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“BOWMAN”: A versatile simulation program for high-resolution NMR multipulse experiments
JOURNAL
OF MAGNETIC
RESONANCE
70,290~294 (1986)
NOTES “BOWMAN”:
A Versatile Simulation Program for High-Resolution NMR Multipulse Experiments
DENIS PIVETEAU, MARC ANDRE DELSUC, AND JEAN-YVES
LALLEMAND
Laboratoire de RtfsonanceMagnktique Nucliaire, Imitut de Chimie des SubstancesNaturelles, CNRS, 91190 Gif-sur- Yvette, France Received December 3, 1985; revised June 26, 1986
In the NMR laboratory, the need for a simulation program is very often encountered (I, 2), especially for the study of complex phenomena which are tedious to compute algebraically, such as relaxation effects,m o d ifications and artifacts due to unusual flip angles, or random delay obeying a given distribution law. Such a program may also be useful for the optimization of parameters in routine 1D or 2D experiments, for the choice of convenient schemesof composite pulses, and, last but not least, for pedagogical purposes. This prompted us to develop a new NMR simulation program fitting these requirements which we called BOWMAN (bidimensional simulator operating on weakly coupled systemsfor m u ltipulse sequenceson any NMR software) (3, 4). The version we developed features the following: (i) simulation of the evolution of a 2- or 3-spin-j system, systematically supposedto be in the weak coupling approximation, (ii) numeric simulation of the system through step-by-stepintegration of the differential equation of evolution, (iii) full implementation of relaxation processes (NOE effects: seeF ig. 2), (iv) capacity to take into account the imperfections of pulses (flip angle, off-resonanceeffects:seeF ig. I), (v) no sequencelim itation (except the fact that a four-phase-axis receiver is assumed), (vi) ease of use for those who are not computer specialists, (vii) drawing of the observable magnetization evolution in the rotating frame during the experiment: see F ig. 1B. In the NMR experiment, the spin system can be described by the density operator p, the evolution of which follows the Liouville equation: iW/dt
= lx,
PI + &P - PO)
111
where Z is the Hamiltonian of the system, and R the Redfield matrix which includes components of the evolution such as relaxation, and renders the differential equation not directly integrable. W e have chosen to represent this density matrix operator in the basis built from the direct product of Pauli matrices (product operators (5-7)). For an n-spin-f system, this basis is a 4”-term family including the identity matrix, the latter not being taken into account since its coefficient remains constant in the classical experiments. W ith this representation,the solution of the Liouville equation for elementaryevents 0022-2364186 $3.00 Copyright Q 1986 by Academic FVes, Inc. All rights of reproduction in any form reserved.
290
NOTES
291
(strong or selective pulses, evolution) has a simple expression, permitting some gain in computation time as compared to certain other possible representations. The experiment to be simulated is broken down into elementary events, and the evolution of the density matrix operator is computed stepwisethroughout the sequence. Each step of the computation is driven either exactly or in the first-order approximation, depending on the nature of the elementary event. The program thus constructs one or more “pseudo-FIDs,” corresponding to an acquisition either in the simple detection mode or in the quadrature detection mode (8). The sequenceis described through a sequenceinterpreter which can represent nearly all kinds of experiments, notably through the use of nested loops. For instance, composite pulses, composite decoupling, 2D experiments with all phase programs of any 1e:ngthcan be directly carried out. Furthermore this interpreter allows starting a new simulation without any change of the code, and thus avoids a new compilation process.
A
FIG. 1. Simulated MLEV-4 experiment, demonstrating off-resonance effects on broadband decoupling (Fig. 1A). The lower diagram (Fig. 1B) shows the trajectory of proton magnetization for a 3 kHz o&et of the composite inversion pulse and a B, strength of frequency equivalence 5 kHz.
292
NOTES
The spin system itself is described by the frequencies, spin-spin couplings and initial intensities for each spin, along with the relaxation parameters. When the drawing option is called upon, the magnetization components for each elementary event are stored and consecutively drawn, according to the Euler projection angles previously chosen. Some examples of simulated spectra featuring the BOWMAN program’s possibilities are shown in Figs. 1 to 3. Figure 1 shows the simulation of a MLEV-4 broadband decoupling experiment (9) on a 13C-lH two-spin system with various decoupling field offsets. The simulated sequence is a 10 s MLEV-4 presaturation delay on the ‘H spin (to produce the NOE enhancement), followed by an excitation pulse and detection of 13Cwith MLEV4 decoupling. The simulation is performed at 400 MHz for the proton, with an interspin distance of 1.1 A and a correlation time of 0.25 ns (extreme narrowing case). Parameters ofthe system are the following: T,(‘H) = T#H) = 4 s, TI(13C) = T2(13C)= 6 s (omitting the dipolar contribution which is calculated by the program itself), and J value of I25 Hz. Figure 1A shows spectra of the r3C spin for various offsets of the carrier of the decoupling field B, from the frequency of the ‘H spin, with a B, field strength of 5 kHz. The asymmetry of peaks for large offsets is a consequence of imperfect MLEV presaturation due to off-resonance effects. The maximum NOE enhancement found for an offset of zero is 1.64, which differs from the theoretical maximum value of y( ‘H)/2r(r3C) corresponding to a system in which the dipolar interaction would act as the only relaxation mechanism (this “pure” dipolarrelaxing system could be simulated with the initial values T, = T, = co). Figure IB displays the trajectory of the ‘H magnetization (darker trace) during one elementary MLEV inversion (90”-240”-90” cluster) for an offset of 3 kHz and B1 field strength of 5 kHz. Figure 2 shows the simulation of two “phase-sensitive” 2D-NOESY experiments. The left part of Fig. 2 shows the 2D-NOESY spectrum in which the distorted shape of the peaks can be attributed to the superposition of undesired zero-quantum cor-
/ -12
-6
F2tHz)
6
12
FIG. 2. Phase-sensitive ZD-NOESY simulated experiments, performed on a proton spin system, in the spin-diffusion limit. (Left) Without elimination of zeroquantum coherences. (Right) With elimination of zeroquantum coherence%by a variable mixing time.
293
NOTES
re:lations. The right part of Fig. 2 shows the same spectrum with zero-quantum coherences suppressed by varying the m ixing time about the initial value (10). The following parameters are used: Tr( 1st spin) = Tz( 1st spin) = 4 s, Tr(2nd spin) = T2(2nd spin) = 3 s (without dipolar contribution), spin frequencies of 400 MHz, interspin distance of 2.5 A and correlation time of 25 ns (spin diffusion case). The NOE m ixing time is 300 ms in the first experiment, and is varied around the same value in the second. Data matrices of 128 X 256 pts. with 8 scans for each experiment were simulated. Figure 3 displays the simulation of a “phase-sensitive” COSY-45 spectrum with the three-spin program. Note that the three spin-spin coupling constants are not chosen with the same sign; note also that the weak coupling approximation is made, irrespective of the actual 6/J ratio. The following parameters are used: WA= -46.2 Hz, W M= 0.3 HZ, wx = 34.8 HZ, JAM = 8.0 HZ, JMX = - 11.6 HZ, JAx = 3.5 HZ, and for all spins jr = 10 s and T2 = 8 s. A data matrix of 128 X 256 points with one level of zero fkng in each dimension was simulated, with two scans for each experiment and a relaxation delay of one hour between each scan. The BOWMAN program is written in PASCAL. Data of the simulated experiment are generated on ties containing the FIDs so they may be processed by any standard NMR program. The set of procedures used for the drawing of the magnetization is written in a portable form, the version actually used being written for a Watanabe plotter. The two-spin version of the program will be provided along with a compre-
WA
F2 FIG. 3. COSY-45 spectrum simulated for a three-spin system.
294
NOTES
hensive manual, on a standard 1600 bpi magnetic tape, upon request to the authors, together with a 600 foot magnetic tape sent to the address given above. Distribution for the Bruker Aspect system is also available through the Bruker user group ABACUS (I I). ACKNOWLEDGMENTS We thank Dr. A. D. Bain for stimulating discussions, and Dr. C. Her& du Penhoat for her careful reading of the manuscript. REFERENCES 1. B. K. JOHN AND R. E. D. MCCLUNG, J. Magn. Reson. 58,47 (1984). 2. A. D. BAIN, “SIMPLTN,” 7th International RCS meeting on NMR Spectroscopy, Cambridge, July 1985. 3. M. A. Dtxsuc, D. PNETEAU, AND J. Y. LALLEMAND, “BOWMAN,” 7th International RCS meeting on NMR Spectroscopy, Cambridge, July 1985. 4. D. PIVETEAU,ThZse de Dootorat d’Etat, Universiti Paris-Sud, Centre d’orsay, June 1985. 5. F. M. J. VAN DE VEN AND C. W. HILBERS,J. Magn. Reson. 54,512 (1983). 6. 0. W. WRENSEN, G. W. EICH, M. H. LEV~T, G. BODENHAUSEN,AND R. R. ERNST, Frog. Magn. Reson. 16, 163 (1983). 7. M. A. DELSUC,Th&e de Dootorat d’Etat, Universitk Paris-Sud, Centre d’Orsay, January 1985. 8. A. G. REDFIELD AND S. D. KUNZ, J. Magn. Reson. 19,250 (1975). 9. M. H. LEVITT AND R. FREEMAN,J. Magn. Reson. 43,502 (1981). 10. S. MACURA, Y. HUANG, D. SUTER,AND R. R. ERNST,J. Magn. Reson. 43,259 (1981). II. “ABACUS,” Bmker Spectrospin A.G., Industriestrasse 26, CH-8117, Pallanden, Switzerland.
OF MAGNETIC
RESONANCE
70,290~294 (1986)
NOTES “BOWMAN”:
A Versatile Simulation Program for High-Resolution NMR Multipulse Experiments
DENIS PIVETEAU, MARC ANDRE DELSUC, AND JEAN-YVES
LALLEMAND
Laboratoire de RtfsonanceMagnktique Nucliaire, Imitut de Chimie des SubstancesNaturelles, CNRS, 91190 Gif-sur- Yvette, France Received December 3, 1985; revised June 26, 1986
In the NMR laboratory, the need for a simulation program is very often encountered (I, 2), especially for the study of complex phenomena which are tedious to compute algebraically, such as relaxation effects,m o d ifications and artifacts due to unusual flip angles, or random delay obeying a given distribution law. Such a program may also be useful for the optimization of parameters in routine 1D or 2D experiments, for the choice of convenient schemesof composite pulses, and, last but not least, for pedagogical purposes. This prompted us to develop a new NMR simulation program fitting these requirements which we called BOWMAN (bidimensional simulator operating on weakly coupled systemsfor m u ltipulse sequenceson any NMR software) (3, 4). The version we developed features the following: (i) simulation of the evolution of a 2- or 3-spin-j system, systematically supposedto be in the weak coupling approximation, (ii) numeric simulation of the system through step-by-stepintegration of the differential equation of evolution, (iii) full implementation of relaxation processes (NOE effects: seeF ig. 2), (iv) capacity to take into account the imperfections of pulses (flip angle, off-resonanceeffects:seeF ig. I), (v) no sequencelim itation (except the fact that a four-phase-axis receiver is assumed), (vi) ease of use for those who are not computer specialists, (vii) drawing of the observable magnetization evolution in the rotating frame during the experiment: see F ig. 1B. In the NMR experiment, the spin system can be described by the density operator p, the evolution of which follows the Liouville equation: iW/dt
= lx,
PI + &P - PO)
111
where Z is the Hamiltonian of the system, and R the Redfield matrix which includes components of the evolution such as relaxation, and renders the differential equation not directly integrable. W e have chosen to represent this density matrix operator in the basis built from the direct product of Pauli matrices (product operators (5-7)). For an n-spin-f system, this basis is a 4”-term family including the identity matrix, the latter not being taken into account since its coefficient remains constant in the classical experiments. W ith this representation,the solution of the Liouville equation for elementaryevents 0022-2364186 $3.00 Copyright Q 1986 by Academic FVes, Inc. All rights of reproduction in any form reserved.
290
NOTES
291
(strong or selective pulses, evolution) has a simple expression, permitting some gain in computation time as compared to certain other possible representations. The experiment to be simulated is broken down into elementary events, and the evolution of the density matrix operator is computed stepwisethroughout the sequence. Each step of the computation is driven either exactly or in the first-order approximation, depending on the nature of the elementary event. The program thus constructs one or more “pseudo-FIDs,” corresponding to an acquisition either in the simple detection mode or in the quadrature detection mode (8). The sequenceis described through a sequenceinterpreter which can represent nearly all kinds of experiments, notably through the use of nested loops. For instance, composite pulses, composite decoupling, 2D experiments with all phase programs of any 1e:ngthcan be directly carried out. Furthermore this interpreter allows starting a new simulation without any change of the code, and thus avoids a new compilation process.
A
FIG. 1. Simulated MLEV-4 experiment, demonstrating off-resonance effects on broadband decoupling (Fig. 1A). The lower diagram (Fig. 1B) shows the trajectory of proton magnetization for a 3 kHz o&et of the composite inversion pulse and a B, strength of frequency equivalence 5 kHz.
292
NOTES
The spin system itself is described by the frequencies, spin-spin couplings and initial intensities for each spin, along with the relaxation parameters. When the drawing option is called upon, the magnetization components for each elementary event are stored and consecutively drawn, according to the Euler projection angles previously chosen. Some examples of simulated spectra featuring the BOWMAN program’s possibilities are shown in Figs. 1 to 3. Figure 1 shows the simulation of a MLEV-4 broadband decoupling experiment (9) on a 13C-lH two-spin system with various decoupling field offsets. The simulated sequence is a 10 s MLEV-4 presaturation delay on the ‘H spin (to produce the NOE enhancement), followed by an excitation pulse and detection of 13Cwith MLEV4 decoupling. The simulation is performed at 400 MHz for the proton, with an interspin distance of 1.1 A and a correlation time of 0.25 ns (extreme narrowing case). Parameters ofthe system are the following: T,(‘H) = T#H) = 4 s, TI(13C) = T2(13C)= 6 s (omitting the dipolar contribution which is calculated by the program itself), and J value of I25 Hz. Figure 1A shows spectra of the r3C spin for various offsets of the carrier of the decoupling field B, from the frequency of the ‘H spin, with a B, field strength of 5 kHz. The asymmetry of peaks for large offsets is a consequence of imperfect MLEV presaturation due to off-resonance effects. The maximum NOE enhancement found for an offset of zero is 1.64, which differs from the theoretical maximum value of y( ‘H)/2r(r3C) corresponding to a system in which the dipolar interaction would act as the only relaxation mechanism (this “pure” dipolarrelaxing system could be simulated with the initial values T, = T, = co). Figure IB displays the trajectory of the ‘H magnetization (darker trace) during one elementary MLEV inversion (90”-240”-90” cluster) for an offset of 3 kHz and B1 field strength of 5 kHz. Figure 2 shows the simulation of two “phase-sensitive” 2D-NOESY experiments. The left part of Fig. 2 shows the 2D-NOESY spectrum in which the distorted shape of the peaks can be attributed to the superposition of undesired zero-quantum cor-
/ -12
-6
F2tHz)
6
12
FIG. 2. Phase-sensitive ZD-NOESY simulated experiments, performed on a proton spin system, in the spin-diffusion limit. (Left) Without elimination of zeroquantum coherences. (Right) With elimination of zeroquantum coherence%by a variable mixing time.
293
NOTES
re:lations. The right part of Fig. 2 shows the same spectrum with zero-quantum coherences suppressed by varying the m ixing time about the initial value (10). The following parameters are used: Tr( 1st spin) = Tz( 1st spin) = 4 s, Tr(2nd spin) = T2(2nd spin) = 3 s (without dipolar contribution), spin frequencies of 400 MHz, interspin distance of 2.5 A and correlation time of 25 ns (spin diffusion case). The NOE m ixing time is 300 ms in the first experiment, and is varied around the same value in the second. Data matrices of 128 X 256 pts. with 8 scans for each experiment were simulated. Figure 3 displays the simulation of a “phase-sensitive” COSY-45 spectrum with the three-spin program. Note that the three spin-spin coupling constants are not chosen with the same sign; note also that the weak coupling approximation is made, irrespective of the actual 6/J ratio. The following parameters are used: WA= -46.2 Hz, W M= 0.3 HZ, wx = 34.8 HZ, JAM = 8.0 HZ, JMX = - 11.6 HZ, JAx = 3.5 HZ, and for all spins jr = 10 s and T2 = 8 s. A data matrix of 128 X 256 points with one level of zero fkng in each dimension was simulated, with two scans for each experiment and a relaxation delay of one hour between each scan. The BOWMAN program is written in PASCAL. Data of the simulated experiment are generated on ties containing the FIDs so they may be processed by any standard NMR program. The set of procedures used for the drawing of the magnetization is written in a portable form, the version actually used being written for a Watanabe plotter. The two-spin version of the program will be provided along with a compre-
WA
F2 FIG. 3. COSY-45 spectrum simulated for a three-spin system.
294
NOTES
hensive manual, on a standard 1600 bpi magnetic tape, upon request to the authors, together with a 600 foot magnetic tape sent to the address given above. Distribution for the Bruker Aspect system is also available through the Bruker user group ABACUS (I I). ACKNOWLEDGMENTS We thank Dr. A. D. Bain for stimulating discussions, and Dr. C. Her& du Penhoat for her careful reading of the manuscript. REFERENCES 1. B. K. JOHN AND R. E. D. MCCLUNG, J. Magn. Reson. 58,47 (1984). 2. A. D. BAIN, “SIMPLTN,” 7th International RCS meeting on NMR Spectroscopy, Cambridge, July 1985. 3. M. A. Dtxsuc, D. PNETEAU, AND J. Y. LALLEMAND, “BOWMAN,” 7th International RCS meeting on NMR Spectroscopy, Cambridge, July 1985. 4. D. PIVETEAU,ThZse de Dootorat d’Etat, Universiti Paris-Sud, Centre d’orsay, June 1985. 5. F. M. J. VAN DE VEN AND C. W. HILBERS,J. Magn. Reson. 54,512 (1983). 6. 0. W. WRENSEN, G. W. EICH, M. H. LEV~T, G. BODENHAUSEN,AND R. R. ERNST, Frog. Magn. Reson. 16, 163 (1983). 7. M. A. DELSUC,Th&e de Dootorat d’Etat, Universitk Paris-Sud, Centre d’Orsay, January 1985. 8. A. G. REDFIELD AND S. D. KUNZ, J. Magn. Reson. 19,250 (1975). 9. M. H. LEVITT AND R. FREEMAN,J. Magn. Reson. 43,502 (1981). 10. S. MACURA, Y. HUANG, D. SUTER,AND R. R. ERNST,J. Magn. Reson. 43,259 (1981). II. “ABACUS,” Bmker Spectrospin A.G., Industriestrasse 26, CH-8117, Pallanden, Switzerland.