Radial sampling for fast NMR: Concepts and practices over three decades.

For almost as long as three- and four-dimensional NMR experiments have been used, NMR spectroscopists have been devising ways to speed them up. Indeed, the publication that is often cited as the very first to demonstrate a 3-D NMR experiment commented, “it has been thought that…high-resolution 3-D NMR experiments are impracticable because of huge data matrices and long measurement times,” but went on to “suggest a technique for reduction of data matrices” using selective excitation pulses [1]. Although both spectrometers and computers have advanced considerably since this 1987 publication, somewhat changing the definitions of huge and long, the fundamental problem of measurement time continues to limit the experiments that can be carried out in practice using conventional multidimensional NMR methodology, and significant effort is still devoted to alleviating this restriction. The problem arises from the very nature of Fourier transform (FT) NMR, which involves the systematic sampling of a signal over time followed by the calculation of the spectrum using the FT [2, 3]. Traditionally, an n-dimensional (n-D) experiment is obtained through the sampling of the time domain on a complete n-D Cartesian grid; since the number of points in an n-D grid grows exponentially with the number of dimensions n, the measurement time needed to record the experiment becomes considerable even for small n. Yet NMR spectra are generally only sparsely populated with signals, suggesting that there is no statistical need for so many observations (other than signal accumulation for sensitivity in some cases), and that a suitable alternative approach to sampling and/or processing might significantly reduce the time requirement, while generating the same or more spectral information. Two trends in biomolecular NMR research have given particular impetus to these efforts: the drive for increased throughput in studies of small proteins—for example in structural genomics—which requires running today’s routine experiments more quickly, and the increasing attention given to large and challenging systems, which require more dimensions and higher resolution than conventional experiments can offer. A variety of methods have been introduced in these efforts to reduce the amount of data needed for multidimensional NMR, but a surprising number of them share in common that they sample the indirect dimensions of the time domain along radial spokes. The measurement of a radial spoke simply means collecting data samples along a line in the time domain that passes through the origin. If multiple radial spokes are sampled, the resulting dataset is equivalent to recording the NMR experiment in cylindrical coordinates or their higher-dimensional equivalent (cylindrical rather than spherical because the directly observed dimension is always sampled conventionally). The potential advantage of this arises from the fact that one can arrange the radial sampling points so as to obtain higher resolution information than with conventional Cartesian sampling, for the same or a smaller number of samples. Depending on the processing method used to extract spectral information from the data, this approach may lead to artifacts or ambiguities—but it also has the potential to provide complete spectral information in considerably less time than required for conventional NMR. The purpose of this article is to review the long history of radial sampling in NMR, from its initial introduction in the “accordion spectroscopy” experiments, through reduced-dimensionality and G-matrix Fourier Transform (GFT) spectroscopy, and on to the projection spectroscopy and projection-reconstruction techniques. Because all of these methods share a common mathematical foundation—despite their sometimes differing vocabularies—we first explain these underlying concepts. We then continue with a chronological survey of the different approaches, describing how they were developed, how they work and how they have been put to use. Particular attention is given to how these methods can be used to reduce the measurement time of the experiment, including the theoretical basis for the time savings and the practical tradeoffs that can result. It is important to note that while this review describes a number of techniques for reducing NMR measurement time, it does not attempt to describe the many methods that have been introduced recently for that purpose which do not use radial sampling. These include random sampling [4–11], concentric ring or shell sampling [12, 13] and other unconventional sampling approaches (e.g. spiral [14]); filter diagonalization analysis to extract high-resolution information from low-resolution conventionally sampled data [15–17]; the measurement of a spectrum in a single scan through the encoding of the spectroscopic frequency information spatially within the sample [18, 19]; Hadamard encoding to measure signal intensities at a small number of directly excited frequencies [20–22]; covariance spectroscopy, which enhances the resolution in the indirect dimensions through a statistical symmetrization with the directly observed dimension [23–26]; and the “minimal sampling” procedure, which involves calculating the possible correlations between the signals on the “first planes” of a multidimensional experiment and resolving any ambiguities by measuring a single additional sampling point [27, 28]. We touch on processing methods such as multidimensional decomposition [7, 29] and maximum entropy reconstruction [30] only to the limited extent that they have been applied to radial sampling experiments. Additionally, we do not discuss methods for reducing experiment time by optimizing the longitudinal relaxation rate to allow a much shorter interscan delay, which could be applicable to any type of sampling [31–34].