Axis-matching excitation pulses for CPMG-like sequences in inhomogeneous fields
Graphical abstract
Introduction
The Carr–Purcell–Meiboom–Gill (CPMG) [1], [2] sequence plays a key role in many NMR applications ranging from quantum information processing [3], [4], [5], [6], magnetometry [7], [8], [9] to applications in inhomogeneous fields that include stray-field NMR [10] and oilfield logging [11]. In these cases, the inhomogeneity of B0 across the sample typically exceeds the strength of B1. As a consequence, is very short and comparable to the pulse duration. To overcome this rapid signal decay, the CPMG sequence is used with a pulse spacing as short as possible. This generates the maximum number of echoes per unit time and can be used to improve the effective signal-to-noise ratio [12]. The short echo spacing also allows the monitoring of relaxation properties over a wide range of time scales.
The spin dynamics of the standard CPMG sequence in such grossly inhomogeneous fields is well understood [13], [14], [15], [16], [17]. With standard rectangular pulses and in a static magnetic field characterized by a gradient, the main contributions to the signal comes from frequency offsets within ±ω1 around the carrier frequency of the rf pulses. This range of frequency offsets can be increased with the use of composite, shaped, chirped or adiabatic pulses [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Much progress has been recently made in the development of broadband excitation and refocusing pulses that are also robust with respect to variation of the rf field strength and other constraints [25], [28], [29], [30], [31], [26]. These developments took advantage of the availability of new algorithms for pulse sequence design based on methods of optimal control [32], [33]. These algorithms make it feasible to find new sequences in an efficient manner in a high-dimensional parameter space.
In the general case, the performance of the standard 180° refocusing pulse is improved by constructing a pulse with sophisticated phase- and/or amplitude modulation. Such pulses are more complex and typically much longer than the standard 180° pulse. This increase in pulse duration has the undesirable consequence that it entails an increase of the minimum echo spacing and an increase in the power consumption per refocusing pulse. This is of particular concern for mobile and field applications such as NMR well logging. For this reason, we investigated in [27] the potential of very short refocusing pulses. We searched for pulses under the constraint that the overall duration and maximum amplitude do not exceed those of the standard 180° refocusing pulse. We showed that it is possible to generate larger echoes by modulating the phase within a 180° refocusing pulse. The optimal refocusing pulse of duration t180 was found to be of the form α−xβ+xα−x with α ≃ 27° and β ≃ 126° [27]. This refocusing pulse is an example of a symmetric phase-alternating pulse, a class of pulses that was first described by Shaka et al. [34]. In the rest of this paper, we refer to the 27°−x126°+x27°−x pulse as the SPA pulse. In a gradient field, a modified CPMG sequence based on such SPA refocusing pulses and a perfect broadband excitation pulse generates echoes with a ratio of signal to noise power (SNR) that is 2.38 times larger compared to the standard CPMG sequence. With practical excitation pulses that are no longer than 20 times the duration of the refocusing pulses and without an increased B1 amplitude, we demonstrated an improvement in SNR of 1.52 [27].
Here we show that it is possible to substantially improve the performance of the CPMG sequence in inhomogeneous fields by replacing the excitation pulse with a so-called axis matching excitation pulse, or AMEX pulse for short. This approach is also effective for more general CPMG-like sequences that contain broadband refocusing pulses (such as SPA pulses). The new excitation pulses take the limitations of the refocusing pulses explicitly into account and are specific for a given refocusing pulse and echo spacing. Rather than trying to generate transverse magnetization over the widest possible bandwidth, AMEX pulses generate magnetization that at each offset frequency aligns with the axis characterizing the refocusing cycle from echo to echo. We present practical AMEX pulses for a number of simple refocusing pulses that were obtained using methods of optimal control [33]. We show that this results in a significant increase of the SNR. Using AMEX pulses optimized for SPA refocusing pulses, we found experimentally an increase in SNR by over a factor of 3 compared to the standard CPMG sequence. Remarkably, this improvement substantially exceeds the theoretical limit achievable with a perfect broadband 90° excitation pulse and the same SPA refocusing pulse.
We first summarize the theory underlying the relevant spin dynamics and formulate the condition for the AMEX pulses. We also discuss the necessary modifications to the phase cycling. The search algorithm used to find the AMEX pulses is presented in Section 3. In Section 4, we present a number of AMEX pulses that were optimized for a number of different refocusing pulses, including the SPA pulse and several rectangular refocusing pulses. Experimental results are presented in Section 5. We finish with the conclusion and outlook for further work.
Section snippets
Theory
We start with a brief review of the relevant spin dynamics for a collection of uncoupled spins 1/2 placed in inhomogeneous B1 and B0 fields. This problem has been studied extensively and further details can be found in the literature [13], [14], [15], [24], [31], [17], [35], [27]. We consider a generalized CPMG-like sequence that consists of an initial excitation pulse followed by a long train of equally-spaced, identical refocusing pulses. The excitation and refocusing pulses can be simple
Search algorithm
To find suitable AMEX pulses, we take advantage of recent progress in optimal control methods and the development of powerful algorithms for NMR applications [33]. We began our search for AMEX pulses by using the GRAPE algorithm [32] to design an excitation pulse with 100 segments, each of length 0.1 × t180, for the SPA refocusing pulse and an echo spacing tE = 7t180. The pulse segments were allowed to have variable amplitudes between 0 and ω1,max = π/t180 and arbitrary phases. In the simulation,
Results: performance of CPMG sequence with AMEX pulses
Using the optimal control methods described in the previous section, we derived practical AMEX pulses for four different refocusing pulses, namely the rectangular pulses 124°x, 135°x, 180°x, and the SPA pulse 27°−x126°x27°−x, all optimized for tE = 7t180. Here we denote the rectangular pulses by their nutation angle on resonance. The 124° refocusing pulse produced the highest SNR of all rectangular pulses with on-resonance nutation angles between 90° and 180°. The AMEX pulses were designed to be
Experimental results
We have experimentally verified the performance of the new pulse sequences. We used a Bruker Avance-II spectrometer and a superconducting magnet (Oxford) operating at the proton frequency of 42.57 MHz. The spectrometer was equipped with an RF amplifier that can deliver 1 kW, and gradient amplifiers that can supply a maximum of 200 A. The sample was placed inside a cylindrical NMR tube with diameter of 5 mm and length of 15 cm, or a larger tube with diameter of 10 mm and length of 23 cm. The tube was
Conclusion
We have shown that AMEX pulses greatly improve the performance of the CPMG sequence in inhomogeneous fields and lead to significantly larger echo amplitudes and a reduced transient. The AMEX pulses are designed to transform the magnetization, initially pointing in the longitudinal direction, to a direction pointing along the axis of the refocusing cycle. This can be considered to be the generalized CPMG condition. At any offset frequency, magnetization pointing along this axis is fully
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