Denoising of Hyperpolarized ¹³C MR Images of the Human Brain Using Patch-based Higher-order Singular Value Decomposition

Abstract

Purpose

To improve hyperpolarized ¹³C (HP-¹³C) MRI by image denoising with a new approach, patch-based higher-order singular value decomposition (HOSVD).

Methods

The benefit of using a patch-based HOSVD method to denoise dynamic HP-¹³C MR imaging data was investigated. Image quality and the accuracy of quantitative analyses following denoising were evaluated first using simulated data of [1-¹³C]pyruvate and its metabolic product, [1-¹³C]lactate, and compared the results to a global HOSVD method. The patch-based HOSVD method was then applied to healthy volunteer HP [1-¹³C]pyruvate echo-planar imaging (EPI) studies. Voxel-wise kinetic modeling was performed on both non-denoised and denoised data to compare the number of voxels quantifiable based on SNR criteria and fitting error.

Results

Simulation results demonstrated an 8-fold increase in the calculated signal-to-noise ratio (SNR) of [1-¹³C]pyruvate and [1-¹³C]lactate with the patch-based HOSVD denoising. The voxel-wise quantification of k_PL (pyruvate-to-lactate conversion rate) showed a 9-fold decrease in standard errors for the fitted k_PL after denoising. The patch-based denoising performed superior to the global denoising in recovering k_PL information. In volunteer datasets, [1-¹³C]lactate and [¹³C]bicarbonate signals became distinguishable from noise across captured timepoints with over a 5-fold apparent SNR gain. This resulted in >3-fold increase in the number of voxels quantifiable for mapping k_PB (pyruvate-to-bicarbonate conversion rate) and whole brain coverage for mapping k_PL.

Conclusions

Sensitivity enhancement provided by this denoising significantly improved quantification of metabolite dynamics and could benefit future studies by improving image quality, enabling higher spatial resolution, and facilitating the extraction of metabolic information for clinical research.

1. Introduction

Hyperpolarization (HP) of ¹³C nuclei via dynamic nuclear polarization provides a significant MR signal enhancement of 4 to 5 orders of magnitude,¹ which has enabled non-invasive measurements of real-time energy metabolism of hyperpolarized substrates and downstream metabolites.^2,3 Various HP metabolic precursors, including carbon-13 enriched pyruvate,^4,5 glucose,⁶ α-ketoglutarate⁷ and others,⁸ have been shown to serve as probes for assessing different metabolic pathways in the brain that would otherwise be difficult to access. Among those precursors, HP-¹³C pyruvate is the only probe translated into clinical use to date.^9,10 In clinical research of multiple brain tumors, HP [1-¹³C]pyruvate has been used to report on the upregulated glycolytic metabolism of [1-¹³C]pyruvate-to-[¹³C]lactate conversion in lesions (Warburg effect),^11,12 measure [¹³C]bicarbonate production^13–15, and assess metabolic changes induced by treatment response.¹⁶ Recent studies in healthy volunteers have also shown region-specific and age-dependent lactate production in the brain using HP [1-¹³C]pyruvate.^17,18

Despite the enormous gain in MR signal by hyperpolarization as well as its capability of accessing multiple metabolic pathways, a limitation of this method comes from the non-renewable decay of hyperpolarization via spin-lattice nuclear spin relaxation (T₁). Typically, T₁ relaxation of ¹³C spins is on the order of several tens of seconds, which limits the temporal window for acquiring HP signals to approximately 1 – 2 min. This limits the monitoring of the kinetics of conversions to downstream metabolites such as [¹³C]bicarbonate when using HP [1-¹³C]pyruvate due to slow pyruvate-to-bicarbonate conversion and short T₁ of bicarbonate. To tackle this problem, various fast acquisition strategies have been developed to efficiently utilize HP signals in the limited time available including specialized pulse sequences^19–22 such as variable flip angle schemes²³ and variable resolution imaging approaches²⁴ as well as hardware improvements such as parallel imaging using multi-channel receive arrays for ¹³C imaging.^25,26

Image denoising also can improve HP-¹³C MR image quality following data acquisition. Multiple algorithms have been proposed to denoise multiparametric MRI data. These methods can be roughly divided into two groups. The first treats the dataset as a stack of images and denoises each one individually, generally assuming a similarity between neighboring voxels (non-local means and patch-based methods)^27–29 or by truncating a succession of increasingly accurate approximations to the image (wavelet and low-rank methods).^30–33 The second assumes a similarity between the same spatial image acquired at different time points or with different parameters (T₁-, T₂-, CEST, and diffusion imaging).^34,35 These methods generally employ higher-order singular value decomposition (HOSVD), which is a generalization of matrix SVD for higher dimensions,^36,37 to globally approximate the multidimensional dataset in terms of a combination of lower dimensional features. The HOSVD technique is particularly powerful when applied to dynamic MR spectral data, as the spatial variance in the signal is typically much higher than the spectral variance (chemical shifts do not change greatly with time or throughout the image). Using this technique, it was possible to improve quantification of low-SNR metabolites that were otherwise undetectable in both preclinical³⁸ and clinical conditions.³⁹

For HP MRI, the two methods can be combined by performing the HOSVD at the patch level by taking into account similarity within patches in nearby regions in the image.³⁷ By applying the HOSVD transform to a group of similar local patches, the similarity within the group, in addition to all dimensions of multidimensional data, can be exploited for effective noise removal. This patch-based HOSVD method showed superior performance over the global HOSVD when applied to diffusion-weighted ¹H MR images.³⁵ However, HP-¹³C EPI images are acquired at lower resolution than conventional MRI (typically 16×16 versus 128×128 and higher), and it was uncertain if a patch-based method would be effective. In this study, we compared the denoising performance of global and patch-based HOSVD methods using simulated dynamic ¹³C metabolic data, and then investigated patch-based HOSVD denoising in healthy volunteer brain HP-¹³C EPI data acquired using multichannel arrays and a single channel quadrature head coil.

2. Methods

2.1. HOSVD-based denoising for dynamic HP-¹³C MR images

2.1.1. Patch-based HOSVD denoising method

The patch-based HOSVD denoising method implemented in this study was adapted from the global-local HOSVD (GL-HOSVD) denoising algorithm demonstrated on diffusion-weighed ¹H MRI data by Zhang et al.³⁴ Figure 1 illustrates the GL-HOSVD algorithm for denoising dynamic HP-¹³C EPI data. It consists of two main stages. The first stage involves prefiltering noisy images by applying the HOSVD transform globally to the whole dynamic data, which was shown to alleviate stripe-like artifacts in denoised images with low SNR that occur when pure patch-based HOSVD was used. For multi-slice acquisitions, each slice of the dynamic HP-¹³C MR imaging data comprises two spatial (x and y) and one temporal dimension, which can be represented as a 3D tensor. The HOSVD transform for this 3D tensor of size (d₁ x d₂ x d₃), defined as I, can be written as

$$I=\left[S; U^{\left(1\right)},U^{\left(2\right)},U^{\left(3\right)}\right]=S{\times }_1{U}^{\left(1\right)}{\times }_2{U}^{\left(2\right)}{\times }_3{U}^{\left(3\right)}$$ (Eq. 1)

where S is the core tensor of size (d₁ x d₂ x d₃) consisting of transform coefficients; U^(k) (k = 1, 2, 3) is a factor matrix of size (d_k x d_k) for the k^th dimension; and x_k is a mode-k product of the tensor I by a matrix U^(k).³⁶ The S and U^(k) are first computed by performing SVD to the mode-k unfolding of the tensor I. The pre-filtered images are then obtained by using hard thresholding of the core tensor comprised of small transform coefficients contributing noise to the data. This threshold is determined based on the noise level of the original data and the size of tensor, scaled by a parameter k_global to effectively suppress the noise.⁴⁰ The pre-filtered images are reconstructed by the inverse HOSVD transform. In the second stage, the local HOSVD begins by grouping similar 3D patches in the pre-filtered images into stacks. The similar patches are searched in a search window of size (N_sw × N_sw) based on the sum of squared differences between the intensities of two input patches, one of which is a reference. Next, the selected patches are arranged into a 4D tensor (G_p) based on the similarity to the reference patch. The tensor G_p is then decomposed into the core tensor S_p and four factor matrices U_p^(k) (k = 1, 2, 3, 4), as expressed by

$$G_p=\left[S_p; U_p^{\left(1\right)},U_p^{\left(2\right)},U_p^{\left(3\right)},U_p^{\left(4\right)}\right]$$ (Eq. 2)

Similarly, G_n is constructed from the original data using the patches at the same coordinates as those selected from the pre-filtered data. Subsequently, the core tensor S_n for the original data is reconstructed using the factor matrices U_p:

$$S_n=\left[G_n; {U_p^{\left(1\right)}}^T,{U_p^{\left(2\right)}}^T,{U_p^{\left(3\right)}}^T,{U_p^{\left(4\right)}}^T\right]$$ (Eq. 3)

Through a hard-threshold operation on the transform coefficients in S_n, a noise-suppressed core tensor ${\widehat{S}}_n$ can be obtained. A scaling factor k_local is used to manipulate the statistically derived threshold for an effective noise removal at the patch level. The denoised tensor ${\widehat{G}}_n$ can be estimated by performing.

$${\widehat{G}}_n=\left[{\widehat{S}}_n; U_p^{\left(1\right)},U_p^{\left(2\right)},U_p^{\left(3\right)},U_p^{\left(4\right)}\right]$$ (Eq. 4)

This patch-wise denoising of G_p is carried out across the image domain by moving the search window with a step size of N_step. The final denoised images are then reconstructed by aggregating the multiple estimates at each pixel. The code for GL-HOSVD was modified based on existing scripts and is available online (https://github.com/XinyuanZhang719/gl-hosvd).

Figure 1. Diagram of the GL-HOSVD algorithm.

The GL-HOSVD algorithm includes a prefiltering stage (left vertical) followed by the patch-based denoising. From the pre-filtered images, similar patches are searched and grouped into a 4D stack, G_p. Factor matrices U_p obtained from HOSVD of G_p are utilized to reconstruct HOSVD transform coefficients of G_n (S_n). By hard thresholding S_n, the denoised stacks are estimated by inverse HOSVD, and the multiple estimates at each pixel are aggregated to obtain final denoised images. The figure was adapted for HP-¹³C MRI from Zhang et al.³⁴ including dynamic HP [1-¹³C]lactate images of a brain slice as an example. Denoising was performed on dynamic data on a slice-by-slice basis and was done separately for each metabolite to avoid signal crosstalk between metabolites.

2.1.2. TRI Denoising Method

We used the TRI (Tensor Rank truncation-Image enhancement) method³⁹ to denoise metabolite images in the simulations for a comparison of the denoising performance to the GL-HOSVD. The TRI method applies the HOSVD transform (Eq. 1) to the entire data series as in the first stage of GL-HOSVD. Regardless of the global or patch-based approach to denoising, the major difference between the two algorithms lies in how small transform coefficients are removed via manipulation of the tensor core. The TRI method employs a low-rank approximation (fixed number of non-zero values in S_n) while the GL-HOSVD method utilizes hard thresholding to eliminate small values in the core tensor S_n. In practical terms, this means GL-HOSVD can be parameterized directly from the noise level of the image while TRI requires prior knowledge or extensive simulations to estimate the rank.

2.2. Simulations

2.2.1. Designing a simulation framework

A brain phantom was created using gray and white matter extracted from a normal brain model accessible from the BrainWeb Simulated Brain Database.⁴¹ The matrix size (N_x × N_y) was 32 × 32 (spatial resolution of 6.8 × 6.8 mm²) and was chosen to provide sufficient structural detail in the simulated ¹³C MR dataset. Dynamic signals of pyruvate and lactate in the brain phantom were simulated using an irreversible, two-site kinetic model utilizing the hyperpolarized-mri-toolbox from GitHub (https://github.com/LarsonLab/hpyperpolarized-mri-toolbox/).⁴² The rate of pyruvate-to-lactate conversion (k_PL) for gray and white matter was respectively set to 0.015 and 0.0075 s⁻¹, similar to the k_PL values observed from our own data^16,43 and those reported by Grist et al.¹⁸ The T₁ relaxation times for [1-¹³C]pyruvate and [1-¹³C]lactate were set as 30 s and 25 s, respectively, in simulations. Other input parameters included flip angles ⍺_pyruvate/⍺_lactate = 20°/30°, 3 s TR and 20 timeframes (N_t). This kinetic modeling utilized the measured pyruvate signals as the input at each timepoint and accounted for the signal loss via T₁ relaxation and RF pulses. The maximum intensity of the dynamic lactate signals was normalized to 1. To vary the SNR in the simulated datasets, zero-mean random Gaussian noise of σ_n = 0.2, 0.3 and 0.4 was added to the noise-free dynamic images of pyruvate and lactate, resulting in peak dynamic SNR of 14.6 to 29.2 for pyruvate and 2.5 to 5 for lactate. At the highest noise level (σ_n = 0.4), the peak signal intensity of lactate was within 3σ_n of the mean of noise distribution. This SNR range is close to or lower than that of in-vivo data acquired in our study. Using the noise-added image sets, the parameters for GL-HOSVD and TRI were determined empirically from simulation based on the root-mean-square error (RMSE) between the ground truth and denoised images and the SNR gain. The five GL-HOSVD parameters were the patch size p × p of 5 × 5; search window size N_sw of 11; step length N_step of 2; scales k_global of 0.4 and k_local of 0.8. For TRI, a tensor rank of (20, 20, 4) was chosen from a data matrix size (N_x, N_y, N_t) = (32, 32, 20). To avoid cross-contaminating signals between slices and metabolites, the denoising was performed independently on different slices of the brain images and also on different metabolites. For each denoising method, a total of 500 simulations were conducted per noise level, and k_PL values were determined voxel-by-voxel using the kinetic modeling designed to generate the metabolic phantom.

2.2.2. Assessment of denoising performance analysis

The sensitivity enhancement was evaluated by comparing the mean SNR of metabolic signals between denoised and non-denoised images. The image quality of the denoised data was assessed using structural similarity index metric (SSIM)⁴⁴ with respect to the noise-free data. A brain mask was applied to the images in order to exclude background voxels in the SSIM calculation. To assess the accuracy of k_PL quantification introduced by denoising, k_PL errors were calculated as |true k_PL − estimated k_PL|/true k_PL × 100 (%), and k_PL precision was determined by the relative standard error calculated as ${ \sigma }_{k_{PL}}/k_{PL}\times 100 \left(\%\right)$, where ${ \sigma }_{k_{PL}}$ is a standard deviation obtained from non-linear least-square fitting of k_PL.

2.3. Brain studies in healthy volunteers

2.3.1. MR experiments

The HP-¹³C metabolic data of the brain (N = 6) in this study were acquired from two healthy volunteers using a 2D multi-slice EPI sequence,⁴⁵ following the injection of 250 mM HP [1-¹³C]pyruvate. The pulse sequence consisted of a single-band spectral-spatial RF pulse that sequentially excited [1-¹³C]pyruvate, [1-¹³C]lactate, and [¹³C]bicarbonate resonances with 20°, 30°, 30° flip angles, respectively. Data were acquired with an in-plane spatial resolution of 1.5 × 1.5 cm² (24.0 × 24.0 cm² FOV, 16 × 16 matrix) for 8 slices with 1.5 cm thickness, with 62.5 s TR, 21.7 ms TE, and 1.03 ms echo-spacing. Twenty timeframes were acquired per metabolite with a 3 s temporal resolution, yielding a total scan time of 60 s. Hyperpolarization of [1-¹³C]pyruvate was achieved by polarizing the samples for at least 2 hrs using a 5T SPINlab operating at 0.8 K (GE Healthcare, Chicago, IL). For anatomic reference, proton T₂-weighted and 3D T₁-weighted images were acquired. All experiments were performed on a 3T clinical MR scanner (MR750, GE Healthcare, Waukesha, WI) equipped with multi-nuclear imaging capability. For HP-¹³C MRI, either a 32-channel custom-built receiver (UCSF-MGH)²⁵ (datasets HV-01, HV-04) or a 24-channel receiver (Rapid Biomedical, Rimper, Germany) (datasets HV-02, HV-03, HV-05) were utilized in combination with birdcage RF transmit coils. A custom-built volume transmit/receive birdcage coil (UCSF-MGH) was used to obtain dataset HV-06 (see Table S1). Transmit RF power was calibrated on a head-shaped phantom containing unenriched ethylene glycol (HOCH₂CH₂OH, anhydrous, 99.8%, Sigma Aldrich, St. Louis, MO) prior to the volunteer study. All human studies were approved by an institutional review board requiring informed consent and followed Food and Drug Administration investigational new drug application (FDA IND)-approved protocols.

2.3.2. Data analysis

All EPI datasets were reconstructed using the Orchestra toolbox (GE Healthcare). First, phase coefficients from the reference scan were applied to the raw k-space data to correct for EPI Nyquist ghost artifacts. For multi-channel data, k-space data were pre-whitened and then coil-combined using estimated coil weights derived from the pyruvate data.⁴⁶ Complex data were phased to maximize the real part of the image, which was used for denoising and kinetic analysis, and retained the Gaussian noise distribution. To create brain masks for kinetic modeling, the T₁-weighted images were first aligned by FLIRT⁴⁷ to the T₂-fast spin echo images acquired prior to the ¹³C examination, and then white/gray matter segmentation was performed on the aligned T₁-weighted images using the FSL FAST algorithm.⁴⁸ The ¹³C data voxels containing > 10% of white or gray matter were selected to generate whole brain masks, and kinetically modeled to obtain k_PL and k_PB maps. After fitting, only those voxels satisfying that the SNR_auc (SNR of temporally summed metabolic signals) is > 3 for pyruvate and lactate/bicarbonate in the raw data and that only voxels where ${\sigma }_{k_{PX}}<0.3{\bullet k}_{PX}$ (X = L or B) in the fit were included in the kinetic maps.⁴³ The SNR cutoff was determined from the k_PL error analysis in the numerical simulations to obtain reasonable kinetic rate maps. Other fit parameters include the T₁ relaxation time constants that were chosen to optimally fit the data. The number of voxels with k_PL or k_PB fulfilling the SNR and fitting error criteria were counted from the whole brain to calculate the spatial coverage in k_PL or k_PB maps. The total number of the ¹³C data voxels was determined from the brain mask.

2.3.3. Simulations based on the experimental data

To investigate the performance of GL-HOSVD on experimental data with added noise, a total of 10 simulated pyruvate and lactate datasets were generated from one healthy volunteer dataset by deteriorating the data with zero-mean random Gaussian noise of σ = 0.8, 1.2 and 1.6. With the additive noise, the noise level of the simulated datasets was 3, 4 and 5 times higher than that of the original data. From the noise-added and GL-HOSVD denoised images, k_PL was determined as described in the following sections. All data analysis and simulation work (https://github.com/ykim-hmtrc/hp13c_EPI-hosvd_denoising) were performed in MATLAB R2019a (The MathWorks Inc., Natick, MA).

3. Results

3.1. Simulated denoising results

3.1.1. Image quality

In Figure 2A, the images in the first two columns show the simulated [1-¹³C]pyruvate and [1-¹³C]lactate ground truth (noise-free) images and those with additive 30% random Gaussian noise (σ_n = 0.3) from a central slice of the brain phantom at an early (t = 12 s) and later (36 s) timepoint. At this noise level, the peak dynamic SNR was respectively 19.5 and 3.3 for pyruvate and lactate. The images in the third column were produced by applying GL-HOSVD to the noise-added images. The gray and white matter is well separated in the denoised pyruvate images, where the signal intensities from the two compartments are similar. It is also apparent that the denoised lactate images, with signal intensities that are near the noise floor in the noise-added data, closely resemble the noise-free images. For comparison, the Tensor Rank truncation-Image enhancement (TRI) method was applied to reduce noise from the same noise-added images, and the resultant images are shown in the last column. This TRI method was previously demonstrated to provide a sensitivity improvement in HP-¹³C MR spectroscopic data by an order of magnitude and a moderate SNR gain for image data.^38,39 The relative mean SNR values from the noise-added and denoised lactate signals ($\mathit{\text{Rel}}.\overline{SN{R}_{\mathit{\text{Lac}}}}$), plotted in Figure 2B, demonstrate that GL-HOSVD outperforms TRI in terms of the SNR enhancement, which was almost 3 times greater than that of TRI, providing an overall a >8-fold SNR increase with respect to the noise-added images. Similar SNR enhancements were observed from the denoised pyruvate signals as well as at different noise levels. Figure 2C depicts mean SSIM values calculated from the 500 simulated lactate images. The more similar the denoised and noise-free images are, the closer the SSIM is to 1. The GL-HOSVD images are close to 1 at all time points, consistently higher than those from the TRI images. As the signals diminish due to T₁ relaxation and RF utilization, the SSIM values deviate from unity more rapidly with TRI than GL-HOSVD. These observations indicate that GL-HOSVD better preserves image details than TRI, especially in low-SNR images.

Figure 2.

Comparative Results Between the GL-HOSVD and Tensor Rank Truncation-Image Enhancement (TRI) methods for Denoising Image Data. (A) Representative images from the simulated pyruvate and lactate data with 30% Gaussian noise (σ_n = 0.3) at an early (t =12 s) and late (t = 36 s) timepoint before and after denoising. In the top two rows, noise-free, noise-added, and denoised pyruvate images are displayed from left to right. The lactate images are shown in the bottom two rows; (B) Relative mean values of the signal-to-noise ratio (SNR) of the noise-added and denoised lactate signals from 500 simulated datasets. The peak $\overline{SN{R}_{\mathit{\text{Lac}}}}$ of the noise-added data was normalized to 1. The $\mathit{\text{Rel}}.\overline{SN{R}_{\mathit{\text{Lac}}}}$ values showed 2–3 times higher SNR gain by using GL-HOSVD versus TRI; (C) Structural Similarity Index Metric (SSIM) values calculated for the noise-added and denoised lactate images as a function of time. The error bars indicate standard deviations from 500 simulations. High SSIM values of the GL-HOSVD images compared to the TRI images indicate higher similarity of the denoised images to the noise-free images especially in the late scans.

3.1.2. Accuracy of k_PL quantification

Evaluating the effect of denoising on the voxel-wise quantification of metabolic conversion enabled a further comparison of GL-HOSVD versus TRI approaches. Figure 3 shows examples of estimated k_PL maps from the simulated pyruvate and lactate images with increasing amounts of noise. The left and right image sets are from a central and superior slice of the brain phantom. The reference k_PL map for each slice is shown on the top. The image set below displays the k_PL maps from the non-denoised images and the corresponding GL-HOSVD and TRI images in the presence of Gaussian noise of σ_n = 0.2, 0.3, and 0.4. Comparisons between the k_PL maps from the GL-HOSVD images (second column) and those from the TRI images (third column) show the superior performance of GL-HOSVD in recovering the reference k_PL information relative to TRI, especially in the white matter that was simulated with a lower k_PL than the gray matter. The k_PL values from the GL-HOSVD images were found to be more uniform within a compartment, while those from the TRI images had greater variability. This can be clearly seen from the uniformity of white matter in the superior slice processed with GL-HOSVD (Figure 3b,e,h). From 500 replicate simulations, the mean of voxel-by-voxel k_PL errors (i.e. bias) was evaluated as displayed in Figure 4. Overall, smaller k_PL errors were obtained from GL-HOSVD, and they showed 20% reduction compared to TRI on average. Also, the errors were relatively greater from the white matter versus gray matter as expected from the lower k_PL value of white matter. The plot of mean k_PL errors from the white and gray matter at different noise levels against the original SNR_auc of lactate showed that when SNR_auc < 3, the k_PL error from GL-HOSVD becomes higher than 20% (Figure S2). Elevated k_PL errors from GL-HOSVD were also observed at the boundary of the gray and white matter compartments at the highest noise levels, as a result of the binary gray and white matter masks used for the simulation. These errors indicated that the k_PL values from GL-HOSVD from the voxels at the boundary between the gray and white matter were mostly overestimated while those from voxels at the boundary between the gray matter and background were underestimated. Overall, the mean k_PL value decreased by 2 – 3 % at the highest noise level simulated. The k_PL precision displayed in Figure 5 shows that k_PL values from the GL-HOSVD data were more precise. On average, the mean standard errors from GL-HOSVD were 3 times smaller than those from TRI. Compared to the non-denoised data, k_PL quantification was approximately 9 – 10 times more precise with the GL-HOSVD data. This improvement in k_PL precision translated into an increase in the number of reliable voxels comprising kinetic rate maps, after application of SNR threshold and k_PL fitting error criteria. The k_PL error and precision maps for the superior slice of the brain phantom are displayed in Figure S1. Similar to the k_PL accuracy, the accuracy of lactate-to-pyruvate ratio from temporally summed metabolite signals (AUC ratio) was also improved by 20% compared to TRI (Figure S3).

Figure 3. Simulated Estimation of Pyruvate-to-Lactate Conversion Rate (k_PL).

Top row: Ground-truth k_PL maps from a noise-free central slice (left) and a superior slice (right). The k_PL values of 0.015 and 0.0075 s⁻¹ were assigned to the k_PL for gray and white matter, respectively. For the central slice, from left to right: A–C: k_PL maps obtained from the simulated data with the Gaussian noise of σ_n = 0.2 and those from the corresponding GL-HOSVD and TRI images; D–F: k_PL maps from the simulated data with the Gaussian noise of σ_n = 0.3 and G–I with σ_n = 0.4, respectively. For the superior slice, the k_PL maps from the noise-added and denoised data are displayed with the same order as for the central slice. The k_PL maps from the GL-HOSVD data showed less error than the TRI k_PL maps but can exhibit poor edge sharpness and over-smoothing at high noise levels.

Figure 4. Simulated Assessment of k_PL Accuracy.

The error images showing the mean percent differences (bias) in k_PL estimates between noise-added and noise-free images (a,d,g); the corresponding GL-HOSVD processed and noise-free images (b,e,h); and the TRI processed and noise-free images (c,f,i), calculated from 500 simulated datasets. The standard deviation of the random Gaussian noise contained in the noise-added data is indicated on the left. Compared to the TRI data, the GL-HOSVD data show lower k_PL errors on average, but relatively higher errors at the boundary of the gray and white matter regions.

Figure 5. Simulated Assessment of k_PL Precision.

The error images showing the mean standard errors of the fitted k_PL values from 500 noise-added simulated data (a,d,g), and corresponding GL-HOSVD (b,e,h) and TRI images (c,f,i) at different noise levels as indicated on the left. Approximately 2 – 3 times higher k_PL precision was obtained from GL-HOSVD versus TRI.

3.2. Denoising results for in-vivo data using GL-HOSVD

3.2.1. Sensitivity improvement

Application of GL-HOSVD to dynamic HP-¹³C EPI scans of six healthy human brain volunteers demonstrated the practical performance of this method. Figure 6A displays the original non-denoised and denoised pyruvate, lactate, and bicarbonate images from one volunteer dataset. The original metabolite images, shown in the first row for each metabolite, were observed with the peak dynamic SNR of 569, 25, and 6.5 for pyruvate, lactate, and bicarbonate, respectively. Background noise was successfully reduced by applying the GL-HOSVD method, yielding the denoised images in the second row for each metabolite. The most substantial improvement in image quality was observed for the bicarbonate images. The bicarbonate signals were barely distinguishable from the noise of the original images, especially at later time points, but were readily observable (apparent SNR > 10) after denoising. For lactate, the edges were better defined in the denoised images than the original data. Also, spatially and temporally different lactate distribution remained preserved in the denoised images without noticeable distortion which can be caused by the dynamics of high SNR signals. High similarity between the denoised and original pyruvate images suggests minimal bias and corruption of the originally high-SNR resonances. In Figure 6B, the sensitivity improvement in the denoised metabolite images throughout the entire course of the acquisition is illustrated with mean SNR relative to the maximum value from the original data. The apparent SNR gain for the first 10 timeframes was determined as 5.4 ± 0.3, 4.8 ± 0.2, and 5.8 ± 1.6 for pyruvate, lactate, and bicarbonate, respectively.

Figure 6. Dynamic Hyperpolarized Imaging of [1-¹³C]Pyruvate Metabolism.

(A) Dynamic HP-¹³C EPI data of [1-¹³C]pyruvate, [1-¹³C]lactate, and [¹³C]bicarbonate signals from a selected slice in a healthy brain volunteer (HV-01), acquired with the whole-brain multi-slice EPI dynamic sequence with 3 s temporal resolution. The acquisition started 5 s following the end of the saline injection at time zero. Data are shown before (‘Orig.’; original) and after applying the global-local higher-order singular value decomposition (‘DN’; denoised). The images have been zero-filled 4-fold for display. For lactate and bicarbonate, the upper window-level was set to the overall maximum intensity voxel. For pyruvate, the upper window level was adjusted to 20% of the overall maximum intensity voxel. For anatomic reference, a ¹H T₂-FSE (fast spin echo) image is shown on the top; (B) apparent SNR timecourse from pre- and post-denoised images of pyruvate, lactate, and bicarbonate (from left to right). Relative values of mean signal-to-noise ratio ($\mathit{\text{Rel}}.\overline{SNR}$) of metabolite signals from original (‘o’) and denoised images (‘x’), displayed in (A), are plotted as a function of time with respect to the peak $\overline{SNR}$ of the original images. The highest apparent SNR gain of 7 was observed in the denoised bicarbonate signals.

In Figure 7, pre- and post-denoised pyruvate, lactate, and bicarbonate images from 5 central slices of another healthy volunteer (HV-02) are shown. These images represent the signals at the early and late time points, obtained 14 and 38 s after the saline flush following the pyruvate injection. The late time scans demonstrate well the sensitivity enhancement in the denoised images of all three metabolite signals. In particular, the brain became more clearly defined in the denoised bicarbonate images than the original data, where the signal intensities were on the same order of magnitude as the noise. The multi-slice images from the first volunteer data (HV-01; Figure 6) are presented in Supporting Information (Figure S4). Figure 8 displays the denoising of the volunteer dataset acquired using a volume coil (HV-06) and illustrates the performance of GL-HOSVD at higher noise levels. This dataset exhibited the lowest SNR among the 6 datasets analyzed in this study (see Table S1) due to the reduced sensitivity of the single channel birdcage coil. The denoised images demonstrated significantly improved sensitivity for all metabolites, although the denoised bicarbonate images showed potentially limited reliability for this denoising method at exceptionally low SNR. Examples of the original and denoised dynamic data of the metabolites from a single voxel are shown for each of three volunteer datasets in Figure S5. This figure shows that the late-time, low-intensity pyruvate signals can be well preserved after denoising although the pyruvate signal intensities at later timepoints are 5–10 times smaller than the peak intensity. Also, the denoised bicarbonate traces follow the original traces well despite high noise at the initial timeframes.

Figure 7. Denoising Results Using the GL-HOSVD for the Multi-slice, Dynamic HP-¹³C EPI Data from a Healthy Brain Volunteer.

The original (Orig.) and denoised (DN) metabolite images from an early and late timepoint (t = 14 and 38 s, corresponding to the 4^th and 12^th scans, respectively) are displayed for 5 central slices of a healthy brain volunteer (HV-02). The effect of denoising was most apparent for bicarbonate images at both timepoints; whereas the noise reduction in pyruvate and lactate images was most visible at the later time point. White arrows in the bicarbonate images indicate the signal from the [¹³C]urea phantom. For anatomic reference, ¹H T₂-FSE images are shown on the left.

Figure 8. Denoising Results of GL-HOSVD for the Multi-slice, Dynamic HP-¹³C EPI Data Acquired from a Healthy Brain Volunteer Using a Volume Coil.

The original (Orig.) and denoised (DN) metabolite images from the early and later time points (t = 14 and 38 s, corresponding to the 4^th and 12^th scans, respectively) are displayed for 5 axial slices of a healthy brain volunteer (HV-06). The images have been zero-filled 4-fold for display. The upper window level was adjusted to 55 % (pyruvate) or 110 % (lactate and bicarbonate) of the maximum intensity voxel in the 5 slices of the denoised images. For anatomic reference, ¹H T₂-FSE images are shown on the left. Noise reduction for the pyruvate images can be clearly seen in the later time point, and at both time points for the lactate images. For bicarbonate, the denoised image quality is also greatly improved.

3.2.2. Quantitative analysis

The improvement of fit quality in kinetic analysis and spatial coverage for the k_PL and k_PB maps using GL-HOSVD was verified on in vivo datasets from the healthy volunteers. The SNR and error criteria were applied to generate these maps as described in Methods 2.3.2. Figure 9 shows representative k_PL and k_PB values overlaid on the anatomical images and their distributions. The brain coverage of k_PL maps from the GL-HOSVD data (bottom row, Figure 9A) was improved from 69 to 94%. There was no noticeable difference in the image contrast between the original and denoised maps, which can be also seen from similar k_PL distributions in both the original and GL-HOSVD data. This was supported by good voxel-by-voxel agreement of k_PL values between the original and denoised maps (Figure S6A). A larger impact on the spatial coverage was observed from the k_PB maps following denoising, with nearly a 3-fold increase from 12 to 37% (Figure 9B). The k_PB distributions illustrated that newly estimated k_PB values from the denoised data were mostly lower than the average value of k_PB from the original data. Voxel-by-voxel comparison between the original and denoised k_PB values showed sightly decreased values of the denoised data to 85 ± 6% of the original data on average (Figure S6B), similar to the bias observed in numerical simulations of high noise. As a result, the average k_PB value from the denoised data was decreased by 1.6 times as compared to that of the original data. Figure 9C shows the kinetic fits of the denoised and original traces of bicarbonate signals from the selected voxels indicated by white arrows. The noise removal enabled k_PB quantification from the left voxel, which would otherwise be neglected from the k_PB mapping because of poor fit quality and unsatisfied error criteria. From the second voxel on the right, the k_PB precision was improved by 2 times in the denoised data, with less than 15 % of the relative standard error. Additionally, k_PL fits from four selected voxels with varying lactate SNR (SNR_auc = 2.4~17) were compared (Figure S7). Here, we included a voxel with SNR_auc < 3 for raw lactate signal (SNR criteria for kinetic analysis) to show the data and fit quality. From the voxels with lactate SNR of 14 and 17, the fitted k_PL values from the original and denoised data showed good agreement while the k_PL precision was much higher with the denoised data. With lower lactate SNR, the k_PL values estimated from the original data were not reliably calculated due to the elevated noise floor, in agreement with numerical simulations. The relative standard k_PL error was 42% from the voxel with SNR_auc of 2.4, which improved to 6% after denoising. Table S1 summarizes the improvement in SNR_auc and spatial coverage of k_PL and k_PB maps, and the mean voxel-wise k_PL and k_PB values from the evaluated brain volume from the six healthy volunteers.

Figure 9. k_PL and k_PB Maps from a Healthy Volunteer Before and After Denoising.

(A) Left: k_PL maps generated from the original (top row) and the GL-HOSVD denoised (bottom row) metabolite images of a healthy volunteer (HV-01). Right: Distributions of quantified k_PL values from the original and denoised 32-channel images; (B) Left k_PB maps generated from the original data (top row) and the GL-HOSVD denoised data (bottom row). Right: Histograms of voxels with quantified k_PL values from the original and denoised images; both k_PL and k_PB maps have been zero-filled 4-fold for display. (C) Traces of the original and denoised bicarbonate signals in an arbitrary unit (a.u.) from 2 selected voxels, indicated by an arrow in the k_PB maps (B). The fitted curves from the denoised and the original data are displayed, and the corresponding k_PB values are denoted in the plots (k_PB,Orig: k_PB from the original data, k_PB,DN: k_PB from the denoised data). The denoised images show improved spatial coverage of k_PL and k_PB maps, particularly for white matter. For k_PB, the spatial coverage with acceptable fit quality is increased threefold after denoising.

4. Discussion

In this study, we implemented and tested the patch-based denoising method of GL-HOSVD in the processing pipeline for dynamic HP-¹³C MR images of the human brain. The performance of this method for improving SNR and the accuracy of quantified kinetic rates was first evaluated with numerical simulations, where GL-HOSVD denoising was shown to be superior to TRI. GL-HOSVD denoising was then applied to HP-¹³C EPI datasets from healthy brain volunteers, where it improved the detection of bicarbonate and enhanced quantification of k_PL and k_PB using a 32- or 24-channel array (Figures 6,7, and 9) and a single-channel volume coil (Figure 8) that is used for studies at multiple sites.²⁶

Simulation studies showed increased k_PL error in the GL-HOSVD denoised data at the boundary between the white and gray matter regions of the brain phantom which manifested as blurring of the periphery of white matter in k_PL maps. The sharp signal discontinuity at the interface between the white and gray matter of the metabolic phantom likely contributed to these high k_PL errors. A smaller patch size can alleviate this problem but comes at the cost of less effective noise reduction (Figure S8); nevertheless, this artifact would be less apparent for human studies due to the realistic EPI PSF and blurring related to T₂* decay and B₀ inhomogeneity.

The matrix size of the simulated data was twice as large as the healthy human carbon-13 data for in-plane dimensions with a two-fold finer spatial resolution. This was to represent detailed features like gray/white matter contrast in the simulated brain phantom for testing the denoising performance. No adjustments were needed to the GL-HOSVD parameters determined from the simulation such as k_local, k_global, patch size, and step size when denoising the low-resolution human data. Similar to aforementioned effects of patch size, the smoothing artifacts were reduced by lowering k_local and k_global parameters, but at the expense of SNR gain. A variation in step size and size of the search window made no appreciable change in the outputs from the simulation. The denoising parameters may need to be optimized for other data types such as of different organs or spatial resolution using relevant numerical phantoms.

In our study, we expressed the 5D dynamic multi-slice, multi-frequency HP-¹³C EPI dataset as a 3D tensor per each frequency and slice, and denoised each metabolite on a slice-by-slice basis. With this approach, no crosstalk between different metabolites can occur. When performing the GL-HOSVD denoising on the whole 4D metabolite dataset, a slight increase in the apparent SNR was observed from simulations. However, crosstalk between different slices was apparent in the denoised images (data not shown). This was likely due to the small number of slices (8) in the z-dimension and may be obviated by acquiring more slices or using a smaller patch size in that dimension.

As described in Figure S2, the simulation results indicated that the accuracy of kinetic rate quantification from the GL-HOSVD denoised images depends on the SNR_auc of lactate in the raw data. These results can be interpreted in terms of minimum SNR needed to accurately quantify metabolism rather than just being relevant for the pyruvate-to-lactate conversion. For instance, while the simulations did not explicitly include bicarbonate, it would have similar dynamics modeled by the same two-site unidirectional kinetic model, and the SNR_auc can be used to help guide interpretation and analysis of denoised data for both k_PL and k_PB. Based on the simulations, kinetic rates can be estimated with less than 20% error from any voxels with SNR_auc of 3 or higher. While kinetic rates can still be estimated at lower SNR, care must be taken when interpreting this data due to the potential for increased bias. These simulations also did not include other potential confounders, such as recirculation or motion. However, the in-vivo datasets suggested that the timecourse of pyruvate recirculation can be preserved after denoising (Figure S5).

More than fivefold improvement in SNR following denoising of the volunteer datasets, indicates the possibility of acquiring metabolic data at higher spatial resolution in future clinical studies, thereby mitigating partial volume effects and facilitating extraction of metabolic information for improved clinical management. It could be particularly beneficial for other image-based ¹³C studies using HP [2-¹³C]pyruvate, [¹³C]urea and [1-¹³C]⍺-ketoglutarate, exhibiting lower SNR compared to [1-¹³C]pyruvate due to shorter T₁ or lower polarization.

5. Conclusion

A patch-based approach for denoising images was investigated and applied in human HP-¹³C brain studies to enhance the sensitivity of metabolite signals from HP [1-¹³C]pyruvate MRI data. With sensitivity enhancements of 5+ fold, low SNR metabolites were better visualized, and k_PL and k_PB maps with greater brain coverage were obtained, especially when using a volume head coil. This demonstrated the benefit of patch-based denoising to improve the quality of HP-¹³C images and spatial resolution for future studies.

Supplementary Material

tS1,fS1-S8

Figure S1. Simulated assessment of k_PL accuracy and precision. (A) The error images showing the mean percent differences (bias) in k_PL estimates between noise-added and noise-free images (a,d,g); the corresponding GL-HOSVD processed and noise-free images (b,e,h); and the TRI processed and noise-free images (c,f,i), calculated from 500 simulated datasets. The standard deviation of the random Gaussian noise contained in the noise-added data (σ_n = 0.2) is indicated on the left; (B) Mean standard errors for the fitted k_PL values from 500 noise-added simulated data (a,d,g), and corresponding GL-HOSVD (b,e,h) and TRI images (c,f,i) at different noise levels.

Figure S2. Plot of mean k_PL error (%) obtained from denoised k_PL maps versus the original SNR_auc of lactate. Mean k_PL errors calculated from white and gray matter of k_PL maps obtained from 500 GL-HOSVD processed datasets simulated with 20, 30, and 40, and additionally 50, 80, 100, and 150 % random Gaussian noise, and plotted over the original (non-denoised) SNR_auc (SNR of temporally summed metabolic signals) of lactate from the corresponding brain region. The error bars indicate the standard deviation of the mean k_PL error.

Figure S3. Simulated assessment of accuracy of AUC ratio (Lac_auc/Pyr_auc). Error images showing the mean percent differences (bias) in lactate-to-pyruvate area under the curve (AUC) ratio estimates between noise-added and noise-free images (a,d,g); the corresponding GL-HOSVD processed and noise-free images (b,e,h); and the TRI processed and noise-free images (c,f,i), calculated from 500 simulated datasets. The standard deviation of the random Gaussian noise contained in the noise-added data is indicated on the left.

Figure S4. Denoising results of GL-HOSVD for the multi-slice, dynamic HP-¹³C EPI data acquired from a healthy brain volunteer using a multi-channel coil. Multi-slice images from the data presented in Figure 6. The original (Orig.) and denoised (DN) metabolite images from the early and later time points (t = 14 and 38 s, corresponding to the 4^th and 12^th scans, respectively) are displayed for 5 axial slices. The images have been zero-filled 4-fold for display. The upper window level was adjusted to 55 % (pyruvate) or 110 % (lactate and bicarbonate) of the maximum intensity voxel in the 5 slices of the denoised images. For anatomic reference, ¹H T₂-FSE images are shown on the left. Noise reduction for the pyruvate images can be clearly seen in the later time point, and for both time points in the lactate and bicarbonate images.

Figure S5. Representative traces of original and denoised [1-¹³C]pyruvate, [1-¹³C]lactate, and [¹³C]bicarbonate signals from a selected voxel of the healthy volunteer data. The selected voxel for each dataset (HV-01, HV-02, and HV-06) is marked on the anatomical ¹H image on the left. The original and denoised data points are plotted with blue circles and orange crosses, respectively. The SNR_auc for the raw pyruvate, lactate, and bicarbonate data prior to denoising respectively for HV-01, HV-02, and HV-06 are [50, 24, 7], [58, 31, 9], and [20, 12, 3].

Figure S6. Comparison of k_PL and k_PB values obtained from the original and denoised in vivo data. (A) Correlation plots of k_PL estimates from the raw images and those of the corresponding voxels from the GL-HOSVD denoised images (DN) of the dataset HV-01 at five different SNR_auc ranges of raw lactate signals. (B) Correlation plots of k_PB estimates from the raw images and those of the corresponding voxels from the GL-HOSVD denoised images (DN) of the dataset HV-01 at three different SNR_auc ranges of raw bicarbonate signals. The red diagonal lines indicate when the ratio of k_PL(raw) to k_PL(DN) is 1. The average of the ratios is indicated in each plot.

Figure S7. Comparison of original and denoised metabolite dynamic data. Traces of the original and denoised metabolite signals in arbitrary units (a.u.) from 4 selected voxels, indicated by green boxes in the k_PL maps with numbered labels, are shown. For each voxel, pyruvate and lactate data points before (‘Orig.’; blue) and after denoising (‘DN’; orange) are displayed. The SNR_auc for the raw lactate signals are denoted in each plot. In the lactate graphs, the curves from k_PL fitting of the original and denoised data are also shown with dashed black lines and solid orange lines, respectively. The corresponding fitted k_PL values are plotted in the bottom graphs with error bars representing standard deviations of the fitted k_PL. The fitted curves of the original and denoised lactate signals from voxels #1 and 4 showed good agreement while the k_PL values from the denoised data were determined with higher precision than those from the original data as can be seen from the k_PL error bars. From voxels #2 and 3 where the lactate signals were relatively low, the initial signal rise was not accurately captured for the original data due to fluctuating initial data points, resulting in high k_PL with low precision.

Figure S8. Effect of patch size on the performance of GL-HOSVD. Three different patch sizes (3×3, 5×5, 7×7) were applied in GL-HOSVD to denoise a dynamic series of pyruvate and lactate images with random Gaussian noise σ_n = 0.4, and k_PL maps were calculated from the resulting images. (A–C) Ground truth k_PL maps for the whole brain slice, gray matter, and white matter (top to bottom); the k_PL maps from the denoised data using the patch size of 3×3 (D–F), 5×5 (G–I), and 7×7 (J–L) for the whole brain slice, gray matter, and white matter (top to bottom). As the patch size increases, the k_PL maps contain less noise but lose finer details, as manifested by the blurred edges.

Table S1. Summary of metabolite SNR_auc; spatial coverage in k_PL and k_PB maps; and the mean voxel-wise k_PL and k_PB values from the original (Orig.) and denoised (DN) dynamic HP-¹³C EPI images of the brain acquired from 6 healthy volunteers.