Cross-encoded magnetic resonance imaging in inhomogeneous fields

ArticleinJournal of Magnetic Resonance 201(2):199-204 · September 2009with96 Reads
Impact Factor: 2.51 · DOI: 10.1016/j.jmr.2009.09.008 · Source: PubMed
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 inhomogeneous fields where the breadth of the water resonance extends over several kHz.

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Available from: Raphaël Paquin
Cross-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; zÞ¼kxz 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; zÞ½exp ik
x
ðtÞx þ ik
z
ðtÞz½dxdz 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; zÞ¼k
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; zÞ¼k
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
¼2k 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
ð1Þ
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
ð1Þ
n
D3
ðt
0
Þ
dt
0
ð22Þ
with n
D3
ðtÞ¼n
D1
ðt
0
Þ so that
x
k
c
ðf
D3
ðt
k
ÞT
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
ÞT
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