Accelerating 2D NMR Relaxation Dispersion Experiments Using Iterated Maps

Abstract

NMR relaxation dispersion experiments play a central role in exploring molecular motion over an important range of timescales (Walinda et al, 2018), and are an example of a broader class of multidimensional NMR experiments that probe important biomolecules (Palmer et al, 2001; Loria et al, 1999; Neudecker et al, 2009). However, resolving the spectral features of these experiments using the Fourier transform requires sampling the full Nyquist grid of data, making these experiments very costly in time. Practitioners often reduce the experiment time by omitting 1D experiments in the indirectly observed dimensions, and reconstructing the spectra using one of a variety of post-processing algorithms. In prior work, we described a fast, Fourier-based reconstruction method using iterated maps according to the Difference Map algorithm of Veit Elser (DiffMap). Here we describe coDiffMap, a new reconstruction method that is based on DiffMap, but which exploits the strong correlations between 2D data slices in a pseudo-3D experiment. We apply coDiffMap to reconstruct dispersion curves from an R₂ relaxation dispersion experiment, and demonstrate that the method provides fast reconstructions and accurate relaxation curves down to very low numbers of sparsely-sampled data points.

Introduction

High resolution spectra from pulsed Fourier transform spectroscopy provide a window into molecular structure and dynamics of important biomolecules (Palmer et al, 2001; Neudecker et al, 2009; Walinda et al, 2018). An increasing variety of information is being gathered by observing spectra from 2D, 3D, and higher dimensional experiments (Kanelis et al, 2001; Levitt, 2008). However, these experiments can be very time intensive, since the time to acquire nD data is directly proportional to the number of samples in the indirect dimensions. To acquire an nD spectrum using a discrete Fourier transform (FT), the sampled time points must be on a Cartesian grid with uniform intervals (“dense sampling”); the intervals must be small enough to satisfy the Nyquist-Shannon sampling theorem for the width of the spectrum along each dimension, and extend to long enough times to resolve the features in the spectrum.

To accelerate acquisition, we may instead sample fewer points in the indirect dimension by skipping certain 1D experiments altogether, creating a series of “gaps” in the nD data. We refer to this as “sparse sampling,” also called nonuniform sampling (NUS). The resulting data set (with gaps replaced by zeros) will yield a distorted spectrum upon applying an FT (Maciejewski et al, 2009), so an alternative method is required to reconstruct sparsely sampled data.

A number of reconstruction methods exist, and operate by imposing some form of constraint. Examples include iterative methods like Iterative Soft Thresholding (IST) (Stern et al, 2007), iteratively reweighted least squares (IRLS) (Schlossmacher, 1973; Kazimierczuk and Orekhov, 2012), and Spectroscopy by Integration of Frequency and Time domain information (SIFT) (Matsuki et al, 2009, 2010, 2011); as well as other non-Fourier techniques like compressed sensing (Kazimierczuk and Orekhov, 2011; Bostock and Nietlispach, 2018), maximum entropy (MaxEnt) (Hoch and Stern, 1996; Mobli and Hoch, 2008; Hyberts et al, 2010; Paramasivam et al, 2012), or MultiDimensional Decomposition (MDD) (Orekhov and Jaravine, 2011). These methods and others are part of a broader study into sampling schemes and reconstruction algorithms to speed up the acquisition of information-rich multidimensional NMR experiments (Mobli et al, 2012; Stanek and Koźmiński, 2010; Stanek et al, 2012; Kazimierczuk et al, 2010; Kazimierczuk and Orekhov, 2012; Mobli and Hoch, 2014; Palmer et al, 2015; Billeter, 2017; Ying et al, 2017; Zambrello et al, 2018; Hoch, 2018; Rovnyak and Schuyler, 2018). In particular, several recent papers have studied how non-uniform sampling approaches can accelerate relaxation experiments while retaining quantitatively accurate results (Stetz and Wand, 2016; Linnet and Teilum, 2016; Urbańczyk et al, 2017).

In the search for a reconstruction method that would work well with our 3D MRI of solids images (Frey et al, 2012), we developed a method that uses iterated maps arranged according to the “Difference Map” algorithm of Veit Elser (Elser, 2003; Elser et al, 2007; Fienup, 1982), and which exploits the computational speed of the FFT (Frey et al, 2013). This method (which we refer to as “DiffMap”) is related to SIFT (Matsuki et al, 2009, 2010, 2011), as well as to POCS (Haacke et al, 1991). We recently reported results using this method applied to 2D NMR of liquids spectra, a 2D NMR of solids spectrum, and a 3D ³¹P MRI of solids image (Frey et al, 2013). For one case in that paper (see Fig. 2 and Fig. 3 from Frey et al, 2013), we focused on a single 2D spectrum from a pseudo-3D R₂ relaxation dispersion experiment, using a “top-down” approach to achieve globally excellent results down to a sampling fraction of 75/128. Below this sampling fraction, global errors began to accumulate quickly. While this was an improvement over dense sampling, it was still above the fundamental limit we suspected was possible for that data, based on its sparsity.

In a companion paper (Blum et al, 2019) we present a new “bottom-up” approach to finding the lowest number of sparse samples required for the DiffMap method. There, we focus on reconstructing single peaks in particular 1D spectra taken from 2D slices of the pseudo-3D data set, and show that it is indeed possible to reach a minimum sampling number very close to the fundamental limit of sparse sampling for DiffMap, significantly lower than the sampling limits reached previously (Frey et al, 2013). Using a better understanding of the relationship between signal, noise, and sampling fraction, we provide detailed metrics for predicting and reaching nearly the minimum number of sampled points for each individual peak, one (or a few) at a time. However, when we try to extend the method to reconstruct many (or all) of the peaks in the data set at once, the lowest-attainable sampling fraction quickly increases to the original limits (Frey et al, 2013). In other words, when reconstructing entire data sets rather than individual peaks, we were unable to use our improved understanding of the DiffMap method (from Blum et al, 2019) to beat our previous minimum sampling limit (from Frey et al, 2013).

Here we demonstrate a new method to overcome the barrier encountered in that bottom-up approach. We do this by using additional information during the reconstruction, gained by examining the entire pseudo-3D experiment at once. This is in the same vein as two recent efforts to reconstruct relaxation data, which have used the information available in the third dimension of a pseudo-3D experiment: co-processing by MDD (coMDD) (Linnet and Teilum, 2016), and an inverse Laplace-Fourier transform (joint FT-ILT) for exponential relaxation (Urbańczyk et al, 2017). Here, we directly exploit the correlations that exist among the multiple 2D slices of a pseudo-3D experiment. Not only does this allow us to reconstruct all the peaks at once, but the additional information also allows us to push to a much lower sampling fraction than was achieved previously (Frey et al, 2013), sometimes even reaching below the fundamental limits of the original DiffMap applied to individual 2D slices (Blum et al, 2019) — this is possible because we are adding new information to the reconstruction problem.

Since we build on DiffMap while taking advantage of the correlations in a pseudo-3D experiment, we refer to our new method as coDiffMap. The coDiffMap approach enables the reconstruction of all the features in a spectrum at once, rather than just one or a few. It does not require exponential relaxation, is reasonably robust to noisy data, and takes advantage of the knowledge available in a densely sampled reference spectrum, which may be obtained as part of a sparsely-sampled pseudo-3D data set. We illustrate coDiffMap by reconstructing dispersion curves in an R₂ relaxation dispersion experiment, obtaining reconstructions that closely match the dense data in tens of seconds of processing time.

Background

In this section, we first describe the characteristics of our pseudo-3D data, and the relevant parts of the original DiffMap method. We then briefly reiterate the expected sparse sampling limits that are relevant for DiffMap, before describing the coDiffMap method itself.

Characteristics of our Pseudo-3D data set

For each 2D slice of the pseudo-3D experiment, we start with densely sampled “States”-like data, which is typically used to fill out only a single (t₁ ≥ 0, t₂ ≥ 0) quadrant of 2D data (States et al, 1982). Instead, we use this data to fill out all four quadrants of a 2D time domain grid to produce a complex M × N vector T(t) (Fig. 1a), which we construct with Hermitian symmetry (see Supplementary Information) (Frey et al, 2013; Mayzel et al, 2014; Blum et al, 2019). Here, we include the argument t = (t₁, t₂) in our notation to clarify that T is the time-domain data; we will use T (t) to refer to the complex value of T at coordinate t. In the densely sampled data, we have $N_{t_1}^{\text{dense}}$ points along the indirect dimension t₁. Because this four-quadrant data has Hermitian symmetry, T(t) = T*(−t), it produces an entirely real M × N spectrum after applying a discrete Fourier transform (FT) and proper phasing (Ph), $\tilde{T}(f)=\text{Ph}(\text{FT}[T(t)])$, where f = (f_1,f₂) (Fig.1b). Notice that $M=2N_{t_1}^{\text{dense}}$ because of our data preparation method. In a pseudo-3D experiment these steps are repeated to acquire multiple 2D spectra, where some parameter is varied from spectrum to spectrum. Here, we consider an R₂ relaxation dispersion experiment, with a series of twelve 2D spectra corresponding to twelve different τ_cp times. We label each “slice” (each 2D spectrum) of the pseudo-3D experiment by an index $i_{{\tau }_{\text{cp}}}$, in increasing order of τ_cp. Thus we label our full pseudo-3D data set $T\left(t_1,t_2,i_{{\tau }_{\text{cp}}}\right)$ (Fig. 1a), with corresponding spectra $\widetilde{\mathbf{\text{T}}}\left(f_1,f_2,i_{{\tau }_{\text{cp}}}\right)$ (Fig. 1b).

Fig. 1.

Overview of a pseudo-3D experiment. a The original 2D (t₁ ≥ 0, t₂ ≥ 0) data, extended to all four quadrants (see Supplementary Information). In the indirect dimension, there are $N_{t_1}^{\text{dense}}$ points in the densely sampled data. A different 2D spectrum is acquired for each τ_cp; each 2D “slice” is labeled by an index $i_{{\tau }_{\text{cp}}}$. b Spectra from a FT of each time-domain slice, with zeros of the $\widehat{P_1}$ support mask shown in gray. We treat each 1D “column” independently (red box). c An example 1D spectrum along f₁, with the positive mask support region shown in pink, where positive signal is allowed to remain. An artifact support mask is used for features with unpredictable sign. For a 1D column at f₂ index n, the value K(n) is taken from the total number of frequency points with positive, negative, or artifact support masks in that column

In a sparse-sampling experiment, we sample only $N_{t_1}\le N_{t_1}^{\text{dense}}$ points along t₁ by skipping certain 1D experiments altogether, resulting in a series of gaps in the 2D data. Since in this study we have access to the full set of densely sampled data T, we imitate sparse sampling using an operator ${\widehat{P}}_0$ that sets the dense data to zero for a chosen set of skipped t₁ values, resulting in the sparse data set $S^0(t)={\widehat{P}}_{0}T(t)$. Our goal is then to reproduce the dense data from this sparsified data, using reconstruction. Because we always sample densely along the direct dimension t₂, we may reduce each 2D spectrum to a series of 1D spectra at particular f₂ values (Fig. 1c), which we treat independently during the DiffMap reconstruction. Defining an f₂ index n = {0, 1, …, N − 1}, we refer to the 1D spectrum at a given n as ${\tilde{T}}_n\left(f_1,i_{{\tau }_{\text{cp}}}\right)$, with corresponding 1D time-domain data $T_n\left(t_1,i_{{\tau }_{\text{cp}}}\right)=\text{IFT}\left({\text{Ph}}^{-1}\left[{\tilde{T}}_n\left(f_1,i_{{\tau }_{\text{cp}}}\right)\right]\right)$, and similarly for the 1D sparse data $S_n^0\left(t_1,i_{{\tau }_{\text{cp}}}\right)$.

Original DiffMap

DiffMap is an iterative reconstruction method that relies on two projection operators, one in the frequency domain and one in the time domain, similar to SIFT (Matsuki et al, 2009, 2010, 2011).

In the frequency domain, the ${\widehat{P}}_1$ operator sets the imaginary part of the spectrum to zero, and acts on the real part of the spectrum using a predefined mask function mask (f), which can take one of four values at each frequency point:0, +1, −1, or 2. For mask(f)=0, the signal is zeroed always. For mask(f)=+1 or −1, the signal is zeroed only if it is negative or positive, respectively (Fig. 1c, red shaded background). For mask(f) = 2, the signal is always left alone; this is an “artificial support” region, where we are unsure of the sign (Fig. 1c, black hatched background). We define mask(f) by using a densely sampled spectrum $\tilde{T}(f)$ and establishing a positive threshold τ; we let mask (f) =(0) whenever $|\text{Re}(\tilde{T}(f))|<\tau$, let mask (f) = +1 or −1 whenever $\text{Re}(\tilde{T}(f))\ge \tau$ or $\text{Re}(\tilde{T}(f))\le -\tau$ (respectively), and manually choose mask(f) = 2 or whenever we are unsure of the sign. In the time domain, the ${\widehat{P}}_2$ operator resets measured points to their known values. Rather than applying these two operators back and forth, the Difference Map combines these operators into a composite operator $\widehat{D}=\mathbb{1}+{\widehat{P}}_1\cdot \left(2{\widehat{P}}_2-\mathbb{1}\right)-{\widehat{P}}_2$, where $\mathbb{1}$ is the identity operator (Elser, 2003; Elser et al, 2007; Frey et al, 2013). Fourier transforms and phase corrections are applied as needed to reversibly convert between time and frequency domains. The Difference Map operator $\widehat{D}$ is applied to the initial sparse data for i iterations, $S^i={\widehat{D}}^i{S}^0$, after which a single ${\widehat{P}}_2$ operation is applied to produce the final output, $F^i={\widehat{P}}_2{S}^i$. We then compare our final output spectrum ${\tilde{F}}^i$ to the dense target spectrum $\tilde{T}$. For a full description of DiffMap, see the Supplementary Information, Frey et al (2013), and Blum et al (2019).

For this original DiffMap method, it is possible to use general arguments to estimate the expected fundamental limits of sparse sampling. In the next section we briefly reiterate these expected sparse sampling limits, since this will help benchmark the sampling fractions we reach using coDiffMap applied to a full pseudo-3D data set below.

Limits to Sparse Sampling

The achievable sampling number for a given spectrum depends on the number of significant spectral features, K (Hyberts et al, 2014; Shchukina et al, 2017). We determine the number of significant (relative to our threshold τ) features in the n th 1D column, K (n), by counting the number of frequency points with positive, negative, or artifact ${\widehat{P}}_1$ support masks in that column, shaded in Fig. 1c. Because we prepare our data Hermitian-symmetrically around the t₁ = 0 point, the relevant number of sampled points to compare to K(n) is $\text{2}{N}_{t_1}$, where $N_{t_1}$ is the number of complex t₁ ≥ 0 points sampled. A result from compressed sensing (CS) suggests that a high quality spectral reconstruction requires a minimum number of $N_{t_1}$ points, given by (see Eq. 9.40 of Foucart and Rauhut, 2013)

$$2N_{t_1}>2K\text{ln}(M/K)(CS\mathbf{\text{ limit }}).$$ (1)

We may hope to beat the CS limit using DiffMap if we know where spectral features ought not to appear, since this constraint provides more information about the spectrum than is assumed by CS (i.e., that the spectrum is sparse). This was accomplished using DiffMap for a full 2D spectrum by Frey et al (2013). Ideally, however, we may hope to reach as far as the limit imposed by linear algebra (LA): that the number of “constraints” M − K (n) must be at least equal to the number of “unknowns” $M\text{ - 2}{N}_{t_1}$:

$$2N_{t_1}\ge K(LA\mathbf{\text{ limit }}).$$ (2)

Defining a “compression ratio” $\mathcal{C}\text{ = 2}{N}_{t_1}\text{/ }K\text{( }n\text{ )}$ (see Blum et al, 2019), one could hope to push down to $\mathcal{C}=1$ using DiffMap. This is precisely the limit approached in the companion paper (Blum et al, 2019), where a “bottom-up” approach yielded a mean compression factor ${\overline{C}}_{\text{bu}}\approx 1.3$, achieved by analyzing individual features in particular 1D spectra. In the next section, we show that we may improve results globally, reconstructing an entire pseudo-3D experiment, by adding information to our reconstruction method, allowing us to push to (and sometimes even below) $\mathcal{C}\approx 1$.

Method

In a companion paper (Blum et al, 2019), we shifted our analysis from the entire 2D spectrum down to specific peaks in particular 1D columns of the 2D spectrum. Here, in our search for more information to help constrain the reconstruction problem, we instead move in the other direction, broadening our scope from a single 2D spectrum to the full pseudo-3D relaxation dispersion experiment. Looking at the full series of 2D data slices across $i_{{\tau }_{\text{cp}}}$, we notice that a strong correlation exists between the time-domain data values from one $i_{{\tau }_{\text{cp}}}$ value to the next (Fig. 2a–c). The correlation is so strong that it is often difficult to see the differences between the time domain data, as the set of data points at each t₁ are clustered in a “neighborhood.” However, when we compare this to the reconstructed values achieved by the original DiffMap at very low $N_{t_1}$ (black crosses in Fig. 2a–c), we see that the reconstructed values often land far outside the neighborhood, by many times the neighborhood size. The tight clustering of the time-domain data points across the $i_{{\tau }_{\text{cp}}}$ slices is not factored into the original DiffMap approach, which treats each 2D slice separately. Thus, to improve the reconstruction at very low $N_{t_1}$, we should try to include information from the full pseudo-3D data set about the neighborhood of points at each t₁. However, we cannot do this by sampling each 2D spectrum in the same way, since we would gain no knowledge about the data in the unsampled “gaps” in t₁. With this “normal” sampling, missing t₁ values are not sampled at any $i_{{\tau }_{\text{cp}}}$ (Fig. 2d, Normal sampling).

Fig. 2.

a Real parts of 1D data along t₁ at fixed f₂ (circles), taken from 11 spectra at different τ_cp values of a relaxation-dispersion experiment (time data correspond to the spectrum with relaxer Ile73 δ¹). Each $i_{{\tau }_{\text{cp}}}$ uses a different color circle. Crosses are the DiffMap reconstructed values from sparsely-sampled data in one $i_{{\tau }_{\text{cp}}}$ slice (using $N_{t_1}=30$). b,c A close look shows the strong correlation among the data sets because the spectra do not change dramatically across $i_{{\tau }_{\text{cp}}}$. However, the reconstructed values (black crosses) can deviate significantly using the original reconstruction method at such low $N_{t_1}$. This suggests that the correlation among the data sets is useful information. d Illustration of staggered sampling. In normal sampling, the same points are sampled (dark blue) at each $i_{{\tau }_{\text{cp}}}$, so that an unsampled t₁ point at one $i_{{\tau }_{\text{cp}}}$ (red) is never sampled for any $i_{{\tau }_{\text{cp}}}$ (pink). In staggered sampling, we stagger the set of sampled t₁ points for each $i_{{\tau }_{\text{cp}}}$, to guarantee each t₁ is sampled for at least one $i_{{\tau }_{\text{cp}}}$ (green). e The original ${\widehat{P}}_2$ operator only acts on data at sampled t₁, pushing the time domain point (black cross) directly to its sampled “target” value (circle). The new ${\widehat{P}}_2^{*}$ operator also acts on unsampled data, pushing the data not to the target (which is unknown), but towards a proxy value from a different $i_{{\tau }_{\text{cp}}}$ (green circle)

We can gain information about these unsampled points by choosing a different sampling scheme for each $i_{{\tau }_{\text{cp}}}$, such that for each value of t₁, we have a sampled data point for at least one $i_{{\tau }_{\text{cp}}}$ value in the neighborhood of points (Fig. 2d, Staggered sampling). When reconstructing an unsampled time point in one data set (at a particular $i_{{\tau }_{\text{cp}}}$), a sampled time point from another data set (at a different $i_{{\tau }_{\text{cp}}}$) can help constrain the allowed reconstruction values. For example, if we did not sample the (t₁, $i_{{\tau }_{\text{cp}}}$) point, we might sample the (t₁, $i_{{\tau }_{\text{cp}}}+1$) point, and make use of the fact that these two points should be of comparable value. Pushing the reconstructed value of an unsampled point towards the neighborhood of a sampled value at a different τ_cp will greatly suppress the size of the solution space for the final reconstructed image. This improves our ability to arrive at a correct value using the Difference Map operators. In this way, we are able to turn the high correlation among data sets into useful information; we do this by altering both our sampling regimen and our reconstruction algorithm, changing both $\widehat{P_0}$ and $\widehat{P_2}$. These two changes will define coDiffMap.

The new coDiffMap operators, ${\widehat{P}}_0^{*}$ and ${\widehat{P}}_2^{*}$

To access the correlation information using our sampling, we alter our sampling regimen $\widehat{P_0}$ in order to sample each $i_{{\tau }_{\text{cp}}}$ data set according to a different pattern, defining a new sampling operator ${\widehat{P}}_0^{*}\left(i_{{\tau }_{\text{cp}}}\right)$ that now depends on $i_{{\tau }_{\text{cp}}}$. In choosing a pattern, we must ensure that each unsampled data point has a corresponding sampled data point at a different τ_cp value. The distribution of sampled points we use is quasi-even in the first data set, and shifts and wraps for each subsequent data set (Fig. 2d). The actual sampling schedule is provided in the Supplementary Information. The data point closest to t₁ = 0 (at δt₁/2 in this data) is always sampled, for every slice, to help establish the correct integral of the frequency spectrum. This staggered sampling schedule was the minimal change to the schedule that had worked well before (Blum et al, 2019), but it is possible that other sampling schedules could perform comparably, so long as every t₁ has still been sampled for at least one $i_{{\tau }_{\text{cp}}}$.

To use the correlation information during our reconstruction, we alter the time-domain projection operator $\widehat{P_2}$ used during the Difference Map algorithm. The original $\widehat{P_2}$ operation resets data at sampled t₁ to their known values, but does not operate on data at unsampled t₁ values (Fig. 2e, Sampled t₁). Here, we add an operation acting on the data at unsampled t₁ values (Fig. 2e, Unsampled t₁). For a complex data point at an unsampled t₁ in a given f₂ column of a particular $i_{{\tau }_{\text{cp}}}$, $S_{\text{U}}\left(t_1,f_2,i_{{\tau }_{\text{cp}}}\right)$, the staggered sampling pattern guarantees that we have access to a “proxy” sampled data point at the same t₁ and f₂, but at a different $i_{{\tau }_{\text{cp}}}:S_{\text{ proxy }}\left(t_1,f_2,i_{{\tau }_{\text{cp}}}^{\prime }\right)$, with $i_{{\tau }_{\text{cp}}}^{\prime }\ne i_{{\tau }_{\text{cp}}}$. This sampled data point will act as a proxy to the correct data value, since it should lie in the same neighborhood $\left(T_n\left(t_1,i_{{\tau }_{\text{cp}}}^{\prime }\right)\approx T_n\left(t_1,i_{{\tau }_{\text{cp}}}\right)\right)$. If there is more than one available proxy, we choose the one with the most-similar τ_cp available from the sampling pattern. Instead of leaving the unsampled data points alone (as in the original ${\widehat{P}}_2$ operator), the new ${\widehat{P}}_2^{*}$ operator “pushes” the unsampled point towards the proxy value:

$$S_{\text{U}}\stackrel{{\widehat{P}}_2^{*}}{\to }{S}_{\text{U}}+p\left(S_{\text{proxy}}-S_{\text{U}}\right),$$ (3)

where 0 ≤ p ≤ 1 is a “push” parameter determining the strength of the nudge towards the proxy point (Fig. 2e), and S_proxy − S_U ≡ d is the distance from the unsampled point to the proxy point (notice that ${\widehat{P}}_2^{*}$ acts on the real and imaginary parts of the data separately). Thus, in the new ${\widehat{P}}_2^{*}$ operation, the sampled data points are reset to their sampled values, and the unsampled points are pushed a fraction p of the distance d to a different sampled value, which we know to be comparable to the correct value. Note that ${\widehat{P}}_2^{*}$ is really ${\widehat{P}}_2^{*}\left(i_{{\tau }_{\text{cp}}}\right)$ since the set of sampled and proxy values varies from slice to slice. This operation helps correctly establish the overall shape of the time domain data and prevents errors caused by drastic deviations in the reconstruction of unsampled data points.

Although the original derivation of the Difference Map algorithm assumed the use of projection operators, the ${\widehat{P}}_2^{*}$ operator is no longer a projection operator, since ${\left({\widehat{P}}_2^{*}\right)}^2\ne {\widehat{P}}_2^{*}$ (Elser, 2003). We tried incorporating the same information in the form of different operators which satisfied the projection requirement, but all that we tried gave worse results. Nevertheless, ${\widehat{P}}_2^{*}$ as defined above is only one example of a way to incorporate the information about correlations present in a pseudo-3D NMR experiment, and it is possible that other, even better implementations can be found.

To apply coDiffMap to the properly sampled data, we simply replace all instances of ${\widehat{P}}_2$ in the DiffMap method with ${\widehat{P}}_2^{*}$ to arrive at the reconstructed image $F^i={\widehat{P}}_2^{*}{\left({\widehat{D}}^{*}\right)}^i{S}^0$, where ${\widehat{D}}^{*}=\mathbb{1}+{\widehat{P}}_1\cdot \left(2{\widehat{P}}_2^{*}-\mathbb{1}\right)-{\widehat{P}}_2^{*}$.

Choosing the free parameters

Before using coDiffMap, we must first choose values for the two free parameters: the push parameter p and the number of iterations i. To choose the best values for p and i in a given experiment, we assume that we have access to one densely sampled 2D spectrum, which we use as a guide. After completing the sparse sampling experiment at a particular $N_{t_1}$, but before reconstruction, this “guide spectrum” may be used to optimize both p and i. This is shown in detail in the Supplementary Information, but there are a few important results worth mentioning here.

First, to find an optimal push parameter p, we must first pick an iteration number (here we choose i = 100), then reconstruct the target spectrum for a range of p values. It is advisable to use the longest-τ_cp spectrum as the guide spectrum, since this spectrum is the most different from the others, and therefore suffers the most if the push is too strong (as p → 1). In Fig. 3a, we show the RMS error in reconstructed amplitude for all 114 peaks in the densely sampled guide spectrum – the minimum error occurs at p = 0.92. Second, coDiffMap does not have the same dependence on i as the original DiffMap. Instead, it changes much more rapidly over the first several iterations, quickly reaching a plateau where the solution is often better than the DiffMap solution for very low $N_{t_1}$ (Fig. 3b). The results change very little over a broad plateau in i, before they eventually drift very slowly (over ≫ 10³ iterations) towards the same results obtained using the original DiffMap (Fig. 3c). We do not yet completely understand this behavior. However, studying the action of DiffMap and coDiffMap on the real NMR data, as well as on simplified toy models, reveals a consistent pattern. Namely, coDiffMap rapidly pushes the reconstructed spectrum into the “neighborhood” of the proxy data. Once there, it very slowly drifts towards a solution that approaches the original DiffMap solution at large iteration numbers. Additional insights may be gained by following this process in the time domain. The nontrivial dynamics of individual points as a function of iteration number may be seen in the Supplementary video.

Fig. 3.

a RMS error for all 114 peaks in the densely sampled guide spectrum $i_{{\tau }_{\text{cp}}}=12$ for $N_{t_1}=30$, i = 100, and many push parameters p. Slice $i_{{\tau }_{\text{cp}}}=12$ corresponds to the first point in a relaxation curve (inset). In the inset, R₂ are calculated using R₂ (1/τ_cp) =−(1/T)ln[I(1/τ_cp) / I_ref](Eq. 4). The minimum error occurs at p = 0.92. b Convergence of coDiffMap and DiffMap reconstructions for the residue Leu10 δ¹ at $N_{t_1}=30$ and p = 0.92. At this $N_{t_1}$, which is below the linear algebra limit for this peak (K(n)/2 = 37.5), and in a “danger zone” as defined by the metrics in a companion paper (Blum et al, 2019), the DiffMap-reconstructed peak amplitude (red) converges slowly, and in fact does not reach the correct, densely sampled amplitude (dashed line). In contrast, using coDiffMap (blue) the amplitude rapidly plateaus near the correct amplitude. c After exponentially many iterations, coDiffMap eventually drifts towards the incorrect DiffMap solution

To find a value for i with a safe minimum in error, we track the effect of varying iteration number on the guide spectrum, similar to how we chose p (see Supplementary Information). Finally, notice that p and i are not independent, in that the best value for p may depend on i, and vice versa; however, we found that performing this parameter-selection process once was sufficient to give good results. In our detailed application of coDiffmap below, we use i = 50 iterations and a push parameter p = 0.92. All coDiffMap calculations were performed in Igor Pro 8. We have also ported the code to Python 3, which is available on GitHub at https://github.com/robbyblum/DiffMap-coDiffMap-Python, and will also be available on the NMRbox platform at http://NMRbox.org (Maciejewski et al, 2017).

Application to Relaxation Dispersion Data

We apply our coDiffMap method to a series of twelve liquid-state 2D NMR spectra, from a ¹³CH₃ multiple-quantum CPMG relaxation dispersion experiment that was previously described in Frey et al (2013). Briefly, the ²H, ¹³C-methyl labeled Isoleucine, Leucine, Valine NMR sample of imidazole glycerol phosphate synthase (IGPS) from T. maritima was prepared as described previously (Lipchock and Loria, 2010; Lisi et al, 2018). A ¹³CH₃ multiple-quantum CPMG relaxation dispersion experiment was collected at 14.1 T and 30 °C with 120 t₁ increments and 1889 t₂ points, using the sequence described by Korzhnev et al (2004). In both dimensions, the data are zero-filled to the nearest power of 2, and cosine-bell apodization is used. The ¹H carrier was centered at 0.75 ppm (not 4.7 ppm as incorrectly stated in (Frey et al. 2013)) with a spectral width of 8500 Hz, while the ¹³C carrier was centered at 19.5 ppm with a spectral width of 3200 Hz.

The enzyme IGPS is a 51 kDa heterodimer (Lipchock and Loria, 2009), providing a reasonably complicated spectrum for a detailed, “deep-dive” analysis of our method (Blum et al, 2019). The twelve spectra correspond to τ_cp values of 0, 0.4167, 0.5, 0.625, 0.7682, 1.0, 1.4286, 2.0, 2.5, 3.333, 5.0, and 10.0 ms. We will refer to these below by their respective indices $i_{{\tau }_{\text{cp}}}=1,2,\dots ,12$. After filling the four quadrants of the time domain by Hermitian symmetry as discussed above (and in the Supplementary Information), each 2D spectrum is M × N = 256 × 4098. The first of these spectra, with τ_cp = 0, is a “reference” spectrum used to calculate R₂ rates below (this spectrum was also shown in Figs. 2 and 3 of Frey et al (2013)). In our data set, the reference spectrum $i_{{\tau }_{\text{cp}}}=1$ is quite different from the others, so we instead choose to make our ${\widehat{P}}_1$ mask using the $i_{{\tau }_{\text{cp}}}=2$ spectrum, although the mask could have been made using any of the $i_{{\tau }_{\text{cp}}}=2-12$ spectra. In the following, we assume that the reference spectrum $\left(i_{{\tau }_{\text{cp}}}=1\right)$ has been sampled densely, so that our reconstruction is completed on the remaining 11 spectra $\left(i_{{\tau }_{\text{cp}}}=2-12\right)$.

The original experiment acquired all $N_{t_1}^{\text{dense}}=128$ rows of data, so we mimic a sparse-sampling experiment by zeroing some of the original data with the ${\widehat{P}}_0^{*}$ operator, until we are left with until we are left with $N_{t_1}$ time domain points along t₁. First, we reconstruct the peak amplitudes and relaxation curves using $N_{t_1}=28$ points, comparing our results to the original densely sampled data sets. Then, we examine the behavior of the reconstructions as a function of $N_{t_1}$ and motivate the choice of $N_{t_1}=28$, showing that even lower $N_{t_1}$ are possible with certain trade-offs.

Results

In Fig. 4 we compare the percent error in peak amplitudes using DiffMap and coDiffMap, showing that coDiffMap improves upon the original DiffMap when pushing to low $N_{t_1}$. For coDiffMap, we use staggered sampling for all 11 slices, i = 50 iterations, and a push p = 0.92. We implement DiffMap using the same quasi-even sampling plus jitter (QUEST) pattern for all 11 slices (see Supplementary Information of Blum et al. 2019), i = 100 (Fig. 4a) or i = 10⁴ (Fig. 4b) iterations, and $\widehat{P_2}$ rather than ${\widehat{P}}_2^{*}$. We use both methods to reconstruct the columns containing all 114 peaks in each 2D spectrum, then make a histogram of the percent errors in the peak amplitudes. Figure 4a shows the good performance of the original DiffMap method at $N_{t_1}=75$ (which was the limit reached in the original study for this data set (Frey et al, 2013)), while Fig. 4b shows that the original DiffMap method has much poorer performance at $N_{t_1}=28$. If we instead switch to coDiffMap at $N_{t_1}=28$, the performance is greatly improved (Fig. 4c). In fact, 93% of the 1254 reconstructed peak amplitudes are within 5% of the dense values. The reconstruction is also very fast, requiring only 6.5 seconds to reconstruct 114 1D spectra, using a MacBook Pro laptop.

Fig. 4.

a-c Histogram of the errors in reconstructed peak amplitude for 114 peaks of varying quality and SNR across 11 data sets (1254 reconstructions). a The original method (DiffMap with the same QUEST sampling in each $i_{{\tau }_{\text{cp}}}$ slice, and i = 100) performs well at $N_{t_1}=75$. b Reducing $N_{t_1}$ to 28, the performance of DiffMap degrades. Note that at this low $N_{t_1}$, DiffMap can occasionally require ~10⁵ iterations (or many more) to converge, although we used i = 10⁴ here. c coDiffMap with staggered sampling across all slices and i = 50 performs much better at $N_{t_1}=28$. For 1254 reconstructions, 93% of the fractional errors are less than 5%

Next, we use these reconstructed peak amplitudes to produce R₂ data, to investigate whether the high quality amplitude reconstructions translate to high quality relaxation data. Relaxation rates R₂ are calculated as a function of 1/τ_cp according to

$${\text{R}}_2\left(\frac{1}{{\tau }_{\text{cp}}}\right)=-\frac{1}{T}\text{ln}\left[\frac{I\left(1/{\tau }_{\text{cp}}\right)}{I_{\text{ref}}}\right],$$ (4)

where I(1/τ_cp) is the peak intensity in the corresponding spectrum, I_ref is the intensity in the (densely-sampled) reference spectrum, and T is the total CPMG time, 40 ms in these experiments (Lipchock and Loria, 2010; Mulder et al, 2001). Here, we calculate I(1/τ_cp) and I_ref by looking at a grid of 9 frequency points centered on the peak of interest, and taking a simple sum of their amplitudes. Notice that this is the only context in which we use “peak intensity” I; The rest of our analysis refers to the “peak amplitude” (the amplitude at a particular (f₁,f₂) point). The leftmost point in a relaxation curve (as seen, for instance, in Fig. 5a) comes from the spectrum with the longest τ_cp value (i.e., $i_{{\tau }_{\text{cp}}}=12$ for our data set), and is the most different from its neighbors. For this reason, and because it is useful as a “guide” spectrum to determine p (as discussed above), as well as for other reasons discussed further below, we found that it is best to sample this spectrum densely. This allows us to fix the first point on each relaxation curve, protecting against the most significant adverse effects of a too-low sampling number $N_{t_1}$ or a too-high push parameter p

Fig. 5.

a-g Dense (open circles) and reconstructed (red open triangles) relaxation curves for seven residues with ΔR₂ > 2 s⁻¹, using $N_{t_1}=28$. The dense (thick black lines) and reconstructed (thin red lines) fits are very similar. The reconstructed fits use the densely sampled value for the first point (solid triangles) instead of the reconstructed first point, since we assume slice $i_{{\tau }_{\text{cp}}}=12$ was sampled densely. Uncertainties in the dense values (gray shading) are propagated from assumed amplitude uncertainties of 4σ_q for each amplitude, where σ_q is the quiescent noise far from the region of signal in the 2D spectrum. All fits incorporate this uncertainty. h,i Flat curves with ΔR₂ < 2 s⁻¹ also maintain their shape. j,k Reconstructed fit values for k_ex and p_Ap_BΔω² compared to their dense values, corresponding to the curves in a-g

We fit both the original and reconstructed data to the function (Palmer, 2004)

$${\text{R}}_2\left(\frac{1}{{\tau }_{\text{cp}}}\right)={\text{R}}_2^0+\frac{p_A{p}_B\mathrm{\Delta }{\omega }^2}{k_{\text{ex}}}\left(1-\frac{2\text{tanh}\left(k_{\text{ex}}{\tau }_{\text{cp}}/2\right)}{k_{\text{ex}}{\tau }_{\text{cp}}}\right).$$ (5)

Although strictly not appropriate for multiple quantum relaxation data, Eq. 5 accurately reproduces k_ex and the amplitude of the dispersion curves (Lipchock and Loria, 2009). The limited number of fit parameters in this expression allows us to better assess the quality of our coDiffMap reconstructions. Of the three fitting parameters ${\text{R}}_2^0$, k_ex, and p_Ap_BΔω², we focus on the latter two below. We show the reconstructed R₂ and corresponding fits using $N_{t_1}=28$ in Fig. 5, for seven residues with $\mathrm{\Delta }{R}_2\equiv R_2\left(1/{\tau }_{\text{cp}}^{\text{max}}\right)-R_2\left(1/{\tau }_{\text{cp}}^{\text{min}}\right)>2{\text{s}}^{-1}$ (Fig. 5a–g), as well as for two residues with little to no relaxation dispersion (Fig. 5h,i). Although we show the reconstructed values for the first R₂ point, the fits are all done using the densely sampled value for this point, as would be the case in an actual experiment using our protocol. Even though the sampling number $N_{t_1}=28$ is at or below the linear algebra limit K(n)/2 (Eq.2) for four of the seven relaxing residues shown in Fig. 5a–g, corresponding to a compression ratio at or below $\mathcal{C}=1$, the fit parameters k_ex and p_Ap_BΔω are comparable between the reconstructed and dense data (Fig. 5j–k). We explore the $N_{t_1}$ -dependence of these results below, including the justification for our choice of $N_{t_1}=28$.

Dependence on sampling fraction

Compared to the ${\widehat{P}}_2$ used in the original DiffMap (Frey et al, 2013; Blum et al, 2019), the ${\widehat{P}}_2^{*}$ used in coDiffMap has an important new property. Namely, for ${\widehat{P}}_2^{*}$ every data point in the time domain has some operation acting on it. This seems to help protect against the significant errors that arise from intraband aliasing, which are described further in a companion paper (Blum et al, 2019) and elsewhere (Maciejewski et al, 2009; Frey et al, 2013). These errors from aliasing caused problems for sampling below $N_{t_1}=75$ (Frey et al, 2013), and stopped us from being able to extend the methods developed in our companion paper (Blum et al, 2019) to many peaks at once. The fact that coDiffMap mitigates these errors is an important factor in its success. However, coDiffMap works under the assumption that the data in a pseudo-3D experiment are similar from one 2D slice to the next; if this is not true, then that introduces a new source of reconstruction error.

In Fig. 6 we show an example of this kind of error, using the 1D spectrum at the f₂ coordinate of the residue Val79 γ² (Fig. 6a, showing overlaid spectra for $i_{{\tau }_{\text{cp}}}=2-12$). Our data set includes artifacts at the edges of the spectra, which alternate in sign from one $i_{{\tau }_{\text{cp}}}$ to the next, rather than varying smoothly as coDiffMap assumes (Fig. 6a, inset). We demonstrate the effect of these artifacts by examining the point spread function (PSF) for the quasi-even sampling pattern used for $i_{{\tau }_{\text{cp}}}=2$, centered at the location of the peak of interest (Maciejewski et al, 2009). As explained further in a companion paper (Blum et al, 2019), the intraband aliases of this PSF will indicate which frequency points “communicate” with the peak of interest during reconstruction (Fig. 6b). When $N_{t_1}=26$ or 27, the aliases in the PSF overlap with the fast-fluctuating artifacts at the edge of the spectrum (Fig. 6b, circled regions). These “artifact collisions” tend to cause errors in the reconstruction, as shown in the reconstruction in Fig. 6d. The reconstructions work well at $N_{t_1}=28$ and $N_{t_1}=25$, above and below the artifact collision (Fig. 6c,e). In Fig. 6f, we indicate the $N_{t_1}$ values for which Val79 γ² (dark blue) is predicted to have intraband aliases with artifact collision, as well as the $N_{t_1}$ values for which any one of the seven relaxers in Fig. 5a–g is predicted to have at least one artifact collision. We predict these $N_{t_1}$ values by adapting the mask collision metric from Blum et al (2019), but only checking for intraband alias collisions with artifact regions. We calculate the alias overlaps centered only on the location of the peak of interest rather than repeating the calculation for each of the 9 points that define I, but this seems sufficient to predict significant errors. It is worth emphasizing the key difference between this result for coDiffMap and the results using DiffMap: whereas reconstructions using DiffMap tend to have significant errors when aliases overlap with any mask(f) ≠ 0 region (see Blum et al, 2019), reconstructions using coDiffMap tend to show errors only when aliases overlap with the artifact support (mask(f)=2) regions, which is where the signal changes sign from one $i_{{\tau }_{\text{cp}}}$ to the next in this data set. Further, the artifact collision errors for reconstructions with coDiffMap tend to be smaller than the aliasing errors for reconstructions with DiffMap. Finally, these errors may be removed entirely by acquiring data in a way that avoids these kinds of fluctuating artifacts, in order to better match the assumptions underlying the ${\widehat{P}}_2^{*}$ operator.

Fig. 6.

Effect of artifact variations. a Eleven 1D spectra containing the peak from the residue Val79 γ² (marked with a red arrow), which is discussed in Fig. 5b. Both edges of each spectrum have artifacts regions (only the left edge is shown), which change sign from one $i_{{\tau }_{\text{cp}}}$ to the next, such that odd $i_{{\tau }_{\text{cp}}}$ have positive artifacts and even $i_{{\tau }_{\text{cp}}}$ have negative artifacts (inset). b Absolute value of the point spread function for a delta-function peak at the same location as the peak of interest. Intraband aliases of the peak of interest land in the artifact region around $N_{t_1}=27$. c-e The reconstructions are reasonable except when an alias collides with an artifact region, for instance at $N_{t_1}=27$ (d), demonstrating the dangers for data which contradict the assumption of slow variation across 2D spectra. f $N_{t_1}$ for which there is at least one artifact collision, for any of the seven relaxers in Fig. 5a–g (pink), or for just Val79 γ² (dark blue)

These artifact collisions can cause sharp errors at particular $N_{t_1}$, but they are not the only reasons for errors as N_t1 is lowered, and they are not the most important source of errors for coDiffMap. As mentioned above, the staggered sampling regimen chooses a proxy value from the slice with the most-similar τ_cp available from the staggered sampling, since spectra acquired using more similar τ_cp are more similar for this kind of pseudo-3D data set (except for artifacts). As $N_{t_1}$ is lowered and the sampling becomes more sparse, the proxy value must be drawn from a slice at an increasingly distant $i_{{\tau }_{\text{cp}}}$, where the data are more likely to be different. This problem will be most pronounced for the longest-τ_cp value spectrum ($i_{{\tau }_{\text{cp}}}=12$ here) since the spectra change more rapidly at these τ_cp values. This is another reason to sample this spectrum densely, in addition to the reasons discussed above. The next most vulnerable spectrum is $i_{{\tau }_{\text{cp}}}=11$; for this spectrum, we show the furthest-$i_{{\tau }_{\text{cp}}}$ jump required to find enough proxy values in Fig. 7a, with the jumps illustrated in the inset. This “furthest-required jump” does not change linearly with $N_{t_1}$, but rather remains steady from $N_{t_1}=127$ down to $N_{t_1}=44$, below which it begins to increase more and more rapidly. In the “stair-step” between $N_{t_1}=32$ and $N_{t_1}=27$ (Fig. 7a, shaded region) the required jumps still extend to less than half the data, but have begun to extend beyond the region of significant R₂ change. It is below this region of $N_{t_1}$ that we begin to see significant deviations from the dense data, which we will explore further below.

Fig. 7.

a The proxy values used by ${\widehat{P}}_2^{*}$ come from increasingly different 2D spectra as $N_{t_1}$ is lowered (inset), increasing the furthest jump in τ_cp required for $i_{{\tau }_{\text{cp}}}=11$ (blue steps). For $27\le N_{t_1}\le 32$ (gtray striped region), the required jumps have become significant relative to the size of the region where R₂ is changing fastest. b-e The dense (open circles) and reconstructed (open triangles) R₂ values and fits (thick black lines and thin red lines respectively), for Leu153 δ² for the four values of $N_{t_1}$ indicated in a. The reconstructed fits always use the densely sampled R₂ value for the first point (solid triangles). As $N_{t_1}$ is lowered, the first few points are “dragged down” by their neighbors. Eventually, the entire curve is severely flattened. f-g Errors in reconstructed k_ex and p_Ap_BΔω² across $N_{t_1}$ for the seven relaxers shown in Fig. 5a–g, relative to the uncertainties from the dense fits σ_Dense. Horizontal black lines indicate ±2σ_Dense, and RMS values are shown in bold lines for clarity. The furthest-$i_{{\tau }_{\text{cp}}}$ jumps for $i_{{\tau }_{\text{cp}}}=11$ (from a) are included for comparison (blue shading). Small errors steadily accumulate in the reconstructed fit parameters until $N_{t_1}~25$, below which errors increase dramatically

In Fig. 7b–e, we show the resulting relaxation curve for the residue Leu153 δ² at four values of $N_{t_1}$ (chosen for reasons discussed below). Generally, as one might expect, the leftmost reconstructed R₂ point changes most rapidly with dropping $N_{t_1}$ (this is part of the motivation for sampling that one point densely, and using that dense R₂ value in the fits); as $N_{t_1}$ drops further, the entire reconstructed dispersion curve eventually flattens. At $N_{t_1}=44$ (Fig. 7b), the reconstruction is not identical to the dense data, but it is very close. At $N_{t_1}=28$ (Fig. 7c), the leftmost reconstructed R₂ points have noticeably dropped, being “dragged down” by their neighbors (open red triangles in Fig. 7c). At $N_{t_1}=22$ (Fig. 7d), the errors in the leftmost points have become even larger, while the rest of the reconstructed R₂ values are still close to the dense data. Eventually, at the lowest $N_{t_1}$ where staggered sampling can sample every t₁ for at least one $i_{{\tau }_{\text{cp}}}$($N_{t_1}=13$, Fig. 7e), the entire R₂ curve is severely flattened, making $N_{t_1}=13$ clearly too low a sampling number to be useful. The reconstructed fits are done to a curve with the densely sampled leftmost R₂ $\left(i_{{\tau }_{\text{cp}}}=12\right)$, and the reconstructed R₂ everywhere else. This helps the reconstructed fits to match the dense fits even when the leftmost reconstructed R₂ values have begun to degrade (Fig. 7d).

In Fig. 7f we examine the reconstructed exchange parameter k_ex as a function of $N_{t_1}$ for the seven relaxers shown in Fig. 5a–g, by showing the error in the fit value $\epsilon \equiv k_{\text{ex}}^{\text{recon}}-k_{\text{ex}}^{\text{dense}}$, scaled relative to the uncertainty from the dense fit value, σ_Dense. We repeat this procedure for p_Ap_BΔω² in Fig. 7g, and include the RMS values (ε/σ_Dense)_RMS for clarity. These relative errors increase gradually until crossing below $N_{t_1}~25$, where they begin to deviate significantly. This jump in errors occurs as the number of “connected” spectra grows, as shown in Fig. 7a and as replicated in Fig. 7f–g for comparison (blue shaded region).

We have shown that sharp errors can occur for particular $N_{t_1}$ when there are artifact collisions, and that systematic errors occur as $N_{t_1}$ drops too low for the proxy data to be accurate. We combine our metrics for both of these effects in Fig. 8, for the seven relaxing residues shown in Fig. 5a–g. At each $N_{t_1}$, we indicate not only the furthest-required $i_{{\tau }_{\text{cp}}}$ jump for slice $i_{{\tau }_{\text{cp}}}=11$ (as in Fig. 7a), but also whether any of the seven relaxers experience at least one artifact collision (as in Fig. 6f). This combination of metrics informs our decision of $N_{t_1}$; however, the final choice of sampling fraction depends on the details of the experiment, and how much time savings is required. The most conservative sampling fraction is about one third ($N_{t_1}=44$ here), since this ensures each spectrum only appeals to nearest-neighbors for proxy values during the reconstruction (like Fig. 7b). It is possible to push below this limit, to a region of $N_{t_1}$ where the connected slices are not too different; for instance, $N_{t_1}=28$ avoids the artifact collisions between $N_{t_1}=27$ and 32 (Fig. 8) to achieve good results (Fig. 7c). It is possible to push even further down in $N_{t_1}$ to a region where some of the data has begun to degrade but the fits are still reasonable, for instance at $N_{t_1}=22$ in our data (Fig. 7d), because of a densely-sampled first τ_cp value. Pushing all the way down to the minimum possible sampling fraction ($N_{t_1}=13$ in our case) is inadvisable, since this requires every slice to pull proxy values from all other slices, over-correlating the data and flattening the relaxation curves (Fig. 7e).

Fig. 8.

The furthest required $i_{{\tau }_{\text{cp}}}$ jump to obtain proxy values for $i_{{\tau }_{\text{cp}}}=11$, as shown in Fig 7a. In addition, at certain $N_{t_1}$ values (red bars), the seven relaxing residues shown in Fig. 5a–g encounter alias collisions with regions of the spectra containing artifacts. We avoid these regions in our choices for $N_{t_1}$ (arrows), except for $N_{t_1}=13$

Error Estimation

As we change $N_{t_1}$, errors can appear in a few ways. First, for a given peak, we have seen that sharp errors occur when there are alias overlaps between the peak of interest and the artifact support region. Second, we have also seen that errors can grow rapidly when $N_{t_1}$ is too low for the staggered sampling regimen to provide accurate proxy data. We now discuss a third kind of error: the slow accumulation of noise-like errors with lower $N_{t_1}$. In Fig. 9, we explore these errors using $i_{{\tau }_{\text{cp}}}=12$ as an example. We start by examining the histogram of absolute errors in peak amplitude for all 114 peaks at each $N_{t_1}$ ($N_{t_1}=28$ is shown in Fig. 9a). We fit the histogram at a given $N_{t_1}$ to a Gaussian, and study the offsets μ and standard deviations σ from the fit (Fig. 9a, inset). In Fig. 9b, we plot μ$(N_{t_1})$ (open circles) and σ$(N_{t_1})$ (closed circles) as functions of $N_{t_1}$ for the spectrum $i_{{\tau }_{\text{cp}}}=12$.

Fig. 9.

a Histogram of the errors in reconstructed peak amplitudes, for all 114 peaks at $N_{t_1}=28$ in spectrum $i_{{\tau }_{\text{cp}}}=12$. We fit Gaussians to these histograms and extract standard deviations and offsets σ and μ (inset), which we plot in b as functions of $N_{t_1}$. Predictions from the simplified model for the offset μ_E (dashed blue line) and standard deviation σ_E (solid red line) roughly bound most of the errors, without matching them closely. c Peak errors (black circles) at $N_{t_1}=28$ for all 114 peaks in slice $i_{{\tau }_{\text{cp}}}=12$, and prediction bands from the simplified model. Shaded regions indicate artifact collisions. We show the expected average error μ_E = μ_ν (128/28−1) (blue line) with prediction bands ${\mu }_E\pm 2{\sigma }_{\nu }\sqrt{128/28-1}$ (red lines), where μ_ν and σ_ν are the average and standard deviation of the signal outside the ${\widehat{P}}_1$ support mask for each peak in spectrum $i_{{\tau }_{\text{cp}}}=12$. Errors for $i_{{\tau }_{\text{cp}}}=2-11$ are shown in open circles

We will compare these errors to a simple model that seemed to work for the reconstruction errors of the original DiffMap (see Blum et al, 2019). Although that model did not incorporate the operation of the new ${\widehat{P}}_2^{*}$ operation, we try adopting it here to see whether it can still be useful for anticipating the reconstruction errors of coDiffMap. In that simple model, we assume that the reconstruction errors depend on the average μ_v and the standard deviation σ_v of the signal outside the ${\widehat{P}}_1$ support mask (not to be confused with parameters of the Gaussian fits, μ and σ). The average μ_v of the signal outside the ${\widehat{P}}_1$ mask may be nonzero since it contains not only noise, but also tails of spectral features, which are net positive in our data. As we go down in $N_{t_1}$, we expect the noise and tails outside the mask to increasingly affect our reconstruction errors. If the features outside the mask support region in a given 1D spectrum have standard deviation σ_v and average value μ_v, the model predicts (see Blum et al (2019) for details) a standard deviation ${\sigma }_E\left(N_{t_1}\right)={\sigma }_v\sqrt{N_{t_1}^{\text{dense}}/N_{t_1}-1}$ (Fig. 9b, solid red line), and a mean ${\mu }_E\left(N_{t_1}\right)={\mu }_v\left(N_{t_1}^{\text{dense}}/N_{t_1}-1\right)$ (Fig. 9b, dashed blue line).

In Fig. 9b, we see a few key differences between this simple model (and typical DiffMap performance) and the results from coDiffMap, in both μ and σ. First, μ does not grow as quickly as the model μ_E, as the ${\widehat{P}}_2^{*}$ operator leads to less folding of the signal outside the mask onto the peaks, consistent with our picture of how DiffMap works (Blum et al, 2019). While this is true for i = 50, recall that as i → ∞ we expect the coDiffMap solution to approach the DiffMap solution, so the errors are likely to approach this model. Second, although σ_E fits the prediction at high $N_{t_1}$, it seems to begin leveling off around $N_{t_1}=64$, approaching a smaller value than the model. We explore this behavior using simulations in the Supplementary Information. Because of these two differences, the simple noise model for DiffMap seems to bound rather than describe the coDiffMap results. In Fig. 9c, we compare the results of the coDiffMap to the simple model for all 114 peaks. The prediction bands ${\mu }_E\left(N_{t_1}\right)\pm 2{\sigma }_E\left(N_{t_1}\right)$ (Fig. 9c, red solid lines) bound most of the errors for all 11 τ_cp slices. These prediction bands were calculated by drawing μ_v and σ_v from a single densely sampled spectrum ($i_{{\tau }_{\text{cp}}}=12$), and could be used to predict the expected reconstruction errors before choosing $N_{t_1}$ for a sparse sampling experiment. The peaks which have alias artifact collisions at $N_{t_1}=28$ are indicated in red shading, which were predicted just by knowing the peak locations and artifact locations. While there are some deviations from the predicted error bands, at least half of the largest deviations occur for the peaks which are predicted to have artifact collisions for this $N_{t_1}$ (red shading), which can lead to worse reconstructions. Nevertheless, notice that all of the errors that we do see here are small relative to the size of the peaks themselves, as can be seen from the histogram of the relative errors at this same $N_{t_1}$, shown in Fig. 4c.

Discussion

We began our sparse sampling work in the search for a method that would work well with our ³¹P MRI of solids data (Frey et al, 2012), achieving good results across an entire 2D spectrum (Frey et al, 2013) using the newly-developed DiffMap, and eventually pushing towards the fundamental limits of the DiffMap method by analyzing individual features from a single 2D spectrum (Blum et al, 2019). We were able to extend these results to reconstructing an entire pseudo-3D experiment by developing coDiffMap, which makes use of the correlations that exist among 2D slices of the pseudo-3D data set. This new method achieves sparse sampling numbers close to or slightly better than the fundamental limits of the original DiffMap, while still being able to provide high-quality reconstructions globally.

An accelerated relaxation dispersion experiment using coDiffMap requires four steps: 1. densely sampling at least one 2D spectrum (to create a mask, and act as a guide for the free parameters), 2. choosing an $N_{t_1}$ and sparsely sampling the remaining 2D spectra (using a sampling pattern which cumulatively samples all t₁ values), 3. determining best values for p and i using the guide spectrum, and 4. reconstructing by repeatedly applying the coDiffMap operator ${\widehat{D}}^{*}$. We now detail these four steps.

First, at least one 2D spectrum is sampled densely, as usual. After processing (proper phasing and Fourier transforming), the densely sampled spectra will be used to identify peaks, create a ${\widehat{P}}_1$ support mask where signal is expected to appear, and optimize the free parameters (in step 3 below). We choose to densely sample the $i_{{\tau }_{\text{cp}}}=12$ slice in our data for three reasons: (a) to choose the most cautious value for p, since an overly strong push will affect the $i_{{\tau }_{\text{cp}}}=12$ slice first (Fig. 3a), (b) to protect the reconstructed fits from errors in the leftmost points of the dispersion curve (Fig. 5), and (c) to protect our reconstructions from a too-low $N_{t_1}$, which affects the leftmost points first (Fig. 7). This spectrum could also be used to create the ${\widehat{P}}_1$ mask although we used a densely sampled $i_{{\tau }_{\text{cp}}}=2$ slice to make the ${\widehat{P}}_1$ mask for our data.

Second, all remaining 2D spectra are sparsely sampled. The sampling fraction $N_{t_1}$ depends on the goal of the experiment; the trade-off between reconstruction quality and experiment speed is discussed in the prior two sections. Each spectrum would be sampled according to a slightly different scheme, such that the union of sampling patterns across τ_cp fills the dense grid of time data along t₁.

Third, the push parameter p and iteration number i may be optimized using the densely-sampled guide spectrum (ideally, this is the longest-τ_cp spectrum, which corresponds to the first, most sensitive R₂ value in a relaxation curve). This is done as described in the Supplementary Information, by artificially sparse-sampling the guide spectrum and reconstructing it with different values of p and i after sparsely sampling the other spectra. This process can be automated, and would be fast because of the speed of our algorithm.

Fourth and finally, the other sparsely-sampled spectra are reconstructed using the coDiffMap algorithm with the optimized p and i values. This requires the ${\widehat{P}}_2^{*}$ operator, which acts on a given slice by making use of the sampled data from other slices.

Some considerations about choosing an $N_{t_1}$ are particular to the data we used in this study. In some contexts it is not unusual to acquire many tens of 2D slices, far more than the ~ 10 being sparsely sampled here. Further, the spectra here change gradually and (typically) monotonically across the 3rd dimension, justifying the assumption that there will be correlations between nearby $i_{{\tau }_{\text{cp}}}$. Depending on the suddenness and magnitude of the changes between the 2D slices in a pseudo-3D experiment, it is important that the number of sampled 2D slices be high enough to resolve these changes. Generally, the attainable $N_{t_1}$ will depend on the number of available 2D slices relative to the changes between slices.

This algorithm allows us to selectively reconstruct individual 1D “columns” (constant f₂) of interest, rather than dealing with the entire 2D spectrum at each iteration. This, combined with the FFT-basis of the algorithm, makes our method very fast, requiring only 6.5 seconds to reconstruct 114 1D spectra with i = 50 iterations each. The speed of the algorithm also enables rapid determination of the free parameters p and i. Further, this method is potentially useful whenever highly correlated spectra are repeatedly acquired, for instance in CEST (van Zijl and Yadav, 2011; Vallurupalli et al, 2012; Long et al, 2015).

To characterize the performance of DiffMap and coDiffMap using a more objective metric, we have compared the number of sampled points to the number of features in the spectrum using the compression ratio $\mathcal{C}$, where DiffMap achieved a mean compression ratio ${\overline{\mathcal{C}}}_{\text{bu}}\approx 1.3$ (Blum et al, 2019) when reconstructing just one or a few peaks at a time, and coDiffMap achieved compression ratios near and sometimes below $\mathcal{C}=1$ by exploiting the correlations among 2D slices. For reconstructions of relaxation data using MDD, Linnet and Teilum (2016) provide a very useful comparison between their achieved sampling fractions and sparsity of their data, which helpfully allows us to provide another reference point for our results as follows. Defining sparsity as the fraction of data points in a restricted part of the spectrum with intensities below six times the noise level, the sparsities of four different spectra reconstructed by Linnet and Teilum (2016) were $\mathcal{S}=92$, 86, 78, and 71%, compared to the corresponding sampling fractions (“coverages”) of c = 25, 35, 60, and 80% for MDD. We summarize these results using an effective compression ratio ${\mathcal{C}}_{\text{LT}}=c/(1-\mathcal{S})$, (similar to $\mathcal{C}$ but defined in 2D) which for the four spectra used by Linnet and Teilum has a mean value ${\overline{\mathcal{C}}}_{\text{LT}}\approx 2.8$ for MDD. The corresponding compression ratio for coDiffMap comes out to ${\mathcal{C}}_{\text{cDM}}=2N_{t_1}/\overline{K}$ (see Blum et al, 2019), where $\overline{K}$ is the average K (n) calculated between 0.6 and 1.1 ppm. In our data, $\overline{K}/M\approx 23\%$ (a sparsity of 77%), so that the effective compression ratio for coDiffMap is ${\mathcal{C}}_{\text{cDM}}\approx 1$ for $N_{t_1}=28$. This compression ratio drops further if we use $N_{t_1}=22$, where the fits are still reasonable; the LA limit $\mathcal{C}\ge 1$ is not a fundamental limit for coDiffMap as it is for DiffMap (Blum et al, 2019), again because we are adding information to the reconstruction problem by using the correlations among 2D slices in the pseudo-3D data set. While ${\mathcal{C}}_{\text{cDM}}\approx 1$ for coDiffMap is better than ${\overline{\mathcal{C}}}_{\text{LT}}\approx 2.8$ reported for MDD, the two methods are not directly comparable, since here we are taking advantage of the correlations among 2D slices. This shows the benefit of making use of the correlations available in pseudo-3D experiments.

The coDiffMap method lowers the number of required sparse samples to—or even below—the expected DiffMap limit of $2N_{t_1}\approx K(n)$ or $\mathcal{C}\approx 1$. For an individual sparsely-sampled 2D spectrum, $N_{t_1}=28$ corresponds to about 22% sampling fraction $\left(N_{t_1}/N_{t_1}^{\text{dense}}\right)$. Since we have assumed we have access to the dense data from $i_{{\tau }_{\text{cp}}}=1$ (for the R₂ calculations), $i_{{\tau }_{\text{cp}}}=2$ (for the $\widehat{P_1}$ mask), and $i_{{\tau }_{\text{cp}}}=12$ (for optimization), the total effective sparse sampling rate across the whole experiment is about 41% (or 35% if we had made the $\widehat{P_1}$ mask using $i_{{\tau }_{\text{cp}}}=12$ instead). However, this number is subject to change for other applications of this method depending on the number of 2D slices, the sparsity of the spectrum, and other factors like how many 2D slices are densely sampled in step 1 above. Depending on these factors, the effective sampling rate for the whole experiment may be pushed closer to the sampling rate achieved for individual slices. Nevertheless, even the most conservative result of 41% sampling still represents a 1.4× reduction from prior results, for global reconstructions available in seconds. By exploiting information available in the correlated time-domain data in higher dimensions, we show that coDiffMap is a powerful new approach to accelerate NMR relaxation dispersion experiments.