Exploring the limits of broadband 90° and 180° universal rotation pulses
Graphical abstract
Highlights
► A systematic study of performances of universal 90° and 180° pulses is presented. ► Pulses were studied with respect to bandwidth, pulse length, tolerance to B1-inhomogeneity. ► The majority of time-optimal pulses belongs to one of two specific symmetry classes. ► BURBOP pulses found are shown to perform better than previously published pulses. ► UR 180° pulses perform better than two consecutive PP inversion pulses in INEPT steps.
Introduction
The physical limits of pulse performance are of fundamental and practical importance for NMR spectroscopy. A thorough characterization of these limits allows one to choose the best pulse for a given application.
Currently, hard 90° and 180° pulses are the workhorses of modern NMR spectroscopy. They are commonly used in the majority of pulse sequences, but the development of spectrometers with magnetic field strengths on the order of 1 GHz and the large chemical shift ranges of nuclei like 13C, 19F, or 31P pose serious challenges to their utility. A simple hard pulse provides effective excitation or inversion only over a limited range of resonance offsets that cannot be increased significantly due to pulse power constraints.
For the many applications requiring uniform excitation or inversion over resonance offsets that are greater than the maximum available field, computer-optimized pulse shapes can provide an effective broadband solution. They can also be designed with a defined tolerance to field inhomogeneity.
Such pulses typically belong to the class of point-to-point (PP) pulses (also called class B2 pulses [1]) designed to transfer a single initial magnetization component, for example , to a specific target component, e.g. for excitation and for inversion. Whenever the transfer of a single magnetization component is desirable in a pulse sequence, broadband PP pulses will be the adequate replacement for hard pulses.
In many situations, however, more than one magnetization component must be transformed via a universal rotation (UR) capable of rotating any orientation of the initial magnetization about the same axis.
UR pulses (also denoted as class A pulses [1], constant rotation pulses [2], general rotation pulses, plane rotation pulses, or simply universal pulses) are characterized by a well-defined rotation axis and rotation angle. Initial vectors pointing along the x, y and z direction are rotated by UR pulses to well-defined new orientations, e.g. along the -z, y and x direction, respectively, for a UR () pulse (c.f. Fig. 1a). In contrast, PP pulses are only required to rotate a single initial vector (e.g. pointing along the z axis) to a desired final orientation, e.g. along the x axis for a PP() pulse (c.f. Fig. 1b). In this example, initial x and y magnetization vectors are allowed to end up anywhere in the y–z plane. Each PP pulse can be realized by an infinite set of different combinations of rotation axes and rotation angles [3]. For example, a PP() pulse can be realized using arbitrary rotation angles between 90° (with rotation axis (0, 1, 0), i.e. the y axis) and 180° (with rotation axis , corresponding to the bisecting line of the angle between the x and z axis). As UR pulses transform all three axes in a controlled way while only one axis is controlled in PP pulses, UR pulses can always replace corresponding PP pulses in a pulse sequence but not vice versa.
We have previously characterized the performance limits of broadband PP excitation and inversion pulses [4], [5]. In addition, direct UR optimizations have provided UR pulse shapes with exceptional performance for specific problems [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Pulse sequences in need of 90° UR pulses include the mixing step in COSY [16], ADEQUATE [17], and INADEQUATE-type [18], [19], [20] sequences, or the sensitivity enhancement [21], [22], [23] in HSQC-type experiments. The crucial refocusing of chemical shift evolution requires a 180° UR pulse. Ideally, this refocusing pulse should provide the same effective universal rotation for a given bandwidth as an on-resonant hard 180° pulse.
In this article, we extend our work on optimal pulse performance and consider the physical bounds of broadband 90° and 180° UR pulses for the first time. A major optimization effort begun years ago [8] has been applied to the de novo design of UR pulses to yield important new results, giving the best UR pulse performance as functions of pulse length, bandwidth, and B1-tolerance using the GRAPE algorithm [6], one of the most efficient optimization tools currently known to the authors.
Although UR pulse design is generally considered to be more difficult than optimizing PP pulses, traditional pulse design methods have yielded a number of composite or shaped UR pulses in the last decades [1], [2], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. More recently, a principle has been derived which allows the construction of a UR pulse with flip angle based on PP pulses with effective flip angle starting with initial magnetization along the y-axis [3]. The open question is the degree to which existing UR pulses achieve the best possible performance. Given the crucial role of UR pulses in the wide variety of important coherence transfer applications in NMR, a thorough characterization of the performance limits of UR pulses is highly desirable.
After the introduction of a general pulse nomenclature, we introduce and evaluate the convergence criteria of two important classes of cost function. The evaluation of these classes turns out to be of utmost importance for the vast number of individual optimizations performed thereafter and will be of general interest for future optimization protocols. We then provide time-optimal curves for various bandwidths and tolerances to rf-scaling and discuss and classify the large number of optimal pulse shapes obtained. Different symmetries emerge among the pulse shapes, which are optimized without any symmetry constraints. Most noteably, one symmetry relates to the aforementioned construction principle [3]. The performance of these classes of symmetric pulse shapes are characterized and tested with respect to time-optimality. Finally, optimal UR pulse performance is compared to previously published UR pulses, showing the significant performance improvements revealed by the present study. We also determine whether a single UR 180° pulse can provide superior performance in INEPT-type transfer elements compared to the well-known use of two PP inversion pulses with suitable delays [40], [41]. Mathematical derivations of symmetry and cost function relations used in the main text are provided in an appendix.
Section snippets
Pulse nomenclature
As with BEBOP [4], [42], [43], [44], [45], BIBOP [4], power-BEBOP [5], power-BIBOP [5], RC-BEBOP [46] and ICEBERG [47] pulses, the universal rotation pulses presented in this article describe a whole family of pulses with transfer properties close to the physical limits for a given set of rf amplitude, bandwidth, and time-constraints. For the accurate description of corresponding pulses we therefore would like to introduce a common nomenclature for BURBOP (Broadband Universal Rotations By
Performance functions and GRAPE algorithm for UR-pulses
Optimal control theory is a mathematical field that is concerned with control policies that can be deduced using optimization algorithms [48], [49]. In other words, this theory provides a way to find such controls for a given dynamical system, which drive it from an initial state to a target state in a most efficient way. It is based on a classical Euler–Lagrange formalism, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s.
Instead of optimizing a point-to-point problem with
Phase factors and convergence of the algorithm
Although, as noted in the previous section, the global phase of the propagator representing a given pulse has no effect on the final state , global phase factors do play an important role in the optimization of such unitary operators [7], [8], [10]. As discussed in Appendix A.2, for a single spin 1/2, only two global phase factors are possible: or . Optimizations based on the performance function as defined in Eq. (8) do not distinguish between these two possibilities for
Limits of 90° and 180° universal rotation pulses with respect to offset bandwidth
A systematic study of pulse performance, similar to the ones performed previously for PP excitation pulses [4], [5], was performed for UR pulses using the described algorithm based on optimal control theory. Sets of and rotations were calculated for different bandwidths ranging from 10 kHz to 50 kHz with the rf amplitude limited to 10 kHz in all cases. Optimizations were performed using the performance function (Eq. (5)) with the two target propagators for the case of 90° UR
Directly optimized UR pulses vs. constructed UR pulses
Considering the high degree of symmetry of most optimized BURBOP pulses, it is an interesting question as to how far such symmetric pulses represent the global optimum. Since broadband UR pulses can be constructed from symmetry principles out of two PP pulses with half the effective flip angle [3], we consider the relative merits of directly optimized UR pulses compared to constructed pulses, as in [14] for 180° UR pulses, before looking into more general symmetry principles.
For a comparison,
Symmetry properties of universal rotation pulses
For a systematic examination of the symmetries found empirically in the optimized UR pulses, we consider the following four types of pulse symmetries:
- •
the fully symmetric case, with both x and y component symmetric with respect to the center of the pulse (also referred to as type I pulse or ),
- •
antisymmetric x and symmetric y component (type II pulse or ),
- •
symmetric x and antisymmetric y component (type III pulse or ),
- •
and the fully antisymmetric case (type IV pulse or ).
B1 inhomogeneity
UR and PP pulses in this article were optimized so far without considering variations in rf amplitude as, for example, imposed by B1 inhomogeneities in a real experimental setup. For a systematic study of the effect of rf-variations and for best comparability with previous results on excitation and inversion pulses [4], [5], we focused on optimizations for a fixed bandwidth of 20 kHz with varying constraints on the compensation of rf-variations expressed by the relative range of allowed rf
Discussion
Modern high resolution NMR spectroscopy with magnetic field strengths of up to 1 GHz results in large offset ranges that have to be uniformly covered in corresponding pulse sequences. Especially for frequently used heteronuclei like 13C, 19F, 15N, and 31P, their large chemical shift ranges pose serious problems for conventional probeheads using hard pulses. High frequencies, in addition, restrict the available maximum rf amplitudes due to technological limitation. As a result, the use of phase-
Conclusion
In this article, a systematic study of bounds for the physical performance limits of broadband 90° and 180° universal rotation pulses with respect to pulse length, bandwidth, and compensation for rf-variations is presented, which can be regarded as an extension to our previous work on point-to-point pulses [4], [5]. The study is based on optimizations using principles of optimal control.
The main results are the achievable qualities of broadband 90° and 180° UR pulses for a large variety of
Acknowledgments
T.E.S. acknowledges support from the National Science Foundation under Grant CHE-0943441. S.J.G. acknowledges support from the DFG (GL 203/6-1), SFB 631 and the Fonds der Chemischen Industrie. B.L. thanks the DFG (Heisenberg Program LU 835/2,3,4,7 and Forschergruppe FOR 934) and the Fonds der Chemischen Industrie.
References (78)
Composite pulses
Prog. Nucl. Magn. Reson. Spectrosc.
(1986)- et al.
Construction of universal rotations from point to point transformations
J. Magn. Reson.
(2005) - et al.
Exploring the limits of broadband excitation and inversion pulses
J. Magn. Reson.
(2004) - et al.
Exploring the limits of broadband excitation and inversion pulses: II. RF-power optimized pulses
J. Magn. Reson.
(2008) - et al.
Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms
J. Magn. Reson.
(2005) - et al.
Application of optimal control to CPMG refocusing pulse design
J. Magn. Reson.
(2010) - et al.
Robust slice-selective broadband refocusing pulses
J. Magn. Reson.
(2011) - et al.
Design and application of robust RF pulses for toroid cavity NMR spectroscopy
J. Magn. Reson.
(2011) - et al.
New strategies for designing robust universal rotation pulses: application to broadband refocusing at low power
J. Magn. Reson.
(2012) - et al.
Designing optimal universal pulses using second-order, large-scale, non-linear optimization
J. Magn. Reson.
(2012)
ADEQUATE a new set of experiments o determine the constitution of small molecules at natural abundance
J. Magn. Reson. A
Sensitivity improvement in proton-detected 2-dimensional heteronuclear correlation NMR-spectroscopy
J. Magn. Reson.
Composite pulses without phase distortion
J. Magn. Reson.
Symmetric phase-alternating composite pulses
J. Magn. Reson.
Iterative schemes for phase-distortionless composite pulses
J. Magn. Reson.
Band-selective radiofrequency pulses
J. Magn. Reson.
Symmetrical pulses to induce arbitrary flip angles with compensation for RF inhomogeneity and resonance offsets
J. Magn. Reson.
Optimization of shaped selective pulses for NMR using a quaternion description of their overall propagators
J. Magn. Reson.
180° Refocusing pulses which are insensitive to static and radiofrequency field inhomogeneity
J. Magn. Reson.
Derivation of broadband and narrowband excitation pulses using the Floquet formalism
J. Magn. Reson. Ser. A
Broadband narrowband and passband composite pulses for use in advanced NMR experiments
J. Magn. Reson. A
Robust refocusing pulses of limited power
J. Magn. Reson.
Use of composite refocusing pulses to form spin echoes
J.Magn. Reson.
Evaluation of non-selective refocusing pulses for 7 T MRI
J. Magn. Reson.
Application of optimal control theory to the design of broadband excitation pulses for high-resolution NMR
J. Magn. Reson.
Reducing the duration of broadband excitation pulses using optimal control with limited RF amplitude
J. Magn. Reson.
Tailoring the optimal control cost function to a desired output: application to minimizing phase errors in short broadband excitation pulses
J. Magn. Reson.
Optimal control design of constant amplitude phase-modulated pulses: application to calibration-free broadband excitation
J. Magn. Reson.
Optimal control design of excitation pulses that accommodate relaxation
J. Magn. Reson.
Linear phase slope in pulse design: application to coherence transfer
J. Magn. Reson.
Second order gradient ascent pulse engineering
J. Magn. Reson.
Optimal experiments for maximizing coherence transfer between coupled spins
J. Magn. Reson.
Symmetric phase-alternating composite pulses
J. Magn. Reson.
Time-optimal coherence-order-selective transfer of in-phase coherence in heteronuclear IS spin systems
J. Magn. Reson.
Broadband geodesic pulses for three spin systems: time-optimal realization of effective trilinear coupling terms and indirect SWAP gates
J. Magn. Reson.
Optimal control of spin dynamics in the presence of relaxation
J. Magn. Reson.
P.E.-HSQC: a simple experiment for the simultaneous and sign-sensitive measurement of () and () couplings
J. Magn. Reson.
The CLIP/CLAP-HSQC: pure absorptive spectra for the measurement of one-bond couplings
J. Magn. Reson.
Variable angle NMR spectroscopy and its application to the measurement of residual chemical shift anisotropy
J. Magn. Reson.
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These authors contributed equally to this work.