Nonlinear sampling in ultrafast Laplace NMR Journal of Magnetic Resonance

Ultrafast Laplace NMR (UF-LNMR) reduces the experiment time of multidimensional relaxation and dif- fusion measurements to a fraction. Here, we demonstrate a method for nonlinear (in this case logarithmic) sampling of the indirect dimension in UF-LNMR measurements. The method is based on the use of frequency-swept pulses with the frequency nonlinearly increasing with time. This leads to an optimized detection of exponential experimental data and signiﬁcantly improved resolution of LNMR parameters. (cid:1) 2019 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
NMR relaxation and diffusion experiments provide detailed information about molecular rotational and translational motion as well as chemical resolution complementary to frequency spectra [1][2][3]. The experiments result in exponentially decaying or increasing data, and the distribution of relaxation times or diffusion coefficients is solved by the inverse Laplace transform [4]. Therefore, the relaxation and diffusion experiments are called Laplace NMR.
Like in traditional NMR spectroscopy, the resolution and information content of LNMR can be increased substantially by a multidimensional approach [1][2][3]. However, the approach leads to very long experiment times due to need for repetitions with an incremented evolution time or gradient strength. Furthermore, the need for repetitions significantly hinders the use of modern hyperpolarization methods [5][6][7][8], which otherwise could improve the experimental sensitivity up to five orders of magnitude.
In this communication, we describe a method for nonlinear sampling of the indirect dimension in UF-LNMR experiments. This leads to an optimized detection of the exponential experimental data and significantly improved resolution of LNMR parameters.

Methods
If an observable, which is exponentially dependent on time, is sampled with a constant time step (linear sampling), then the beginning of the data is sampled in a non-optimal way, as the change of the observable is the biggest in the beginning. If the data is a sum of several exponential components, the situation is even worse; in the extreme case, the fastest changing component is not sampled at all. In traditional LNMR experiments, this problem is overcome by using a logarithmic sampling, i.e., by keeping the difference of natural logarithms of successive sampling point times constant. Consequently, the rapidly changing initial part of the data is more densely sampled, leading to more optimal detection of both single-and multi-exponential data. [27] In all the UF-LNMR experiments based on continuous spatial encoding, the spatial encoding is realized by applying an adiabatic frequency-swept pulse with frequency linearly increasing (or decreasing) with time along with a magnetic field gradient [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. This leads to a linear (non-optimal) sampling of the indirect dimension; the evolution time s or spatial frequency q is linearly increasing (or decreasing) with the position.
For example, the pulse sequence for UF T 1 -T 2 correlation experiment is shown in Fig. 1a [9]. The initial magnetic field gradient pulse G se makes the Larmor frequency of nuclei linearly dependent At the end of the pulses, the spins in the top have almost fully recovered back to the thermal equilibrium, and the magnetization profile along the z axis corresponds to the traditional inversion recovery curve. The profile is read in the CPMG loop by using a read gradient like in magnetic resonance imaging (MRI). Because the recovery time is linearly dependent on the position, the approach leads to linearly sampled T 1 data.
A simple way to change from the linear to logarithmic sampling of the indirect dimension in UF-LNMR experiments is to make the frequency sweep nonlinear. For example, in the UF T 1 -T 2 correlation experiment, the frequency of the adiabatic pulse, m F , should vary logarithmically with time t (see Section 3 in Supporting Information): Here, Dm = m i À m f is the width of the frequency sweep, m i and m f are the initial and final frequencies of the sweep, s min and s max are the minimum (see Fig. 1a) and maximum recovery times, t F = s max À s min is the length of the frequency-swept p pulse, and Àt F /2 t +t F /2. Consequently, the recovery time s is exponentially increasing with position z (see Section 3 in Supporting Information): Here, c is the gyromagnetic ratio, G is the gradient strength, and the spatial encoding region is ÀpDm/cG z +pDm/cG. This analysis is based on the instantaneous inversion approximation, i.e., we assume that the spins in certain layer are inverted instantaneously when their Larmor frequency matches with the frequency of the adiabatic pulse. This has turned out to be a good approximation [18]. The power of the adiabatic pulse has to be high enough so that the adiabatic condition is satisfied even at the end of the pulse, when the sweep rate is fastest (once the power is high enough, the adiabatic pulse is inverting the magnetization independently of the pulse power) [18]. Alternatively, the amplitude of the frequency-swept pulse may increase with time to fulfil the adiabatic condition. The validity and limitations of instantaneous excitation and inversion as well as adiabatic condition are discussed in detail elsewhere [28].

Results and discussion
To validate that the nonlinear sampling method works, we performed UF inversion recovery (UF-IR) measurements of single-and double-tube water samples visualized in Fig. 2a. The experiments were performed with Bruker Avance III 400 MHz spectrometer (see experimental details in Section 1 in Supporting Information). Water was doped with copper chlorite so that, according to the traditional IR reference measurements, T 1 in the inner compartment of the double-tube sample was about ten times longer (332 ± 2 ms) than in the outer compartment as well as in the single-tube sample (33.5 ± 0.2 ms). The UF-IR pulse sequence was otherwise identical to the UF T 1 -T 2 correlation shown in Fig. 1a except that only one echo was measured in the CPMG loop. A frequency-swept p pulse with the pulse length t F of 1 s and frequency nonlinearly increasing from À5000 to +5000 Hz (see Fig. 2b) was used in the experiment for the spatial encoding. For comparison, linearly sampled data was also measured with a similar frequency-swept pulse with frequency linearly increasing with time (shown also in Fig. 2b; for more details of the pulse, see Section 2 in Supporting Information).
The magnetization profiles (i.e. the Fourier transforms of the UF-IR data sets in magnitude) of the single-tube sample are shown in Fig. 2c. The spatially encoded IR curves are visible in the sweep region indicated in the figure. In the case of the nonlinear sweep, the minimum around z = À3 mm corresponds to the zero intensity of the IR curve. The other minimum around z = À4 mm is an artefact, because the adiabatic pulse does not work perfectly right in the beginning and end of the frequency sweep [18]. The linearly sampled data does not reach zero intensity at all, because the shortest recovery delays are not observed. The magnetization profiles of the double tube sample in Fig. 2d show also that the IR curve is much better probed in the case of the nonlinear sweep. Fig. 2e and f show the ultrafast IR curves extracted from the magnetization profiles. In the processing, the profiles were subjected to the coil sensitivity profile correction and the data close to the beginning and end of the frequency sweep were removed [9]. Then the position axis z was converted into recovery time axis s by using Eqs. (S5) and (S23) shown in Supporting information for the linearly and nonlinearly sampled data, respectively. The figures visualize clearly much more optimal sampling of the IR curves with nonlinear sweep than with linear; much shorter recovery times are observed and therefore the beginning part of the IR curve is much better sampled. The observed recovery times varied from 102 to 910 ms and from 18 to 747 ms for the linear and nonlinear sweep, respectively. The T 1 values resulted from single-or doubleexponential fits shown in the figures are in good agreement with the reference experiments within the error bars, while in the case of linear sampling the deviations are bit larger (see also difference traces between the data and fits in Fig. S9 in Supporting  Information).
We performed also full T 1 -T 2 correlation experiments for the double-tube sample using the pulse sequence shown in Fig. 1a. The correlation maps resulting from the 2D Laplace inversion [29,30] of the data measured using linear and nonlinear sweep are visible in Fig. 2g and 2h, respectively. Both maps show two components corresponding to the two compartments, and they are in a good mutual agreement, proving that the nonlinear sampling concept works also in 2D LNMR experiments. In the nonlinear map, the low relaxation time signal is about 40% narrower in the T 1 dimension than in the linear map due to the more optimal sampling (see Fig. S10 in Supporting Information), resulting in improved resolution of the components (note that the line widths depend also on the parameters used in the Laplace inversion).
As a final demonstration of the feasibility of the nonlinear ultrafast sampling, we measured ultrafast and reference T 1 -T 2 correlation maps of water in Silica gel 60 porous material. The results are shown in Fig. 3. All the maps show two dominant peaks, one from water inside the pores (nominal pore size 60 Å) characterized by the shorter relaxation times and another from bulk water in between the particles (particle size 63-200 lm) of the porous material. However, in the UF-linear map the signals are in a significantly narrower region in the T 1 dimension than in the other maps due to the non-optimal sampling. Furthermore, in the UF-nonlinear map the signals are in a slightly narrower region in the T 1 dimension than in the normal reference map. Most probably, this is a consequence of the effect of magnetic field inhomogeneity in the porous material on the spatial encoding. The linewidth of the water in porous material is about 110 Hz, which is a significant portion of the frequency range (30 kHz) used in the spatial encoding. Therefore, a certain point in the observed magnetization profile does not anymore correspond to a single recovery delay, but it represents contributions of a range of delays. This leads to an averaging effect; a narrower range of T 1 values is observed. The averaging effect might be reduced by using larger gradient amplitude and sweep width in the spatial encoding.
Regardless of the above mentioned issues, the two water sites are visible in the ultrafast maps, and T 2 range as well as the average values of T 1 are in good agreement with the reference map. In fact, the coordinates of the tops of the peaks in the non-linearly sampled ultrafast and reference maps are close to each other, and, if the peak widths are taken as experimental error, the maps can be considered to be in good agreement with each other.
As the T 1 values of the components observed from the Silica sample are much longer than the T 2 values, the diffusion driven exchange of water between the pore and bulk sites has more significant averaging effect on the T 1 than T 2 values. On the other hand, the effect of exchange should be the same for the ultrafast and reference experiments, as the both types of experiments rely on the observation of similar exponential data.
It is noteworthy that the ultrafast experiments with 4 accumulated scans took only for 30 s while the reference experiment with the same number of scans lasted for 16 min. Naturally, the signalto-noise ratio (SNR) in the ultrafast experiments was lower than in the reference experiment due to splitting the sample into layers in the spatial encoding. However, the observed SNR decreases have been relatively small (factors of 2-4), which is not a problem with the high concentration samples, and the ultrafast experiment can be repeated several times during a single traditional experiment, which may lead to even improved SNR per unit time [10,13]. Furthermore, the single-scan approach makes it possible to increase the sensitivity by several orders of magnitude by hyperpolarization [10][11][12]14].

Conclusions
We demonstrated a method for nonlinear sampling of indirect dimension in the UF IR and T 1 -T 2 correlation experiments. The method is based on the use of frequency-swept pulses with the frequency nonlinearly increasing with time. This leads to more optimal detection of single-exponential data and much more optimal detection of multi-exponential data with T 1 values differing significantly. Therefore, it results in significantly improved resolution of T 1 distribution. The maximum T 1 value observable in the ultrafast experiments is defined by the length of the frequency-swept pulse used in spatial encoding, and there is no fundamental technical limitation for the length. As shown in this Communication, much shorter T 1 values can be observed by using the non-linear sampling instead of the linear. Adiabatic condition and instantaneous excitation approximation sets additional limits for the minimum observable T 1 value [18,28]. Here, we focused on logarithmic sampling, but other kinds of non-linear sampling schemes can be readily obtained with appropriate modifications of the sweep function of the frequency-swept pulse. The method can be extended to many other multidimensional ultrafast Laplace NMR experiments including indirect T 1 encoding or experiments correlating T 1 with spectroscopic data [21,22]. Furthermore, it can be utilized for nonlinear sampling of spatial frequency (q) space in ultrafast diffusion experiments as well as other multidimensional ultrafast NMR experiments including indirect diffusion encoding [9][10][11][12][13][14][15][16][17][18][19][20]. The optimized multidimensional UF-LNMR experiments have lots of interesting potential applications in various disciplines, including chemistry, biochemistry, geology, archaeology, and medicine [1][2][3][9][10][11][12][13][14][15][31][32][33][34][35].