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 data sets. The SD of the random Gaussian noise contained in the noise-added data (σ_n = 0.2) is indicated on the left; (B) Mean SEs 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 data sets 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 SD of the mean k_PL error

FIGURE S3 Simulated assessment of accuracy of under the curve ratio (Lac_auc/Pyr_auc). Error images showing the mean percent differences (bias) in lactate-to-pyruvate area under the curve 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 data sets. The SD 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 4th and 12th scans, respectively), are displayed for 5 axial slices. The images have been 0-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 data set (HV-01, HV-02, and HV-06) is marked on the anatomic 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 before 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 data set HV-01 at 5 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 data set HV-01 at 3 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 SDs of the fitted k_PL. The fitted curves of the original and denoised lactate signals from voxels no. 1 and 4 showed good agreement whereas 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 no. 2 and 3 where the lactate signals were relatively low, the initial signal rise was not accurately captured for the original data because of 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