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Q-Ball Imaging

David S. Tuch1,2*

Magnetic resonance diffusion tensor imaging (DTI) provides a

powerful tool for mapping neural histoarchitecture in vivo. How-

ever, DTI can only resolve a single fiber orientation within each

imaging voxel due to the constraints of the tensor model. For

example, DTI cannot resolve fibers crossing, bending, or twist-

ing within an individual voxel. Intravoxel fiber crossing can be

resolved using q-space diffusion imaging, but q-space imaging

requires large pulsed field gradients and time-intensive sam-

pling. It is also possible to resolve intravoxel fiber crossing

using mixture model decomposition of the high angular resolu-

tion diffusion imaging (HARDI) signal, but mixture modeling

requires a model of the underlying diffusion process.

Recently, it has been shown that the HARDI signal can be

reconstructed model-independently using a spherical tomo-

graphic inversion called the Funk–Radon transform, also known

as the spherical Radon transform. The resulting imaging

method, termed q-ball imaging, can resolve multiple intravoxel

fiber orientations and does not require any assumptions on the

diffusion process such as Gaussianity or multi-Gaussianity. The

present paper reviews the theory of q-ball imaging and de-

scribes a simple linear matrix formulation for the q-ball recon-

struction based on spherical radial basis function interpolation.

Open aspects of the q-ball reconstruction algorithm are

discussed. Magn Reson Med 52:1358–1372, 2004. © 2004

Wiley-Liss, Inc.

Key words: diffusion MRI; diffusion tensor imaging; high angu-

lar resolution diffusion imaging; tractography

Magnetic resonance diffusion tensor imaging (DTI) maps

the orientational architecture of neural tissue by measur-

ing the water diffusion tensor within each voxel of the MR

image (1–3). The orientational architecture of the underly-

ing tissue can be inferred from the eigenstructure of the

diffusion tensor. For example, the major eigenvector of the

diffusion tensor gives the mean fiber direction within the

voxel, and the minor eigenvector indicates the sheet nor-

mal vector (4). In cerebral white matter, the anisotropy of

the diffusion tensor also provides a putative marker of

myelination degree and fiber density (5,6).

DTI has a significant limitation, however, in that the

technique can only resolve a single fiber direction within

each voxel (7,8). This shortcoming is significant since hu-

man cerebral white matter possesses considerable intra-

voxel structure at the millimeter resolution typical of MRI.

The tensor model is consequently deemed inadequate for

resolving neural architecture in regions with complex fiber

patterns.

The inability of DTI to resolve intravoxel orientational

heterogeneity has a number of consequences. This limita-

tion presents a significant obstacle for efforts to trace white

matter pathways from diffusion MRI (see Ref. (9) for re-

view). The fiber crossing confound also complicates inter-

pretation of diffusion anisotropy in regions of intravoxel

heterogeneity. In cerebral white matter, the anisotropy of

the diffusion tensor is typically expressed through the

fractional anisotropy (FA) metric. In voxels containing

intravoxel orientational heterogeneity, a decrease in the

FA of an individual fiber population may result in a par-

adoxical increase in the overall FA (4,10).

DTI’s inability to resolve intravoxel orientational heter-

ogeneity stems from the constraints of the tensor model,

which implicitly assumes a single Gaussian diffusion com-

partment within each voxel. The Gaussian function has

only a single directional maximum and therefore cannot

adequately describe diffusion functions with multiple

maxima. Multimodal diffusion may arise when the fiber

populations within a voxel possess different orientations

and the diffusion between the populations is in slow ex-

change (7,8,11).

The fiber crossing confound in DTI has prompted efforts

to develop diffusion imaging methods capable of resolving

intravoxel fiber crossing (7,8,12–17). Using q-space imag-

ing (QSI), investigators have measured the microscopic

diffusion function directly and have found that in regions

of fiber crossing the diffusion function possesses signifi-

cant multimodal structure (12,14,18). QSI employs the

Fourier relation between the diffusion signal and the dif-

fusion function to measure the diffusion function directly,

without recourse to a model of the diffusion process (19).

QSI is also referred to as diffusion spectrum imaging (DSI)

(12,14,18), diffusion displacement imaging, or dynamic

NMR microscopy (19).

QSI measures the diffusion function directly by sam-

pling the diffusion signal on a three-dimensional Cartesian

lattice. The QSI technique suffers from two practical weak-

nesses however. The technique requires gradient sampling

on a three-dimensional Cartesian lattice, which is time-

intensive. Further, QSI requires large pulsed field gradi-

ents to satisfy the Nyquist condition for diffusion in nerve

tissue.

To address the sampling burden of QSI, investigators

have proposed an alternative approach based on sampling

on a spherical shell (or combination of shells) in diffusion

wavevector space. The spherical sampling approach re-

ferred to as high angular resolution diffusion imaging

(HARDI) (8,11,15–17). In theory, the efficiency gain of

HARDI would stem from the need to sample only on a

spherical shell as opposed to the three-dimensional Carte-

sian volume required by QSI. By selecting a sampling shell

1358

1Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts Gen-

eral Hospital, NMR Center, Charlestown, Massachusetts.

2Harvard–MIT Division of Health Sciences and Technology, Cambridge, Mas-

sachusetts.

Grant sponsor: NINDS; Grant number: NS46532; Grant sponsor: NCRR;

Grant number: RR14075; Grant sponsor: Glaxo Smith Kline; Grant sponsor:

Athinoula A. Martinos Foundation; Grant sponsor: Mental Illness and Neuro-

science Discovery (MIND) Institute.

*Correspondence to David S. Tuch, MGH-NMR Center, 149 13th Street,

Room 2301, Charlestown, MA 02129. E-mail: dtuch@nmr.mgh.harvard.edu

Received 23 April 2004; revised 1 July 2004; accepted 9 July 2004.

DOI 10.1002/mrm.20279

Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 52:1358–1372 (2004)

© 2004 Wiley-Liss, Inc.

Page 2

of a particular radius the acquisition could also be targeted

toward specific lengthscales of interest.

Notwithstanding the potential advantages of HARDI,

widespread application of the technique has been limited

by the unavailability of a model-independent reconstruc-

tion scheme for HARDI data. Various models and numer-

ical fitting procedures have been proposed to relate the

spherical diffusion signal to the underlying diffusion func-

tion (7,8,15–17), but a model-free inversion has remained

elusive.

Recently, we described a completely model-free recon-

struction scheme for HARDI (12). The reconstruction is

based on a spherical tomographic inversion called the

Funk–Radon transform, also known as the spherical Ra-

don transform or simply the Funk transform. The resulting

method, called q-ball imaging (QBI), has a number of ben-

efits over previous HARDI reconstruction approaches in-

cluding model-independence, linearity in the signal, an

image resolution framework, and computational simplic-

ity. In the present paper we review the theoretical basis of

the QBI method, provide a simple linear matrix formula-

tion for the QBI reconstruction, and demonstrate the tech-

nique’s ability to resolve intravoxel white matter fiber

architecture.

THEORY

Background

In this section we review the theoretical relationship be-

tween the diffusion signal and the diffusion function. We

then describe the theory for inversion of the diffusion

signal using the Funk–Radon transform (FRT), which

forms the basis of the QBI method.

The diffusion function can be described generally by the

conditional diffusion probability density function P(x,x0).

The conditional probability density function describes the

probability for a spin to displace from position x0to posi-

tion x in the experimental diffusion time ? (19–21). The

conditional probability density function is referred to in

other contexts as the diffusion Green’s function, the diffu-

sion propagator, or the diffusion van-Hove self-correlation

function.

In MR, the observed signal is generated from an average

over all spins in the voxel. The resulting ensemble-average

is written P(r) ? ?P(x,x0)?(x0) dx0, where r ? x ? x0is the

relative spin displacement and ?(x0) is the initial spin

density (19,21). With some abuse of nomenclature, we will

refer to the ensemble average of the conditional probability

density function as simply the probability density func-

tion (PDF) or diffusion function and denote it P(r). In the

notation that follows, the diffusion function P(r) therefore

denotes the ensemble-average probability for a spin to

undergo a relative displacement r in the experimental

diffusion time ?.

The diffusion PDF P(r) is related to the measured MR

diffusion signal by the Fourier relationship,

P?r? ? ??E?q??,

where ? denotes the Fourier transform with respect to the

diffusion wavevector q (19–22). The diffusion wavevector

is defined as q ? (2?)-1??g, where ? is the gyromagnetic

ratio for the nucleus of interest, ? is the diffusion gradient

duration, and g is the diffusion gradient vector. The diffu-

sion wavevector q is the reciprocal vector to the relative

spin displacement vector r.

The Fourier relation between the diffusion function and

MR diffusion signal enables direct reconstruction of the

diffusion function by Fourier transformation of a three-

dimensional lattice sampling of q-space. Reconstruction of

the diffusion PDF by Fourier transform of the diffusion

signal forms the basis of the QSI method (19). QSI has been

employed to measure the diffusion PDF in nonbiological

materials with complex microstructure (19,23), as well as

small animals in vivo (14,24–26). QSI has also been ap-

plied in humans to map the one-dimensional (27,28) and

three-dimensional PDF (12,18).

In in vivo applications, reconstructing the diffusion PDF

using the complex Fourier transform is not feasible since

the phase of the signal is corrupted by biological motion,

primarily due to cardiac pulsation. Instead, the diffusion

function can be reconstructed using the modulus Fourier

transform P(r) ? ?[?E(q)?]. Using the modulus FT as op-

posed to the full complex FT does not sacrifice any infor-

mation since the diffusion signal is real and positive. The

reality and positivity of the diffusion signal entails that the

modulus FT and complex FT of the diffusion signal are

equivalent. The reality of the diffusion signal follows from

the symmetry of the diffusion propagator, and the positiv-

ity, which is not trivial, is a consequence of the positive

definiteness of the diffusion propagator (18).

It should be noted that Fourier transformation of the

diffusion signal only gives the diffusion PDF exactly when

there is no appreciable diffusion during the diffusing en-

coding period. This condition requires that the diffusion

mixing length associated with the diffusion encoding time

is smaller than a characteristic diffusion restriction size of

the material. The requirement for short diffusion pulses is

referred to as the “narrow-pulse condition” (29). It has

been shown that when the pulse duration is finite the

resulting PDF can be described as a center-of-mass propa-

gator, which is a spatially contracted form of the true PDF

(29).

While the three-dimensional PDF provides invaluable

information on the tissue microstructure, for the purposes

of mapping the orientational architecture of tissue the

primary object of interest is the orientational structure of

the diffusion function. The orientational structure of the

diffusion function can be described through the diffusion

orientation distribution function (ODF). The diffusion

ODF ?(u) is defined as the radial projection of the diffusion

function,

Z?

??u? ?1

0

?

P?ru?dr, [1]

where Z is a dimensionless normalization constant. The

ODF framework is widely used in materials science to

describe the orientational composition of polymers, liquid

crystals, and grain composites (30,31).

The normalization constant in Eq. [1] ensures that the

ODF is properly normalized to unit mass. Even though the

Q-Ball Imaging1359

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PDF is normalized, the ODF obtained by radial projection

is not guaranteed to be normalized since the ODF is a

distribution on the radial projections and not on the true

sphere. To define the ODF over the proper sphere would

require an integral over solid angle elements.

Using Eq. [1] we can derive the ODF for Gaussian diffu-

sion. The PDF for anisotropic Gaussian diffusion is

P?r? ? ?4????3/2?D??1/2exp??rTD?1r/?4???,

where ?.? is the determinant. Integrating over radius as

described by Eq. [1] gives the ODF,

Z?

??u? ?1

??

uTD?1u, [2]

where Z is a normalization constant.

The ODF can be derived from the diffusion PDF mea-

sured by QSI. Deriving the ODF from QSI has a number of

limitations, however. Extracting the ODF from the PDF

requires explicitly calculating the radial projection. The

mapping between Cartesian and spherical coordinates sys-

tems may introduce Cartesian artifacts in the ODF. Carte-

sian coloration of the ODF may be a particular problem at

the coarse Cartesian resolution typically used for QSI.

Further, the radial projection is highly inefficient since the

projection discards a considerable fraction of the acquired

data. The efficiency of QSI is also hampered by the strong

pulsed field gradients needed to satisfy the Nyquist con-

dition for the diffusion PDF in cerebral white matter.

It is substantially more efficient to measure the diffusion

ODF by directly sampling the diffusion signal on a spher-

ical shell in diffusion reciprocal space. This approach

forms the basis of QBI (12). Reconstructing the ODF di-

rectly using spherical sampling and reconstruction has a

number of advantages. First, both the sampling and the

reconstruction are both performed on the sphere so the

reconstruction is immune to Cartesian reconstruction bias.

With a spherical sampling scheme, there is also a natural

framework for calculating the angular resolution, whereas

it is not clear how to define the angular resolution for a

Cartesian scheme. Last, the acquisition can be targeted to

specific spatial frequency bands of interest by specifying

the radius of the sampling shell. In the next section we

review the theory underlying the QBI reconstruction.

Reconstruction

The QBI reconstruction is based on the FRT, also known

as the spherical Radon transform or simply the Funk trans-

form (32). The FRT is an extension of the planar Radon

transform to the sphere. The FRT is a transform from the

sphere to the sphere. Given a function on the sphere f(w),

where w is a unit direction vector, the FRT is defined as

the sum over the corresponding equator, i.e., the set of

points perpendicular to w. The FRT ? for a direction u can

be written

??f?w???u? ??

w?u?

??f?w???wTu?dw,

f?w?dw

where ? is the Dirac delta function.

While the original FRT is defined as a transform from the

sphere to the sphere, here we extend the definition of the

FRT to map from three-dimensional Cartesian space to the

sphere. The extended FRT is defined as the FRT evaluated

at a particular radius r?. Given a three-dimensional func-

tion f(x), where x is a three-dimensional vector, the FRT at

a particular radius r? is written

??f?x???u, r?? ??f?x???xTu????x? ? r??dx.

For notational simplicity we denote the above transform as

simply ?r?. In general, we also denote transforms F[f(x)](y)

as simply F[f(x)] where the final argument is implied.

Recently, we have shown that the FRT of the diffusion

signal gives a strong approximation to the ODF, that is,

??u? ?1

Z?q??E?q??,[3]

where q? is the radius of the sampling shell and Z is a

normalization constant (12). This remarkable relationship

entails that the sum of the diffusion signal over an equator

approximately gives the diffusion probability in the direc-

tion normal to the plane of the equator. Consequently, to

estimate the diffusion probability in a particular direction

all that is needed is to sum the diffusion signal along the

equator around that direction. This provides a model-free

approach for estimating the diffusion probability from the

spherically sampled diffusion signal (12).

We note that FRT of the diffusion signal has been de-

scribed previously by Zavada and colleagues in the con-

text of isotropic diffusion (33). We also note that the FRT

bears a strong resemblance to the infinite anisotropy inver-

sion model described by Behrens et al. (13). The latter has

an additional constant term for the spherical fit. It may be

possible to show that the two inversions are equivalent.

Equation [3] represents an approximation. The exact

relationship between the ODF and the FRT can be written

as follows. We write the PDF in cylindrical coordinates as

P(r,?,z). Without loss of generality, we take the z-axis to be

along the direction of interest u. In Appendix A we prove

that the ODF and the FRT of the diffusion signal are related

according to

??u? ? ?q??E?q??

? 2?q??P?r, ?, z?J0?2?q?r?rdr d? dz,[4]

where J0is the zeroth-order Bessel function (12). This

relationship states that the FRT of the diffusion signal

gives the radial projection of the PDF, except that instead

of the projection being along an infinitely thin line the

projection is along a Bessel beam with a width defined by

the width of the zeroth-order Bessel function (Fig. 1). The

Bessel beam projection resembles the true radial projec-

tion to the extent that the mass of the zeroth-order Bessel

function is concentrated at the origin.

1360David S. Tuch

Page 4

Equation [3] can be written in a simpler form by recalling

the Hankel transform ?[ f(r,?,z)] ? ? f(r,?,z)J0(2?kr)rdr

and the X-ray transform (also known as the planar Radon

transform) X[f(r,?,z)] ? ? f(r,?,z) dz. We then have

??u? ? ?q??E?q?? ? 2?q????X?P??d?.

The above relation states that the FRT of the diffusion

signal is proportional to the Hankel transform of the X-ray

transform of the diffusion function. Note that the X-ray

transform evaluated at the origin is equivalent to the radial

projection described by Eq. [1]. (It is important to note that

the 2?q? term in Eq. [3] arises from the circle integral and

so is equal to unity if the great circle sampling density is

independent of the sampling wavevector q?).

To understand Eq. [4] it is helpful to consider the inte-

gral in parts. The X-ray transform projects the diffusion

function onto r?-plane, which is the tangent plane to the

direction of interest z. Evaluating the X-ray transform at

the origin r ? 0 would give the radial projection exactly.

Instead, the X-ray projection is multiplied by J0through

the Hankel transform. The integral over the plane dr and

FIG. 1. Schematic illustration of the relationship

between the Funk–Radon transform and the

Bessel beam projection. The q-space sampling

scheme is indicated by the blue spherical lattice.

The white arrow gives the direction of interest.

The light blue circle indicates the equator around

the direction of interest. Integration of the

q-space signal along the equator defines a pro-

jection beam, which is shown by the dot pattern.

The projection beam (i.e., Bessel beam) falls off in

intensity according to the zeroth-order Bessel

function. The intensity of the Bessel beam is in-

dicated by the density of the green and yellow

dots. The green dots indicate the positive signal

contribution and the yellow dots the negative

contribution.

FIG. 2. Reconstruction of the diffusion ODF from the diffusion signal using the FRT. The diffusion data are taken from a single voxel from the

data set described under Methods. The sampling and reconstruction schemes are also described under Methods. (a) Diffusion signal sampled

on fivefold tessellated icosahedron (m ? 252). The signal intensity is indicated by the size and color (white ? yellow ? red) of the dots on the

sphere. (b) Regridding of diffusion signal onto set of equators around vertices of fivefold tessellated dodecahedron (k ? n ? 48 ? 755 ? 36240

points). (c) Diffusion ODF calculated using FRT. (d) Color-coded spherical polar plot rendering of ODF. (e) Min–max normalized ODF.

Q-Ball Imaging1361

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d? then sums the weighted signal in the plane. To the

extent that J0is concentrated near the origin r ? 0, Eq. [4]

gives the true ODF. For example, for illustrative purposes,

if we take J0(r) ? ?(r) then we obtain the true radial pro-

jection. The main consequence of Eq. [4] is that to estimate

the diffusion probability in a particular direction we sim-

ply need to add the diffusion signal along the equator

around that direction. Appendix A provides a detailed

derivation of Eq. [4].

Implementing the FRT in practice requires a numerical

procedure for calculating the equator integral. Since the

equator points will not coincide with the diffusion

wavevector sampling points it is necessary to regrid the

diffusion data onto the equatorial circles. The regridding

can be implemented using a form of spherical interpola-

tion called spherical radial basis function (sRBF) interpo-

lation, which we describe in the following section (34).

Algorithm

In this section we describe a simple matrix implementa-

tion of the FRT. We ultimately derive a matrix relationship

of the form ? ? (1/Z) Ae where the reconstruction matrix

A implements the FRT. The following describes how to

derive the FRT reconstruction matrix A and the normal-

ization constant Z.

The diffusion signal for a diffusion wavevector q is

denoted E(q). We are given a set of m diffusion measure-

ments, which we denote by the signal vector e ? [E(q1)

E(q2) . . . E(qm)]T. The measurements were acquired with

the m diffusion wavevectors {q} ? {q1, q2, . . . , qm}. The

diffusion wavevectors are also written as the 3 ? m col-

umn matrix Q ? [q1q2. . . qm]. For notational simplicity,

the diffusion wavevectors are normalized to unit length,

i.e., qi4 qi/q.

We specify a set of n diffusion directions of interest

{u} ? {u1, u2, . . . , un} onto which we wish to reconstruct

the ODF. The reconstruction directions are also denoted

by the 3 ? n column matrix U ? [u1u2. . . un]. We wish to

reconstruct the ODF vector ? ? [?(u1) ?(u2) . . . ?(un)]T

using the FRT.

To compute the FRT we need to specify the equator of

points for each reconstruction direction ui. The points

comprising each equator can be specified as follows. We

construct a circle of k equally spaced points in the xy-

plane. The points are denoted by the 3 ? k matrix C ? [cos

? sin ? 0k]Twhere ? ? (2?/k)[1

1 vector of zeros. For each ui, we then rotate the circle so

that the normal to the circle-plane points in the direction

ui. The rotation matrix is the matrix which rotates z into ui,

which is given by

2 . . . k]Tand 0kis a k ?

Rz?ui? ??z ? ui??z ? ui?T

?zTui? 1?

? I.

The equator points for a uican then be written as Rz(ui)C.

We would like to represent all of the equator points for all

of the reconstruction directions as a single matrix. To do so

we form the 3 ? (kn) matrix S ? ?i?1

Rz(u2)C. . . Rz(un)C] where ? is the matrix concatenation

operator.

n

Rz(ui)C ? [Rz(u1)C

The points on the equator do not correspond to sampling

points so it is necessary to interpolate the data. We per-

form the interpolation by regridding the original sampling

scheme onto the set of equators. The regridding is imple-

mented using sRBF interpolation (34). The main idea of

sRBF interpolation is to fit the signal with a linear combi-

nation of positive definite kernels on the sphere (35) and

then use the kernel fit to evaluate the function at the

interpolation points.

In order to implement the sRBF interpolation it is nec-

essary to specify a basis function and a distance metric on

the sphere. The distance metric d is taken to be the mini-

mum angle between the direction vectors, i.e., d(n1,n2) ?

cos-1?n1

and ?.? denotes the absolute value. For the interpolation

kernel we choose the spherical Gaussian ?(?) ? exp(??2/

?2)where ? ? d(n1,n2) and ? is a width parameter. Other

common basis functions for sRBF interpolation include

the inverse multiquadric function ?(?) ? (?2? ?2)?1/2,

thin-plate spline ?(?) ? ?2?log(?), and the ultraspherical

(Gegenbauer) polynomial ?(?) ? Cn

of sRBF interpolation, some other common basis functions

for the ODF include the spherical harmonics (30) and the

Wigner polynomials (36).

In sRBF interpolation the interpolation kernel width ?

controls the tradeoff between the accuracy and the stabil-

ity of the spherical interpolation; a small width will pro-

vide high accuracy but low stability and reciprocally for a

large width. The accuracy–stability tradeoff is specified by

the condition number of the interpolation matrix H, so that

the optimal tradeoff is achieved when the condition num-

ber is minimal (37). The optimal kernel width can be

derived numerically knowing only the wavevector sam-

pling scheme. It should also be possible to derive an ana-

lytical expression for the optimal kernel width (in a least

upper-bounded sense) based on existing analytical upper

bounds on the condition number of the spherical interpo-

lation matrix (37).

The kernels are centered on a set of p specified unit

vectors {v} ? {v1, v2, . . . , vp}, which can be taken to be the

sampling directions, the reconstruction points, or any

other set of unit vectors. The basis function centers are also

denoted by the 3 ? p column matrix V ? [v1v2. . . vp]. The

diffusion signal can then be expressed as a convolution of

the spherical basis functions, e ? Hw, where w is the

coefficient vector and H is the m ? p convolution matrix H

? [Hij] ? [?(d(qi, vj))] ? ?(cos-1?QTV? ). From the mea-

sured signal e we can estimate the weight vector as w ˆ ?

H?e where H?? (HTH)-1HTis the Moore–Penrose pseudo-

inverse of H. If the noise is independent and identically

distributed (iid) additive Gaussian noise, then noise regu-

larization can be implemented using the noise-regularized

pseudoinverse H?? (HT?-1H)-1HT?-1where ? is the noise

covariance matrix for the signal vector e.

The diffusion signal estimate for the equator points is

given by Gw where G ? [Gij] ? [?(d(si, vj))] ?

?(cos?1?STV?) is the (kn) ? p convolution matrix from the

basis function centers to the equator points. Here, siand vj

are the column vectors of S and V, respectively. Equating

w and w ˆ and substituting gives the (kn) ? m matrix GH?.

To compute the sum over the equators, we define the

summation matrix ? ? (InR 1k

Tn2? where n1and n2are unit direction vectors,

?(?) (34). Independent

T) where R is the matrix

1362David S. Tuch