kalvis m jansons and daniel c alexander- persistent angular structure: new insights from diffusion...
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8/3/2019 Kalvis M Jansons and Daniel C Alexander- Persistent angular structure: new insights from diffusion magnetic resona…
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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.
0266-5611/03/051031+16$30.00 © 2003 IOP Publishing Ltd Printed in the UK 1031
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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.
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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
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1034 K M Jansons and D C Alexander
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
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Persistent angular structure 1035
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
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1036 K M Jansons and D C Alexander
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.
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Persistent angular structure 1037
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
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1038 K M Jansons and D C Alexander
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.
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Persistent angular structure 1039
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
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1040 K M Jansons and D C Alexander
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
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Persistent angular structure 1041
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,
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1042 K M Jansons and D C Alexander
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
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Persistent angular structure 1043
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
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1044 K M Jansons and D C Alexander
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.
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Persistent angular structure 1045
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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