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Three-Dimensional NMR Imaging Using Large Oscillating Field Gradients
JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.
Series A 119, 105–110 (1996)
0057
Three-Dimensional NMR Imaging Using Large Oscillating Field Gradients M. J. D. MALLETT, S. L. CODD, M. R. HALSE, T. A. P. GREEN,
AND
J. H. STRANGE
Physics Laboratory, University of Kent, Canterbury CT2 7NR, United Kingdom Received June 19, 1995; revised October 24, 1995
A simple gradient-echo technique is described which uses large oscillating field gradients to produce three-dimensional NMR images of small (20 mm) moulded plastic and rubber components. The in-plane resolution is 0.5 mm. The echo time is 128 ms. The method is suitable for imaging soft solid materials having linewidths up to Ç3 kHz. The large gradients are used to dominate the homogeneous dipolar broadening, while the sinusoidal waveform helps overcome the limit imposed by the rise times of the gradient coils. Phase encoding and frequency encoding in a manner similar to conventional 3DFT enables k space to be sampled in three dimensions. An essential feature of the method is that the frequency of the two sinusoidal gradients is half the frequency of the read gradient. The use of sinusoidal gradients gives rise to a nonuniform coverage of k space, and suitable computer correction of the nonlinearly sampled k-space data is therefore required. Many different approaches have been adopted for the NMR imaging of solids and soft solids. A comprehensive review of the various techniques of solid-state imaging has been given by Veeman and by Kimmich, Demco, and Hafner, and by others in (1). The reader is also referred to the more recent 3D magic-echo phase-encoding scheme proposed by Hafner, Barth, and Kuhn (2) and the STRAFI stray field technique of Zick (3). Probably the simplest approach has been described by Cottrell et al. (4) who use large oscillating field gradients to dominate the line broadening and back projection to produce a two-dimensional image (5). A method has been described by Mat Daud and Halse (6) for obtaining a two-dimensional image by using sinusoidal read and phase-encode gradients in a manner similar to standard 2DFT (7, 8). In this Note, we have generalized their method to three dimensions by introducing a second sinusoidal phase-encode gradient. The timing diagram of the 3D imaging sequence is shown in Fig. 1. In the experiment, the 907 RF pulse is synchronized to occur at the zero crossing of the read gradient, Gread (t) Å GO read sin( vt),
produce an echo. The data-sampling window extends from T/2 to 3T/2. For 3D imaging, this pulse sequence is repeated in the presence of two phase-encoding gradients, Gphase1 (t, n2 ) Å
( 0 Np1 /2 £ n2 õ Np1 /2) Gphase2 (t, n3 ) Å
* G(t * )dt *. t
k(t) Å g
[4]
0
To perform the Fourier transform from k space to real space requires a linearly sampled data set based on Cartesian coordinates [see Kumar, Welti, and Ernst (11)]. If the gradients vary during the data-sampling period, k space is nonlinearly sampled. In order to correct for the nonlinear sampling, it is necessary to consider how the trajectory in k space for oscillating gradients differs from that obtained when using constant gradients. For the usual 3D gradient-echo imaging sequence, using constant read and phase-encode gradients, gread , gphase1 , and gphase2 , the k-space data set is given by
S
k(m1 , m2 , m3 ) Å m1 ggread
S
[1]
/ m2 g
magas
[3]
where the frequency of the phase-encoding gradients is arranged to be half the frequency of the read gradient. The trajectory in k space [Callaghan (8), Mansfield (9), and Ljunggren (10)] is given by
105
02-13-96 20:00:32
SD
[2]
2n3 vt GO phase2 sin Np2 2
( 0 Np2 /2 £ n3 õ Np2 /2),
and, after one period of T Å 2p / v, the spins refocus to
m4834$0805
SD
2n2 vt GO phase1 sin Np1 2
D
T kread Nsp
D
2 T gphase1 kphase1 Np1 2
1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
AP: Mag Res
106
NOTES
the read window ( 0 Nsp /2 £ n1 õ Nsp /2,), then the sampling of k space is given by k(n1 , n2 , n3 ) Å
S
g
/
/
FIG. 1. Timing diagram of the oscillating-field-gradient method. The data-capture window extends from point A at time t Å T/2 to point C at time t Å 3T/2. The echo center, point B, occurs at a time t Å T.
S
/ m3 g
D
2 T gphase2 kphase2 , Np2 2
TGO read 2p
S S
t
g
0
GO read sin t
/
g
0
t
/
g
0
[5]
S S
g
n3T pn1 GO phase2 1/ cos pNp2 Nsp
DD DD
kphase1
kphase2 .
[7]
The coverage in k space after the first sequence of phase encodes, but without the second phase-encode gradient (i.e., n3 Å 0), is shown in Fig. 2a. Note that since the read gradient is reversed halfway through the data-sampling window, the trajectory only covers half of k space. Full coverage is obtained by making use of the conjugate nature of the data r(k); i.e., [8]
Thus by taking the complex conjugate of the data collected after the echo center, t Å T, and swapping over the appropriate quadrants, the trajectory can be made to cover all of k space (see Fig. 2b). If the relationship between the two sets of k-space sample points is determined, it is possible to obtain a data set that is equivalent to that which would be obtained if the usual constant-gradient imaging sequence had been used. Comparing Eqs. [5] and [7], the maximum excursion in each kspace direction using oscillating gradients can be made equal to that obtained with constant gradients provided that GO read Å ( p /2)gread
[9a]
GO phase1 Å ( p /4)gphase1
[9b]
GO phase2 Å ( p /4)gphase2 .
[9c]
[6]
S
gTgread 4
S
S
1 0 cos
Å m1 ggread
If the data are sampled at times t Å T / n1T/Nsp , during
02-13-96 20:00:32
kread
Again comparing Eqs. [5] and [7], in general we require that
2n2 pt * GO phase1 sin dt * kphase1 Np1 T
m4834$0805
DD
n2T pn1 GO phase1 1/ cos pNp1 Nsp
2pt * dt * kread T
2n3 pt * GO phase2 sin dt * kphase2 . Np2 T
2pn1 Nsp
g
k(t, n2 , n3 )
S* S D D S* S D D S* S D D
1 0 cos
r( 0k) Å r*( /k).
where Np1 and Np2 are the number of steps for the two phaseencode gradients, Nsp is the number of sample points in the read direction, T is the duration of the read window, and m1 , m2 , and m3 are integers such that 0 Nsp /2 £ m1 õ Nsp / 2, 0 Npl /2 £ m2 õ Np1 /2, 0 Np2 /2 £ m3 õ Np2 /2. These sample points cover k space on a uniform 3D grid with unit vectors kread , kphase1 , and kphase2 . For the oscillating-gradient sequence, the trajectories in k space are given by substituting Eqs. [1] – [3] into Eq. [4]. Thus
Å
S
magas
AP: Mag Res
T Nsp
2pn1 Nsp
D
DD [10a]
107
NOTES
FIG. 2. Trajectories in k space. The dark trajectory, ABC, corresponds to the data-capture window as shown in Fig. 1. (a) The k-space map demonstrates how only half of k space is covered when oscillating gradients are used, each echo being symmetric. (b) Modified k-space map shows the trajectory, ABC, after taking the complex conjugate of the signal after the echo center.
S
g
S
n2T pn1 gphase1 1 / cos 2Np1 Nsp
S
Å m2 g
S
g
2 T gphase1 Np1 2
D
S
S
2 T gphase2 Np2 2
D
n3 Å
[10b]
n3T pn1 gphase2 1 / cos 2Np2 Nsp
Å m3 g
DD DD
.
[10c]
These relationships can be used to calculate the values of n1 , n2 , and n3 in terms of m1 , m2 , and m3 . Thus
n1 Å
n2 Å
Å
S S DD S YS S DD r S YS DD Nsp 4m1 cos 01 1 0 2p Nsp
2m2
1 / cos
2m2
1/
m4834$0805
[11a]
pn1 Nsp
10
2Ém1É Nsp
02-13-96 20:00:32
[11b]
magas
Å
S YS S YS 2m3
2m3
1 / cos
S DD p n1 Nsp
r 1/
10
2Ém1É Nsp
DD
.
[11c]
Since in general n1 , n2 , and n3 will be nonintegers, linear interpolation is used within the acquired data set. In practice, linearization is first performed along the read direction by applying Eq. [11a] to the data. For example, if m1 Å 1 and the number of samples Nsp Å 128, then n1 Å 5.1. Hence the correct datum for the first point in k space along the read direction is estimated by linearly interpolating between the fifth and sixth data points collected after the center of the echo. If this is repeated for all values of m1 , then a data set is obtained that samples k space as shown in Fig. 3a. Linearization is then carried out in a similar manner in the two phase-encode directions by applying Eqs. [11b] and [11c] to the new data. Note that if n2 ú Np1 /2 or n3 ú Np2 / 2, then there is no data to interpolate between, and a value of zero is assumed for this data point (see Fig. 3b). Equal spatial resolution of the image in all three dimensions can be obtained by ensuring that the maximum k-space excursion is the same in all three directions. Thus, from Eq.
AP: Mag Res
108
NOTES
FIG. 3. (a) Trajectories in k space after linearizing along kread by interpolating appropriately between the sample points in Fig. 2. (b) The trajectories in k space after linearization in the phase 1 direction; the crosses in the corners of k space indicate the regions where interpolation is impossible and the data array is filled with zeroes.
[6], the maximum value of kread occurs at t Å T/2 and is given by kread (T/2) Å gGO readT/ p.
[12]
The maximum value of kphase1 occurs when Én2É Å Np1 /2 and t Å T and is given by
S
kphase1 T,
Np1 2
D
Å gGO phase1T/2p.
[13]
Similarly the maximum value of kphase2 is given by
S
kphase2 T,
Np2 2
D
Å gGO phase2T/2p.
[14]
Hence we require GO read Å
GO phase1 GO phase2 Å . 2 2
[15]
The experiments were carried out using a 0.73 T superconducting magnet corresponding to a proton frequency of 31.2 MHz. A small RF birdcage coil was used for ‘‘transmit’’ and ‘‘receive.’’ The gradient coils were constructed on a design obtained from Magnex Ltd. (12) and consist of a
m4834$0805
02-13-96 20:00:32
magas
Maxwell pair for the z gradient and two sets of Golay pairs for the x and y gradients. The gradient strengths were measured to be 15, 13, and 30 mT m01 A 01 for the x, y, and z gradients, respectively. Each gradient coil was connected to a Techron amplifier working in controlled-current mode. A PC-compatible computer containing hardware provided by SMIS Ltd. (13) was used to generate the sinusoidal gradient waveforms from a pair of two-channel 12 bit DACs. The SMIS hardware included a pulse programmer which controlled the whole of the imaging sequence. The pulse programmer was interfaced to a Bruker CXP spectrometer to provide the necessary RF frequency generator, transmitter, amplifier, and receiver. The NMR data were acquired by two 12 bit Datel ADCs sampling at 500 kHz. The data were linearized on a DEC Alpha workstation and then Fourier transformed to a three-dimensional data set using standard SMIS reconstruction software. The final images (Fig. 4) were produced from a general surface-rendering software package written by one of us (T.A.P.G.). The sample used to demonstrate the technique was an ABS plastic Lego brick. The T 2 relaxation curve displayed two significant components in which 90% of the signal had a T 2 of 760 ms and the remaining 10% had a T 2 of 6 ms. By imaging at an echo time of 128 ms it was possible to view both components present in the sample. Figure 4a shows a three-dimensional surface-rendered image of the sample. Figures 4b and 4c show cross-sectional views through the sample. The images were acquired using 64 samples in fre-
AP: Mag Res
109
NOTES
FIG. 4. (a) A three-dimensional surface-rendered image of an ABS plastic Lego brick whose size is approximately 15 1 15 1 11 mm. (b, c) Crosssectional views through the same sample.
quency-encoding direction and 64 phase-encoding steps in the other two directions. The peak gradient strength in the frequency-encode direction was 1.2 T m01 (40 A peak) while the two phase-encode gradients reached a peak of {600 mT m01 (40 and 46 A peaks for Gx and Gy , respectively). The frequency-encoding gradient oscillated at 7.8 kHz, and the two phase-encoding gradients oscillated at 3.9 kHz. The data-capture window therefore extended from 64 to 192 ms, with the center of the gradient echo occurring at 128 ms. The T 1 of the sample was on the order of 150 ms, and a repetition time of 100 ms was found to be satisfactory. A total of 32 averages were acquired in a time of 32 1 64 1 64 1 0.1 s, i.e., 3 hours 38 minutes. The field of view for the images is 32 1 32 1 32 mm. In summary, the application of oscillating gradients to an imaging system previously used for liquids proved to be straightforward and allowed us to reduce the echo time from 2 ms to 128 ms. Although 3D back-projection schemes based on capturing the FID have similar capabilities in terms of
m4834$0805
02-13-96 20:00:32
magas
imaging soft solids, the 3DFT method presented here relies on the generation of an echo and therefore has the advantage that image artifacts associated with missing k-space data around k Å 0 due to receiver deadtime are avoided. On the other hand, it should be remembered that all echo schemes which rely on phase encoding suffer from unavoidable signal decay due to T 2 relaxation during the phase-encoding period. In the present oscillating-gradient scheme, this period can be minimized for a given gradient amplifier performance by using a series-resonant gradient-coil circuit. Finally it should be noted that a worthwhile increase in S/N should be possible by using a FLASH (14, 15) version of the present technique. ACKNOWLEDGMENTS This work was supported by an EPSRC grant. We thank SMIS Ltd. for their support in setting up the imaging system software and the Molecular Dynamics Group at UKC for the use of their DEC Alpha workstation.
AP: Mag Res
110
NOTES
REFERENCES 1. B. Blumich and W. Kuhn (Eds.), ‘‘Magnetic Resonance Microscopy,’’ VCH, Weinheim, 1992. 2. S. Hafner, P. Barth, and W. Kuhn, J. Magn. Reson. A 108, 21 (1994). 3. K. Zick, Nondestr. Test. Eval. 11, 255 (1994). 4. S. P. Cottrell, M. R. Halse, and J. H. Strange, Meas. Sci. Technol. 1, 624 (1990). 5. R. A. Brooks and G. Di Chiro, Phys. Med. Biol. 21, 689 (1976). 6. Y. Mat Daud and M. R. Halse, Physica B 176, 167 (1992). 7. G. Johnson, J. M. S. Hutchison, T. W. Redpath, and L. M. Eastwood, J. Magn. Reson. 54, 375 (1983).
m4834$0805
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magas
8. P. T. Callaghan, ‘‘Principles of Magnetic Resonance Microscopy,’’ Chap. 3, Oxford Univ. Press, Oxford, 1991. 9. P. Mansfield and P. G. Morris, ‘‘NMR Imaging in Biomedicine,’’ Academic Press, London, 1982. 10. S. Ljunggren, J. Magn. Reson. 54, 338 (1983). 11. A. Kumar, D. Welti, and R. R. Ernst, J. Magn. Reson. 18, 69 (1975). 12. Magnex Scientific Limited, 21 Blacklands Way, Abingdon, Oxon, UK, private communication. 13. SMIS (Surrey Medical Imaging Systems Ltd.), Alan Turing Road, Guildford, Surrey, UK. 14. A. Haase, J. Frahm, D. Matthaei, W. Hanicke, and K-D. Merboldt, J. Magn. Reson. 67, 258 (1986). 15. P. J. McDonald, K. L. Perry, and S. P. Roberts, Meas. Sci. Technol. 4, 896 (1993).
AP: Mag Res
Series A 119, 105–110 (1996)
0057
Three-Dimensional NMR Imaging Using Large Oscillating Field Gradients M. J. D. MALLETT, S. L. CODD, M. R. HALSE, T. A. P. GREEN,
AND
J. H. STRANGE
Physics Laboratory, University of Kent, Canterbury CT2 7NR, United Kingdom Received June 19, 1995; revised October 24, 1995
A simple gradient-echo technique is described which uses large oscillating field gradients to produce three-dimensional NMR images of small (20 mm) moulded plastic and rubber components. The in-plane resolution is 0.5 mm. The echo time is 128 ms. The method is suitable for imaging soft solid materials having linewidths up to Ç3 kHz. The large gradients are used to dominate the homogeneous dipolar broadening, while the sinusoidal waveform helps overcome the limit imposed by the rise times of the gradient coils. Phase encoding and frequency encoding in a manner similar to conventional 3DFT enables k space to be sampled in three dimensions. An essential feature of the method is that the frequency of the two sinusoidal gradients is half the frequency of the read gradient. The use of sinusoidal gradients gives rise to a nonuniform coverage of k space, and suitable computer correction of the nonlinearly sampled k-space data is therefore required. Many different approaches have been adopted for the NMR imaging of solids and soft solids. A comprehensive review of the various techniques of solid-state imaging has been given by Veeman and by Kimmich, Demco, and Hafner, and by others in (1). The reader is also referred to the more recent 3D magic-echo phase-encoding scheme proposed by Hafner, Barth, and Kuhn (2) and the STRAFI stray field technique of Zick (3). Probably the simplest approach has been described by Cottrell et al. (4) who use large oscillating field gradients to dominate the line broadening and back projection to produce a two-dimensional image (5). A method has been described by Mat Daud and Halse (6) for obtaining a two-dimensional image by using sinusoidal read and phase-encode gradients in a manner similar to standard 2DFT (7, 8). In this Note, we have generalized their method to three dimensions by introducing a second sinusoidal phase-encode gradient. The timing diagram of the 3D imaging sequence is shown in Fig. 1. In the experiment, the 907 RF pulse is synchronized to occur at the zero crossing of the read gradient, Gread (t) Å GO read sin( vt),
produce an echo. The data-sampling window extends from T/2 to 3T/2. For 3D imaging, this pulse sequence is repeated in the presence of two phase-encoding gradients, Gphase1 (t, n2 ) Å
( 0 Np1 /2 £ n2 õ Np1 /2) Gphase2 (t, n3 ) Å
* G(t * )dt *. t
k(t) Å g
[4]
0
To perform the Fourier transform from k space to real space requires a linearly sampled data set based on Cartesian coordinates [see Kumar, Welti, and Ernst (11)]. If the gradients vary during the data-sampling period, k space is nonlinearly sampled. In order to correct for the nonlinear sampling, it is necessary to consider how the trajectory in k space for oscillating gradients differs from that obtained when using constant gradients. For the usual 3D gradient-echo imaging sequence, using constant read and phase-encode gradients, gread , gphase1 , and gphase2 , the k-space data set is given by
S
k(m1 , m2 , m3 ) Å m1 ggread
S
[1]
/ m2 g
magas
[3]
where the frequency of the phase-encoding gradients is arranged to be half the frequency of the read gradient. The trajectory in k space [Callaghan (8), Mansfield (9), and Ljunggren (10)] is given by
105
02-13-96 20:00:32
SD
[2]
2n3 vt GO phase2 sin Np2 2
( 0 Np2 /2 £ n3 õ Np2 /2),
and, after one period of T Å 2p / v, the spins refocus to
m4834$0805
SD
2n2 vt GO phase1 sin Np1 2
D
T kread Nsp
D
2 T gphase1 kphase1 Np1 2
1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
AP: Mag Res
106
NOTES
the read window ( 0 Nsp /2 £ n1 õ Nsp /2,), then the sampling of k space is given by k(n1 , n2 , n3 ) Å
S
g
/
/
FIG. 1. Timing diagram of the oscillating-field-gradient method. The data-capture window extends from point A at time t Å T/2 to point C at time t Å 3T/2. The echo center, point B, occurs at a time t Å T.
S
/ m3 g
D
2 T gphase2 kphase2 , Np2 2
TGO read 2p
S S
t
g
0
GO read sin t
/
g
0
t
/
g
0
[5]
S S
g
n3T pn1 GO phase2 1/ cos pNp2 Nsp
DD DD
kphase1
kphase2 .
[7]
The coverage in k space after the first sequence of phase encodes, but without the second phase-encode gradient (i.e., n3 Å 0), is shown in Fig. 2a. Note that since the read gradient is reversed halfway through the data-sampling window, the trajectory only covers half of k space. Full coverage is obtained by making use of the conjugate nature of the data r(k); i.e., [8]
Thus by taking the complex conjugate of the data collected after the echo center, t Å T, and swapping over the appropriate quadrants, the trajectory can be made to cover all of k space (see Fig. 2b). If the relationship between the two sets of k-space sample points is determined, it is possible to obtain a data set that is equivalent to that which would be obtained if the usual constant-gradient imaging sequence had been used. Comparing Eqs. [5] and [7], the maximum excursion in each kspace direction using oscillating gradients can be made equal to that obtained with constant gradients provided that GO read Å ( p /2)gread
[9a]
GO phase1 Å ( p /4)gphase1
[9b]
GO phase2 Å ( p /4)gphase2 .
[9c]
[6]
S
gTgread 4
S
S
1 0 cos
Å m1 ggread
If the data are sampled at times t Å T / n1T/Nsp , during
02-13-96 20:00:32
kread
Again comparing Eqs. [5] and [7], in general we require that
2n2 pt * GO phase1 sin dt * kphase1 Np1 T
m4834$0805
DD
n2T pn1 GO phase1 1/ cos pNp1 Nsp
2pt * dt * kread T
2n3 pt * GO phase2 sin dt * kphase2 . Np2 T
2pn1 Nsp
g
k(t, n2 , n3 )
S* S D D S* S D D S* S D D
1 0 cos
r( 0k) Å r*( /k).
where Np1 and Np2 are the number of steps for the two phaseencode gradients, Nsp is the number of sample points in the read direction, T is the duration of the read window, and m1 , m2 , and m3 are integers such that 0 Nsp /2 £ m1 õ Nsp / 2, 0 Npl /2 £ m2 õ Np1 /2, 0 Np2 /2 £ m3 õ Np2 /2. These sample points cover k space on a uniform 3D grid with unit vectors kread , kphase1 , and kphase2 . For the oscillating-gradient sequence, the trajectories in k space are given by substituting Eqs. [1] – [3] into Eq. [4]. Thus
Å
S
magas
AP: Mag Res
T Nsp
2pn1 Nsp
D
DD [10a]
107
NOTES
FIG. 2. Trajectories in k space. The dark trajectory, ABC, corresponds to the data-capture window as shown in Fig. 1. (a) The k-space map demonstrates how only half of k space is covered when oscillating gradients are used, each echo being symmetric. (b) Modified k-space map shows the trajectory, ABC, after taking the complex conjugate of the signal after the echo center.
S
g
S
n2T pn1 gphase1 1 / cos 2Np1 Nsp
S
Å m2 g
S
g
2 T gphase1 Np1 2
D
S
S
2 T gphase2 Np2 2
D
n3 Å
[10b]
n3T pn1 gphase2 1 / cos 2Np2 Nsp
Å m3 g
DD DD
.
[10c]
These relationships can be used to calculate the values of n1 , n2 , and n3 in terms of m1 , m2 , and m3 . Thus
n1 Å
n2 Å
Å
S S DD S YS S DD r S YS DD Nsp 4m1 cos 01 1 0 2p Nsp
2m2
1 / cos
2m2
1/
m4834$0805
[11a]
pn1 Nsp
10
2Ém1É Nsp
02-13-96 20:00:32
[11b]
magas
Å
S YS S YS 2m3
2m3
1 / cos
S DD p n1 Nsp
r 1/
10
2Ém1É Nsp
DD
.
[11c]
Since in general n1 , n2 , and n3 will be nonintegers, linear interpolation is used within the acquired data set. In practice, linearization is first performed along the read direction by applying Eq. [11a] to the data. For example, if m1 Å 1 and the number of samples Nsp Å 128, then n1 Å 5.1. Hence the correct datum for the first point in k space along the read direction is estimated by linearly interpolating between the fifth and sixth data points collected after the center of the echo. If this is repeated for all values of m1 , then a data set is obtained that samples k space as shown in Fig. 3a. Linearization is then carried out in a similar manner in the two phase-encode directions by applying Eqs. [11b] and [11c] to the new data. Note that if n2 ú Np1 /2 or n3 ú Np2 / 2, then there is no data to interpolate between, and a value of zero is assumed for this data point (see Fig. 3b). Equal spatial resolution of the image in all three dimensions can be obtained by ensuring that the maximum k-space excursion is the same in all three directions. Thus, from Eq.
AP: Mag Res
108
NOTES
FIG. 3. (a) Trajectories in k space after linearizing along kread by interpolating appropriately between the sample points in Fig. 2. (b) The trajectories in k space after linearization in the phase 1 direction; the crosses in the corners of k space indicate the regions where interpolation is impossible and the data array is filled with zeroes.
[6], the maximum value of kread occurs at t Å T/2 and is given by kread (T/2) Å gGO readT/ p.
[12]
The maximum value of kphase1 occurs when Én2É Å Np1 /2 and t Å T and is given by
S
kphase1 T,
Np1 2
D
Å gGO phase1T/2p.
[13]
Similarly the maximum value of kphase2 is given by
S
kphase2 T,
Np2 2
D
Å gGO phase2T/2p.
[14]
Hence we require GO read Å
GO phase1 GO phase2 Å . 2 2
[15]
The experiments were carried out using a 0.73 T superconducting magnet corresponding to a proton frequency of 31.2 MHz. A small RF birdcage coil was used for ‘‘transmit’’ and ‘‘receive.’’ The gradient coils were constructed on a design obtained from Magnex Ltd. (12) and consist of a
m4834$0805
02-13-96 20:00:32
magas
Maxwell pair for the z gradient and two sets of Golay pairs for the x and y gradients. The gradient strengths were measured to be 15, 13, and 30 mT m01 A 01 for the x, y, and z gradients, respectively. Each gradient coil was connected to a Techron amplifier working in controlled-current mode. A PC-compatible computer containing hardware provided by SMIS Ltd. (13) was used to generate the sinusoidal gradient waveforms from a pair of two-channel 12 bit DACs. The SMIS hardware included a pulse programmer which controlled the whole of the imaging sequence. The pulse programmer was interfaced to a Bruker CXP spectrometer to provide the necessary RF frequency generator, transmitter, amplifier, and receiver. The NMR data were acquired by two 12 bit Datel ADCs sampling at 500 kHz. The data were linearized on a DEC Alpha workstation and then Fourier transformed to a three-dimensional data set using standard SMIS reconstruction software. The final images (Fig. 4) were produced from a general surface-rendering software package written by one of us (T.A.P.G.). The sample used to demonstrate the technique was an ABS plastic Lego brick. The T 2 relaxation curve displayed two significant components in which 90% of the signal had a T 2 of 760 ms and the remaining 10% had a T 2 of 6 ms. By imaging at an echo time of 128 ms it was possible to view both components present in the sample. Figure 4a shows a three-dimensional surface-rendered image of the sample. Figures 4b and 4c show cross-sectional views through the sample. The images were acquired using 64 samples in fre-
AP: Mag Res
109
NOTES
FIG. 4. (a) A three-dimensional surface-rendered image of an ABS plastic Lego brick whose size is approximately 15 1 15 1 11 mm. (b, c) Crosssectional views through the same sample.
quency-encoding direction and 64 phase-encoding steps in the other two directions. The peak gradient strength in the frequency-encode direction was 1.2 T m01 (40 A peak) while the two phase-encode gradients reached a peak of {600 mT m01 (40 and 46 A peaks for Gx and Gy , respectively). The frequency-encoding gradient oscillated at 7.8 kHz, and the two phase-encoding gradients oscillated at 3.9 kHz. The data-capture window therefore extended from 64 to 192 ms, with the center of the gradient echo occurring at 128 ms. The T 1 of the sample was on the order of 150 ms, and a repetition time of 100 ms was found to be satisfactory. A total of 32 averages were acquired in a time of 32 1 64 1 64 1 0.1 s, i.e., 3 hours 38 minutes. The field of view for the images is 32 1 32 1 32 mm. In summary, the application of oscillating gradients to an imaging system previously used for liquids proved to be straightforward and allowed us to reduce the echo time from 2 ms to 128 ms. Although 3D back-projection schemes based on capturing the FID have similar capabilities in terms of
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imaging soft solids, the 3DFT method presented here relies on the generation of an echo and therefore has the advantage that image artifacts associated with missing k-space data around k Å 0 due to receiver deadtime are avoided. On the other hand, it should be remembered that all echo schemes which rely on phase encoding suffer from unavoidable signal decay due to T 2 relaxation during the phase-encoding period. In the present oscillating-gradient scheme, this period can be minimized for a given gradient amplifier performance by using a series-resonant gradient-coil circuit. Finally it should be noted that a worthwhile increase in S/N should be possible by using a FLASH (14, 15) version of the present technique. ACKNOWLEDGMENTS This work was supported by an EPSRC grant. We thank SMIS Ltd. for their support in setting up the imaging system software and the Molecular Dynamics Group at UKC for the use of their DEC Alpha workstation.
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8. P. T. Callaghan, ‘‘Principles of Magnetic Resonance Microscopy,’’ Chap. 3, Oxford Univ. Press, Oxford, 1991. 9. P. Mansfield and P. G. Morris, ‘‘NMR Imaging in Biomedicine,’’ Academic Press, London, 1982. 10. S. Ljunggren, J. Magn. Reson. 54, 338 (1983). 11. A. Kumar, D. Welti, and R. R. Ernst, J. Magn. Reson. 18, 69 (1975). 12. Magnex Scientific Limited, 21 Blacklands Way, Abingdon, Oxon, UK, private communication. 13. SMIS (Surrey Medical Imaging Systems Ltd.), Alan Turing Road, Guildford, Surrey, UK. 14. A. Haase, J. Frahm, D. Matthaei, W. Hanicke, and K-D. Merboldt, J. Magn. Reson. 67, 258 (1986). 15. P. J. McDonald, K. L. Perry, and S. P. Roberts, Meas. Sci. Technol. 4, 896 (1993).
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