Dynamic Imaging of Glucose and Lactate Metabolism by 13C-MRS without Hyperpolarization

Metabolic reprogramming is one of the defining features of cancer and abnormal metabolism is associated with many other pathologies. Molecular imaging techniques capable of detecting such changes have become essential for cancer diagnosis, treatment planning, and surveillance. In particular, 18F-FDG (fluorodeoxyglucose) PET has emerged as an essential imaging modality for cancer because of its unique ability to detect a disturbed molecular pathway through measurements of glucose uptake. However, FDG-PET has limitations that restrict its usefulness in certain situations and the information gained is limited to glucose uptake only.13C magnetic resonance spectroscopy theoretically has certain advantages over FDG-PET, but its inherent low sensitivity has restricted its use mostly to single voxel measurements unless dissolution dynamic nuclear polarization (dDNP) is used to increase the signal, which brings additional complications for clinical use. We show here a new method of imaging glucose metabolism in vivo by MRI chemical shift imaging (CSI) experiments that relies on a simple, but robust and efficient, post-processing procedure by the higher dimensional analog of singular value decomposition, tensor decomposition. Using this procedure, we achieve an order of magnitude increase in signal to noise in both dDNP and non-hyperpolarized non-localized experiments without sacrificing accuracy. In CSI experiments an approximately 30-fold increase was observed, enough that the glucose to lactate conversion indicative of the Warburg effect can be imaged without hyper-polarization with a time resolution of 12s and an overall spatial resolution that compares favorably to 18F-FDG PET.

the excitation process is not entirely specific and analysis of glucose uptake is also complicated by the strong pH dependence of the exchange rate 11 . The imaging time can also be quite long if adequate spatial resolution is desired.
A more direct and specific way to track in vivo metabolism is to follow the metabolism of exogenous tracers by magnetic resonance spectroscopy or spectroscopic imaging (MRS/MRSI). The breakdown of labeled exogenous metabolic tracers can be tracked non-invasively and precisely by 13 C MRS. This has the advantage over static techniques, which measure the equilibrium state that may stem from many processes. However, widespread use of 13 C MR in the clinic has been limited by poor sensitivity stemming from three fundamental reasons. First, even in the best-case scenario, the effective concentration of an exogenous contrast agent is at least 1000 times lower than that of the water signal detected in conventional MRI. Further complicating the issue, 13 C has a low gyromagnetic ratio, which translates into a relative sensitivity only ~2-6% that of 1 H for an equivalent number of nuclei. Finally, metabolic processes are inherently transient. The usual approach to overcoming low sensitivity is to simply acquire the signal for longer and average the scans. Often, however it is the metabolic flux, which is of interest rather than the steady-state concentrations as enzymatic rates are a direct link to the activity of metabolic enzymes that are potential targets in cancer and other pathologies. Long signal averaging makes non-invasive interrogation of the rate impossible.
Dissolution dynamic nuclear polarization (dDNP) was developed to enhance signal to noise in 13 C MR and make rapid dynamic imaging of 13 C labeled substrates and their metabolic products possible. Dynamic nuclear polarization makes use of the fact that unpaired electrons in a paramagnetic molecule can be aligned to a magnetic field to a much greater extent than the atomic nuclei with spins detected by MRI. This alignment can then be transferred to the atomic nuclei for detection. Since the signal in MRI is proportional to the degree of alignment, this polarization transfer results in a very large increase in the MRI signal, >10,000 times in favorable circumstances 12 . This process occurs most efficiently at low temperatures near ~1 K. By rapidly dissolving the frozen tracer and bringing it quickly to room temperature, a polarization sufficient to conduct metabolic MRI can be realized. After dissolution, the polarization decays by the liquid state spin lattice relaxation time of the target nuclear spin.
As a technique, dDNP is an impressive technical achievement and has demonstrated tremendous potential for metabolic imaging in vivo 13,14 but it has its limitations, especially in a clinical setting. Hyperpolarization is limited to a small set of molecules whose relaxation time is long enough that the enhanced polarization is not lost before the kinetics can be determined. Many key metabolites, such as glucose, have short relaxation times, and are difficult, or impossible, to image with dDNP for this reason. Since one of the fundamental limitations of 13 C MRI is noise, it is logical that a method that reduces noise can greatly extend the utility of the technique. We show here that by considering the natural sparsity of the signal matrix, it is possible to get an order of magnitude improvement in SNR using rank reduction of the signal tensor by tensor factorization. The increase in signal to noise is large enough that it may be possible perform dynamic 13 C tracer imaging of some 13 C tracers without DNP.

SVD Based Low Rank Denoising for Dynamic Single Voxel Spectroscopy
We start by considering the simplest example, the dynamic non-localized pulse-acquire experiment (the same method can also be applied to dynamic localized spectroscopic imaging), as it can be described by basic concepts of linear algebra 15 . A key observation is that after the injection of a tracer, the intensities of peaks change as the tracer is broken down to its metabolites but the chemical shifts are largely invariant. The time independence of the chemical shifts suggests the spectral and kinetic information are separable; an n x m signal matrix M can therefore be written as a linear combination of a small number of vectors representing spectra (u) multiplied by an equal number of vectors representing kinetics (v): The singular value decomposition (SVD) theorem guarantees any matrix can be reconstructed fully from n vectors (where n < m) in this manner and the weights σ are equal to the square roots of the eigenvalues of M T M. In reality, the true signal is corrupted by noise. Noise reduction is achieved by reducing the rank of M, which is the vector space spanned by the columns (spectra) of M. More intuitively, the rank of M is the number of independent spectra n in Eq. 1. A rank of 1 implies the signal can be completely described as a single spectrum which decays with a single time profile. The spectra and time profile can be any arbitrary shape, but it is uniform in this case -no individual peak in the spectra decays at a different rate than any other. A signal matrix with a rank of 2 can be described as a linear combination of two spectra, with two distinct time profiles. A rank 3 matrix is a linear combination of three spectra with three distinct time profiles and so on. It is easy to see from this definition that we expect the rank (r) of the noise free matrix to be approximately equal to the number of metabolites in the sample, since noise is random and not correlated in either frequency or time. To get an approximation to M of rank r, we simply set the lowest r-n diagonal entries in Σ in Eq. 1 equal to zero. This result is guaranteed by the Eckart-Young-Mirsky theorem to be the best low rank approximation in the least squares sense to the original signal matrix M 16 . Since the noise-free solution is inherently low rank by the biochemistry of the problem, this solution is also likely to be an excellent approximation to the noise-free signal.
This method was first tested on data obtained from 41 mice by following the metabolism of a hyperpolarized 13 C tracer in a single pulse (not spatially resolved) MRI experiment. A volume of 300 µL of a 98 mM solution of hyperpolarized [1-13 C]pyruvate was injected into the tail vein of nude mice bearing tumor xenografts in the left leg. Cancer cells exhibit the Warburg effect and have higher lactate than the normal tissue. The dissolution process involved in making hyperpolarized pyruvic acid is not always perfect, resulting in spectra of varying quality. Under optimized conditions, the signal is strong enough after DNP that the main pyruvate (173 ppm) to lactate (185 ppm) conversion can be easily quantified without additional signal processing. In others, the signal is barely detectable. Since the biochemistry is the same in each of these cases, we know a priori the peak positions www.nature.com/scientificreports www.nature.com/scientificreports/ and approximate kinetics in the noisy data. This property makes the pyruvate dDNP experiment an excellent test platform for the accuracy of signal reconstruction. sVD Based Low Rank Denoising Gives an order of Magnitude Improvement in sNR in DNp 13 C Tracer Experiments Figure 1A shows an example of noisy pyruvate and lactate peaks in a dDNP experiment. The smaller peaks corresponding to alanine and pyruvate hydrate are completely buried within the noise. Even averaging 5 scans together is insufficient to accurately quantify the lactate or to detect the minor peaks. Averaging also results in a significant loss of time resolution (Fig. 1A). Rank reduction by SVD gives a substantial (9.3 fold) improvement in signal to noise (Fig. 1C). If we define signal to noise as the maximum value of the most intense peak divided by the standard deviation of the signal in a region of the where no peak is expected, signal to noise increased by nearly an order of magnitude (mean = 9.4, median = 7.3) after rank reduction to a rank of 5 (Fig. 1D). Figure 1. SNR improvement in noisy data using rank reduction. (A) Dynamic data from a 13 C pyruvate tracer single pulse dynamic nuclear polarization experiment with low signal to noise. (B) Signal from A averaged over 5 scans. Even with averaging, the signal is too weak to be quantified accurately. (C) Signal reconstruction using rank reduction to a rank of 5. The peaks corresponding to the two main metabolic products pyruvate and lactate are now clearly visible in the reconstructed spectra, along with two minor peaks corresponding to alanine and pyruvate hydrate side products. (D) Histogram of the SNR improvement using rank reduction (r = 3) over 41 mice. SNR is defined here as the intensity of the maximum signal divided by the standard deviation in a 40 point region of the spectrum where signal is known not to be present. (E,F) Slice from A and C showing the 9 fold SNR enhancement after reconstruction.

Previously Undetectable Peaks can be Quantified by Low Rank Denoising
This level of noise reduction enables some measurements that were previously considered difficult. The bicarbonate provides information on the balance between the glycolytic and oxidative metabolic pathways 17,18 and in combination with the CO 2 signal can serve as a measure of intracellular pH 19 . The bicarbonate signal is difficult to detect in the raw spectra even in hyperpolarized experiments ( Fig. 2A,C) 17 . After rank reduction by SVD, the bicarbonate signal becomes evident (Fig. 2B,D) and breakdown of pyruvate through the citric acid cycle in the oxidative phosphorylation pathway can be followed (Fig. 2F). The detection of the bicarbonate peak at the expected position shows that rank reduction in the kinetic domain gives an actual increase in sensitivity, as opposed to the cosmetic improvement that eliminates both noise and true weak signals from some other noise suppression techniques 20 .

Higher sNR translates to More precise Kinetic Fitting
The performance seen in Figs 1 and 2 may not be universal as noise reduction algorithms are often sensitive to the characteristics of the signal. In particular, large dynamic ranges are often problematic for many noise reduction algorithms 20 . To confirm the generality of the method, we also tested the method using hyperpolarized [2-13 C] pyruvate. C-1 labeled pyruvate restricts analysis to the first steps of the TCA cycle as the 13 C label is released as 13 C-bicarbonate. Hyperpolarized [2-13 C]pyruvate allows tracking further downstream in the TCA cycle and into other metabolic pathways 17,21 . The data using hyperpolarized [2-13 C]pyruvate shows a dominant C-2 pyruvate peak at 208 ppm and weak peaks near the noise level corresponding to the downstream metabolites 5-glutamate and 1-acetyl-carnitine at 184 ppm and 175 ppm, among other peaks (Fig. S1B). The downstream metabolite peaks Only the pyruvate, lactate, and pyruvate hydrate signals are quantifiable. Right Rank 5 reconstruction of the raw data. In the raw data (A,C), the bicarbonate peak at 162.5 ppm is not detectable even though the overall signal to noise is high. Using the low rank approximation (B,D) a kinetic profile of bicarbonate metabolism can be obtained (F). No prior information about the bicarbonate signal or kinetics was used.
www.nature.com/scientificreports www.nature.com/scientificreports/ are less than 1% of the substrate signal. Rank reduction to a rank of 5 independent species substantially improves the signal to noise of later time points where the weaker downstream metabolite peaks are no longer visible in the raw signal (compare Figs S1C to S1E). The signal to noise increase translates to a corresponding increase in the accuracy of the kinetic reconstruction. The 8.3-fold increase in SNR translates to a 50 fold increase in the precision of the influx rate k 1 corresponding to the conversion of pyruvate to glutamate through the TCA cycle and a 200 fold increase in the precision of the loss rate k 2 corresponding to signal loss through relaxation and transformation to other metabolites. (compare Figs S1H to S1G). The same data was also tested with another state of the art denoising technique, wavelet shrinkage. While wavelet shrinkage using soft thresholding on individual spectra was unsuccessful in significantly reducing the noise (Fig. S1D), SVD rank reduction (Fig. S1E) was able to detect and quantify the weak peaks by using the additional information present in the entire time course.

Denoising Does Not Bias Relative Intensities or Kinetics
Rank reduction could introduce bias into curve-fitting of the kinetics, limiting its usefulness as a noise reduction technique. One method to test bias is to simulate realistic spectra and time courses and test the accuracy of the method by kinetic modeling in the presence of increasing amounts of noise. The pyruvate DNP experiment of Fig. 1 is used as a test case. Nonlinear regression is sensitive to small levels of noise and curve fitting was imprecise with even modest noise levels (Fig. S2). Since potential bias was difficult to measure with curve-fitting when the measurements are so imprecise, we used an alternate technique, the Area Under the Curve (AUC) approach, which is more robust against noise as the kinetic constants are derived from a ratiometric sum over all time points. Any potential bias introduced by SVD can therefore be measured at high levels of noise by this method even when traditional curve-fitting fails. The drawback is that transporter uptake cannot be quantified and the method is difficult to apply to models more complicated than three site exchange.
The accuracy of SVD reconstruction depends on the SNR of the metabolite peak. When the noise level is significantly less than the metabolite peak (SNR of the metabolite peak >10), no bias is introduced by SVD; the kinetic constants of all metabolites can be recovered accurately by the AUC method when SVD is used with a rank of 5 or higher (Fig. S2). Since the SNR of the pyruvate and lactate peaks is almost always above 10 in DNP experiments, any effect of SVD on these metabolites will be minimal in most cases. The kinetics of bicarbonate can also be recovered without bias at all levels of noise ( Fig. S3A). At higher levels of noise; however, there is a slight tendency for the alanine signal to drift towards the kinetics of the stronger pyruvate signal (Fig. S3B). The pyruvate to alanine conversion rate is overestimated by ~ 10% when SVD reconstruction. The origin of this error is that at low SNR the 4 th and 5 th kinetic eigenvectors that differentiate the kinetics of alanine from the other signals become indistinguishable from noise. While this suggests the error could be corrected by using a larger rank in the reconstruction, adding more eigenvectors decreases the precision to the point that the method becomes ineffective. However, the effect is modest and confined to the alanine signal under ordinary conditions. Overall, the simulations suggest rank reduction by SVD does not introduce a significant bias into measurements of either kinetics or intensities in DNP experiments under normal conditions. Synthetic data is less reliable for testing denoising algorithms than actual data due to the assumed idealities in the simulation. Noise in real data may show a frequency or time dependence that may differ from an ideal Gaussian white noise distribution 22 , and kinetics often do not exactly follow simple models. To test the accuracy of rank reduction on a more realistic sample, we performed a 13 C MRI DNP experiment using hyperpolarized [1-13 C]pyruvate, with excitation pulses alternating between a 10° pulse on odd scans and 2° pulses on even scans to generate high and low noise datasets from the same sample (Fig. 3). SVD reduction of the low flip angle signal to a rank of five gave a 3-fold increase in signal to noise. The kinetic profiles of the reconstructed pyruvate, pyruvate hydrate, and lactate signals, which have relatively strong signals that could be accurately measured even at low flip angles, closely match the high flip angle signal when normalized for intensity ( Fig. 3B-D). The reconstructed signal of alanine also nearly exactly matches the high-power signal despite the significant noise in the low power signal (Fig. 3E). Only the very weak bicarbonate signal, which is completely unrecognizable in the raw signal (Fig. 3F), shows a slight error in reconstruction.

Dynamic Single Voxel Spectroscopy of Glucose Metabolism without DNP
The results from DNP experiments encouraged us to try the SVD processing on molecules less amenable to hyperpolarization than pyruvate. Increased glucose uptake to meet the increased energetic and synthetic demands of increased cell growth is common feature of many cancers 23,24 , which makes glucose a primary target for metabolic imaging through both PET 25 and MRI techniques like CEST 9,26 . Unfortunately, the short T1 of glucose causes rapid relaxation of the hyperpolarized signal, meaning most of the signal is lost before 13 C glucose enters the glycolytic pathway 27 . Figure S4 shows the results from a tail vein injection of 50 µL of 100 mM uniformly 13 C labeled glucose, deuterated at all non-exchangeable protons to increase T1 (see Table S1), into a mouse with a leg xenograft from the Hs766t metastatic pancreatic carcinoma cell line. Even with deuteration, most of the hyperpolarized signal is lost within the first ten seconds and unavailable for detection of downstream metabolic products (Fig. S4). Nothing besides the glucose peak can be detected in the raw signal (Fig. S4). In the signal rank reduced by SVD; however, a very faint signal of approximately 0.5% the intensity of the glucose peak can be seen at 184 ppm corresponding to the 1-position of lactate. Like the glutamate and bicarbonate signals before, SVD brings the weak lactate signal up to detectability.
With hyperpolarized glucose, the DNP scans were of limited value due to the rapid signal decay. The success of SVD rank reduction in suppressing noise encouraged us to image uniformly labeled glucose solutions without hyperpolarization. The first studies were non-localized spectroscopy experiments similar to the ones described above. Figure 4 shows the results before and after SVD rank reduction from a tail vein injection of 350 µL of 555 mM uniformly 13  www.nature.com/scientificreports www.nature.com/scientificreports/ breakdown to lactate and alanine, reflecting transport limited uptake and the primarily aerobic glycolysis in this particular cell line 28 . Every peak of both the αand β anomers of glucose were in the expected positions and could be identified and quantified (Fig. 4D-F). Glucose-6-phosphate was identified as possible metabolite by a shoulder of the 6′ carbon of glucose with distinctly different kinetics 29 (Fig. 4E,F). In addition to alanine, lactate, glucose, and glucose-6-phosphate, a weak peak belonging to glycogen at 99 ppm 29 was identified in some of the samples with high signal to noise. As a control, the experiment was repeated in a mouse without a tumor xenograft. No glucose peaks could be seen in the absence of a tumor (Fig. S5), confirming the signal originates from the enhanced uptake and retention of glucose in tumors. The overall time-scale of glucose metabolism approximately matched previous 13 C measurements but with greatly increased time resolution 30 . Previous measurements were limited by signal-to-noise to taking one spectra every 5 minutes, while a quantifiable signal can be acquired using rank reduction every 3.2 s. The increased temporal resolution allowed measurement of the rate of uptake of glucose, which was challenging with the previous 13 C measurements, and difficult even with PET imaging 31,32 . The experiments in Fig. 4 were made on a 9.4 T scanner, which is not clinically widely available. With an eye towards eventual clinical translation, we made corresponding measurements at 3 T (Fig. S6). Although the lactate signal was not resolved, possibly because decoupling was not implemented, both glucose uptake and metabolism could be clearly quantified.

Extending SVD into Higher Dimensions through Tensor Decomposition
SVD is a strictly two-dimensional matrix method and cannot be used directly on images of this type. In order to adapt this method to higher dimensional data typical of dynamic, volumetric, and spectroscopic medical imaging experiments, a different method is proposed. The individual voxels can be treated independently and denoised by SVD but the fewer time points are acquired for each voxel in an imaging experiment. Because noise reduction power of SVD varies with the matrix size by Q / r, where Q is the smallest matrix dimension and r is the predicted rank 33 , using SVD to denoise individual voxels is not powerful enough for imaging experiments (see below). www.nature.com/scientificreports www.nature.com/scientificreports/ Treating voxels independently ignores the correlation between voxels and the low rank structure of the overall image 34 . A multidimensional analog of the SVD, the Tucker Decomposition, can be used to find a low rank reconstruction of the entire image. Similar to the SVD, the Tucker Decomposition factorizes an n-dimensional tensor X into a core tensor G of the same dimensions as the original data array and a set of n factor matrices {A, B, C,…} 35 : where ⊗ refers to the tensor outer product. While SVD decomposes a data matrix into a linear combination of vectors, the Tucker Decomposition decomposes a higher dimensional data tensor into a multilinear combination of matrices corresponding to images and time-dependent spectra (see Fig. 5). Like the SVD, the core tensor can be truncated to suppress noise while retaining as much of the signal as possible. The use of a more complex basis allows the underlying structure of the data to be represented in a more compact and natural way, which translates to more effective denoising when the distribution of signals is not random 36 .  Figure 6 shows the results from dynamic chemical shift imaging of a 50 mg bolus of [U-13 C]glucose injected into the tail vein of a mouse with a leg xenograft from the MiaPaca cell taken at 9.4 T before and after tensor decomposition. One image was taken every 48 s. The raw image is predominantly noise (Fig. 6C) with no discernable signal detectable in the voxel (Fig. 6E,F). A very weak glucose signal can be detected in some voxels by using voxel by voxel SVD with a high threshold for rank reduction (Fig. 6G,H), a roughly 3-fold improvement in SNR compared to the raw signal (see Fig. 6M). More aggressive rank reduction distorts the image by suppressing the signal in some of the voxels. The multi-dimensional nature of the tensor decomposition is essential to its success, unfolding the data into a 2D matrix and then using SVD low rank reconstruction (similar to the initial step of the LORA technique (see methods) 37 , (Fig. 5B) resulted in a 3-fold less improvement in SNR on this data set (Fig. 6I,J). Only by tensor factorization is a clear glucose signal detectable in each of the time traces, giving on average a ~31 fold improvement in SNR (Fig. 6M). The improvement in SNR is large enough that the SNR for each time point is actually higher than the SNR obtained by averaging over the entire time course of 90 min (450 scans) and sufficient to image the lactate signal in addition to the lipid and glucose signals. The time traces from the CSI images show relatively rapid uptake of glucose within the tumor that reaches a plateau within approximately 10 min, a similar time scale as the non-localized experiment. Glucose uptake is almost entirely localized within the tumor with a clear separation from lipid signal originating from the gluteal fat pad in the upper leg (Fig. 6A). 13 C lactate builds up slowly as glucose is broken down (Fig. 6C,D). To our knowledge, this represents the first successful dynamic imaging of tracer metabolism in vivo through 13 C MRSI without dynamic nuclear polarization 27 .

Noise Reduction Without a Spectral Dimension
Spectroscopic imaging is difficult in hyperpolarization experiments due to the irreversible loss of hyperpolarization, which sets a limit on how many scans can be acquired before the signal becomes undetectable. One method of rapidly acquiring a metabolite specific image is to use a fast imaging sequence like EPI in combination with spectrally selective pulses switching the excitation profile on alternate scans to center on different metabolites. The result is a series of images that correspond to the distribution of the metabolite in time.
To test the usefulness of the low rank approximation on these types of experiments, as well as other dynamic MRI experiments where spectral information is not available, we tested low rank reconstruction on spectrally selective EPI images of lactate metabolism after the delivery of a bolus of hyperpolarized 13 C-1-pyruvate. In the raw data, only the kidneys and part of the liver can be seen in the individual time points making the goal of following pyruvate metabolism at the organism level difficult. Using tensor reconstruction, the signal to noise is improved by a factor of ~3 (Fig. 7). With the increase in signal to noise, the heart became visible and the liver and www.nature.com/scientificreports www.nature.com/scientificreports/ kidneys were more clearly defined. With the reconstructed model, it is possible to trace the production of lactate through the body, detected in the heart by frame 3 and being retained in the liver and kidneys until frame 12. While the increase in signal to noise is less impressive than in spectral images, it does show that tensor factorization may be a viable technique for increasing the SNR in low signal, low complexity images.

Discussion
The possibility of using 13 C MRI for molecular imaging was investigated early on in the history of MRI but is not used routinely once it became apparent that the SNR was insufficient for imaging. If this limitation can be overcome, 13 C MRI with uniformly labeled 13 C glucose may become an alternative/complementary technique to 18 F-FDG PET. We show here that this limitation is not as formidable as may have once seemed. The key to this development is a relatively simple but efficient and robust post-processing noise suppression technique that takes advantage of the common underlying structure in metabolic imaging experiments, particularly those with a kinetic component. There have been a number of attempts at denoising individual magnetic resonance spectra, either specifically in the context of MRSI or more generally throughout the NMR field. The methods can be divided into those that attempt to denoise the free induction decay (FID) signal in the time domain (Cadzow reduction 38,39 aka HLSVD 39,40 , HTLS [41][42][43] and Pade transform methods) and those that attempt to directly denoise the MR spectrum in the frequency domain. Time domain methods make the assumption that the signal www.nature.com/scientificreports www.nature.com/scientificreports/ originates from a small set of ideal Lorentzian peaks with well-defined frequencies, intensities, and line widths 44 . From this assumption, a variant of the rank reduction through SVD procedure is employed to distinguish true peaks from false noise peaks, using a sliding window to convert the FID signal into a matrix. In ideal conditions, this results in a perfect, noise-free spectrum. However, the true rank of the noise-free signal is difficult to estimate a priori from the FID. Imperfect shimming, magnetic susceptibility artifacts, or chemical exchange processes can cause a deviation from ideal line-shapes. Any peak with a non-ideal line-shape manifests as multiple Lorentzian peaks in the fitting procedure. For example, the residual water peak in 1 H MR can only be modeled by at least 4 Lorentzian peaks 45 . In many cases, weak peaks like the bicarbonate peak in Fig. 2 are suppressed by this procedure in the presence of strong peaks since most of the degrees of freedom are used in modeling more accurately the line-shape unless the rank is set to a fairly large number, which has the undesirable consequence of creating spurious peaks. Quantitation may also be difficult in many cases unless the spectrum is largely noise free 37 . To surmount these problems, one variant of the LORA technique 37,46-51 uses a low rank approximation in the spatial domain before attempting to denoise the FID of individual spectra. The method is not perfect in suppressing artifact peaks and necessarily results in a loss of spatial resolution.
Other methods denoise the MR spectrum directly. The total variation approach uses a smoothing operator to dampen sharp discontinuities that likely reflect noise 52 . Wavelet methods seek to decompose the signal into sub-signals of increasing complexity, with the highest complexity signals, which likely represent noise, thrown out 53,54 . Maximum entropy methods take a similar approach form the perspective of information theory. These methods have the disadvantage of low sensitivity in the sense that they tend to flatten weak peaks 20 and may eliminate other fine details of the spectra.
Despite their differences, all of these methods operate in the level of individual spectra. Low rank denoising methods use the collective information in the entire image to achieve a higher SNR than denoising individual spectra alone 37,[46][47][48][49][50][51][55][56][57] . By considering rank reduction in the kinetic domain specifically, where the signal matrix is particularly sparse and possesses a natural connection to the underlying chemistry, it is possible to achieve an order of magnitude or more improvements in SNR without sacrificing resolution or accuracy. This level of enhancement can detect dynamic signals in the presence of large amounts of noise (Fig. 1) and quantify weak signals even in the presence of much stronger signals (Fig. 2). For dDNP experiments, this opens up the possibility of using lower flip angles and doses than are currently being used. Since the transverse magnetization after the Figure 7. Denoising of Image Data Using Tensor Decomposition. Top Left One slice from a 20 time point image set of lactate production following the injection of hyperpolarized pyruvate centered on the heart. The data was acquired using a spectral selective pulse sequence that yields a series of images for each metabolite, one image for each time point. Top Right The same data using the Tensor Decomposition; reducing the rank from [32,32,10,20] to [16,12,8,4]. Bottom The corresponding images from a slice centered on the liver and kidneys. In both cases a roughly 3 fold improvement in signal to noise is observed. improvements in SNR coupled with low flip angles may enable the detection of metabolites further downstream than is currently possible. Higher signal-to-noise is also useful in expanding the range of chemical probes amenable to DNP. Substrates with low polarization efficiencies or short T1 relaxation times like glucose (Figs 4 and 6) can be quantified more easily using SVD or tensor factorization. Finally, higher SNR may open up the possibility of off-site hyperpolarization through the brute force approach 58,59 , which would eliminate the largest barrier for clinical use.
Some challenges associated with the translation of 13 C glucose imaging into a clinical technique remain to be overcome. The 2 g/kg dose used in this study is about three times larger than the maximum tolerated IV dose 60 . Whole body imaging as done in 18 F-FDG PET/CT will be difficult with the technique. To get adequate resolution in humans, the number of k-space points will need to be larger, around 64 × 64 for a brain scan with a FOV of approximately 190 mm and 3 mm voxels, for example. This will result in unacceptably long acquisition time for each scan if conventional CSI imaging is used since each k-space point is phase encoded and requires a separate RF excitation and data acquisition. These are not insurmountable difficulties and could possibly be addressed with a faster imaging sequence, improved 1 H decoupling, and parallel imaging. Typical polarization efficiencies were around 20%. After 40-60 min, the hyperpolarized sample was rapidly dissolved in 4.5 mL of a superheated HEPES based alkaline buffer. The dissolution buffer was neutralized with NaOH to pH 7.4. The hyperpolarized [1-13 C]pyruvate solution (96 mM) was intravenously injected through a catheter placed in the tail vein of the mouse (12 μL/g body weight). Hyperpolarized 13 C MRI studies were performed on a 3 T scanner (MR Solutions, Guildford, UK) using a home-built 13 C solenoid leg coil. After the rapid injection of hyperpolarized [1-13 C]pyruvate, spectra were acquired every second for 240 s using a single pulse acquire sequence with a sweep width of 3.3 kHz and 256 FID points. Dynamic 13 C Glucose MRS without DNP. Magnetic resonance spectroscopy was performed on either a 9.4 T Biospec 94/30 horizontal scanner or a MR Solutions 3 T horizontal scanner. The coil assembly for the mouse leg consists of 3 16 mm independent wire loops that are each terminated with a double balanced tune/ match network to the 50 Ohm characteristic impedance of the coaxial cable. The two 13 C coils are geometrically decoupled because they have their radiofrequency field orthogonally oriented. Those coils are fed with radiofrequency currents with a 90° phase difference. One of the coils has a solenoidal shape and the other is constructed by two saddle loops with 120° arcs (quasi-Helmholtz pair), arranged coaxially to the solenoidal coil. The 1 H coil is a double sized surface coil coaxially arranged to the solenoidal coil. The proton coil has integrated 13 C frequency traps and the 13 C coils have integrated 1 H frequency traps to minimize coupling between them. A bandpass filter was used to minimize contamination of the 13 C signal by the 1 H decoupling pulse.

Mouse
Each mouse was anesthetized during imaging with isoflurane 1.5-2.0% administered as a gaseous mixture of 70% N2 and 30% and kept warm using a circulating hot water bath. Both respiration and temperature were monitored continuously through the experiment and the degree of anesthesia adjusted to keep respiration and body temperature within a normal physiological range of 35-37 °C and 60-90 breaths per min. Anatomical images were acquired with a RARE fast spin echo sequence 61 with 15 256 × 256 slices of 24 mm × 24 mm × 1 mm size with 8 echoes per acquisition, a 3 s repetition time, and an effective sweep width of 50,000 Hz. Samples were shimmed to 20 Hz on the 9.4 T with first and second order shims using the FASTMAP procedure 62 . Non-localized spectra of glucose without DNP at 9.4 T were acquired with the NSPECT pulse-acquire sequence using maximum receiver gain, a repetition time of 50 ms, Ernst Angle excitation of 12°, 256 FID points, a sweep width of 198.6 ppm, 16 averages per scan, and 4500 scans for a total acquisition time of 1 hour. MLEV16 decoupling 63,64 was applied during acquisition using -20 dB of decoupling power and a 0.2 ms decoupling element. The decoupling pulse was centered on the main proton lipid resonance at 1.3 ppm. Data at 3 T was acquired similarly except decoupling could not be applied efficiently on this scanner and was omitted.
Signal processing. For non-localized (two dimensional) experiments, the first 67 points of the FID in the time dimension were removed to eliminate the distortion from the group delay corresponding to the 13 ms dead time of the Bruker 9.4 T 65 . The FID was Fourier transformed and the phase estimated by the entropy minimization method of Chen et al. 66 , as implemented in MatNMR 67 . The baseline was estimated by a modification of the Dietrich first derivative method to generate a binary mask of baseline points 68 , followed by spline interpolation using the Whittaker smoother 69 to generate a smooth baseline curve 70  www.nature.com/scientificreports www.nature.com/scientificreports/ "forward-backward" linear prediction method of Zhu and Bax 71 . Fourier transforming the FID of the transients from each voxel individually generated the final spectrum. The method proved difficult to apply to the chemical shift imaging experiments and no preprocessing was applied. Spectra for chemical shift imaging experiments are shown in magnitude mode. Wavelet shrinkage was tested using using Daubechies S8 wavelets 72 with soft minimax thresholding 73 on individual spectra.

Low Rank Reconstruction.
For the two-dimensional signal matrices generated by non-localized pulse acquire experiments, SVD was applied to the FID in the time domain after the removal of the group delay artifact. The rank reduced signal was generated from the domain by truncating the SVD by setting the N-r diagonal values of the singular value matrix S to 0, where N is the number of rows in S and r is the predicted rank. The predicted rank was set to 5 unless otherwise specified, which is equal to the number of independent species in the pyruvate DNP experiment. For the four-dimensional imaging experiments, three methods were tried as described in Fig. 5. The magnitude signal was used in each case. (1) Applying truncated SVD to each voxel independently (2) Unfolding the four-dimensional tensor into a two-dimensional matrix and applying truncated SVD (a simplified version of LORA 37 ) (3) Factorizing the four-dimensional tensor by tensor decomposition. The predicted rank was set to 10 for voxel-by-voxel SVD and 16 for the matricization technique, which is the smallest rank which does not introduce substantial distortions in the time averaged image (for example, the inappropriate bleed through of the lipid signal into the tumor mass) as measured by a comparison of the reconstructed and raw time averaged signals. Tensor decomposition was achieved through higher order orthogonal iteration 74 in the Matlab NWay package 75 . A tensor rank of 8 in the temporal and spatial dimensions and 6 in each spatial dimension was used for the glucose and CSI images.