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Multiple modulation multiple echoes: A one-shot method
Magnetic Resonance Imaging 23 (2005) 301 – 303
Multiple modulation multiple echoes: A one-shot methodB Yi-Qiao SongT Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, CT 06877, USA
Abstract This paper reviews a recently reported NMR method capable of determining the diffusion constant within milliseconds and without the need of multiple scans. The method can be used with static or pulsed magnetic field gradients. D 2005 Elsevier Inc. All rights reserved. Keywords: Multiple modulation; Molecular diffusion; NMR
1. Introduction
2. Multiple modulation multiple echoes
The molecular diffusion constant is often measured by NMR experiments using static or pulsed magnetic field gradients. The pulse sequences for such measurements are based on the work of Hahn [1] and Stejskal and Tanner [2]. The basic idea is to first use the field gradient to create a spatial sinusoidal modulation of spin magnetization in the sample, then diffusion causes a decay in the amplitude to the modulation. This decay is measured by the amplitude of the echo. For a single diffusion constant (D), the amplitude often exhibits an exponential decay:
The pulse sequences considered in this article consist of a train of N pulses, and s l is the time period between the lth and the (l+1)th pulses. A coherence pathway is characterized by a series of N+1 numbers, Qu( q 0, q 1,. . ., q N ), where q 0 = 0 is the magnetization state before the first pulse. The contribution of each coherence pathway can be written as a product of three factors
S~expð bDÞ;
ð1Þ
Here, b is determined by the field gradient pattern and the coherence pathway. The diffusion constant is obtained by a series of measurements with different values of b and for each b potentially several scans in order to select the desired coherence pathway. The current paper reviews a recently introduced one-scan diffusion acquisition to acquire data at a series of b [3]. It requires no additional scans for phase cycling or for different values of b. It is thus a truly fast measurement.
MQ ¼ AQ BQ CQ
where A Q is the frequency spectrum of the resulting signal and it depends only on the RF pulses. B Q and C Q are due to diffusion and spin relaxation, respectively, and they are independent of frequency. In order to understand B Q, it is useful to introduce for each coherence pathway the instantaneous wave vector k(t) Rt [4,5]: kðtÞ ¼ cg 0 qðtVÞdtV, where g is the field gradient, q(tV) is the instantaneous value of q that is piecewise constant between pulses. The attenuation induced by unrestricted diffusion for a given coherence pathway can then generally be written as [2] BQ ¼ exp D
B
This paper reviews a recently reported NMR method capable of determining the diffusion constant within milliseconds and without the need of multiple scans. The method can be used with static or pulsed magnetic field gradients. T Tel.: +1 2034315417; fax: +1 2034383819. E-mail address: [email protected]. 0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2004.11.033
ð2Þ
Z
T
2
kðtÞ dt ;
ð3Þ
0
where D is the bulk diffusion constant, and time t = 0 is defined as the beginning of the sequence and T is the echo time. A modified equation [6] can be used to evaluate the diffusion decay for restricted diffusion. We will describe the
302
Y.-Q. Song / Magnetic Resonance Imaging 23 (2005) 301–303
resulting B Q for a specific example of the multiple modulation multiple echoes (MMME) sequences. Consider a static magnetic field gradient and a train of three pulses with tipping angles a 1, a 2, and a 3, and the time spacing between them to be s 1, s 2, a1 s1 a2 s2 a3 acquisition
For each coherence pathway Q, the diffusion and the relaxation factors can be evaluated: ð6Þ BQ ¼ exp bQ Dc2 g2 s3 s1 s1 : ð7Þ CQ ¼ exp c1Q c2Q T1 T2
ð4Þ
The diffusion number b Q and relaxation numbers c 1Q and c 2Q can be calculated for all coherence pathways, and the results for the above example sequence are listed in the table above. For arbitrary s 1 and s 2, analytical expressions for the diffusion decay rate (b Q) can be evaluated for all echoes:
The nutation angles of the pulses are not necessarily multiples of 908. Neglecting recovery due to T 1 relaxation for the moment, the above sequence will allow a total of five coherence pathways to be observed and thus create five signals, echo number 0 1 2 3 4
q0 q1 0 0 0 1 0 1 0 0 0 1
q2 0 0 1 1 1
q3 bQ c1Q 1 0 0 1 11=3 3 1 6 0 1 18 1 1 128=3 0
2s31 þ s21 s2 3 2s2 3s21 3s1 s2 þ s22 2: 3
ð8Þ
echo 1 :
c2Q 0 2 6 6 8
3: ð5Þ
ð9Þ
2s32 3
ð10Þ
4 : 2ðs1 þ s2 Þ3 =3
q 3 =1 for the detection period as is the convention. The first coherence pathway (0,0,0,1) gives rise to an FID signal and all others produce echoes. The echoes appear at time s 3 after the last pulse: s 3 =q 1s1+q 2s 2 and all echoes can be separated and equally spaced in the time domain when s 2 = 3s 1. The symbols b Q, c 1Q and c 2Q will be explained later. Different coherence pathways create different spatial phase modulation and they yield echo signals at different times, completely separable. As a result, in one scan of the sequence, five different modulations can be measured. Thus, we name this class of sequences MMME, pronounced M-M-Me.
ð11Þ
Because the diffusion effect (the b value) is different for each echo, in one scan of the sequence, several different modulations can be measured. The above example of three-pulse sequence can be extended to a general methodology with N pulses, called MMMEN. For example, with four pulses (MMME4), 13 echoes can be recorded, providing 13 different modulations. With five pulses, MMME5 will have 40 echoes. This is one of the unique advantages of the MMME technique that the number of modulations is exponential with respect to the number (N) of RF pulses. One of the possible ways to use MMME sequences to determine diffusion is by repeating the sequence, for 1
(A)
(B)
Amplitude ratio
Echo amplitude
1000
100
10
0.1
Diffusion constant obtained: 2.1*10-5 cm2/s
0.01
τ= 0.5ms τ= 3ms τ= 4ms
0
2
4
3/0.5 4/0.5
6
8
echo number
10
12
14
0
1
b
2
(105
3
s/cm2)
Fig. 1. MMME4 data obtained for bulk water at room temperature. (A) Plot of the MMME echoes measured with s = 0.5 (circles), 3 (squares) and 4 ms (crosses). Magnetic field gradient used is 2 G/cm. Notice the progressive decay of the corresponding echoes as s increases. (B) Plot of amplitude ratio of the corresponding echo signals as a function of the bQc 2g 2b Q(sV3s 3). Two sets of ratio are shown between the data with s = 0.5 and 3 ms (solid circles), and those with s = 0.5 and 4 ms (open squares). The single exponential decay of the data points is consistent with the single diffusion constant of water. The slope of the decay determines the diffusion constant to be 2.1105 cm2/s.
Y.-Q. Song / Magnetic Resonance Imaging 23 (2005) 301–303
instance, MMME4, with two sets of s i with s 1 =s and sV. Then, the shapes of the corresponding echoes will be identical and their amplitude ratio can be found to be
SQ ðsÞ ¼ exp bQ Dc2 g 2 s3 sV3 : SQ ðsVÞ
coherence pathways are very similar to that of the stimulated echo. 4. Conclusions
ð12Þ
Each pair of the echoes will give one data point for the diffusion decay, and a total of 13 data points can be obtained by the two acquisitions, as shown in Fig. 1. 3. Stimulated MMME: MMME as a unit in stimulated-echo-type sequence The stimulated echo technique with pulsed-field gradient is commonly used for measuring diffusion. In a typical experiment, the spin magnetization is rotated into the transverse plane by a 908 pulse, and it then precesses in a field gradient. A second 908 pulse turns the spatially modulated magnetization back to the z-axis. After a time period to allow diffusion, a third 908 pulse rotates spins into the transverse plane to form a stimulated echo. Between the first and second pulses, normally only one coherence pathway is selected and hence one modulation is utilized. The MMME idea can be incorporated into the stimulated echo sequence to create multiple modulations within one scan in order to retrieve diffusion information from several coherence pathways. For example, MMME can be used in the initial modulation period, followed by a long waiting period, then a detection period: MMME D1 h1 D2 h2 : : :
303
ð13Þ
Acquisitions of echo signals can be made during the multiple D periods. Here, the MMME segment is used as the encoding unit to create multiple modulations in contrast to the conventional stimulated echo with one modulation. In the first acquisition period after h 1, two groups of echoes will be observed. The first group appears after the h 1 pulse in the order of the same MMME sequence, for instance, the 13 echoes of MMME4. The coherence pathways of these echoes follow the form of ( Q MMME, 0, 1), where Q MMME is the corresponding coherence pathway of the MMME sequence, the 0 and 1 are the coherence states during D 1 and D 2. In sequences with additional h pulses, similar echoes appear after each h pulse and their coherence pathways are of the form ( Q MMME, 0,. . ., 0, 1) where q = 0 occurs during all of the intervening D periods. Such
The MMME methodology is reviewed for truly rapid measurement of diffusion. This method uses a few RF pulses and static or pulsed field gradient to create and observe multiple coherence pathways. Signals from all coherence pathways are well separated in time, and no phase cycling is needed. This method is closely related to the previous lowresolution imaging methods of PREVIEW [7] and QUEST [8] and the methods for diffusion measurements by small tipping angle pulses, such as BURST-type sequences, for example, in Refs. [9–11] and the DIFFTRAIN technique [12,13]. It has been shown that the current method is advantageous in more efficient use of the available signal (thus, better signal-to-noise ratio) and not limited to small tipping angles. References [1] Hahn EL. Spin echoes. Phys Rev 1950;80:580. [2] Stejskal EO, Tanner JE. Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. J Chem Phys 1965; 42:288. [3] Song Y-Q, Tang X. A one-shot method for measurement of diffusion. J Magn Reson 2004;170:136. [4] Callaghan PT. Principles of nuclear magnetic resonance microscopy. New York7 Oxford University Press; 1993. [5] Sodickson A, Cory DG. A generalized k-space formalism for treating the spatial aspects of a variety of NMR experiments. Prog Nucl Magn Reson Spectrosc 1998;33:77. [6] Zielinski LJ, Sen PN. Restricted diffusion in grossly inhomogeneous fields. J Magn Reson 2003;164:145. [7] Counsell CJ. PREVIEW: A new ultrafast imaging sequence requiring minimal gradient switching. Magn Reson Imaging 1993;11:603. [8] Heid O, Deimling M, Huk W. QUEST — a quick echo split NMR imaging technique. Magn Reson Med 1993;29:280. [9] Peled S, Tseng CH, Sodickson AA, Mair RW, Walsworth RL, Cory DG. Single-shot diffusion measurement in laser-polarized gas. J Magn Reson 1999;140:320. [10] Lowe IJ, Wysong RE. DANTE ultrafast imaging sequence (DUFIS). J Magn Reson B 1993;101:106. [11] Doran SJ, De´corps M. A robust, single-shot method for measuring diffusion coefficients using the bBurstQ sequence. J Magn Reson A 1995;117:311. [12] Stamps JP, Ottink B, Visser JM, van Duynhoven JP, Hulst R. Difftrain: a novel approach to a true spectroscopic single-scan diffusion measurement. J Magn Reson 2001;151:28. [13] Buckley C, Hollingsworth KG, Sederman AJ, Holland DJ, Johns ML, Gladden LF. Applications of fast diffusion measurement using Difftrain. J Magn Reson 2003;161:112.
Multiple modulation multiple echoes: A one-shot methodB Yi-Qiao SongT Schlumberger-Doll Research, 36 Old Quarry Road, Ridgefield, CT 06877, USA
Abstract This paper reviews a recently reported NMR method capable of determining the diffusion constant within milliseconds and without the need of multiple scans. The method can be used with static or pulsed magnetic field gradients. D 2005 Elsevier Inc. All rights reserved. Keywords: Multiple modulation; Molecular diffusion; NMR
1. Introduction
2. Multiple modulation multiple echoes
The molecular diffusion constant is often measured by NMR experiments using static or pulsed magnetic field gradients. The pulse sequences for such measurements are based on the work of Hahn [1] and Stejskal and Tanner [2]. The basic idea is to first use the field gradient to create a spatial sinusoidal modulation of spin magnetization in the sample, then diffusion causes a decay in the amplitude to the modulation. This decay is measured by the amplitude of the echo. For a single diffusion constant (D), the amplitude often exhibits an exponential decay:
The pulse sequences considered in this article consist of a train of N pulses, and s l is the time period between the lth and the (l+1)th pulses. A coherence pathway is characterized by a series of N+1 numbers, Qu( q 0, q 1,. . ., q N ), where q 0 = 0 is the magnetization state before the first pulse. The contribution of each coherence pathway can be written as a product of three factors
S~expð bDÞ;
ð1Þ
Here, b is determined by the field gradient pattern and the coherence pathway. The diffusion constant is obtained by a series of measurements with different values of b and for each b potentially several scans in order to select the desired coherence pathway. The current paper reviews a recently introduced one-scan diffusion acquisition to acquire data at a series of b [3]. It requires no additional scans for phase cycling or for different values of b. It is thus a truly fast measurement.
MQ ¼ AQ BQ CQ
where A Q is the frequency spectrum of the resulting signal and it depends only on the RF pulses. B Q and C Q are due to diffusion and spin relaxation, respectively, and they are independent of frequency. In order to understand B Q, it is useful to introduce for each coherence pathway the instantaneous wave vector k(t) Rt [4,5]: kðtÞ ¼ cg 0 qðtVÞdtV, where g is the field gradient, q(tV) is the instantaneous value of q that is piecewise constant between pulses. The attenuation induced by unrestricted diffusion for a given coherence pathway can then generally be written as [2] BQ ¼ exp D
B
This paper reviews a recently reported NMR method capable of determining the diffusion constant within milliseconds and without the need of multiple scans. The method can be used with static or pulsed magnetic field gradients. T Tel.: +1 2034315417; fax: +1 2034383819. E-mail address: [email protected]. 0730-725X/$ – see front matter D 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.mri.2004.11.033
ð2Þ
Z
T
2
kðtÞ dt ;
ð3Þ
0
where D is the bulk diffusion constant, and time t = 0 is defined as the beginning of the sequence and T is the echo time. A modified equation [6] can be used to evaluate the diffusion decay for restricted diffusion. We will describe the
302
Y.-Q. Song / Magnetic Resonance Imaging 23 (2005) 301–303
resulting B Q for a specific example of the multiple modulation multiple echoes (MMME) sequences. Consider a static magnetic field gradient and a train of three pulses with tipping angles a 1, a 2, and a 3, and the time spacing between them to be s 1, s 2, a1 s1 a2 s2 a3 acquisition
For each coherence pathway Q, the diffusion and the relaxation factors can be evaluated: ð6Þ BQ ¼ exp bQ Dc2 g2 s3 s1 s1 : ð7Þ CQ ¼ exp c1Q c2Q T1 T2
ð4Þ
The diffusion number b Q and relaxation numbers c 1Q and c 2Q can be calculated for all coherence pathways, and the results for the above example sequence are listed in the table above. For arbitrary s 1 and s 2, analytical expressions for the diffusion decay rate (b Q) can be evaluated for all echoes:
The nutation angles of the pulses are not necessarily multiples of 908. Neglecting recovery due to T 1 relaxation for the moment, the above sequence will allow a total of five coherence pathways to be observed and thus create five signals, echo number 0 1 2 3 4
q0 q1 0 0 0 1 0 1 0 0 0 1
q2 0 0 1 1 1
q3 bQ c1Q 1 0 0 1 11=3 3 1 6 0 1 18 1 1 128=3 0
2s31 þ s21 s2 3 2s2 3s21 3s1 s2 þ s22 2: 3
ð8Þ
echo 1 :
c2Q 0 2 6 6 8
3: ð5Þ
ð9Þ
2s32 3
ð10Þ
4 : 2ðs1 þ s2 Þ3 =3
q 3 =1 for the detection period as is the convention. The first coherence pathway (0,0,0,1) gives rise to an FID signal and all others produce echoes. The echoes appear at time s 3 after the last pulse: s 3 =q 1s1+q 2s 2 and all echoes can be separated and equally spaced in the time domain when s 2 = 3s 1. The symbols b Q, c 1Q and c 2Q will be explained later. Different coherence pathways create different spatial phase modulation and they yield echo signals at different times, completely separable. As a result, in one scan of the sequence, five different modulations can be measured. Thus, we name this class of sequences MMME, pronounced M-M-Me.
ð11Þ
Because the diffusion effect (the b value) is different for each echo, in one scan of the sequence, several different modulations can be measured. The above example of three-pulse sequence can be extended to a general methodology with N pulses, called MMMEN. For example, with four pulses (MMME4), 13 echoes can be recorded, providing 13 different modulations. With five pulses, MMME5 will have 40 echoes. This is one of the unique advantages of the MMME technique that the number of modulations is exponential with respect to the number (N) of RF pulses. One of the possible ways to use MMME sequences to determine diffusion is by repeating the sequence, for 1
(A)
(B)
Amplitude ratio
Echo amplitude
1000
100
10
0.1
Diffusion constant obtained: 2.1*10-5 cm2/s
0.01
τ= 0.5ms τ= 3ms τ= 4ms
0
2
4
3/0.5 4/0.5
6
8
echo number
10
12
14
0
1
b
2
(105
3
s/cm2)
Fig. 1. MMME4 data obtained for bulk water at room temperature. (A) Plot of the MMME echoes measured with s = 0.5 (circles), 3 (squares) and 4 ms (crosses). Magnetic field gradient used is 2 G/cm. Notice the progressive decay of the corresponding echoes as s increases. (B) Plot of amplitude ratio of the corresponding echo signals as a function of the bQc 2g 2b Q(sV3s 3). Two sets of ratio are shown between the data with s = 0.5 and 3 ms (solid circles), and those with s = 0.5 and 4 ms (open squares). The single exponential decay of the data points is consistent with the single diffusion constant of water. The slope of the decay determines the diffusion constant to be 2.1105 cm2/s.
Y.-Q. Song / Magnetic Resonance Imaging 23 (2005) 301–303
instance, MMME4, with two sets of s i with s 1 =s and sV. Then, the shapes of the corresponding echoes will be identical and their amplitude ratio can be found to be
SQ ðsÞ ¼ exp bQ Dc2 g 2 s3 sV3 : SQ ðsVÞ
coherence pathways are very similar to that of the stimulated echo. 4. Conclusions
ð12Þ
Each pair of the echoes will give one data point for the diffusion decay, and a total of 13 data points can be obtained by the two acquisitions, as shown in Fig. 1. 3. Stimulated MMME: MMME as a unit in stimulated-echo-type sequence The stimulated echo technique with pulsed-field gradient is commonly used for measuring diffusion. In a typical experiment, the spin magnetization is rotated into the transverse plane by a 908 pulse, and it then precesses in a field gradient. A second 908 pulse turns the spatially modulated magnetization back to the z-axis. After a time period to allow diffusion, a third 908 pulse rotates spins into the transverse plane to form a stimulated echo. Between the first and second pulses, normally only one coherence pathway is selected and hence one modulation is utilized. The MMME idea can be incorporated into the stimulated echo sequence to create multiple modulations within one scan in order to retrieve diffusion information from several coherence pathways. For example, MMME can be used in the initial modulation period, followed by a long waiting period, then a detection period: MMME D1 h1 D2 h2 : : :
303
ð13Þ
Acquisitions of echo signals can be made during the multiple D periods. Here, the MMME segment is used as the encoding unit to create multiple modulations in contrast to the conventional stimulated echo with one modulation. In the first acquisition period after h 1, two groups of echoes will be observed. The first group appears after the h 1 pulse in the order of the same MMME sequence, for instance, the 13 echoes of MMME4. The coherence pathways of these echoes follow the form of ( Q MMME, 0, 1), where Q MMME is the corresponding coherence pathway of the MMME sequence, the 0 and 1 are the coherence states during D 1 and D 2. In sequences with additional h pulses, similar echoes appear after each h pulse and their coherence pathways are of the form ( Q MMME, 0,. . ., 0, 1) where q = 0 occurs during all of the intervening D periods. Such
The MMME methodology is reviewed for truly rapid measurement of diffusion. This method uses a few RF pulses and static or pulsed field gradient to create and observe multiple coherence pathways. Signals from all coherence pathways are well separated in time, and no phase cycling is needed. This method is closely related to the previous lowresolution imaging methods of PREVIEW [7] and QUEST [8] and the methods for diffusion measurements by small tipping angle pulses, such as BURST-type sequences, for example, in Refs. [9–11] and the DIFFTRAIN technique [12,13]. It has been shown that the current method is advantageous in more efficient use of the available signal (thus, better signal-to-noise ratio) and not limited to small tipping angles. References [1] Hahn EL. Spin echoes. Phys Rev 1950;80:580. [2] Stejskal EO, Tanner JE. Spin diffusion measurements: Spin echoes in the presence of a time-dependent field gradient. J Chem Phys 1965; 42:288. [3] Song Y-Q, Tang X. A one-shot method for measurement of diffusion. J Magn Reson 2004;170:136. [4] Callaghan PT. Principles of nuclear magnetic resonance microscopy. New York7 Oxford University Press; 1993. [5] Sodickson A, Cory DG. A generalized k-space formalism for treating the spatial aspects of a variety of NMR experiments. Prog Nucl Magn Reson Spectrosc 1998;33:77. [6] Zielinski LJ, Sen PN. Restricted diffusion in grossly inhomogeneous fields. J Magn Reson 2003;164:145. [7] Counsell CJ. PREVIEW: A new ultrafast imaging sequence requiring minimal gradient switching. Magn Reson Imaging 1993;11:603. [8] Heid O, Deimling M, Huk W. QUEST — a quick echo split NMR imaging technique. Magn Reson Med 1993;29:280. [9] Peled S, Tseng CH, Sodickson AA, Mair RW, Walsworth RL, Cory DG. Single-shot diffusion measurement in laser-polarized gas. J Magn Reson 1999;140:320. [10] Lowe IJ, Wysong RE. DANTE ultrafast imaging sequence (DUFIS). J Magn Reson B 1993;101:106. [11] Doran SJ, De´corps M. A robust, single-shot method for measuring diffusion coefficients using the bBurstQ sequence. J Magn Reson A 1995;117:311. [12] Stamps JP, Ottink B, Visser JM, van Duynhoven JP, Hulst R. Difftrain: a novel approach to a true spectroscopic single-scan diffusion measurement. J Magn Reson 2001;151:28. [13] Buckley C, Hollingsworth KG, Sederman AJ, Holland DJ, Johns ML, Gladden LF. Applications of fast diffusion measurement using Difftrain. J Magn Reson 2003;161:112.