spectral encoding of the spin interactions in

THE JOURNAL OF CHEMICAL PHYSICS 131, 1 共2009兲

Spatial/spectral encoding of the spin interactions in ultrafast 2 multidimensional NMR 1

Yoav Shrota兲 and Lucio Frydmanb兲

3 4

Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel

5

共Received 3 September 2009; accepted 29 October 2009; published online xx xx xxxx兲

6

Two-dimensional nuclear magnetic resonance 共2D NMR兲 spectroscopy provides the means to extract diverse physical, chemical, and biological information at an atomic level. Conventional sampling schemes, however, may result in relatively long 2D experiments; this has stimulated the search for alternative, rapid acquisition schemes. Among the strategies that have been recently proposed for achieving this counts an “ultrafast” approach, relying on the spatial encoding of the indirect domain evolution to provide arbitrary spectra within a single scan. A common feature of all spatial encoding schemes hitherto described is their uniform encoding of a continuous bandwidth of indirect-domain frequencies, regardless of the chemical sites’ spectral distribution within it. These very general conditions, however, are often associated with a number of tradeoffs and compromises in the spectral widths and resolutions that can be achieved for both the direct and indirect domains. This paper proposes a different strategy for single-scan acquisition of 2D spectra, which performs an optimal encoding by employing a priori information regarding the positions of NMR resonances along the indirect domain. We denote this as “spatial/spectral encoding”; the underlying principles of this new approach, together with experimental results based on uni- and multidimensional rf pulse schemes, are presented. © 2009 American Institute of Physics. 关doi:10.1063/1.3266422兴

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

22 I. INTRODUCTION

Two-dimensional nuclear magnetic resonance 共2D NMR兲 spectroscopy serves as a powerful tool to obtain 25 information about structure and dynamics in chemistry and 1,2 This class of experiments is routinely used 26 biochemistry. 27 for identifying the components comprising a sample under 28 investigation as well as for unraveling the scaffolding of un29 known chemical structures. Various methodologies have 3,4 the 30 been devised to extract this variety of information; 31 overwhelming majority of these share a basic scheme based 32 on the general proposal by Jeener and Ernst for the collection 5,6 33 of 2D NMR data,

corresponding frequency axes. These will be given, respec- 48 tively, by 49

23 24

Preparation-Evolution共t1兲-Mixing-Acquisition共t2兲.

34

共1兲

According to this paradigm, the spins evolve as a function of two time variables. The ␯1 indirect domain is monitored via the introduction of a delay parameter t1 that is uniformly incremented in a series of independent experiments. For each value of t1, the spins’ response as a function of t2 关the free induction decay 共FID兲兴 is then monitored along the direct domain t2. Proper rearrangement of the resulting set of FIDs 兵S共t1 , t2兲其, followed by a numerical 2D Fourier transforma43 tion 共FT兲, provides the 2D I共␯1 , ␯2兲 spectrum correlating the 44 spin frequencies that were active during the indirect-domain 45 and direct-domain evolutions. The spectra resulting from this 46 procedure are characterized by independent spectral widths 47 共SW1 , SW2兲 and spectral resolutions 共⌬␯1 , ⌬␯2兲 along the 35 36 37 38 39 40 41 42

a兲

Electronic mail: [email protected]. Tel.: ⫹972-8-9344903. FAX: ⫹ 972-8-9344123. Electronic mail: [email protected].

b兲

0021-9606/2009/131共22兲/1/0/$25.00

共SW1,SW2兲 =

冉

冉

冊

1 1 , ; ⌬t1 ⌬t2

50

冊

1 1 , . 共⌬␯1,⌬␯2兲 = N1⌬t1 N2⌬t2

共2兲 51

Here ⌬t1 is the delay used to increment the indirect-domain time parameter, ⌬t2 is the acquisition dwell time, N1 is the number of distinct t1 increments that were used, and N2 is the number of points collected in the FIDs. Although ⌬t1, ⌬t2, and N2 can usually be set more or less arbitrarily without penalties 共other than for the limits set by the relaxation times T2兲, N1—which linearly defines the total time allocated to the experiment’s duration—may be a taxing parameter to set. Indeed, it follows from Eq. 共2兲 that when not bound by sensitivity considerations, attempting to achieve high ⌬␯1 spectral resolutions while faithfully covering large SW1 bandwidths 共BWs兲 requires N1 to be large, and therefore demands the acquisition of numerous independent scans. Over the years various strategies have been suggested to ameliorate this sampling-imposed limitation, and to minimize the total experimental duration without compromising on spectral resolution and/or the BW of interest. Some of these approaches are “passive,” relying for instance on letting peaks simply fold over along the indirect domain.7,8 A priori information about the indirect-domain spectrum can then be used to provide the exact positions of the peaks along this axis. Another active approach that also exploits prior information about the desired ␯1-spectral lines involves exciting these chemical sites of interest in a selective

131, 1-1

© 2009 American Institute of Physics

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J. Chem. Phys. 131, 1 共2009兲

Y. Shrot and L. Frydman

Spatial Homogeneous Gradient-driven readout encoding mixing 1  2

Initial States in (Mx,My,Mz) Bloch Space

Gz

(a) +Ge

(b)

(c)

S共k,t2兲 ⬀

-Ge

z Site 2 Shift Ω2

z

k1,2 =
C  1,2

fashion.9,10 Given N such sites, one then needs only N onedimensional 共1D兲 FIDs to unambiguously resolve the directdomain spectrum for each of the indirect-domain peaks of 80 interest, and hence for reconstituting the 2D NMR informa81 tion. Other alternatives involve departing from FT process82 ing, and employ nonuniform mapping of the indirect domain 11–15 83 to bypass the strict demands of Eq. 共2兲. Although all 84 these approaches require less repetitions than their conven85 tional counterparts, they may still demand a considerable 86 number of scans when attempting to characterize multiple 87 spreadout sites with high spectral resolution. An alternative general approach to reduce the number of 88 89 experiments of a 2D NMR all the way down to a single scan 16,17 90 is provided by so-called “ultrafast” NMR methods. In 91 these schemes 共Fig. 1兲 all t1 values are spatially encoded into 92 the sample in parallel, thus compressing the whole series 93 involved in a 2D acquisition into a single scan. The corre94 sponding phases of the transverse spin magnetizations at the 95 beginning of the t2 acquisition will be given in these experi17,18 96 ments by spatially dependent factors of the form 77 78 79

␾共z兲 = ⍀1t1共z兲 = ⍀1tmax 1

z − z0 = C⍀1共z − z0兲, L

共3兲

where L is the encoded sample’s length, ⍀1 is the indirectis the effecdomain shift one is trying to measure, and tmax 1 tive overall evolution time. During the FID recording time, the encoded information is acquired while in the presence of ⫾Ga oscillating gradients that map the spins’ response as a function of 关k共t2兲 , t2兴, where k共t兲 ⬅ ␥a兰toGa共t⬘兲dt⬘. Denoting 104 the 2D spectral correlation being sought as I共⍀1 , ⍀2兲 and 98 99 100 101 102 103

␯1

␯2

册

d⍀1d⍀2I共⍀1,⍀2兲eiC⍀1zei⍀2t2 eikz . 共4兲 107

FIG. 1. Principles of ultrafast 2D NMR. Spatial encoding creates a linear relation between position z and evolution time t1; different chemical sites therefore generate different helical patterns depending on their chemical shift. This spatial information is preserved throughout the coherence mixing period. Integrating the contribution of all the spins in the sample at the beginning of the acquisition stage provides a recorded signal of negligible amplitude. But a gradient possessing the same geometry as used in the encoding stage can unwind this spatial profile and generate echoes whose timing varies linearly with the indirect-domain frequency.

97

dz

k/ν1

Site 1 Shift Ω1

z

冕 冋冕 冕 L

Ga

Spatially Encoded States in (Mx,My,Mz )

RF

neglecting for simplicity relaxation effects, the resulting sig- 105 nal is given by 106

The integration over the spatial dimension will provide a negligible signal, except for k-values that match the positions of the indirect domain spectral lines, i.e., k = −C⍀1. Thus, in practice, the k-axis linearly maps the entire ␯1 indirect domain. The ⫾Ga oscillating gradient waveform repeatedly monitors these echo trains and their corresponding train of spectra I共⍀1兲 − 2N2 times as a function of t2. Proper rearrangement of the data and application of a 1D FT along t2 thus provide the full 2D NMR spectrum being sought within a single scan. When setting up the various parameters of such ultrafast 2D NMR acquisitions,19 one needs to consider variables that both match and differ from those arising in conventional 2D experiments. As in Eq. 共2兲, 2Ta = ⌬t2 and N2 will define the FT-derived ␯2-domain axis. SW1 and ⌬␯1, by contrast, will involve the Ga’s amplitude and the timings involved throughout the various stages of the experiment. Given the linear relation in Eq. 共3兲, and taking the acquisition gradient half-cycle period and constant amplitude to be Ta and Ga, respectively, the spectral width to be scanned along the indirect domain will be SW1 = kmax / C = ␥aGaTa / C. As for the indirect domain resolution ⌬␯1, the echo shape as a function of k for a uniform spin distribution of length L is proportional to sinc共kL / 2兲, which provides a typical echo width of the order of L−1.20 When translating this indirect-domain ⌬k ⬇ L−1 resolution from k-terms into frequencies, one obtains that ⌬␯1 = ⌬k / C ⬇ 1 / tmax 1 . It follows from all this that for a given set of SW1, SW2, and ⌬␯1 demands, the acquisition gradient Ga should be set to fulfill

␥ aG aL =

CL · SW1 SW1 · SW2 =2 . Ta ⌬␯1

共5兲

␥aGaL also defines the receiver’s BW that will have to be covered in this kind of experiments. Indeed, by contrast to conventional acquisitions where BW is simply set by SW2, ultrafast experiments will demand an increased BW value for the simultaneous sampling of the two spectral domains. Such demand leads to a degradation of the signal to noise ratio 共SNR兲 on a per-scan basis according to a BW-dependent noise factor of SNR ⬀

1

1

冑BW ⬇ 冑␥aGaL =

冑

⌬␯1 . 2SW1SW2

共6兲

Equations 共5兲 and 共6兲 summarize the main bottlenecks of the ultrafast spatial encoding method. Indeed Eq. 共5兲 indicates that acquiring data spanning large spectral regions while demanding a high ␯1 spectral resolution will require the availability of strong acquisition gradients. This can be a limiting hurdle, especially if the NMR probehead being used is not equipped with customized gradients, or when demanding the high duty cycles that may be necessary to follow

108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136

137 138 139 140 141 142 143 144 145

146 147 148 149 150 151 152 153 154

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Spatial/spectral encoding in ultrafast NMR

J. Chem. Phys. 131, 1 共2009兲

(B) Folded Encoding k

tions the indirect domain into relevant and irrelevant spectral regions. The former are then spatially encoded with the desired linear dephasing, while the latter remains in practice excluded from the spatial encoding. Under such conditions the k-range that one needs to scan for unraveling the spectral information becomes considerably smaller than that required to cover the full SW1. For a given SW2 this then leads to reduced Ga demands, bypassing all the problems listed earlier. In fact an optimal reduction in the k-range to be scanned would result if the spectral regions of interest could be encoded on a site-by-site basis. In such case the ␯1-k mapping could be modified into a discrete, ideal 兵⍀ p ↔ k p其 p=1¯N scheme like the one illustrated in Fig. 2共d兲. Given N sites to be resolved and denoting the minimally resolvable separation along the k-axis as ␦k 共a wave-number on the order of L−1兲, the k-range to be scanned would then be given by an optimal max = N␦k. Assuming N is considerably smaller than value kopt N1 = SW1 / ⌬␯1, the necessary acquisition gradient could then be set significantly smaller than what is needed by conventional ultrafast 2D NMR: a reduction by ⬃N / N1. Such improvement is akin to that involved in Hadamard-based spectroscopy,3,10 except that instead of affecting the minimal number of experiments, it would reduce the gradient and SNR penalties associated with the monitoring of a broad SW1 with high resolution. Several alternative strategies to implement this new spectral/spatial mapping of the indirect domain are discussed in Sec. II, some of them considerably simpler than any encoding approach hitherto suggested. Section III follows by demonstrating the applicability of these approaches and their benefits compared with the conventional, uniform spatial encoding used in 2D ultrafast acquisitions. Section IV concludes by discussing the performance limitations, potential advantages, and possible applications of the new approaches.

191 192

II. SPECTRALLY SELECTIVE SPATIAL ENCODING SCHEMES

225 226

It follows from Sec. I that spatial encoding relies on endowing each ⍀ p site with a spatial winding, such that its corresponding echo will be generated during the acquisition at a different position along the k-axis. Such echoes are considered resolvable if they do not overlap with one another. Denoting this minimally resolvable k-separation by ␦k, an optimal allocation of the winding patterns calls for equally distributing the 兵⍀ p其 p=1¯N peaks to be resolved at distances ␦k apart. Given these conditions, the range to be scanned max = N␦k, during each direct-domain dwell period becomes kopt and our encoding goal can be summarized by the one-to-one allocation

227 228 229 230 231 232 233 234 235 236 237 238

(A) Full Encoding k k max

k

max

1

1

(C) Region-based Encoding k

(D) Site-based Encoding k max kopt

k max

k k

1

1

FIG. 2. Strategies for spatial encoding the indirect spectral domain illustrated for three chemical sites assumed unequally spaced along the frequency axis 共with one far apart from the other two兲. 共a兲 Conventional mapping creating a linear relation between ␯1 and k. This general linear mapping necessitates a strong acquisition readout gradient covering a large k-range. 共b兲 A reduction in the k-range to be scanned is possible by “folding” frequencies into a confined k-region and using multiple scans 共two in this figure兲 to “unfold” the spectrum. 共c兲 Using a priori information about the spectral peaks, a selective encoding can be used to exclude empty ␯1-ranges and encode only the regions of interest. 共d兲 An optimal encoding can be obtained if a discrete set of frequencies is excited and encoded with a minimal winding separation ␦k, leading to a minimal k-range to be scanned.

rapid real-time kinetic acquisitions. Moreover, even if the gradient hardware demands can be fulfilled, Eq. 共6兲 high157 lights the sensitivity penalties that will still arise in such 158 cases due to the noise’s variance increase with the gradient’s 159 amplitude. 160 The limitations just highlighted will often materialize for 161 indirectly detected characterizations of natural abundance or162 ganic samples. In such cases one commonly intends to char13 163 acterize the large SW1 spectral regions associated with C 15 164 or N nuclei, with a high ⌬␯1 resolution. SNR is also a 165 major issue in this kind of experiments. Fortunately, an op166 portunity to alleviate these limitations also arises when deal167 ing with such cases due to the “sparse” nature of the data; 168 i.e., due to the presence of large bands of the indirect do169 mains known a priori to be empty. Uniformly scanning such 170 empty regions with high spectral resolution in the usual spa171 tial encoding mode leading to Eqs. 共5兲 and 共6兲 关Fig. 2共a兲兴 is 172 onerous both in terms of gradients and of sensitivity. A some173 what analogous situation arises in conventional 2D spectros174 copy, where a high number of N1 experiments will be needed 175 关Eq. 共1兲兴 even when dealing with empty ␯1 regions. In con176 ventional 2D NMR this demand can be ameliorated by 7,8 177 folding/unfolding procedures; similarly, folding and un178 folding of the indirect-domain spectral regions can also be 179 used in spatially encoded spectroscopy to alleviate the 180 above-mentioned demands. Examples of such methods have 21,22 181 been demonstrated, whereby different mapping schemes 182 were used to “fold” m different subspectra into a reduced 183 region of k-space 关Fig. 2共b兲兴. At the cost of conducting mul184 tiple scans, these different spectral regions can be subse185 quently separated and the indirect domain faithfully recon186 structed. Although these procedures require reduced gradient 187 amplitudes they demand multiple scans, and may thus be 188 unsuitable for situations where single scans are of essence. In 189 such cases an alternative mapping scheme like that intro190 duced in Fig. 2共c兲 would be preferable. This scheme parti155 156

⍀ p 哫 k p = 共p − N/2兲␦k,

p = 1,2, . . . ,N.

193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224

共7兲 239

Having decided on a minimal ␦k, the objective given by Eq. 共7兲 can then be realized by two different approaches: one uses a strategy based on a discrete, sequential addressing of the frequencies to be encoded; the other addresses all the targeted frequencies simultaneously. We discuss these two cases separately.

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J. Chem. Phys. 131, 1 共2009兲

Y. Shrot and L. Frydman

(A) Serial Spatial/Spectral Encoding (SSE)

(B) Parallel Spatial/Spectral Encoding (PSE)

π/2 ∀{Ωi}i=1…N π

Ω1

RF 2 δk

π 2

Ωp δk

π 2

ΩN RF

共z, ␯1兲 = 兺 exp共ik pz兲Q共␯1 − ⍀ p兲, M desired XY

δk

291 292

293

where Q is the local windowing function introduced in Eq. 共8兲. According to the linear, small-tilt-angle approximation,25,26 the shaped rf pulse capable of delivering this spectral/spatial pattern will also have to fulfill the condition

246 A. Implementing a serial spectral/spatial encoding 247 of the sites

Scheme 1共a兲 illustrates a simple route to implementing the encoding required by Eq. 共7兲. This serial selective encod250 ing 共SSE兲 scheme employs a train of selective pulses that 251 sequentially excite each of the 兵⍀ p其 p=1¯N sites of interest, 252 and interleave these rf manipulations with a train of identical 253 gradient pulses characterized by the minimal wave-number 254 separation ␦k. Every chemical site, being excited at a differ255 ent stage of the encoding period, experiences a different 256 number of gradient pulses; sites become thereby linearly en257 coded by winding patterns of different pitches, as determined 258 by their order of excitation. Assuming a pth selective rf pulse 259 that excites a spectrum characterized by a profile of Q共␯1 260 − ⍀ p兲, the final magnetization arising from Scheme 1共a兲 will 261 be given by 248 249

N

M XY 共z, ␯1兲 ⬀ 兺 I共␯1兲exp关i共N − p + 1兲 · ␦kz兴Q共␯1 − ⍀ p兲, p=1

共8兲

where I共␯1兲 is proportional to the spectral distribution at a shift ␯1. Aiming to address each chemical site selectively 265 requires that the length of each of the constituent rf pulses 266 defining Q共␯兲 be made inversely proportional to the targeted 267 spectral resolution. Although arguably the simplest of all 268 spatial encoding schemes proposed so far, this SSE approach 269 suffers from a number of drawbacks. In particular, if some of 270 the targeted frequencies are close to one another, if dealing 271 with numerous N sites, or if suffering from short T2 trans272 verse relaxation times, this sequential excitation approach 273 may lead to considerably long pulse durations and to ensuing 274 relaxation losses of the various sites over the encoding stage. 275 The origin of this problem lies in the serial way by which the 276 individual sites are encoded; we turn next to an approach that 277 performs the same encoding in a parallel, simultaneous way. 263 264

278 B. Parallel spectral/spatial encoding 279 280 281 282 283 284 285 286 287

289 290

共9兲

p=1

SCHEME 1.

262

288

N

Ge

Ge

water or fat兲 as well as according to their spatial position. In contrast to this pattern, the 2D spatial/spectral pulses designed for the purposes of the present study will have as objective to endow the sample with a response in the ␯1-z plane, according to a targeted winding pattern

The second approach to be discussed uses a strategy whereby all the chemical sites are endowed with their specific dephasing factors simultaneously, making Eq. 共7兲 true for all ⍀ p’s at the same time. Such parallel selective encoding 共PSE兲 关Scheme 1共b兲兴 realizes this kind of manipulations on the basis of so-called 2D spatial-spectral rf pulses.23,24 These rf pulses find ample use in magnetic resonance imaging, where they are employed to selectively excite/invert spins of interest, both according to their chemical shift 共e.g.,

M xy共z, ␯1兲 ⬀

冕

Tpulse

B1共t兲 exp关− i共k共t兲 · z + t␯1兲兴dt.

294 295 296 297 298

共10兲

0

299

On the basis of this demand one can calculate the irradiation field function B1共t兲 delivering the 2D profile in Eq. 共9兲 for arbitrary trajectories in k-space. Indeed, the availability in Eq. 共10兲 of two degrees of freedom 共k and t兲 discriminating the spatial and spectral components defining the spin evolution provides the flexibility needed for imposing the pattern in Eq. 共9兲 from a single continuous pulse. The Appendix gives further details on these spatial/spectral manipulations, including information on how this kind of rf pulses was calculated and implemented in the present study. Among the general features associated with such 2D rf pulses is the fact that the typical spectral resolution of Q will, like the spectral resolution of any selective pulse, be inversely proportional to the pulse’s duration Tpulse. Unlike in the SSE case, however, where the overall preparation time was given by the sum of the individual rf and gradient pulse durations needed to excite the various chemical sites, in the PSE scheme all sites are addressed simultaneously. The total excitation time is thus equal to the preparation time necessary to excite an individual site. The major advantage of this is that the signal losses arising from T2 relaxation effects or from homonuclear J-coupling modulations during the encoding period become considerably ameliorated. On the negative side, the design and applications of these rf waveforms and gradient waveforms are naturally more complex than the very simple encoding steps involved in the SSE. Also, from an experimental performance perspective, the fact that a continuously oscillating gradient is needed in PSE will often require stronger gradients to obtain the same desired spatial phase encoding pattern, as that given by the simple N pulsed-gradient SSE approach. Still, as demonstrated in Sec. III, this does not constitute a limitation for practical PSE applications. For completion, benefits and drawbacks of the various new encoding methods considered in this study vis-à-vis those of the conventional spatial encoding method are summarized in Table I.

300

III. RESULTS

336

A. Experimental methods

337

301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335

In order to test the various considerations just presented, 338 a series of experimental comparisons was made between spa- 339

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J. Chem. Phys. 131, 1 共2009兲

Spatial/spectral encoding in ultrafast NMR

TABLE I. Pros and cons of spatial and spatial/spectral encoding methods.

Pros

Cons

Conventional spatial encoding

Sequential spectral-spatial encoding

Parallel spectral-spatial encoding

General

Simple to use Weak encoding gradients Optimally exploited k-range → Best SNR and weakest Ga

Short encoding times Uniform relaxation weighting Optimally exploited k-range → Best SNR and weakest Ga

Needs a priori information

Needs a priori information

Possibly long encoding time Nonuniform relaxation

Strong encoding gradients Nontrivial pulse design

SW1SW2 ⬀ ␥aGaL ⌬␯1 → Strong acquisition gradients and low SNR for sparse ␯1 domains

tial and spatial/spectral encoding methods. These measurements were carried on a 11.7 T magnet equipped with a HCN ® 342 probe and a Varian iNova NMR console. The pulses and 343 gradient waveforms required by the various experiments 344 were generated via custom-written MATLAB codes 共available 345 upon request兲, so were all the data rearrangement and pro346 cessing manipulations. The experiments implemented for 347 these studies were divided into two groups, seeking to ex348 plore the two main potential advantages of SSE and PSE 349 spatial/spectral encoding. A first set of tests used hetero350 nuclear single quantum correlation 共HSQC兲 2D experiments 351 to explore the potential SNR improvements which could 352 arise upon using the new scheme in the analysis of simple 353 organic molecules. A second set of tests employed homo1 1 354 nuclear H – H J-mediated total correlation spectroscopy 355 共TOCSY兲 experiments to explore the results arising from 356 spatial/spectral encoding in cases where the ␯1 resolution 357 may be dictated by the maximal acquisition gradient ampli358 tudes available. 340 341

Figure 4 illustrates an application of the SSE protocol to this same system. The 13C domain included eight chemical sites 共Fig. 3兲; as the two sites located near 34.7 ppm have a very small gap in between them 共⬃0.05 ppm, 6 Hz兲, no efforts were made to distinguish among these. All remaining 13 C sites could be well separated if addressed independently by a train of reasonably short 共⬃15 ms兲 rf pulses. Following each of these resulting seven selective pulses, gradient pulses characterized by ␦k = 1.2L−1 wave-numbers were applied to evenly distribute their corresponding spatial encodings. This enabled us to set the acquisition kmax to the lowest value compatible with a seven-peak spectrum, thus decreasing the value needed for Ga from 45 to 9.4 G/cm. This decrease enabled in turn a much longer acquisition time 共⬃350 ms兲 (A) Conventional Ultrafast 2D HSQC 3.6

3.6

1H

4

One of the improvements expected from the new spatial/ spectral encoding protocols relates to their need for weaker 363 Ga gradients. This should increase the SNR 关Eq. 共6兲兴, allow364 ing for Ga’s that operate longer acquisition times, and bring365 ing in turn improvements in the ␯2 resolution. To test all this 13 1 366 a series of heteronuclear 2D ultrafast C – H HSQC NMR 367 experiments was performed on a sample containing a mix368 ture of 1-butanol and n-butyl chloride at natural abundance 13 369 dissolved in CDCl3. The 1D C spectrum of this mixture 370 contains eight sites spanning over 50 ppm showing a variety 371 of spacings, making it a good representative of the chal372 lenges mentioned earlier. A purely absorptive ultrafast 2D 373 HSQC of this sample, designed to include all the chemical 374 sites without aliasing, is shown in Fig. 3. The relatively 1 13 375 broad spectral ranges of both H and C in this 2D spectrum 376 demanded the use of relatively strong acquisition gradients: 377 兩Ga兩 ⬇ 45 G / cm. Even under these conditions we had to 378 compromise on a number of factors, including the indirectmax 379 domain evolution time 共t1 of only 8 ms兲 and the total FID max 380 acquisition time 共t2 = 46 ms兲. These choices resulted in re381 duced spectral resolutions along both the indirect and direct 382 domains, as well as in SNR losses due to the broad receiver 383 BW and short acquisition times involved 关Fig. 3共b兲兴.

0.5

9 -21

Gz

359 B. SNR and resolution improvements: Heteronuclear 360 2D NMR correlation examples

3

9

Gx

t

4 1

1

3

13C

0.46

9 21

-9

31

45

-45 100

(B) 1-Butanol + n-Butyl Chloride

361 362

(A) – (B) – (C) – (D) CH3– CH2 – CH2– CH2– Cl

10

(A)+(E)

20

13C

[ppm]

(F)

(B)

30 (G) (C)

40 (H)

50 60

(D)

4

3.5

(E) – (F) – (G)– (H) CH3–CH2– CH2–CH2–OH

3

2.5 2 1H

1.5

1 0.5

[ppm]

FIG. 3. 共a兲 Conventional spatially encoded 2D HSQC experiment on a mixture of 1-chlorobutane and 1-butanol based on a sequence employing two oppositely sweeping ␲ / 2 chirped pulses, separated by a hard ␲ pulse that provides purely absorptive line shapes 共Ref. 34兲. 共b兲 Ultrafast 2D 13 C – 1H correlation spectrum 共real components兲. The acquisition dwell time used here and throughout all the experiments in this paper was 5 ␮s; spectral resolution along the ␯1 axis was 200 Hz. Like throughout all the remaining pulse sequence diagrams in this paper, gradient amplitudes and timings provided in this figure are in units of G/cm and millisecond, respectively; nonselective ␲ / 2 and ␲ pulses are symbolized by thin and thick lines, respectively; arrowed pulses indicate frequency-chirped rf blocks; brackets show events looped over the indicated number of times.

384 385 386 387 388 389 390 391 392 393 394 395 396 397

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J. Chem. Phys. 131, 1 共2009兲

Y. Shrot and L. Frydman (A) Ultrafast PSE 2D HSQC

(A) Ultrafast SSE 2D HSQC 3.6 3 decouple 15

1

H

1 1 20 20


2

13

C 9

1.5

t

0.264

H

-6.3 -6.3

Gz

-40

Gx 9

-9.4

-40

+40

-40

40

-40

9

-40

9.4

-9.4 720

100

9.4

40 7

t

3

3

9

Gz

0.46

1.5

3

Spatial/ Spectral

C

3

3.6

3

1

13

3

{ i }i=1,..,7

Gx

3

3.6

0.46

3.6

720

(B) One Site per k Value

(B) One Site per k Value (A) – (B) – (C) – (D) CH3– CH2 – CH2– CH2–OH

7 6 5

(D)

Site index / k-axis

Site index / k-axis

(A) – (B) – (C) – (D) CH3– CH2 – CH2– CH2– Cl

(B)

4 3

(A) (C) (G)

7 (D)

5 4

(A)

3

(G) (C)

2

2 (H) (E) – (F) – (G)– (H)

1

(F)

3

2 H [ppm]

1

(F)

1

(E)

CH3–CH2– CH2–CH2–OH

4

(H)

0

1

bringing about a better ⌬␯2 resolution and a substantial in399 crease in the signals’ intensities. Notice that despite the long 400 duration taken by the discrete selective pulses in this 401 SSE test 共⬃100 ms兲, the long T2’s characterizing the 1 13 402 H-decoupled C evolution implied only minor losses: peak 403 intensities were thus nearly independent from the order of 404 excitation of the various sites. This fact, coupled to the small 405 encoding wave-number ␦k used 共6.3 G/cm, 0.1 ms兲, implied 406 that neither diffusion nor relaxation should have had much of 407 an influence during the encoding. In cases such as this, the 408 difference in signal intensity arising upon using the SSE or 409 the PSE approaches should be minor. To test this hypothesis, 410 the parallel approach was implemented using a 2D spatial 411 spectral rf pulse designed to endow the various sites in the 412 sample with an encoding pattern identical to that employed 413 for the SSE. The total encoding duration could be thus made 414 considerably shorter, reducing it from ⬃100 to ⬃25 ms, 415 even if at the expense of much stronger oscillatory gradients 416 关Fig. 5共a兲兴. The resulting 2D ultrafast spectrum 关Fig. 5共b兲兴 417 showed, in terms of both SNR and Ga requirements, a similar 418 performance as the results shown in Fig. 4共b兲 upon using the 419 SSE method. Besides a higher efficiency in their use of gradients, the 420 421 new spectral/spatial methods offer a potential encoding flex422 ibility that is just absent in the traditional spatial encoding 423 methods. These variations include, for example, spectral/ 424 spatial encoding profiles that do not necessarily endow each 425 chemical site with a different encoding wave-number: one 426 could then provide certain chemical sites that are known to 427 be resolved in ␯2 with identical k’s wave-numbers. The re-

4

(E) – (F) – (G)– (H) CH3–CH2– CH2–CH2–Cl

3

(E)

0

1

2 1 H [ppm]

(C) Multiple Sites per k Value

3 Site index / k-axis

FIG. 4. 共a兲 Serially selective excitation scheme employing a train of selective rf pulses interleaved with gradient pulses to perform a spatial/spectral encoding. In order to get a narrow-banded response, pulses of 15 ms were designed by the Shinnar–Le Roux algorithm 共Ref. 35兲, providing an excitation BW of ⬃75 Hz. Following each pulse a gradient pulse 共split for 1 H-decoupling purposes兲 endowed the excited sites with uniform increments in their wave-numbers. Acquisition of a 2D spatially encoded HSQC spectrum 共b兲 could thus be achieved using considerably weaker Ga gradients 共display in absolute-value mode兲.

398

(B)

6

(B)

(D)

(A)

2 (G) (C)

1 (F)

(H)

4

(E)

3

2

1

0

1

H [ppm]

FIG. 5. 共a兲 Parallel selective excitation protocol designed with a performance akin to the sequence in Fig. 4 except for its addressing of all the spins simultaneously. The flexibility of the spatial-spectral encoding can be used to either 共b兲 endow every site with a different k wave-number or 共c兲 arrange the sites in fewer, further separated groups, providing a better resolution along the indirect domain. Images are displayed in absolute-value mode and indicate the positions ascribed to each chemical site along the k-axis.

duced number of encoding ␦k wave-numbers could then allow either the use of even weaker acquisition gradients, or a better separation in k / ␯2 by an increase in ␦k without compromising the effective resolving power of the 2D experiment. Figure 5共c兲 illustrates an example of this: it shows the use of an alternative PSE scheme, endowing the eight sites in the mixture with only three k-values in their spatial encodings. Using the same excitation and acquisition gradients one can then spread the indirect domain further apart to ␦k = 3.6L−1 in this example and achieve a correspondingly better resolution without increasing Ga. Another variant of the PSE approach is displayed in Fig. 6共a兲, which introduces a sequence aiming to obtain the same spatial encoding as in Fig. 5共c兲, but operating in a purely absorptive acquisition mode. This can be achieved while retaining the experiment’s singlescan character by employing spatial/spectral pulses as part of both the HSQC’s indirect domain excitation as well as of its storage stages—with a hard 180° pulse inserted in between.

428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445

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J. Chem. Phys. 131, 1 共2009兲

Spatial/spectral encoding in ultrafast NMR

(A) Standard Ultrafast 2D TOCSY (SW1,2≈7ppm)

(A) Purely absorptive ultrafast PSE 2D HSQC 3.6

0.264

0.264

3.6

H

SPSP π/ 2 3

13

C Gx

Gz

20

-10 10 80

20

-10

-20

0.3

40

mix 0.5

1H

9 -20

1.4

0.5

t

SPSP 3 π/ 2

9

Gz

1.5

3

3

1

1.4

0.46

t -10

7.5

7

-7.5

26

-26

9.4 -9.4 720

80

400

(B) Single-scan 2D TOCSY - TyrEE

(B) Experiment – 3 Different k Values 0 1

2



3 H [ppm]

2

1

1

Site index / k-axis

3



4

5 6

H2N



HO

7 4

2 H [ppm]

3

0

1

1

8 7

(C) Cross-section Comparisons Ω1 = 34.7 ppm

* 1.5

2

2.5

Ω1 = 44.3 ppm

2.8

3

3.2

3.4

3.6

3.8

Ω2 [1H shift, ppm] FIG. 6. 共a兲 Pulse sequence encoding purely absorptive spectroscopy based on a variation in the PSE scheme, whereby spatial/spectral pulses are used for both excitation and storage of magnetization. 共b兲 Purely absorptive 2D HSQC spectrum collected using only three k-values along the 13C dimension 关akin to Fig. 5共c兲兴. 共c兲 Normalized ␯2-cross sections extracted for the indicated indirect-domain ppm frequencies, comparing the results arising from the purely absorptive 2D HSQC NMR spectra in 共b兲 and Fig. 3共b兲. An improvement of both SNR and direct domain resolution is evident. The asterisk 共 ⴱ兲 indicates a minor cross-talk caused by an overlap with the adjacent CH2 echoes. 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464



6

5



O O

4 H [ppm]





3

2

1

1

Spatial/Spectral Encoding Spatial Encoding

1



The latter’s role is to cancel any linear chemical shift phase evolution as well as pulse imperfections while the spatial dephasing is accumulated. Due to the “doubled” character of the ensuing encoding, a gradient that is only half the amplitude of the nonsymmetric counterpart 共20 versus 40 G/cm兲 and shorter pulses can provide the same degree of desired encoding with a better line shape and sensitivity 关Fig. 6共b兲 versus Fig. 5共c兲兴. All the spatial/spectral implementations described in Figs. 5 and 6 allowed us to reduce the acquisition gradient vis-à-vis the values required by the original spatial encoding experiment by a factor of ⬃5. This was achieved at no cost in the ␯1 resolution and enabled in turn two concurrent improvements: 共1兲 the use of longer acquisition times, made possible thanks to the weaker Ga’s involved, and 共2兲 lower noise levels, thanks to the accommodation of reduced receiver BWs of ⬃␥aGaL. The combination of these two factors should increase the direct-domain spectral resolution and provide a considerable sensitivity improvement. Figure 6共c兲

FIG. 7. 共a兲 Conventional spatially encoded 1H TOCSY experiment on a sample of tyrosine ethyl ester dissolved in CDCl3. 共b兲 Spectrum displayed in absolute-value mode. In order to enable large direct-domain BW and resolution, a moderate acquisition gradient was necessary and a tradeoff with respect to the indirect-domain resolution had to be made.

evidences these two features by displaying normalized 1D cross sections extracted from both the purely absorptive spectral/spatial PSE encoding scheme and from a traditional spatially encoded 2D purely absorptive HSQC spectrum. The enhancement in resolution 共notice the clear homonuclear J-coupling patterns兲 and decrease in noise are evident. Furthermore, even if SNR is not the limiting constraint in the ultrafast 2D acquisition, one can use the tools afforded by the new methods here presented to improve the performance of other experimental factors—as further described in the following paragraph.

465 466

C. Indirect-domain resolution enhancement: A homonuclear correlation example

476 477

As pointed out earlier 关Eq. 共5兲兴, the mutual relations between the two spectral dimensions of the spatial encoding method often require compromises: for instance, decreasing the indirect-domain resolution and/or spectral width in favor of the direct-domain frequency BW. An example of such tradeoff is shown in Fig. 7 with a homonuclear spatially encoded 2D TOCSY experiment performed on a tyrosine ethyl ester sample. The relatively large SW1, SW2 values 共⬎6 ppm兲 required to cover the full spectral range of this compound demand both a short gradient oscillating period Ta and a large acquisition k-range kmax. In order to achieve these spectral BWs and an acceptable direct-domain resolution, while keeping the experiment compatible with the available gradient’s performance, a tradeoff has to be set with respect to the extent allowed for the indirect-domain evolution. As encoding shown in Fig. 7 this calls for shortening the tmax 1 period to ⬃3 ms, even then, at the expense of demanding

478 479 480 481 482 483 484 485 486

467 468 469 470 471 472 473 474 475

487 488 489 490 491 492 493 494

(A) Single-scan SSE 2D TOCSY TyrEE 15

2

1H

Gz

40 0.08

0.1

0.3

mix 0.1 -10

5 7

-26

26

t

10 400

(B) Encoding High  Low ppm

7

Site index / k-axis

the application of a relatively strong oscillating Ga max 496 ⬇ 26 G / cm for t2 ⬎ 100 ms. Under such low ␯1 resolution 497 conditions, multiple diagonal peaks and cross peaks remain 498 unresolved 关Fig. 7共b兲兴. This example highlights the built-in 499 limitations of a generic approach like spatial encoding: al500 though there are only seven sites to be correlated in this 501 molecule, the kmax of this experiment had to be taken as 502 30L−1 to cover the full spectral width. This is a result of the 503 method’s linear mapping of the ␯1-domain into k-axis ech504 oes: when dealing with a nonuniform spacing of the 505 chemical-shifted peaks, this leads to a nonuniform echo dis506 tribution and a suboptimal use of the gradients. Moreover, 507 even under these demanding Ga conditions, sites that are 508 close together along the indirect domain end up yielding ech509 oes with poor or no resolution. 510 By providing nonlinear ⍀1-shift/ k-echo mapping func511 tions, the spatial/spectral encoding schemes hereby proposed 512 can alleviate this limitation. Even peaks that are close to513 gether in ⍀1 can then be resolved into clearly separated ech514 oes along the k-axis. Examples of this are shown in Fig. 8, 515 which illustrates the use of SSE sequences on tyrosine ethyl 516 ester. Although using identical acquisition conditions as in 517 the conventional 2D ultrafast spectrum of Fig. 7, a much 518 higher ␯1 resolution is achieved in these SSE TOCSY experi519 ments by evenly distributing the peaks so that all echoes are 520 spaced at ⌬k = 3L−1 increments. Such an increased spacing 521 was relatively easy to obtain by moderate gradients, thanks 1 522 to the large gyromagnetic coefficient of H. On the other 523 hand, and unlike in the heteronuclear SSE-based experi524 ments, where sites were characterized by long T2 times and 525 an overall insensitivity to the order of their ␯1 encodings, this 526 homonuclear experiment clearly shows that the order with 527 which sites are addressed can considerably influence their 528 echoes’ amplitudes. Sites which were excited first 共small site 529 index兲 are of smaller intensity than those excited last 共high 530 site index兲, as can be appreciated by comparing the results 531 obtained for different orders of site excitation 关Figs. 8共b兲 and 532 8共c兲兴. This can be attributed to the presence of homonuclear 533 J-couplings, which by amplitude modulating the time534 domain evolution over the course of the encoding act during 535 this time as a sort of an effective T2 decay. 536 By contrast, these weighting heterogeneities should be 537 absent upon implementing the PSE scheme. The 2D spatial/ 538 spectral pulse required for testing this was designed based on 539 oscillatory trapezoidal gradient waveforms; mixing as well 540 as acquisition parameters were taken to be identical to those 541 employed in the SSE experiments 关Fig. 9共a兲兴. The resulting 542 PSE TOCSY 2D spectrum 关Fig. 9共b兲兴 clearly avoids the dif543 ferential site attenuation due to its uniform encoding time. 544 Moreover, when compared with the SSE implementation, it 545 can be seen that the overall duration of the parallel spatial 546 encoding pulse can be made considerably shorter: ⬃25 ms 547 versus ⬎100 ms in the serial case. This in turn can be used 548 to endow the PSE 2D rf pulse with higher shift selectivity in 549 its the ␯1 addressing, as well as with a better average SNR 550 for the final 2D spectrum. 495

J. Chem. Phys. 131, 1 共2009兲

Y. Shrot and L. Frydman

6 5 4 3 2 1

7

6

5

4

3

2

1

2

1

1

H [ppm]

(C) Encoding Low  High ppm 1 2

Site index / k-axis

1-8

3 4 5 6 7

7

6

5

4

3

1

H [ppm]

FIG. 8. 共a兲 Homonuclear SSE-based 2D TOCSY pulse sequence based on an even distribution of the peaks along the k-axis. 关共b兲 and 共c兲兴 2D TOCSY NMR results on the same TyrEE sample as used in Fig. 7—including the same acquisition parameters but a SSE encoding. These two data sets differ by the order of excitation of the chemical sites. Notice 共i兲 the enhanced indirect domain resolution that is obtained; 共ii兲 the fact that due to the long encoding periods, sites that were excited first 共lower k index兲 are also further attenuated due to T2 relaxation and homonuclear J-modulation.

IV. DISCUSSION AND CONCLUSIONS

551

If available, a priori 1D information about the sample under investigation can help to construct more efficient forms of 2D NMR acquisitions. This has been amply documented for conventional 2D NMR experiments, where it is known that selective excitation can reduce the minimum number of acquisitions that are needed. In ultrafast schemes, where the 2D acquisition has already collapsed into a single scan, this kind of information can be traded for the sake of sensitivity or for improving spectral resolution. The present paper demonstrated two strategies that can bring about this flexibility, based on a serial and on a parallel spatial encoding of specific chemical sites. The serial scheme, characterized by maximal simplicity and minimal hardware demands, requires relatively long encoding durations and may thus be susceptible to differential weighting due to the sites address-

552 553 554 555 556 557 558 559 560 561 562 563 564 565 566

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J. Chem. Phys. 131, 1 共2009兲

Spatial/spectral encoding in ultrafast NMR

(A) Single-scan PSE 2D TOCSY TyrEE 40

0.25

1H

SPSP

0.1

-27

10

mix

0.3 0.1 t

Gz

27

-10

26

26

100

400

(B) Example - TyrEE

Site index / k-axis

7 6 5 4 3 2 1

7

6

5

1

4

H [ppm]

3

2

1

FIG. 9. Idem as in Fig. 8 except for the use of a PSE parallel addressing to encode the sites. The shorter overall encoding period prevented the undesired site-dependent attenuation; the shift selectivity is also better than in the SSE experiments given in Fig. 8.

ing order. If this is a limiting factor, an alternative parallel 568 scheme can be employed; although characterized by in569 creased hardware demands and by complex rf/gradient 2D 570 waveforms, considerably shorter encoding periods as well as 571 a uniform relaxation weighting are furnished by this scheme. The key for executing this PSE approach is the design of 572 573 new kinds of 2D spatial/spectral pulses for pure spectro574 scopic applications. Spatial/spectral pulses that exploit a pri575 ori known spectroscopic information have in fact been re576 cently suggested for the realization of single-scan 2D 27 577 Hadamard NMR experiments. While in a conventional 2D 578 Hadamard scheme the selected sites are endowed with a 579 sign-alternating pattern that varies from scan to scan, the 580 underlying principle behind its single-scan counterpart is to 581 encode this set of experiments with a 2D spectral/spatial 582 pulse. From this, distinct spatial regions and chemical sites 583 will be endowed with signals of positive and negative signs. 584 By requiring to excite neighboring spatial regions with op585 posite phases, however, the Hadamard approach puts forth 586 harsher demands on the encoding gradients than the 587 smoother, continuous winding patterns demanded by the PSE 588 encoding. Moreover, the extraction of the spectral informa589 tion requires the acquisition of sharp-edged images followed 590 by a Hadamard-based postprocessing. The novel use of 2D 591 spatial/spectral pulses hereby proposed for the PSE avoids 592 this by choosing the simplest form that is suitable for the 593 extraction of the spectral data: an echo train that in and of 594 itself already represents the spectrum of interest. An important advantage of the spectral/spatial schemes 595 596 when compared with standard, continuous spatial encoding 597 scheme is their insensitivity to diffusion. Indeed, achieving a 598 high ␯1 resolution in the latter case requires relatively long max encoding periods in the presence of constant gradients; 599 t1 600 these conditions may lead to signal degradation due to diffu567

sion effects.28,29 On the other hand, when using either the very weak or the oscillating gradients demanded by the SSE or PSE, these effects become negligible. The spatial/spectral approaches are therefore suitable to deal with single-scan 2D experiments for highly diffusive compounds as well as in cases of microscopic turbulences. The reduced sensitivity to the latter can be a considerable aid to enhance the performance of ultrafast 2D NMR spectroscopy in combination with ex situ dynamic nuclear polarization 共DNP兲, an experiment whereby hyperpolarized compounds are suddenly injected into the NMR spectrometer.30,31 Furthermore, DNPenhanced 2D NMR experiments are usually characterized by wideband, sparse 13C / 15N indirect dimensions, and for them a single-scan acquisition is of essence. The potential of merging into DNP a SSE-like protocol has been very recently described.32 If multiple yet few scans are a possibility worth considering in a 2D acquisition, reductions in Ga going beyond those hereby described can be achieved. One could consider a multiple-scan hybrid scheme that combines a spatial/ spectral ⍀1 / k mapping strategy together with a folded encoding, where two or more spectral regions are put together to overlap within the same reduced k-range. Proper phase cycling may then be used to fully unravel a densely compacted mapping of broadband spectra. Further means to reduce the acquisition gradients could result by assuming sparseness along the direct domain; this may also enable the reconstruction of the full 2D spectrum by partial echo-planar spectroscopic-imaging procedure of minimal acquisition gradients. These and other applications are targets of ongoing studies.

601 602

ACKNOWLEDGMENTS

632

This research was supported by the Israel Science Foundation 共ISF Grant No. 447/09兲, by a Helen and Kimmel Award for Innovative Investigation, and by the generosity of the Perlman Family Foundation. Y.S. acknowledges the Clore Foundation for a Ph.D. fellowship.

633 634

603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631

635 636 637

APPENDIX: DESIGN OF SPATIAL/SPECTRAL PULSES 638

2D spatial/spectral rf pulses are often used in in vivo applications to selectively excite a particular chemical species over an arbitrary geometry in space. The goal of such pulses in the present spectroscopic scenario is slightly different; hence we consider useful to introduce the basic principles of how these combined rf/gradient manipulations were here designed. Our goal was to simultaneously endow multiple targeted sites possessing different shifts with individually tailored spatial winding patterns. Analytically, this can be expressed as seeking a 2D transverse magnetization pattern of the form

639 640 641 642 643 644 645 646 647 648 649

N

M XY 共z, ␯1兲 = 兺 exp关i共N − p + 1兲 · ␦kz兴Q共␯1 − ⍀ p兲. 共A1兲 p=1

650

The method employed for this purpose is based on the linear 651 approximation for pulse design,25 according to which the 652

1-10 653 654

J. Chem. Phys. 131, 1 共2009兲

Y. Shrot and L. Frydman

transverse magnetization created during the course of an excitation period Tpulse is M xy共z, ␯1兲 ⬀

655

冕

Tpulse

B1共t兲e

i␯1共Tpulse−t兲 ik共t兲z

e

共A2兲

dt,

0

where k共t兲 = 兰Tt pulse␥eGe共t⬘兲dt⬘ is defined by the gradient Ge 657 used throughout the excitation. The availability of two de658 grees of freedom, t and Ge共t兲, to “steer” the evolution of the 659 spins during the course of the pulse enables one to use B1共t兲 660 for addressing arbitrary 共z , ␯1兲 values. For a successful de661 sign of this spatial/spectral B1共t兲 pulse, one has first to deter662 mine the oscillatory gradient waveform Ge共t兲 that will suit 663 best the desired magnetization response. The main features 664 to be taken into account in this design include 共1兲 the gradi665 ent’s oscillatory cycle, a period that will be inversely propor666 tional to the maximum ␯1 spectral width that can be ad667 dressed without aliasing, 共2兲 the gradient’s amplitude, which 656

(A) Target Spatial/Spectral Amplitude Response

-1

-0.5

0

0 1.5 3

4.5 6

0.5

ν1

1 [kHz]

k·L

(B) Amplitude of Calculated RF

together with the cycle’s period will determine the spatial resolution of the pulse and, correspondingly, the maximal spatial winding that can be imposed, and 共3兲 the overall duration of the pulse, which will define Q’s spectral resolution. When dealing with samples characterized by large spectral widths, the use of a minimal ⫾Ge gradient oscillation period makes the presence of excitation spectral aliasing unavoidable. A problem will then arise if these harmonic excitation bands coincide with a targeted spectral peak of interest. A natural solution to this issue, akin to that arising in DANTE sequences,26 consists of exploiting “empty” spectral regions where no peaks are known to reside. By slightly modifying the design parameters one can “shift” these bands of harmonics; excitation aliasing can then be set to lie in empty regions, and any undesired spectral contributions are thus avoided. Assuming these various parameters have been properly set, the targeted M XY pattern in Eq. 共A1兲 can be substituted into Eq. 共A2兲, leading to a set of linear equations that relate the desired magnetization response with the required transmission B1 rf as a function of time. In order to derive B1, an inverse non-uniform Fourier transform operation is necessary. In this study we employed a method based on conjugate gradients33 for the evaluation of this FT. The first stage in the resulting iterative search is to transform all the continuous variables involved—including the frequencies as well as the spatial positions and the excitation times—into discrete axes endowed with a sufficiently large number of N⍀, Nz, and Nt points, respectively. Under this discretization, Eqs. 共A1兲 and 共A2兲 can be rewritten in a matrix form as Xគ = Txគ ,

(C) Gradient Waveform

(D) Simulated Spatial/Spectral Amplitude Response

(E) Simulated Spatial/Spectral Phase Response

FIG. 10. Synthesis and typical performance of the 2D spatial/spectral rf pulse design algorithm. 共a兲 Targeted spectral/spatial profile comprising three sites spaced relatively close together 共⌬␯1 = 120 Hz兲 and two additional more distant offsets. The desired spatial pattern was characterized by different k-wave-numbers for each site, spaced ␦k = 1.5L−1 increments apart. The 2D rf was designed using a 48-cycle oscillating gradient waveform with amplitude of ⬃17 kHz/ cm 共corresponding to 4 G/cm for 1H兲 and duration of 23.04 ms 关共b兲 and 共c兲兴. Solving the Bloch equations for spins evolving under the designed rf and gradient waveforms provided a response very similar to the targeted one: 共d兲 shows the amplitude response and 共e兲 shows the phase response of the spins as a function of position for each chemical site.

668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697

共A3兲 698

where 共x兲m = B1共tm兲 with m = 1 ¯ Nt, 共X兲n = M XY 共兵z , ⍀1其n兲 with n = 1 ¯ 共N⍀Nz兲, and Tnm ⬀ ei兵⍀1其ntmeik共tm兲兵z其n. The numerical solution for the rf shape 兵共x兲m其m=1¯Nt of Eq. 共A3兲 was based on the following algorithm.

699 700 701

Given the Linear Relation: X = T · x

702 703

Inputs: X—a vector of length P T—a phasor matrix of size P ⫻ N x0—initial guess for the coefficient vector 共of length N兲 M–the number of iterations

704 705 706 707

Preliminary Calculations: ⌬0 = X − T · x0 p0 = q0 = TH⌬0, where TH is the complex-conjugate of transverse 共T兲.

708 709 710 711

for l = 0 , . . . M do vl = Tpl ␮l = qHl ql / vHl vl xl+1 = xl + ␮l pl ⌬l+1 = ⌬l − ␮lvl ql+1 = TH⌬l+1 H ␰l = ql+1 ql+1 / qHl ql pl+1 = ql+1 + ␰l pl

712 713 714 715 716 717 718 719

end for

720

Output: Vector of coefficients x M

721

Usually, starting with an initial guess of zeroes as the x0 722 vector, the algorithm converged rapidly and provided the de- 723

1-11

sired results within a very few 共1–3兲 iterations. The resulting B1共tm兲 shape was then ported to the spectrometer, and to726 gether with its accompanying Ge共tm兲 gradient waveform 727 clocked out at its predefined ⌬t = tm − tm−1 dwell. An example 728 of the desired and calculated spatial/spectral response for a 729 five-site sample is shown for completion in Fig. 10. 724 725

730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753

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J. Chem. Phys. 131, 1 共2009兲

Spatial/spectral encoding in ultrafast NMR

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