the normalization of ˜ p. The integral in (12) is taken over the unit sphere. Taking a variational

derivative δ I [ ˜ p] and solving δ I [ ˜ p] = 0, we find that the information content, I [ ˜ p], has a

unique minimum at

˜ p(x̂) = exp

λ0 +

N j=1

λ j exp(iq j · r x̂)

, (13)

where λ0 = µ − 1.

We need to solve ˜ p(x̂) exp(iq j · r x̂) dx̂ = A(q j ) (14)

for the λ j , where the integral is taken over the unit sphere.

If we assume x → −x symmetry, which is typically justified, we can simplify the

expression for ˜ p by noting that

˜ p(x̂) = exp

λ0 +

N j=1

λ j cos(q j · r x̂)

(15)

and absorbing the factor of 2 into the definition of λ j , 1 j N . This also helps removesome noise from the numerical analysis.

Note that this is not a maximum entropy reconstruction of p itself, but a method that

extracts the directional information from the MRI data set, which may be small and will often

be restricted to a sphere in Fourier space. Theadvantage of this approach,as shown later, is that

we obtain a robust statistic that corresponds well to the physiological structure of the human

brain. This technique will have applications in other areas where the directional structure on

a microscopic scale is of interest.




5. Implementation

In this section, we discuss the implementation of the fitting procedure.

5.1. Spherical point sets

A spherical point set P is a finite set of points on the unit sphere in three-dimensional space.

The size of P is the number of elements it contains. In the implementation of PAS-MRI, we

use several different types of spherical point set, which we define in this section.

5.1.1. Pseudo-random sphericalpointsets. Pseudo-randomsphericalpointsetshave pseudo-

random elements taken from a uniform distribution on the unit sphere. The spherical point set

U (n, s) is the pseudo-random spherical point set with size n and seed s for the random number

generator.

The spherical point set U P (n, s) is the pseudo-random spherical point set with equal and

opposite pairs. We define

U P (2n, s) = U (n, s) ∪ {−y : y ∈ U (n, s)}. (16)

5.1.2. Electrostatic spherical point sets. The spherical point set E P (n) is the electrostatic

spherical point set with equal and opposite pairs and size n. To choose E P (n), we find local

minima of the electrostatic energy using a Levenberg–Marquardt algorithm while maintaining

theconstraint of equalandoppositepairs. We runthealgorithm with each startingconfiguration

U P (n, i) for 1 i 500, and choose the final configurationwith the lowest energy for E P (n).

The spherical point sets E P (n) exist only for even n.

5.1.3. Covering spherical point sets. Covering spherical point sets have putatively minimal

covering radius. The covering radius C (P ) of a spherical point set P is defined by

C (P

)=

max|x|=1

miny∈P |x

−y

|. (17)

Thesphericalpoint setC I (n) is the coveringsphericalpoint setwith icosahedral symmetry

and size n. These spherical point sets have putatively minimal covering radius subject to the

constraintof icosahedralsymmetry. We choose C I (n) fromtheHardin etal ‘tables of spherical

codes with icosahedral symmetry’ [29] for values of n for which they are available.

5.1.4. Archimedean spherical point sets. We construct what we call Archimedean spherical

point sets using Archimedes’ well-known area-preserving mapping from the unit sphere to the

unit cylinder truncated to just contain the sphere. In the natural cylindrical coordinates, the

mapping takes a point on the sphere and projects it radially to the cylinder. To produce the

spherical point set, we place a point at the centre of each rectangle of a rectangular mesh on

the cylinder and map the points back onto the sphere.

The spherical point set A(k 2) is the Archimedean spherical point set with size k 2. Thespherical point set A(k 2) comes from a regular n = k × k rectangular mesh on the cylinder

that is aligned with the cylinder axis. Thus A(n) exists if and only if n is a perfect square.

5.2. Fitting procedure

In PAS-MRI, we use a Levenberg–Marquardt algorithm [30] to find the PAS, ˜ p, by fitting the

(λ j ) N j=0 in (13) to the data using (14). We model the distribution of the noise on the modulus




of each MRI measurement by N (0, σ 2) to construct the objective function for the algorithm.

We propagate these errors through

A(q j ) = ( ¯ A(0))−1 A(q j )

to obtain

σ j = β

1 +

A(q j )2

M

12

, (18)

where σ j is the standard deviation of A(q j ) and β = ( ¯ A(0))−1σ .

The objective function for the Levenberg–Marquardt algorithm is

χ 2((λi ) N i=0) =

N j=0

A(q j ) − ˜ A(q j ; (λi ) N

i=0)

β−1σ j

2

, (19)

where q0 = 0,

˜ A(q j ; (λi ) N i=0) =

˜ p(x̂) cos(q j · r x̂) dx̂ (20)

and ˜ p implicitly depends on the values of the (λi ) N i=0. In (19), the j = 0 term imposes a soft

normalization constraint on ˜ p. We set A(q0) = 1 and choose the weighting σ 0 using (18).

The constant factor β in (19) simplifies the expression for the weightings of the terms without

affecting the locations of the minima of χ 2.

We observe that reducing σ 0 has the effect of inflating the PAS so that its peaks are less

sharp. This inflation does remove some spurious principal directions that appear in ˜ p when p

is close to isotropic (see section 6.1), but can mask weak but genuine angular structure.

To start theminimizationalgorithm,we setλ0 = − ln(4π) and λ j = 0, j = 1, . . . , N . We

terminate thealgorithmafter an iteration in which thevalueof theobjective functiondecreases,

but the decrease is less than 10−5.

We compute thevalueof theobjective function anditsderivativesby numerical integration

at each iteration of the minimization algorithm. For a spherical point set P , we approximate

the integral of a function f over the unit sphere as follows: f (x̂) dx̂ =

4π

n

y∈P

f (y), (21)

where n is the size of P .

The size of the spherical point set required to approximate integrals using (21) to a given

level of accuracy depends on the integrand, f . We store a database of spherical point sets Pi ,

1 i L , in which the size of the spherical point set increases with the index, i . We first run

the minimization algorithm using the smallest spherical point set, P1. After termination, we

compute the errors between numerical integrals computed using P1 and those computed using

a larger test spherical point set Q. If any of these errors is above a preset threshold, we rerun

the minimization algorithm using the next largest spherical point set. We repeat this procedure

until wefinda sphericalpoint setlargeenough,or flagthefailureto do so. We store sixsphericalpoint sets in the database: P1 = C I (1082), P2 = C I (1922), P3 = C I (4322), P4 = C I (8672),

P5 = C I (15872) and P6 = C I (32672). The test spherical point setQ = A(3502).

5.3. Extraction of principal directions

Once we have found the PAS, ˜ p, from the data, one property of interest is its set of principal

directions, since we expect these to indicate the orientation of white-matter fibres in the brain.




When we assume x → −x symmetry, as we do here, the peaks of ˜ p appear in equal and

opposite pairs. In this case, we do not distinguishx and −x. Principal directions of ˜ p are then

non-directed lines in the directions of the pairs of peaks. In cases where we cannot assume

x → −x symmetry, we would work directly with peak directions of ˜ p.

To find the principaldirections of ˜ p, we compute ˜ p(s) for each point s in a spherical pointset S . We then find the set

M =x ∈ S : ˜ p(x) = max

s∈T (x)˜ p(s)

, (22)

where

T (x) = {s ∈ S : |s − x| < ρ} ∪ {s ∈ S : |s + x| < ρ} (23)

and ρ is a constant, which we set to 0.3. The spherical point set S is the unionof 1000 random

rotations of a set containing a point from each equal and opposite pair of vertices of a regular

icosahedron with vertices on the unit sphere.

The set M contains estimates of the principal directions of ˜ p. We refine these estimates

by searching for local maxima of ˜ p using Powell’s method [30] starting at each element of M.

Finally, we discard tiny principal directions that occasionally appear for which the value of ˜ pis less than the mean of ˜ p.

6. Experiments and results

In this section, we demonstrate PAS-MRI using the common DT-MRI scheme where the q j

for the data are on a sphere. For convenience, we non-dimensionalize all lengths with |q|−1,

which is a natural lengthscale. We also assume x → −x symmetry of p and equation (13)

thus becomes

˜ p(x̂) = exp

λ0 +

N j=1

λ j cos(q̂ j · r x̂)

.

In a more general setting where the q j do not lie on a sphere, a typical or average wavenumberwould be used for the non-dimensionalization.

We test PAS-MRI on synthetic data and then show results from human brain data.

6.1. Synthetic data experiments

Givena function p with Fourier transform F , we generatea syntheticMRImeasurement A(q)

by setting

A(q) = |F (q) + c|, (24)

where the real and imaginary parts of c ∈ C are independent identically distributed random

variables with distribution N (0, σ 2). For a particular test function, we generate M synthetic

A(0) measurements and a single A(q j ) for each j = 1, . . . , N . We then compute ¯ A(0)

and A(q j ). Unless otherwise stated, we emulate the sequence used to acquire the human braindata so that M = 6, N = 54, |q| = 2.0 × 105 m−1, t = = 0.04 s. The q̂ j are those used in

the brain data acquisition.

First, we test the algorithm on a range of simple test functions for p. We use the

simple diffusion model together with the statistical results of Pierpaoli et al [3] to simulate

measurements from the brain. To simulate measurements from dense-white-matter regions we

usethe diffusiontensor10−10 diag(17, 2, 2) m2 s−1, wherediag( X , Y , Z ) is thediagonalmatrix

with leading diagonal elements X , Y and Z . On average, the apparent diffusion tensor in grey




Figure 1. Test functions pi , 0 i 4, plotted over the sphere of radius 2.4|q|−1, together withplots of each ˜ pi (·; r ) for r = 0.4, 1.4, 2.4. To plot a function f over the sphere of radius ρ, weplot f (ρx)x for each x ∈ P , where P is a spherical point set selected from Pi , 1 i L (seesection 5.2).

matter is closer to isotropic than in white matter, but of thesame order of magnitude[3]. We use7 × 10−10

I m2 s−1 to simulate measurements from grey matter. We simulate measurements

from white-matter fibre-crossings using the mixture of Gaussian densities in (8). We use

rotations of 10−10 diag(17, 2, 2) m2 s−1 for the diffusion tensor in each component.

We use the following five test functions to test PAS-MRI:

p0(x) = M1(x;A0; 1; t ), p1(x) = M1(x;A1; 1; t ),

p2(x) = M1(x;A4; 1; t ), p3(x) = M2(x;A1,A2; 12

, 12

; t ),

p4(x) = M3(x;A1,A2,A3; 13

, 13

, 13

; t ),

where Mn is defined in (8),

A0 = 7 × 10−10I m2 s−1, A1 = 10−10 diag(17, 2, 2) m2 s−1,

A2 = 10−10 diag(2, 17, 2) m2 s−1, A3 = 10−10 diag(2, 2, 17) m2 s−1,

A4 = 10−10 diag(9.5, 9.5, 2) m2 s−1.

We use ˜ p(·; r ) for the PAS found with parameter r from data synthesized from a test

function p. Figure 1 shows plots of each pi , 0 i 4, over the sphere of radius r |q|−1 for

r = 2.4 together with ˜ pi (·; r ) found from noise-free data for r = 0.4, 1.4, and 2.4.

We see from (15) that the PAS is the exponential of the sum of N + 1 waves (or basis

functions) over the sphere (including a constant term). The wavelength of the basis functions

of ˜ p decreases with increasing r . At low r , we have insufficient resolution to resolve the




Figure 2. Sixteen examples of ˜ pi (·; 1.4), 0 i 4, found from synthetic data with increasing

noise level. To make each figure in the table, we scale each ˜ pi (·; 1.4) to fit exactly inside a15 × 15 × 15 cube and project onto the xy-plane to form a 15 × 15 square. We pad these squaresto 16 × 16 squares and place them in the image array with no additional gaps.

interesting angular structure. At high r the resolution is too fine to be supported by the data.

As we can see in figure 1, when we set r too low, ˜ p(·; r ) has too few principal directions and

extra principal directions appear in ˜ p(·; r ) when r is set too high. At r = 1.4, the principal

directions of ˜ pi (·; r ), i = 1, 3, 4, match those of the test functions. We observe a range of

values of r over which the principal directions of ˜ pi (·; r ), i = 1,3,4, match those of p.

We say that spherical point sets X and Y are consistent if they satisfy all of the following

conditions:

• X and Y have the same number of elements,

• maxy∈Y (minx∈ X (1 − |x · y|)) < T ,• maxx∈ X (miny∈Y (1 − |x · y|)) < T ,

where T is the consistency threshold .

To test the PAS found from synthetic data, we compare the principal directions of ˜ p(.; r )

with the peaks of p at radius r |q|−1. The spherical point set Y contains one point from each

equal and oppositepair ofpeaks of p at radiusr |q|−1, normalizedto theunit sphere. The spher-

ical point set X is the set of the principal directionsof ˜ p(·; r ) found using themethoddescribed

in section 5.3. We say that ˜ p(·; r ) is consistent if X and Y are consistent with T = 0.05.

We find ˜ pi (·; r ), i = 1, 3, 4, from noise-free data with r at steps of 0.1 from 0.1 up to 2.9

and check the consistency of each ˜ pi (·; r ). We find that ˜ p1(·; r ) is consistent for tested values

of r in [0.1, 1.6], ˜ p3(·; r ) is consistent for tested values of r in [0.6, 1.6] and ˜ p4(·; r ) for tested

values in [1.0, 2.4].

Figure 2 shows how noise in the simulated measurements affects the PAS. We defines = σ −1 F (0) and generate synthetic data for each test function with s ∈ {4, 8, 16, 32, 64}.

For i = 0, . . . , 4, we show 16 examples of ˜ pi (·; 1.4) found from noisy data with different

seeds in the random number generator.

The PAS extracted from the data is determined by both the signal and the noise, though

the choice of r can be used to control the smoothness of the reconstruction to some extent.

We can consider the signal-to-noise ratio of the angular structure, which is distinct from the

notion of signal-to-noise ratio of the MRI measurements. When the angular structure is weak,




as for p0 and p2, its signal-to-noise ratio may be low even if the signal-to-noise ratio of the

MRI measurements is high. When the signal-to-noise ratio of the angular structure is low

the angular structure of the noise dominates the PAS, as we see in figure 2 where spurious

angular structure appears in ˜ p0(·; 1.4) and ˜ p2(·; 1.4). When the angular structure is strong, as

it is for p1, p3 and p4, the noise has less effect. We note that published results from q-spaceimaging appear to showa similar effect. For example, in thefigures shown in chapter 8 of [31],

voxels in fluid-filledand grey-matter regions of the brain contain functions with strong angular

structure, which is probably from noise in the MRI data.

To assess the performance on noisy data further, we compute the consistency fraction C ,

which is the proportion of 256 trials in which ˜ p(·; r ) is consistent. With the noise model

in (24), we compute an estimate σ̃ of σ from the human brain data using measurements in

background regions where the signal is practically zero. We also compute the mean value of

A(0) over two white-matter regions of the human brain data set. In both regions, we find σ̃

is approximately 116

of the mean A(0). For the experimentson noisy synthetic data discussed

in the remainder of this section, we use s = 16 throughout.

We compute C for

r ∈ {0.2, 0.5, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 2.0, 2.3}. (25)For r = 1.4, C > 99% for test functions p1, p3 and p4, with similar results for r = 1.1, 1.2,

1.3, 1.5 where C > 95% for p1, p3 and p4. For the other values of r given in (25), C < 90%

for at least one of p1, p3 and p4. We thus use r = 1.4 for the experiments in the remainder of

this paper.

We investigate the dependence of C on the dimensionless quantity u = |q| R. We have

computed C for

u/u0 ∈ {0.5, 0.7, 0.8, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, 7.0, 10.0}, (26)

where u0 is the value of u for pi , i = 0, . . . , 4, with the value of |q| used in the brain data.

We find that C > 99% for p1, p3 and p4 with u/u0 = 1.0, 1.5, 2.0. For the values in (26)

outside this range, C decreases for all three test functions. This suggests that the choice of |q|

in the data is reasonable for the diffusion time, t . However, in applications where the signal

to noise ratio of the MRI measurements is higher, we expect increasing |q| to reveal more

detailed angular structure.

We also examine the dependence of C on the number, N , of measurements with non-zero

q. We run tests with

N ∈ {10, 12, 14, 16, 18, 20, 30, 40, 50, 60, 70}

and with M = 6. For the q̂ j in each test, we take a point from each equal and opposite pair

in E P (2 N ). For p1, we find that C > 99% for N 12. For p3, C > 99% for N 20. For

p4, C > 99% for only N > 50. To resolve the three orthogonal directions in p4 consistently

the value of N in the human brain data (54) is just large enough. However, to resolve the two

orthogonal directions in p3, we could use fewer measurements and reduce scanning time or

increase image resolution.

We also investigate the effect of changing the mixing parameter in p3, as well as therelative orientation of the two diffusion tensors. We use test functions

p5(x; a) = M2(x;A1,A2; a, 1 − a; t ), (27)

where 0 a 1, and

p6(x; α) = M2(x;A1,R(α)TA2R(α); 1

2, 1

2; t ), (28)

whereR(α) is a rotation matrix fora turn through angle α about the z-axis. We show ˜ p5(·; 1.4)

and ˜ p6(·; 1.4) found fromdata synthesized with s = 16 in figures3 and4, respectively. We see




Figure 3. Examples of ˜ p5(·; 1.4) found from noisy data synthesized with various values of a.

Figure 4. Examples of ˜ p6(·; 1.4) from noisy data synthesized with various values of α.

(a) (b)

Figure 5. Maps of (a) Tr(D) and (b) the fractional anisotropy over a coronal slice of a human braindata set.

that PAS-MRI can still resolve the two principal directions when the weighting between the

components becomes less balanced or the principal directions become closer, but that the

consistency fraction decreases.

6.2. Human brain data experiments

In this section, we show results from PAS-MRI applied to a human brain data set. The

acquisition parameters for the data are given in section 3.

First we useDT-MRI to obtainan estimateD of thediffusiontensor in each voxel. Figure5

shows Tr(D) and the fractional anisotropy [2]over a coronalslice throughthe human brain data

set. The map of Tr(D) highlights fluid-filled regions of the brain and the fractional anisotropy

highlights white-matter regions. We use a threshold of 6σ̃ on ¯ A(0) to remove background

regions of the image.

Figure 6 shows PAS-MRI results over the coronal slice used in figure 5. We have placed

boxes around two regions of interest in figures 5 and 6. The box nearer the bottom of the imageoutlines a region of the pons and the box nearer the top includes part of the corpus callosum.

In the regionof the pons outlined in figure 6, the transpontine tract (left–rightorientation)

crosses the pyramidal tract (inferior–superior orientation). The PAS in this region picks out

the orientations of these crossing fibres.

The fibres of the corpus callosum are approximatelyparallel and run from the left to right

sides of the brain. We can see from figure 6 that, in most voxels within the corpus callosum,

the PAS has a single principal direction along the axis of the fibres. The upper box in figure 6




Figure 6. The PAS in each voxel of a coronal slice through a human brain data set. We scale each˜ p to fit exactly inside a 15 × 15 × 15 cube and project onto the plane of the slice to form a 15 × 15

square. We pad these squares to 16 × 22 rectangles to reflect the voxel dimensions and place themin the image array with no additional gaps.

also contains some voxelsfrom areas of grey matter (the top row in the box) and cerebro-spinal

fluid (the bottom row in the box). In these regions, p is closer to isotropic and the principal

directions that appear in the PAS are noise dominated, as shown by the simulation results in

figure 2.

7. Discussion

We have introduced a new diffusionMRI techniquecalled PAS-MRI. In PAS-MRI, we find the

PAS of the probability density function of particledisplacements, p, from MRI measurements

of theFourier transformof p. We have demonstratedthe techniqueusing data acquired for DT-

MRI using a standard imaging sequence in which the measurements lie on a sphere in Fourier

space. However, PAS-MRI is not limited to data of this kind. In simulation, we find that the

principal directions of the PAS reflect the directional structure of several test functions. In

clinical applications, we aim to use the PAS to determine white-matter fibre directions. Early

results for human brain data are promising.The new technique has the advantage over existing techniques, such as DT-MRI and

three-dimensional q-space imaging, that it can resolve fibre directions at crossings, but still

requires only a modest number of MRI measurements. In clinical applications, the data sets

are likely to remain small for practical use, as the time available for the MRI scan is usually

a significant constraint. This is due not only to the cost, but also the time for which a patient

can be expected to remain in the scanner. In clinical applications, we thus expect tractography

results to improve by using PAS-MRI rather than DT-MRI or q-space imaging.




With DT-MRI, the images can have high resolution, since the method requires a relatively

small number of measurements. However, DT-MRI cannot resolve the directions of crossing

fibres. The q-space imaging technique can resolve the directions of crossing fibres, but

requires many more measurements so that images acquired in the same scanner time have

lower resolution. We note also that q-space imaging can provide information about the radialstructure of p, which PAS-MRI does not attempt to.

The spurious directions in the PAS for noisy data from isotropic regions also appear in

q-space imaging. In these circumstances, methods requiring fewer fitted parameters, such as

DT-MRI, are less sensitive to noise.

Post-processing times are greater for PAS-MRI than for DT-MRI and q-space imaging.

One way to reducecomputation times is to usePAS-MRI with a voxelclassification algorithm,

such as that described in [24], to decide where the simple diffusion model is a poor

approximation. We can use DT-MRI in voxels in which the simple diffusion model is a

good approximation and PAS-MRI where it is not. However, with computer power rapidly

increasing year by year, computational issues will soon become unimportant.

We have restricted attentionto theprincipaldirections of thePAS,which isa robustfeature.

Another useful statistic is thenumber of principaldirectionsin thePAS.We expect this statisticto help, for example, in identifying areas in which white-matter fibres are destroyed by injury

or disease. Other features of the PAS, such as the relative heights and lengthscales of its peaks,

may also provide useful information about the material microstructure. It is not yet clear how

well such features correlate with p and further investigation is necessary. We hope others will

investigate the extent to which PAS-MRI can highlight the early signs of diseases, such as

multiple sclerosis.

In the current implementation, we fit the PAS by searching for a minimum point of χ 2

from a single starting point to keep the computation time manageable. We have compared the

results to solutions closer to the global minimum of the objective function found by running

repeated trials fromdifferent starting points. Apart fromwhen the angular structure in the data

is weak, we find that solutions from the single starting point are consistent with those closer

to the global minimum.

We plan to consider, elsewhere, extensions of PAS-MRI. We believe that one way inwhich the current approach could be improved is to consider the optimal placement of the

observational points, i.e. the set of q j used in the MRI measurements. In particular, the results

are likely to improve for samples not restricted to a sphere in Fourier space.

Acknowledgments

We would like to thank Gareth Barker and Claudia Wheeler-Kingshott at the Institute of

Neurology, UCL, for providing the human brain data used in this work.

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