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High resolution NMR using selective excitation
Journal of Molecular Structure, 266 (1992) 39-51 Elsevier Science Publishers B.V., Amsterdam
39
High resolution NMR using selective excitation Ray Freeman Department of Chemistry, Cambridge University, United Kingdom
Abstract Modern high resolution NMR spectrometers employ the Fourier transform method where the entire spectrum is uniformly excited by a “hard” radiofrequency pulse. Occasionally, information is more easily extracted by an experiment where the excitation is selective in the frequency domain, using a “soft” radiofrequency pulse. We may select a single transition, an entire spin multiplet or an extended range of chemical shift frequencies for the excitation. In the early experiments, simple rectangular-shaped pulses were used, but much better results can be achieved by suitably shaping the envelope of the pulse. Several interesting schemes have been explored for designing pulse shapes, including simulated annealing, computer-simulated neural networks and evolutionary strategies that mimic the processes of natural selection. Selective pulse excitation has been used for relaxation time measurements, solvent peak suppression, resolution enhancement, separation of overlapping twodimensional multiplets, improving the definition of fine structure in twodimensional spectroscopy, and for reducing the time penalty associated with three- or four-dimensional NMR.
1. INTRODUCTION When we use a microscope, we focus our attention on a restricted part of the specimen in order to use high magnification; similarly we may use the zoom feature of a camera to examine fine detail by restricting the field of view. This article examines the analogous procedures in NMR spectroscopy when fine structure information is more important than the global picture. Present-day high resolution spectra have such a high information content that this “zoom” operation is becoming more and more attractive. Routine chemical applications of high resolution NMR excite the entire spectrum of the chosen nuclear species, for example protons or carbon-13. It is 0022-2860/92/$05.00
0 1992 Elsevier Science Publishers B.V. All rights reserved
not normaily feasible to select one small section of the spectrum and reject the rest. This restriction arises because the transient time-domain NMR signal (the “free induction decay”) must be in digital form for the purposes of Fourier transformation. This in turn imposes an all-pervading sampling requirement known as the Nyquist criterion which states that unless there are at least two samples per cycle of the highest frequency component of the free induction decay, there will be ambiguities in the frequencies. Any high-frequency signal is said to be “aliased” and appears not at its characteristic frequency&, but rather at _f, - nfs, where& is the sampling rate and n is an integer. The effect is analogous to the stroboscopic effect, well-known from Western movies, where the wagon wheels appear to rotate at the wrong speed or even go backwards. For serious chemical applications, aliasing of NMR signals would be a disastrous falsification of the data and should be avoided at almost any cost. The aliasing problem can be circumvented by exciting partial NMR spectra. All the responses are confined to a narrow predefined frequency band, leaving the rest of the spectrum completely unperturbed. In this way, only the low-frequency NMR signals need be considered and aliasing is avoided. The radiofrequency pulse is said to be “soft,” whereas a normal pulse is “hard” and affects the entire spectrum. Recent technical innovations make it feasible to generate soft pulses suitable for this “band-selective” excitation.
2. SOFT PULSES At first sight the problem seems quite simple -- by reducing the intensity of a radiofrequency pulse (with a compensating increase in its length) we can make it frequency-selective or soft. If the pulse intensity is Bi tesla, a range of frequencies of the order of yBi/2rr Hz will be excited. Unfortunately the form of the excitation pattern in the frequency domain is far from ideal. There is a central lobe of strong excitation flanked by a set of sidelobes whose amplitude decreases with offset from the centre. These sidelobes originate from the fact that the pulse has sharp leading and trailing edges. The obvious solution is to shape the pulse to smooth out these discontinuities, and considerable progress has been made with pulse envelopes that are Gaussian [l] rather than rectangular. Once the sidelobes have been removed another problem comes to light. Most soft pulses introduce phase gradients, an increasing admixture of dispersion mode signal with increasing offset. Good resolution requires pure absorptionmode signals, so this frequency-dependent phase error has to be corrected. Furthermore, if the spectrum is to be used for quantitative measurements we would like the excitation to be uniform over the desired band of frequencies. Finally, to minimize the intensities of aliased signals, the excitation outside the
41
selected band should be negligible. Of course, we cannot make an instantaneous transition from full to negligible excitation, so we must allow a narrow transition region between the two. The ideal excitation pattern may be sketched schem&icd~~y:
) I I Zero 1
Uniformexcitation
1 vtransition
region
\ \ Zero
absorption
Idealized frequency-domain profile If we think of this behavior in terms of magnetization trajectories it is clear that the pulse must have a focusing action, as in the spin echo experiment, so that all trajectories within the excitation band terminate on the Y axis of the rotating frame. 3. PULSE DESIGN The criteria outlined above are by no means easy to achieve. There is no simple direct way to calculate the shape of the pulse envelope that would generate a particular excitation pattern, and we therefore have recourse to iterative numerical methods. Most of the early work on pulse shaping [2-41 was carried out for the purposes of magnetic resonance imaging where the principal application is “slice selection” by excitation of the sample in an applied static field gradient; high resolution spectroscopists were slower to realize the importance of “designer pulses.” The conventional procedure is to employ a multiparameter optimization routine, but there are some interesting variations, three of which are described below. Simulatedannealing Simulated annealing [5] is a well-established routine that was designed specifically to avoid the “false” solutions that are often found in multiparameter optimization. The differences between the actual excitation pattern and the target pattern define a multidimensional error surface and there may be several minima on this surface. Most of the time the search program behaves like conventional “gradient descent” methods but to avoid becoming trapped in a false minimum the simulated annealing algorithm allows an occasional “uphill” -jump with a
42
probability given by the equivalent of the Boltzmann factor, exp(-AE/kT), where AE is the change in the error function and T is analogous to temperature. At a high enough temperature, the program can escape from any false (“local”) minima and eventually reach the “true” global minimum, which we assume represents the optimum design. The key factor is the annealing schedule, an appropriately slow stepwise decrease in the temperature T. Simulated annealing has been used to design some very effective band-selective soft pulses for highresolution NMR. Neural networks Most optimization programs are time-consuming because of their iterative nature and the asymptotic approach to the solution. One exception is the method of neural networks [6]. This is a simulation (on a digital computer) of the processes by which the brain is believed to operate. A complex network of interconnected neurons is set up, arranged in layers, with an input layer (the retina) at one boundary and an output layer at the other. Each neuron receives voltage impulses vj from many other neurons “upstream” and calculates the weighted sum S=O+
CWjVj
J
where 8 is a bias voltage and wj am “synaptic weights.” The signal S is amplified according to a specific response function (usually a sigmoid curve) and then The common output voltage thus passed to further neurons “downstream”. depends on the signals received by the neuron, and the values of 0 and wj.
0 = bias
w = synaptic weight
When used for pulse design, the network is first operated in a training mode. Frequency-domain excitation patterns are calculated for a set of randomly-generated pulse shapes defined in terms of the coefficients of a finite Fourier series. When these excitation patterns are presented to the retina, the output layer at first produces quite unrelated signals, but the program modifies the bias voltages and synaptic weights to bring these output signals closer to the known Fourier coefficients of the appropriate pulse shape. As training proceeds this correspondence improves. After a protracted training program in which a suitably representative range of pulse shapes has been studied, the network is ready for the operating mode. It incorporates its acquired knowledge in the form of the synaptic weights wj and the bias voltages 8. In the operating mode the desired excitation pattern is presented to the retina and a good approximation to the required pulse shape emerges from the output layer (in the form of the Fourier coefticients). The key advantage is that although the device requires prolonged training, the operating mode works very fast. We might imagine that a young child learns to tie his shoelaces by an analogously slow process, but once it is learned, the execution becomes automatic and rapid. Evolutionary methods The rich proliferation of life forms on this planet has evolved through the powerful process of natural selection, Darwin’s “survival of the fittest.” These ideas have been successfully adapted by Rechenberg [7] to solve practical problems of engineering design. The system is described in terms of a number of physical parameters called “genes” that are incremented or decremented at each stage of the evolution. In this manner, each interim solution behaves as a “parent” and produces several “offspring”. Each new solution is compared with a predefined target and the sum of the squares of the residuals calculated. This allows the computer algorithm to choose one of these to be the parent of the next generation, and the process is repeated. The cumulative effect of many such evolutionary changes is to guide the design towards the target. There is another way to implement the selection step in the evolutionary method [8]. Instead of programming the computer to make the crucial selection, this choice is made by an NMR spectroscopist viewing the calculated excitation profiles on a monitor. This has several important advantages. A human operator does not need a precisely-defined target and there is no need to calculate a least-squares error function. He may modify his goals as the evolution proceeds. Pattern recognition comes quite naturally to the operator whereas it is difficult to write an efficient computer algorithm for this. He may exercise judgement and can readily balance several conflicting requirements.
44
Most intriguing of all, by keeping an open mind, a trained NMR spectroscopist might recognize an evolutionary trend that produces a solution to an entirely different problem. No computer algorithm could be expected to show this kind of versatiiity. Applied to the pulse design problem, the evolutionary algorithm would start with a simple pulse shape and calculate its excitation profile. Pulse shape parameters (genes) would then be incremented and one of the resulting excitation patterns chosen as the most suitable for continuing the iteration.
Calculate excitation pattern (parent)
W
f-l
Increment the Fourier coefficients
Useful solutions are usually obtained rapidly, in only a few generations, since the branching process is similar to the one that applies in a game of chess where only a few moves are required to reach an apparently unprecendented configuration of the pieces on the board. However this mode of operation of the evolutionary method requires the continued presence of the spectroscopist whereas the other techniques mentioned here run unattended.
45
Simulated annealing, neural networks and operator controlled evolution have all been used to design band-selective pulses. The most promising results to date have been achieved by careful application of the simulated annealing protocol 191. An illustrative appiication is shown in Figure 1, where the various spin multiplets in the 400 MHz proton spectrum of leuteinizing hormone releasing hormone (LHRH) have been excited one at a time using band-selective pulses [lo]. Note the high level of suppression of signals outside the excitation bandwidth and the excellent phase behavior. No phase corrections were made during this sequence.
Cc)
(4
““V
Figure 1. Band-selective excitation of the proton NMR spectrum of LHRH. 4. TWO-DIMENSIONAL
SPECTROSCOPY
Two-dimensional Fourier transform NMR is now in widespread use. It allows more complex molecules to be studied by dispersing the information into two orthogonal frequency dimensions, thus reducing the incidence of overlapping peaks and ambiguities of assignment. Nevertheless the Nyquist sampling requirement still limits the ability to examine the very fine structure on
46
NMK lines. This is because the measurements must be repeated n times to give IZsamples across the F1 dimension, necessitating a protracted experiment and a very large data table. The ability to “zoom in” on an interesting feature is facilitated by selective puises, for then a reasonably smaii number of sample points can be concentrated into a narrow frequency range. Closer examination of fine structure is particularly useful when the sample is made up of several This is related components with almost coincident two-dimensional responses. the case of a biosynthetically-generated copolymer [ 1 l] consisting of four barelydistinguishable forms, giving four overlapping patterns (Figure 2). the
I
2CUNl Hz
f
12oHz
>
Figure 2. Zoom operation to examine fine structure of a copolymer spectrum.
5. MULTI-DIMENSIONAL
SPECTROSCOPY
Recent trends have been towards higher dimensionality in high resolution NMR so as to address even more complex molecular structures. This often leads to considerable clarification, but only at the expense of a much longer experiment and enormous data tables. A typical example might be a threedimensional shift correlation experiment with 1000 data points in each frequency dimension. If the time variables tt, tz and ts have typical values of 0.2 seconds then the experiment takes one week to complete and the data table consists of one billion words. Few chemistry departments are prepared to assign one of their high resolution NMR spectrometers for an entire week on a single project. -
Band-selective excitation in the three frequency dimensions relaxes these For example if the excitation bandwidths are reduced to one requirements. third (as shown schematically in the diagram above) then the volume’ under investigation is 27 times smaller, the duration of the experiment is nine times shorter, and the data table is reduced to 37 million words. Only part of the available NMR information is recorded but we can ensure that this is the interesting part by judicious choice of the three excitation ranges. 6. RESOLUTION
ENHANCEMENT
The use of magnetic resonance for medical imaging is founded on a very simple idea -- when a macroscopic sample is excited in a field gradient then the NMR frequency is a function of that spatial coordinate. This concept can be exploited in high resolution spectroscopy to enhance resolution. Normally the resolving power is limited by the spatial inhomogeneity of the magnet. We would therefore expect that the smaller the sample the higher the resolution. In practice this improvement is seldom achieved because the liquid sample has to be confined within a suitable container and the changes in susceptibility at the container walls distorts the magnetic field at the sample, an effect that is particularly marked for very small samples.
48
However we can reduce the efictive volume of the sample by a trick. Soft pulse excitation in a field gradient selects a restricted sample volume.
Since natural field gradients along the Z axis are the most troublesome for high resolution work (because they are unaffected by sample spinning) we would normally choose to excite the sample in a Z gradient, selecting a thin horizontal disc-shaped effective volume (b) instead of the usual large effective volume (a).
7Hz-
Figure 3. Resolution enhancement by excitation in increasingly strong gradients. The applied Z gradient is then extinguished and the free induction decay Resolution is enhapced because the recorded in the natural field gradients. signal arises only from a very homogeneous region of the magnetic field. Sensitivity is of course correspondingly reduced.
49
Figure 3 shows the progressive improvements in resolution for the lowfield ring proton of furan-2-aldehyde in deuterobenzene solution [12]. Under normal high resolution conditions (top) the instrumental linewidth was approximately 0.6 Hz but decreases to approximately 0.08 Hz in the lowest trace. 7. REDUCTION
OF DIMENSIONALITY
Although two-dimensional spectra are very widely used in high resolution NMR, occasionally the requisite information can be obtained by a As a typical example we may take the simpler one-dimensional experiments. two-dimensional chemical shift correlation (COSY) experiment [13-151 which provides valuable connectivity information. In the usual two-dimensional form of the experiment, all correlations are detected. Where only certain interactions are important, it is possible to perform a small number of one-dimensional COSYkxperiments with selective-radiofrequency pulses.
a t
hm.
b t
d
e
n M
7oOHz
*
Figure 4. One-dimensional COSY spectra of a tricyclodecanone derivative r11. This is illustrated for the protons in a tricyclodecanone derivative (Figure 4a). Gaussian-shaped selective pulses were used, at frequencies indicated by the arrows. The remaining responses in traces (b) through (e) are due to coherence transfer and serve to trace out the connectivity of the spin-coupling network.
50 8.
OVERLAPPING
MULTIPLETS
As chemists examine more and more complicated molecules, even twodimensional spectra become overcrowded and beset by problems of overlap. A good example is the two-dimensional correlation spectrum of the biosynthetically-generated copolymer mentioned above, where each twodimensional multiplet is believed to be the superposition of four primitive components with slightly different chemical shifts. This may be proved by exciting each component separately and recording the individual multiplets one at a time. If we can find an exposed part of one spin multiplet pattern it may be excited by a suitable line-selective soft pulse. A subsequent hard 90” pulse spreads this coherence throughout the two-dimensional multiplet but not through the overlapping neighbours. This is basically a line-selective COSY experiment The arrows in Figure 5 indicate the frequencies used for the soft pulses.
lb
00
00
Id
1
I’ Figure 5. Separation of overlapping two-dimensional multiplets [l 11. Figure Sa shows the pattern observed when all four components overlap, while Figures 5c through 5f illustrate the primitive multiplet patterns obtained by These may then be superimposed again (Figure 5b) to selective excitation. match the pattern of Figure 5a.
51
9. DISCUSSIo1v Now that suitable instrumentation is becoming widely available, selective excitation is being extended into most areas of high resolution NMR. It may be applied to relaxation time measurements, the suppression of unwanted solvent peaks, slow chemical exchange studies, nuclear Overhauser experiments and the analogues of most two-dimensional NMR techniques. Pulse shaping enhances the effectiveness of these schemes by avoiding sidelobes in the excitation pattern, by ensuring a uniform response across the excitation band and by eliminating phase gradients. In general, simple shapes are more forgiving of instrumental shortcomings than the more complex patterns; this is probably why the Gaussian pulse has been so popular. No doubt new shaping schemes will emerge with even more valuable properties, but many of the tools for the job are aready available and we expect the field of selective excitation to grow quite rapidly. 1. C. Bauer, R. Freeman, T. Frenkiel, J. Keeler and A. J. Shaka, J. Magn. Reson. 58 (1984) 442-457. 2. T. Ngo and P. G. Morris, Biochem. Sot. Trans. 14 (1986) 1271-1272. 3. J. Murdoch, A. H. Lent, and M. Kritzer, J. Magn. Reson. 74 (1987) 226. 4. W. S. Warren and M. S. Silver, Adv. Magn. Reson. 12 (1988) 247-384. 5. N. Metropolis, A.W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. 6. F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms, Spartan Books, Washington D.C. 1961. 7. 1. Rechenberg, Evolutionsstrategie: Optimierung Technische Systeme nach Principien der Biologischen Evolution, Frommann-Holzboog, 1973. 8. R. Freeman and X. L. Wu, J. Magn. Reson. 75 (1987) 184-189. 9. H. Geen and R. Freeman, J. Magn. Reson. 93 (1991) 93-141. 10. H. Geen, S. Wimperis and R. Freeman, J. Magn. Reson. 85 (1989) 620-627. 11. P. Xu, X. L. Wu and R. Freeman, J. Magn. Reson. 89 (1990) 198-204. 12. A. Bax and R. Freeman, J. Magn. Reson. 37 (1980) 177-181 13. J. Jeener, Ampere International Summer School, Basko Polje, Yugoslavia, 1971 14. W. P. Aue, E. Bartholdi and R. R. Ernst, J. Chem. Phys. 64 (1976) 22292246. 15. A. Bax and R. Freeman, J. Magn. Reson. 44 (198 1) 542-561. Acknowledgment Figures 1 through 5 are reproduced by kind permission of’ Academic Press Inc.
39
High resolution NMR using selective excitation Ray Freeman Department of Chemistry, Cambridge University, United Kingdom
Abstract Modern high resolution NMR spectrometers employ the Fourier transform method where the entire spectrum is uniformly excited by a “hard” radiofrequency pulse. Occasionally, information is more easily extracted by an experiment where the excitation is selective in the frequency domain, using a “soft” radiofrequency pulse. We may select a single transition, an entire spin multiplet or an extended range of chemical shift frequencies for the excitation. In the early experiments, simple rectangular-shaped pulses were used, but much better results can be achieved by suitably shaping the envelope of the pulse. Several interesting schemes have been explored for designing pulse shapes, including simulated annealing, computer-simulated neural networks and evolutionary strategies that mimic the processes of natural selection. Selective pulse excitation has been used for relaxation time measurements, solvent peak suppression, resolution enhancement, separation of overlapping twodimensional multiplets, improving the definition of fine structure in twodimensional spectroscopy, and for reducing the time penalty associated with three- or four-dimensional NMR.
1. INTRODUCTION When we use a microscope, we focus our attention on a restricted part of the specimen in order to use high magnification; similarly we may use the zoom feature of a camera to examine fine detail by restricting the field of view. This article examines the analogous procedures in NMR spectroscopy when fine structure information is more important than the global picture. Present-day high resolution spectra have such a high information content that this “zoom” operation is becoming more and more attractive. Routine chemical applications of high resolution NMR excite the entire spectrum of the chosen nuclear species, for example protons or carbon-13. It is 0022-2860/92/$05.00
0 1992 Elsevier Science Publishers B.V. All rights reserved
not normaily feasible to select one small section of the spectrum and reject the rest. This restriction arises because the transient time-domain NMR signal (the “free induction decay”) must be in digital form for the purposes of Fourier transformation. This in turn imposes an all-pervading sampling requirement known as the Nyquist criterion which states that unless there are at least two samples per cycle of the highest frequency component of the free induction decay, there will be ambiguities in the frequencies. Any high-frequency signal is said to be “aliased” and appears not at its characteristic frequency&, but rather at _f, - nfs, where& is the sampling rate and n is an integer. The effect is analogous to the stroboscopic effect, well-known from Western movies, where the wagon wheels appear to rotate at the wrong speed or even go backwards. For serious chemical applications, aliasing of NMR signals would be a disastrous falsification of the data and should be avoided at almost any cost. The aliasing problem can be circumvented by exciting partial NMR spectra. All the responses are confined to a narrow predefined frequency band, leaving the rest of the spectrum completely unperturbed. In this way, only the low-frequency NMR signals need be considered and aliasing is avoided. The radiofrequency pulse is said to be “soft,” whereas a normal pulse is “hard” and affects the entire spectrum. Recent technical innovations make it feasible to generate soft pulses suitable for this “band-selective” excitation.
2. SOFT PULSES At first sight the problem seems quite simple -- by reducing the intensity of a radiofrequency pulse (with a compensating increase in its length) we can make it frequency-selective or soft. If the pulse intensity is Bi tesla, a range of frequencies of the order of yBi/2rr Hz will be excited. Unfortunately the form of the excitation pattern in the frequency domain is far from ideal. There is a central lobe of strong excitation flanked by a set of sidelobes whose amplitude decreases with offset from the centre. These sidelobes originate from the fact that the pulse has sharp leading and trailing edges. The obvious solution is to shape the pulse to smooth out these discontinuities, and considerable progress has been made with pulse envelopes that are Gaussian [l] rather than rectangular. Once the sidelobes have been removed another problem comes to light. Most soft pulses introduce phase gradients, an increasing admixture of dispersion mode signal with increasing offset. Good resolution requires pure absorptionmode signals, so this frequency-dependent phase error has to be corrected. Furthermore, if the spectrum is to be used for quantitative measurements we would like the excitation to be uniform over the desired band of frequencies. Finally, to minimize the intensities of aliased signals, the excitation outside the
41
selected band should be negligible. Of course, we cannot make an instantaneous transition from full to negligible excitation, so we must allow a narrow transition region between the two. The ideal excitation pattern may be sketched schem&icd~~y:
) I I Zero 1
Uniformexcitation
1 vtransition
region
\ \ Zero
absorption
Idealized frequency-domain profile If we think of this behavior in terms of magnetization trajectories it is clear that the pulse must have a focusing action, as in the spin echo experiment, so that all trajectories within the excitation band terminate on the Y axis of the rotating frame. 3. PULSE DESIGN The criteria outlined above are by no means easy to achieve. There is no simple direct way to calculate the shape of the pulse envelope that would generate a particular excitation pattern, and we therefore have recourse to iterative numerical methods. Most of the early work on pulse shaping [2-41 was carried out for the purposes of magnetic resonance imaging where the principal application is “slice selection” by excitation of the sample in an applied static field gradient; high resolution spectroscopists were slower to realize the importance of “designer pulses.” The conventional procedure is to employ a multiparameter optimization routine, but there are some interesting variations, three of which are described below. Simulatedannealing Simulated annealing [5] is a well-established routine that was designed specifically to avoid the “false” solutions that are often found in multiparameter optimization. The differences between the actual excitation pattern and the target pattern define a multidimensional error surface and there may be several minima on this surface. Most of the time the search program behaves like conventional “gradient descent” methods but to avoid becoming trapped in a false minimum the simulated annealing algorithm allows an occasional “uphill” -jump with a
42
probability given by the equivalent of the Boltzmann factor, exp(-AE/kT), where AE is the change in the error function and T is analogous to temperature. At a high enough temperature, the program can escape from any false (“local”) minima and eventually reach the “true” global minimum, which we assume represents the optimum design. The key factor is the annealing schedule, an appropriately slow stepwise decrease in the temperature T. Simulated annealing has been used to design some very effective band-selective soft pulses for highresolution NMR. Neural networks Most optimization programs are time-consuming because of their iterative nature and the asymptotic approach to the solution. One exception is the method of neural networks [6]. This is a simulation (on a digital computer) of the processes by which the brain is believed to operate. A complex network of interconnected neurons is set up, arranged in layers, with an input layer (the retina) at one boundary and an output layer at the other. Each neuron receives voltage impulses vj from many other neurons “upstream” and calculates the weighted sum S=O+
CWjVj
J
where 8 is a bias voltage and wj am “synaptic weights.” The signal S is amplified according to a specific response function (usually a sigmoid curve) and then The common output voltage thus passed to further neurons “downstream”. depends on the signals received by the neuron, and the values of 0 and wj.
0 = bias
w = synaptic weight
When used for pulse design, the network is first operated in a training mode. Frequency-domain excitation patterns are calculated for a set of randomly-generated pulse shapes defined in terms of the coefficients of a finite Fourier series. When these excitation patterns are presented to the retina, the output layer at first produces quite unrelated signals, but the program modifies the bias voltages and synaptic weights to bring these output signals closer to the known Fourier coefficients of the appropriate pulse shape. As training proceeds this correspondence improves. After a protracted training program in which a suitably representative range of pulse shapes has been studied, the network is ready for the operating mode. It incorporates its acquired knowledge in the form of the synaptic weights wj and the bias voltages 8. In the operating mode the desired excitation pattern is presented to the retina and a good approximation to the required pulse shape emerges from the output layer (in the form of the Fourier coefticients). The key advantage is that although the device requires prolonged training, the operating mode works very fast. We might imagine that a young child learns to tie his shoelaces by an analogously slow process, but once it is learned, the execution becomes automatic and rapid. Evolutionary methods The rich proliferation of life forms on this planet has evolved through the powerful process of natural selection, Darwin’s “survival of the fittest.” These ideas have been successfully adapted by Rechenberg [7] to solve practical problems of engineering design. The system is described in terms of a number of physical parameters called “genes” that are incremented or decremented at each stage of the evolution. In this manner, each interim solution behaves as a “parent” and produces several “offspring”. Each new solution is compared with a predefined target and the sum of the squares of the residuals calculated. This allows the computer algorithm to choose one of these to be the parent of the next generation, and the process is repeated. The cumulative effect of many such evolutionary changes is to guide the design towards the target. There is another way to implement the selection step in the evolutionary method [8]. Instead of programming the computer to make the crucial selection, this choice is made by an NMR spectroscopist viewing the calculated excitation profiles on a monitor. This has several important advantages. A human operator does not need a precisely-defined target and there is no need to calculate a least-squares error function. He may modify his goals as the evolution proceeds. Pattern recognition comes quite naturally to the operator whereas it is difficult to write an efficient computer algorithm for this. He may exercise judgement and can readily balance several conflicting requirements.
44
Most intriguing of all, by keeping an open mind, a trained NMR spectroscopist might recognize an evolutionary trend that produces a solution to an entirely different problem. No computer algorithm could be expected to show this kind of versatiiity. Applied to the pulse design problem, the evolutionary algorithm would start with a simple pulse shape and calculate its excitation profile. Pulse shape parameters (genes) would then be incremented and one of the resulting excitation patterns chosen as the most suitable for continuing the iteration.
Calculate excitation pattern (parent)
W
f-l
Increment the Fourier coefficients
Useful solutions are usually obtained rapidly, in only a few generations, since the branching process is similar to the one that applies in a game of chess where only a few moves are required to reach an apparently unprecendented configuration of the pieces on the board. However this mode of operation of the evolutionary method requires the continued presence of the spectroscopist whereas the other techniques mentioned here run unattended.
45
Simulated annealing, neural networks and operator controlled evolution have all been used to design band-selective pulses. The most promising results to date have been achieved by careful application of the simulated annealing protocol 191. An illustrative appiication is shown in Figure 1, where the various spin multiplets in the 400 MHz proton spectrum of leuteinizing hormone releasing hormone (LHRH) have been excited one at a time using band-selective pulses [lo]. Note the high level of suppression of signals outside the excitation bandwidth and the excellent phase behavior. No phase corrections were made during this sequence.
Cc)
(4
““V
Figure 1. Band-selective excitation of the proton NMR spectrum of LHRH. 4. TWO-DIMENSIONAL
SPECTROSCOPY
Two-dimensional Fourier transform NMR is now in widespread use. It allows more complex molecules to be studied by dispersing the information into two orthogonal frequency dimensions, thus reducing the incidence of overlapping peaks and ambiguities of assignment. Nevertheless the Nyquist sampling requirement still limits the ability to examine the very fine structure on
46
NMK lines. This is because the measurements must be repeated n times to give IZsamples across the F1 dimension, necessitating a protracted experiment and a very large data table. The ability to “zoom in” on an interesting feature is facilitated by selective puises, for then a reasonably smaii number of sample points can be concentrated into a narrow frequency range. Closer examination of fine structure is particularly useful when the sample is made up of several This is related components with almost coincident two-dimensional responses. the case of a biosynthetically-generated copolymer [ 1 l] consisting of four barelydistinguishable forms, giving four overlapping patterns (Figure 2). the
I
2CUNl Hz
f
12oHz
>
Figure 2. Zoom operation to examine fine structure of a copolymer spectrum.
5. MULTI-DIMENSIONAL
SPECTROSCOPY
Recent trends have been towards higher dimensionality in high resolution NMR so as to address even more complex molecular structures. This often leads to considerable clarification, but only at the expense of a much longer experiment and enormous data tables. A typical example might be a threedimensional shift correlation experiment with 1000 data points in each frequency dimension. If the time variables tt, tz and ts have typical values of 0.2 seconds then the experiment takes one week to complete and the data table consists of one billion words. Few chemistry departments are prepared to assign one of their high resolution NMR spectrometers for an entire week on a single project. -
Band-selective excitation in the three frequency dimensions relaxes these For example if the excitation bandwidths are reduced to one requirements. third (as shown schematically in the diagram above) then the volume’ under investigation is 27 times smaller, the duration of the experiment is nine times shorter, and the data table is reduced to 37 million words. Only part of the available NMR information is recorded but we can ensure that this is the interesting part by judicious choice of the three excitation ranges. 6. RESOLUTION
ENHANCEMENT
The use of magnetic resonance for medical imaging is founded on a very simple idea -- when a macroscopic sample is excited in a field gradient then the NMR frequency is a function of that spatial coordinate. This concept can be exploited in high resolution spectroscopy to enhance resolution. Normally the resolving power is limited by the spatial inhomogeneity of the magnet. We would therefore expect that the smaller the sample the higher the resolution. In practice this improvement is seldom achieved because the liquid sample has to be confined within a suitable container and the changes in susceptibility at the container walls distorts the magnetic field at the sample, an effect that is particularly marked for very small samples.
48
However we can reduce the efictive volume of the sample by a trick. Soft pulse excitation in a field gradient selects a restricted sample volume.
Since natural field gradients along the Z axis are the most troublesome for high resolution work (because they are unaffected by sample spinning) we would normally choose to excite the sample in a Z gradient, selecting a thin horizontal disc-shaped effective volume (b) instead of the usual large effective volume (a).
7Hz-
Figure 3. Resolution enhancement by excitation in increasingly strong gradients. The applied Z gradient is then extinguished and the free induction decay Resolution is enhapced because the recorded in the natural field gradients. signal arises only from a very homogeneous region of the magnetic field. Sensitivity is of course correspondingly reduced.
49
Figure 3 shows the progressive improvements in resolution for the lowfield ring proton of furan-2-aldehyde in deuterobenzene solution [12]. Under normal high resolution conditions (top) the instrumental linewidth was approximately 0.6 Hz but decreases to approximately 0.08 Hz in the lowest trace. 7. REDUCTION
OF DIMENSIONALITY
Although two-dimensional spectra are very widely used in high resolution NMR, occasionally the requisite information can be obtained by a As a typical example we may take the simpler one-dimensional experiments. two-dimensional chemical shift correlation (COSY) experiment [13-151 which provides valuable connectivity information. In the usual two-dimensional form of the experiment, all correlations are detected. Where only certain interactions are important, it is possible to perform a small number of one-dimensional COSYkxperiments with selective-radiofrequency pulses.
a t
hm.
b t
d
e
n M
7oOHz
*
Figure 4. One-dimensional COSY spectra of a tricyclodecanone derivative r11. This is illustrated for the protons in a tricyclodecanone derivative (Figure 4a). Gaussian-shaped selective pulses were used, at frequencies indicated by the arrows. The remaining responses in traces (b) through (e) are due to coherence transfer and serve to trace out the connectivity of the spin-coupling network.
50 8.
OVERLAPPING
MULTIPLETS
As chemists examine more and more complicated molecules, even twodimensional spectra become overcrowded and beset by problems of overlap. A good example is the two-dimensional correlation spectrum of the biosynthetically-generated copolymer mentioned above, where each twodimensional multiplet is believed to be the superposition of four primitive components with slightly different chemical shifts. This may be proved by exciting each component separately and recording the individual multiplets one at a time. If we can find an exposed part of one spin multiplet pattern it may be excited by a suitable line-selective soft pulse. A subsequent hard 90” pulse spreads this coherence throughout the two-dimensional multiplet but not through the overlapping neighbours. This is basically a line-selective COSY experiment The arrows in Figure 5 indicate the frequencies used for the soft pulses.
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00
Id
1
I’ Figure 5. Separation of overlapping two-dimensional multiplets [l 11. Figure Sa shows the pattern observed when all four components overlap, while Figures 5c through 5f illustrate the primitive multiplet patterns obtained by These may then be superimposed again (Figure 5b) to selective excitation. match the pattern of Figure 5a.
51
9. DISCUSSIo1v Now that suitable instrumentation is becoming widely available, selective excitation is being extended into most areas of high resolution NMR. It may be applied to relaxation time measurements, the suppression of unwanted solvent peaks, slow chemical exchange studies, nuclear Overhauser experiments and the analogues of most two-dimensional NMR techniques. Pulse shaping enhances the effectiveness of these schemes by avoiding sidelobes in the excitation pattern, by ensuring a uniform response across the excitation band and by eliminating phase gradients. In general, simple shapes are more forgiving of instrumental shortcomings than the more complex patterns; this is probably why the Gaussian pulse has been so popular. No doubt new shaping schemes will emerge with even more valuable properties, but many of the tools for the job are aready available and we expect the field of selective excitation to grow quite rapidly. 1. C. Bauer, R. Freeman, T. Frenkiel, J. Keeler and A. J. Shaka, J. Magn. Reson. 58 (1984) 442-457. 2. T. Ngo and P. G. Morris, Biochem. Sot. Trans. 14 (1986) 1271-1272. 3. J. Murdoch, A. H. Lent, and M. Kritzer, J. Magn. Reson. 74 (1987) 226. 4. W. S. Warren and M. S. Silver, Adv. Magn. Reson. 12 (1988) 247-384. 5. N. Metropolis, A.W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. 6. F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms, Spartan Books, Washington D.C. 1961. 7. 1. Rechenberg, Evolutionsstrategie: Optimierung Technische Systeme nach Principien der Biologischen Evolution, Frommann-Holzboog, 1973. 8. R. Freeman and X. L. Wu, J. Magn. Reson. 75 (1987) 184-189. 9. H. Geen and R. Freeman, J. Magn. Reson. 93 (1991) 93-141. 10. H. Geen, S. Wimperis and R. Freeman, J. Magn. Reson. 85 (1989) 620-627. 11. P. Xu, X. L. Wu and R. Freeman, J. Magn. Reson. 89 (1990) 198-204. 12. A. Bax and R. Freeman, J. Magn. Reson. 37 (1980) 177-181 13. J. Jeener, Ampere International Summer School, Basko Polje, Yugoslavia, 1971 14. W. P. Aue, E. Bartholdi and R. R. Ernst, J. Chem. Phys. 64 (1976) 22292246. 15. A. Bax and R. Freeman, J. Magn. Reson. 44 (198 1) 542-561. Acknowledgment Figures 1 through 5 are reproduced by kind permission of’ Academic Press Inc.