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Available from: Raphaël PaquinCross-encoded magnetic resonance imaging in inhomogeneous ﬁelds

Raphael Paquin

a,b,

*

, Philippe Pelupessy

a,

*

, Geoffrey Bodenhausen

a,b

a

Département de Chimie, associé au CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France

b

Institut des Sciences et Ingénierie Chimiques, Ecole Polytechnique Fédérale de Lausanne, Batochime, 1015 Lausanne, Switzerland

article info

Article history:

Received 29 July 2009

Revised 28 August 2009

Available online 11 September 2009

Keywords:

Magnetic resonance imaging (MRI)

Echo-planar imaging (EPI)

Inhomogeneous magnetic ﬁelds

Adiabatic radio-frequency pulses

Susceptibility effects

Slow motion

abstract

In magnetic resonance imaging (MRI), it is possible to cancel the effects of severe inhomogeneities of the

magnetic ﬁeld even if the ﬁeld proﬁle is unknown. The new ‘cross-encoded’ method is based on adiabatic

frequency-modulated pulses combined with two orthogonal gradients that are applied simultaneously

during encoding and decoding. Undistorted two- and three-dimensional images can be obtained in inho-

mogeneous ﬁelds where the breadth of the water resonance extends over several kHz.

Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction

Magnetic resonance imaging (MRI) is undoubtedly the most

popular application of nuclear magnetic resonance [1], with

wide-ranging implications for medical diagnosis, neurosciences,

metabolism, and material science. Although the objects or living

organisms should ideally be immersed in a magnetic ﬁeld that is

as homogeneous as possible, it is often too difﬁcult or costly to ful-

ﬁll this stringent requirement. In living organisms, the quality of

the images is degraded in the vicinity of voids and surgical im-

plants. Inanimate objects may also suffer from discontinuities of

the magnetic susceptibility. Temporal variations that occur in

stimulated brains mapped by functional MRI [2,3] call for fast

imaging techniques, such as echo planar imaging (EPI) introduced

by Mansﬁeld and co-workers [4]. Inspired by these experiments,

Frydman and co-workers introduced single-scan methods for

‘ultrafast’ spectroscopy [5]. Recently, these ideas were extended

to imaging [6], making it possible to probe ‘real’ Cartesian space di-

rectly without relying on Fourier transformation [7] or back-pro-

jection [1]. These methods were designed to cope with

inhomogeneous ﬁelds with known proﬁles, where the encoding

schemes can be tailored to compensate for the phase dispersion

arising from the inhomogeneity of the static ﬁeld [8]. In this work,

we shall present cross-encoded single-scan methods that are de-

signed to compensate for effects of inhomogeneous ﬁelds in a frac-

tion of a second without requiring any knowledge of their spatial

proﬁles.

2. Principles

2.1. Homogeneous ﬁelds

In Fig. 1, a selective pulse combined with a gradient G

s

y

is ap-

plied along the y axis to excite the magnetization in a {x, z} plane.

In contrast to earlier methods for single-scan spectroscopy [9],

orthogonal G

e

x

encoding gradients of duration T

e

must be combined

with G

e

z

encoding gradients. In the decoding interval, the G

d

z

gradi-

ents unwind the phase accumulated during the G

e

x

encoding gradi-

ents, instead of the chemical shifts in our earlier methods. We

assume that only a single chemical shift

X

= 0 is present. Thus

the precession frequency of the magnetization associated with a

voxel centered on the coordinates

~

r ¼fx; y; zg during the ﬁrst adi-

abatic pulse is

x

~

r

¼

c

~

G

~

r ¼

c

ðG

e

x

x þ G

e

z

zÞð1Þ

The sign of G

e

z

is opposite during the second adiabatic pulse. The

frequency sweep of the adiabatic pulse

D

x

ad

=2 6

x

RF

6

D

x

ad

=2

is deﬁned so that the center of the sweep corresponds to the Lar-

mor frequency in the absence of any gradients. The phase at the

end of the encoding block is:

u

e

ðx; zÞ¼kxz ð2Þ

where

1090-7807/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.jmr.2009.09.008

* Corresponding authors.

E-mail addresses: raphael.paquin@ens.fr (R. Paquin), philippe.pelupessy@ens.fr

(P. Pelupessy).

Journal of Magnetic Resonance 201 (2009) 199–204

Contents lists available at ScienceDirect

Journal of Magnetic Resonance

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

Page 1

k ¼ 4T

e

c

2

G

e

x

G

e

z

=

D

x

ad

ð3Þ

This equation is symmetric with respect to a permutation of the

x and z coordinates. Thus, if the G

e

z

gradient encodes for the x coor-

dinate, the G

e

x

gradient will do the same for the z coordinate. In con-

trast to traditional k-space MRI using Fourier transformations, the

decoding sequence leads to a trajectory in the {x, z} plane [8],

which can be chosen at will. Fig. 1A shows a decoding scheme

D1 based on the simultaneous application of gradients G

d

x

and G

d

z

of duration T

d

. Pairs of such gradients, with alternating signs in

the latter dimension, must be repeated N

d

times. This procedure

gives the ‘zig-zag’ trajectory illustrated in Fig. 1B [8]. The signal

at time t is

SðtÞ¼

ZZ

allx;z

Aðx; zÞexp i

u

e

ðx; zÞ½exp ik

x

ðtÞx þ ik

z

ðtÞz½dxdz ð4Þ

where A(x, z) is the amplitude of the initial longitudinal magnetiza-

tion, and k

x

(t) and k

z

(t) are deﬁned as

k

x

ðtÞ¼

c

Z

t

0

G

d

x

ðt

0

Þdt

0

k

z

ðtÞ¼

c

Z

t

0

G

d

z

ðt

0

Þdt

0

ð5Þ

G

d

x

ðt

0

Þ¼G

d

x

G

d

z

ðt

0

Þ¼ð1Þ

n

D1

ðt

0

Þ

G

d

z

ð6Þ

where n

D1

(t

0

) is the integer part of the fraction t

0

/T

d

, which toggles

between +1 or 1 for positive and negative lobes of the train of

bipolar decoding gradients.

As explained by Shrot and Frydman [6], the overall phase

U

should not vary across a voxel V

k

centered on the coordinates

(x

k

, z

k

)

@

@x

½

Uðx; z; tÞ

ðx

k

;z

k

Þ

¼ 0 and

@

@z

½

Uðx; z; tÞ

ðx

k

;z

k

Þ

¼ 0 ð7Þ

with

Uðx; z; tÞ¼

u

e

ðx; zÞþk

x

ðtÞx þ k

z

ðtÞz ð8Þ

From Eqs. (2), (7), and (8), the relation between the coordinates

(x

k

, z

k

) of the voxel V

k

and the variables k

x

(t) and k

z

(t) is:

x

k

¼

k

z

ðt

k

Þ

k

and z

k

¼

k

x

ðt

k

Þ

k

ð9Þ

Thus at time t

k

in the decoding sequence, an echo signal appears

that reveals the position (x

k

, z

k

) of the voxel V

k

. The x dimension is

decoded by the z gradient and vice versa. While the k

x

‘‘walk” oc-

curs during 2N

d

T

d

due to the continuous application of G

d

x

, the k

z

trajectory is reversed every T

d

when the sign of G

d

z

changes.

In Fig. 1C, an image that could be obtained with the adiabatic

pulse scheme of Fig. 1A is simulated for an ideal rectangular object.

Only minor distortions are apparent around the edges of the object,

in particular near the corners.

2.2. Inhomogeneous ﬁelds

In a static (time-independent) inhomogeneous ﬁeld, the preces-

sion frequency of the magnetization of a voxel V

k

in the encoding

interval is

x

~

r

¼

c

fG

e

x

x þ G

e

z

z þ dB

0

ðx; y; zÞg. The additional term is

due to the (unknown) spatial variation of the static ﬁeld B

0

(x, y,

z). The phase at the end of the simple encoding scheme E1of

Fig. 1 is:

u

E1

ðx; zÞ¼k

E1

xz ð10Þ

where

k

E1

¼ 4T

e

c

2

ðG

e

x

G

e

z

þ G

e

z

dB

0

ðx; y; zÞ=xÞ=

D

x

ad

ð11Þ

Compensation of static inhomogeneities during encoding can be

achieved by the scheme of Fig. 2. After the initial encoding se-

quence E1, a second sequence E2 is inserted, where the signs of

Fig. 1. Characteristics of cross-encoded magnetic resonance imaging (CE-MRI). (A) Excitation of transverse magnetization in a slice {x, z} perpendicular to the y axis is

achieved with a selective radio-frequency (RF) pulse in the presence of a ﬁeld gradient G

s

y

. The simplest encoding sequence E1 comprises two adiabatic RF pulses (rectangles

with diagonal arrows) of duration T

e

, applied in conjunction with a pair of opposite gradients G

e

z

and a pair of identical gradients G

e

x

. In the simplest case, decoding D1is

achieved by applying two identical gradients G

d

x

and a pair of opposite gradients G

d

z

, each of duration T

d

. This scheme is repeated N

d

times. (B) The Cartesian (real) {x, z} space is

probed directly through zig-zag trajectories imposed by the decoding sequence. (C) Numerical simulation of the image of a rectangular two-dimensional object of dimensions

2 30 mm, obtained with the pulse scheme (A). The absolute value of the integral of Eq. (4) has been calculated at intervals

D

x = x

max

/100 and

D

z = z

max

/100 for x

max

/2 < x <

x

max

/2 and z

max

/2 < x < z

max

/2 with the following parameters

c

(

1

H) = 2.6752 rad/(T s), G

e

x

¼ G

e

z

¼ G

d

x

¼ 0:022 T/m, G

d

z

= 0.22 T/m, T

e

= 6 ms, T

d

= 320

l

s,

D

x

ad

=ð2

p

Þ¼40 kHz,

N

d

= 32, resulting in x

max

5.7 mm and z

max

36.4 mm.

Fig. 2. Cross-encoded scheme designed to compensate for unknown inhomogene-

ities of the main ﬁeld. In the second encoding block E2, the signs of all gradients are

inverted with respect to E1. The combined encoding blocks (E1 +E2) yield a phase

of the transverse magnetization where the effects of all inhomogeneities are

cancelled. A delay

s

can be inserted between the two adiabatic pulses in E2to

attenuate the distortions of the trajectories during the decoding sequence.

200 R. Paquin et al. / Journal of Magnetic Resonance 201 (2009) 199–204

Page 2

all gradients are inverted. If the interval

s

=0inE2, the phase

resulting from the encoding unit E2 alone is

u

E2

ðx; zÞ¼k

E2

xz ð12Þ

with

k

E2

¼ 4T

e

c

2

ðG

e

x

G

e

z

G

e

z

dB

0

ðx; y; zÞ=xÞ=

D

x

ad

ð13Þ

Consequently, the effects of inhomogeneous static ﬁelds are

cancelled by concatenating sequences E1 and E2:

u

e

¼2k xz ð14Þ

where k is deﬁned in Eq. (3).

We now focus our attention on the decoding sequence. If we

combine the encoding schemes E1 +E2ofFig. 2 with the decoding

part D1ofFig. 1A, the spatial variation of the static ﬁeld dB

0

(x, y, z)

yields an additional contribution to the overall phase

Uðx; y; z; tÞ¼

u

e

ðx; zÞþk

x

ðtÞx þ k

z

ðtÞz þ

c

dB

0

ðx; y; zÞt ð15Þ

This phase must again fulﬁll the conditions of Eq. (7):

x

k

c

t

k

2k

@ dB

0

ðx; y; zÞ

@z

ðx

k

;y

k

;z

k

Þ

¼

k

z

ðt

k

Þ

2k

z

k

c

t

k

2k

@ dB

0

ðx; y; zÞ

@x

ðx

k

;y

k

;z

k

Þ

¼

k

x

ðt

k

Þ

2k

ð16Þ

The zig-zag trajectory now deviates from the ideal behavior in a

homogeneous ﬁeld because of the terms that are proportional to t

k

and to the derivatives of dB

0

(x, y, z). The signal from the voxel V

k

will be affected by static inhomogeneities if the time t

k

at which

the echo appears is delayed.

During decoding, we can attenuate the effects of static ﬁeld

inhomogeneities but not suppress them completely. A ﬁrst approx-

imate solution (scheme D2inFig. 3A) relies on the insertion of a

composite pulse (

p

/2)

y

(

p

)

x

(

p

/2)

y

[10] between each pair of decod-

ing gradients as can be done in EPI experiments [11,12]. This leads

to an overall phase

Uðx; y; z; tÞ¼

u

e

ðx; zÞþk

x

ðtÞx þ k

z

ðtÞz þ

c

dB

0

ðx; y; zÞf

D2

ðtÞð17Þ

The sign of the term

c

dB

0

(x, y, z)f

D2

(t) is reversed at

t =(2n 1)T

d

with n=1, 2, ..., N

d

. In analogy to k

z

(t) in Eqs. (5)

and (6), we deﬁne:

f

D2

ðtÞ¼

Z

t

0

ð1Þ

n

D2

ðt

0

Þ

dt

0

ð18Þ

where n

D2

(t

0

) is the integer part of t

0

/(2Td) + 1/2, resulting in:

x

k

c

f

D2

ðt

k

Þ

2k

@ dB

0

ðx; y; zÞ

@z

ðx

k

;y

k

;z

k

Þ

¼

k

z

ðt

k

Þ

2k

z

k

c

f

D2

ðt

k

Þ

2k

@ dB

0

ðx; y; zÞ

@x

ðx

k

;y

k

;z

k

Þ

¼

k

x

ðt

k

Þ

2k

ð19Þ

The most severe error in the localization of the voxel of interest

V

k

occurs when t

k

=(2n-1)T

d:

Fig. 3. (A) Comparison of the basic decoding scheme D1ofFig. 1A with two alternative methods D2 and D3. In scheme D2, composite refocusing pulses 90

o

x

180

o

y

90

o

x

ﬂanked

by two gradients G

1

are inserted between each pair of decoding gradients, while in scheme D3 they are inserted after each gradient. (B) Simulated proton spectra and zig-zag

trajectories in homogeneous (black dashed lines) and inhomogeneous ﬁelds (red continuous lines). The proton spectrum arising from a 4 4 20 mm sample shows a full-

width at half maximum of 20 kHz. Scheme D2 entirely refocuses deviations from the ideal trajectory at t =2(n 1)T

d

with n=1, 2, ..., N

d

but leaves errors at t=(2n 1)T

d

.

When a delay

s

= T

d

/2 is inserted in the encoding sequence E2ofFig. 2, scheme D3 results in distorted and ideal trajectories that intersect halfway at x

max

/2 rather than at x

0

,

and the deviations are reduced by a factor two. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this paper.)

R. Paquin et al. / Journal of Magnetic Resonance 201 (2009) 199–204

201

Page 3

D

x

k

¼

c

T

d

2k

@ dB

0

ðx; y; zÞ

@z

ðx

k

;y

k

;z

k

Þ

D

z

k

¼

c

T

d

2k

@ dB

0

ðx; y; zÞ

@x

ðx

k

;y

k

;z

k

Þ

ð20Þ

In the alternative scheme D3inFig. 3A, a composite pulse

(

p

/2)

y

(

p

)

x

(

p

/2)

y

is inserted after each decoding gradient, and a de-

lay

s

= T

d

/2 is used in E2 (see Fig. 2). The overall phase becomes

Uðx; y; z; tÞ¼

u

e

ðx; zÞþk

x

ðtÞx þ k

z

ðtÞz þ

c

dB

0

ðx; y; zÞf

D3

ðtÞ

c

dB

0

ðx; y; zÞ

T

d

2

ð21Þ

where

f

D3

ðtÞ¼

Z

t

0

ð1Þ

n

D3

ðt

0

Þ

dt

0

ð22Þ

with n

D3

ðt’Þ¼n

D1

ðt

0

Þ so that

x

k

c

ðf

D3

ðt

k

ÞT

d

=2Þ

2k

@ dB

0

ðx; y; zÞ

@z

ðx

k

;y

k

;z

k

Þ

¼

k

z

ðt

k

Þ

2k

z

k

c

ðf

D3

ðt

k

ÞT

d

=2Þ

2k

@ dB

0

ðx; y; zÞ

@x

ðx

k

;y

k

;z

k

Þ

¼

k

x

ðt

k

Þ

2k

ð23Þ

The error in the localization of the voxel V

k

is halved compared

to Eq. (20). This error reaches a maximum at intervals t

k

= T

d

.

In Fig. 3B, we simulate the zig-zag trajectories for the three

schemes D1, D2 and D3, describing the intrinsic (time-indepen-

dent) inhomogeneity of the static ﬁeld by second-order

polynomials:

dB

0

ðx; y; zÞ¼ðG

x

x þ F

xx

x

2

; G

y

y þ F

yy

y

2

; G

z

z þ F

zz

z

2

Þð24Þ

If G

x

= G

y

= G

z

= 0.001 T/m and F

xx

= F

yy

= F

zz

= 0.02 T/m

2

, the

breadth of the simulated spectrum of Fig. 3B is around 20 kHz. This

leads to a zig-zag trajectory that deviates from the ideal path dur-

ing the D1 decoding sequence. The D2 scheme allows one to refo-

cus these deviations at t =2(n-1)T

d

with n=1, 2, ..., N

d

(corresponding to the origin of the x axis), leaving signiﬁcant devi-

ations near x

max

. With the D3 scheme combined with a delay

s

= T

d

/2, the distorted and ideal trajectories intersect halfway at x

max

/2

rather than at x

0

, and the errors are attenuated by a factor 2.

2.3. Resolution of CE-MRI in homogeneous ﬁelds

As for spatial encoding using chirped pulses [5], the resolution

of CE-MRI depends only on the encoding conditions. In Fig. 4,we

present numerical simulations of the image of square and rectan-

gular two-dimensional objects obtained with the pulse scheme of

Fig. 1A. In Fig. 4A–C, the overall phase

U

of Eq. (8) was weighted

by factors 0.1, 1 and 10, respectively. This allows one to scale the

encoding parameter k while keeping the same ﬁeld of view

(FOV). The three two-dimensional images and central cross-sec-

tions through the square object at x =0orz = 0 illustrate how the

resolution depends on k. The resolution can be improved by

increasing the strength and/or length of the encoding gradients.

The decoding gradients affect only the FOV but have no inﬂuence

on the resolution.

By comparing the images of the rectangular 2 30 mm object

in Fig. 1C and the square 2 2 mm object in Fig. 4B, one observes

a difference in resolution, although the encoding parameters were

similar. Clearly, the resolution depends on the dimensions of the

object. In Figs. 4D–F, we show that the elongation of the object

Fig. 4. Effect of encoding parameters and dimensions of the object on the resolution. As in Fig. 1C, numerical simulations of the images of square and rectangular two-

dimensional objects were obtained with the pulse scheme of Fig. 1A. The absolute value of the integral of Eq. (4) was calculated at intervals

D

x = x

max

/50 and

D

z = z

max

/100

with the following parameters: G

e

x

¼ G

e

z

¼ 0:022 T/m, G

d

x

¼ 0:0241 T/m and G

d

z

¼ 0:3864 T/m, T

e

= 6 ms, T

d

= 320

l

s,

D

x

ad

=ð2

p

Þ¼40 kHz, N

d

= 16, resulting in x

max

=10mm

and z

max

= 20 mm (FOV = 10 20 mm). From A to C, the overall phase

U

of Eq. (8) was weighted by factors 0.1, 1 and 10, respectively. The 2D images and central cross-

sections (blue lines for vertical cross-sections at x = 0, red lines for horizontal cross-sections at z =0)ofa2 2 mm square object illustrate the dependence of the resolution

on the encoding parameter. Elongation of the object along the z axis to 2 4 mm (D) or 2 8 mm (F) or along the x axis to 4 2 mm (E) increases the resolution in the

perpendicular direction. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this paper.)

202 R. Paquin et al. / Journal of Magnetic Resonance 201 (2009) 199–204

Page 4

in one dimension increases the resolution in the perpendicular

direction. In Fig. 4D and E, the object is elongated either along

the z axis (2 4 mm) or along the x axis (4 2 mm). This leads

to an improvement of the resolution in the x and z dimensions

respectively. The ﬁgures also illustrate that the dimensions of the

FOV have no impact on the resolution. A comparison of the

cross-sections in Fig. 4B, D and F (the latter with a 2 8 mm ob-

ject) highlights a gain in resolution (sharper transitions near the

edges) in the x dimension (red lines) while the resolution in the z

dimension (blue lines) remains the same (same slope in transition

regions).

3. Experimental results

Fig. 5 shows experimental images obtained with a phantom

consisting of a glass capillary (0.9 and 1.6 mm for the inner and

outer diameters) inserted in a sample tube (3.8 mm inner diame-

ter), where the spaces are ﬁlled with water. Various combinations

of encoding and decoding schemes were tested from left to right,

for increasing static ﬁeld inhomogeneities dB

0

(x, y, z) from top to

bottom, characterized by the lineshapes of the water resonance re-

corded in the absence of any pulsed ﬁeld gradients (left margin).

The images are increasingly distorted from top to bottom. Used

in isolation, the encoding {E1 +E2} schemes or the decoding D3 se-

quence cannot cope with very large inhomogeneities. Only the

combination {E1 +E2 +D3} gives nearly undistorted images in

the most inhomogeneous ﬁelds. The combination {E1 +E2 +D2}

(not shown) gives slightly larger distortions.

Fig. 6 (top half) shows applications of the combined scheme

{E1 +E2 +D3} to the same phantom as used in Fig. 5 and (bottom

half) to a different phantom consisting of a glass spiral inserted in

a sample tube (3.8 mm inner diameter). This demonstrates how

the new method yields 3D images that are virtually indistinguish-

able when recorded under dramatically different conditions,

regardless whether the homogeneity is good or severely per-

turbed. The geometrical characteristics of the two phantoms can

be determined quite accurately even when the inhomogeneity of

the static ﬁeld leads to a water resonance with a breadth of about

3000 Hz.

4. Conclusions

The novel cross-encoded MRI schemes, using adiabatic pulses

and alternating gradients in two perpendicular dimensions, can re-

move undesirable effects of inhomogeneous magnetic ﬁelds that

may result from susceptibility effects and prevent blurring due to

slow motion since the images are recorded in a fraction of a sec-

ond. The zig-zag trajectories in the decoding period can be im-

proved by using refocusing pulses. Magnetic resonance images

have been recorded in two and three dimensions that are virtually

indistinguishable in spite of dramatically different static ﬁeld

inhomogeneities.

Fig. 5. Comparison of experimental images of a phantom with various combinations of encoding and decoding schemes for an increasingly inhomogeneous static ﬁeld dB

0

(x,

y, z), characterized by the lineshapes of the water resonance in the absence of any pulsed ﬁeld gradients (left). A glass capillary (0.9 and 1.6 mm inner and outer diameters)

was inserted into a tube with a 3.8 mm inner diameter; the spaces inside the capillary and between the two tubes being ﬁlled with water. The experimental conditions were

G

e

x

¼ 0:022 T/m (for E1) or G

e

x

¼ 0:011 T/m (for E1 +E2), G

e

z

¼ 0:022 T/m, G

d

x

¼ 0:0165 T/m, G

d

z

¼ 0:1925 T/m, T

e

= 6 ms, T

d

= 320

l

s,

D

x

ad

=ð2

p

Þ¼40 kHz, N

d

= 32. All

experiments were performed with a Bruker Avance 600 MHz (14T) spectrometer equipped with a triple-gradient TBI probe.

R. Paquin et al. / Journal of Magnetic Resonance 201 (2009) 199–204

203

Page 5

Acknowledgments

We thank Laetitia Zuccarelli for manufacturing the helical glass

phantom. This work was supported by the European Union (Inte-

grated Infrastucture Initiative, Contract No. RII3-026145, Joint Re-

search Activity JRA1, Contract No. 026145), the Centre National

de la Recherche Scientiﬁque (CNRS, France), the Agence Nationale

pour la Recherche (ANR, France), the Fonds National de la Recher-

che Scientiﬁque (FNRS, Switzerland, No. 200020_124694/1) and

the Commission pour la Technologie et l’Innovation (CTI, Switzer-

land, No. 9991.1 PFIW-IW).

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Fig. 6. Images obtained with the combined scheme {E1 +E2 +D3} of Figs. 2 and 3 with (A) the same glass capillary phantom as in Fig. 5, and (B) with a glass spiral phantom

surrounded by water inserted into a tube with 3.8 mm inner diameter. Both phantoms were immersed either in homogeneous or inhomogeneous ﬁelds characterized by the

proton spectra on the left (blue or red lines, respectively). In the y-dimension, 32 neighboring slices were recorded with a recovery delay of 10 s between scans. Photographs

of the phantoms (black-and-white) are compared with full 3D images obtained by addition of the 32 planes (grey scale) and with 2D images of the central slices (blue scale).

In order to visualize the cross-sections of the phantoms, the different planes of the 3D images are stacked following the experimental zig-zag pattern, like the fold of an

accordion. The experimental conditions were like in Fig. 5: G

e

x

¼ 0:011 T/m, G

e

z

¼ 0:022 T/m, G

d

x

¼ 0:0165 T/m, G

d

z

¼ 0:1925 T/m, T

e

= 6 ms, T

d

= 320

l

s,

D

x

ad

=ð2

p

Þ¼40 kHz, N

d

= 32,

s

= T

d

/2. A Bruker Avance 600 MHz (14T) spectrometer equipped with a triple-gradient TBI probe was used. (For interpretation of the references to color in this ﬁgure

legend, the reader is referred to the web version of this paper.)

204 R. Paquin et al. / Journal of Magnetic Resonance 201 (2009) 199–204

Page 6