Hyperpolarized ¹³C Metabolic Imaging of the Human Abdomen with Spatiotemporal Denoising

Abstract

Purpose:

Improving the quality and maintaining the fidelity of large coverage abdominal hyperpolarized (HP) ¹³C MRI studies with a patch based global-local higher-order singular value decomposition (GL-HOVSD) spatiotemporal denoising approach.

Methods:

Denoising performance was first evaluated using the simulated [1-¹³C]pyruvate dynamics at different noise levels to determine optimal k_global and k_local parameters. The GL-HOSVD spatiotemporal denoising method with the optimized parameters was then applied to two HP [1-¹³C]pyruvate EPI abdominal human cohorts (n=7 healthy volunteers and n=8 pancreatic cancer patients).

Results:

The parameterization of k_global = 0.2 and k_local = 0.9 denoises abdominal HP data while retaining image fidelity when evaluated by RMSE. The k_PX (conversion rate of pyruvate-to-metabolite, X = lactate or alanine) difference was shown to be <20% with respect to ground-truth metabolic conversion rates when there is adequate SNR (SNR_AUC > 5) for downstream metabolites. In both human cohorts, there was >9-fold gain in peak [1-¹³C]pyruvate, [1-¹³C]lactate, and [1-¹³C]alanine apparent SNR_AUC. The improvement in metabolite SNR enabled a more robust quantification of k_PL and k_PA. After denoising, we observed a 2.1 ± 0.4 and 4.8 ± 2.5-fold increase in the number of voxels reliably fit across abdominal FOVs for k_PL and k_PA quantification maps.

Conclusion:

Spatiotemporal denoising greatly improves visualization of low SNR metabolites particularly [1-¹³C]alanine and quantification of [1-¹³C]pyruvate metabolism in large FOV HP ¹³C MRI studies of the human abdomen.

Introduction:

Hyperpolarized (HP) carbon-13 (¹³C) MRI is a rapid, noninvasive investigation of dynamic metabolic processes to detect disease and treatment response by measuring metabolic conversions of HP substrates to downstream metabolites^1,2. Hyperpolarization achieved through the dynamic nuclear polarization (DNP) technique provides unprecedented gains in sensitivity for imaging ¹³C-labeled biomolecules that are endogenous, nontoxic, and nonradioactive ³. HP ¹³C pyruvate is the most widely used molecule for human studies and is a highly biologically relevant probe at a critical branch point of multiple metabolic pathways ¹. Pyruvate is metabolized to lactate and alanine in the cytosol via the enzymes lactate dehydrogenase (LDH) and alanine aminotransferase (ALT), respectively ^4,5. It is also shuttled into mitochondria for ATP production via oxidative phosphorylation. HP ¹³C-pyruvate MRI can measure the increased glycolysis and lactate production mediated by LDH that occurs with aggressive cancers and monitor response to anticancer therapies in both pre-clinical and clinical settings ^6-8.

Despite dynamic nuclear polarization producing >10,000-fold MR signal enhancement, the spatial resolution of HP MRI is limited by the up to 100-fold lower signal-to-noise ratio (SNR) for metabolic products such as [1-¹³C]alanine and [1-¹³C]lactate in HP-¹³C studies with [1-¹³C]pyruvate ^9-14. Noise in HP images can lead to reduced accuracy or imprecise quantification of metabolic biomarkers, such as enzymatic conversion rates of pyruvate to lactate (k_PL) and pyruvate to alanine (k_PA)⁶. Recent evidence has suggested detecting [1-¹³C]alanine metabolism and its reprogramming could benefit the clinical management of abdominal neoplastic and metabolic diseases including pancreatic ductal adenocarcinoma (PDA)^15-19. Low metabolite SNR also poses major challenges for other HP abdominal imaging applications, including hepatocellular carcinoma ²⁰ (HCC), renal cell carcinoma ²¹ (RCC), and body metastatic cancers. The clinical translation of large coverage abdominal HP-¹³C studies faces additional challenges including: large fields of view requiring adequate transmitter and receiver coil coverages, fast imaging sequences to address physiologic motion, and large air-tissue interfaces including lungs and intestines that introduce additional magnetic field inhomogeneities²².

Image denoising post-processing approaches have been shown to improve HP-¹³C MR image quality, including tensor image enhancement ²³, wavelet packet denoising ²⁴, non-local means denoising ^25,26 as well as other super-resolution approaches ^27,28. Higher-order singular value decomposition (HOSVD)—a generalization of matrix SVD in higher dimensions used to approximate multidimensional data as a combination of lower dimensional features—has been used in contexts where there are both spatial and temporal similarities of MR images ^9,10,29,30. Kim et al.⁹ explored the benefits of patch based global-local HOSVD (GL-HOSVD) spatiotemporal denoising to improve image quality and quantification of [1-¹³C]lactate and [¹³C]bicarbonate in dynamic HP-¹³C MR studies in the human brain using echo-planar imaging (EPI)^9,10. Here, this project was designed to investigate this spatiotemporal GL-HOSVD technique for denoising HP [1-¹³C]pyruvate EPI data specialized for imaging the human abdomen and acquired in patients with PDA (n=8) and healthy volunteers (n=7).

Methods:

Numerical Simulations for Spatiotemporal GL-HOSVD Parameter Tuning:

The spatiotemporal GL-HOSVD denoising algorithm was adapted from Kim et al.⁹ and optimized for HP-¹³C MRI studies of the human abdomen. The tradeoff between denoising and image fidelity is primarily affected by the two parameters k_global and k_local. To refine these parameters for abdominal applications, a 32 x 32 (N_x x N_y) abdominal numerical phantom was created by masking a sample HP-¹³C EPI data abdominal dataset. Dynamic [1-¹³C]pyruvate and [1-¹³C]lactate signal within the numerical abdominal phantom were simulated using an irreversible, two-site kinetic model obtained from the open-source hyperpolarized-mri-toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox/) ^6,8. The rate of pyruvate-to-metabolite conversion (k_PX) for the liver, kidney, spleen, and pancreas were respectively initialized at 0.02, 0.015, 0.01, 0.005s⁻¹, similar to the average rate constants observed in healthy volunteers from prior studies ²² A region of 0.03s⁻¹ simulating tumor was added in the liver to evaluate high-metabolic conversion rates areas and changes in apparent resolution after denoising (Figure 1A). The T₁ relaxation times for [1-¹³C]pyruvate and [1-¹³C]lactate were estimated to be 30 seconds and 25 seconds, respectively, in the simulations. Additionally, a fixed flip angle scheme was used for [1-¹³C]pyruvate/[1-¹³C]lactate at 20° and 30° with a 3 second TR for 20 times frames (N_t). The measured [1-¹³C]pyruvate dynamic signals at each timepoint were used as inputs for two-site kinetic modeling which also accounted for signal loss associated with RF excitation and T₁ relaxation. Zero-mean random Gaussian noise was added to the noise-free dynamic [1-¹³C]pyruvate and [1-¹³C]lactate images to achieve different SNR levels. Metabolism was then simulated for 250 iterations at each respective noise level. The sensitivity gain was evaluated by calculating the mean SNR of metabolic signals between original noise-added images and the spatiotemporal denoised images. Denoised image fidelity was assessed using structural similarity index metric (SSIM) with respect to the ground-truth dynamic data^9,31. The optimal parameters for GL-HOSVD algorithm were determined from numerical simulations based on the root-mean square error (RMSE), SSIM and the relative k_PX fit error between noise-free and denoised [1-¹³C]lactate dynamics. A mask was applied around the representative abdominal organs to the images to exclude the background in SNR, SSIM, and RMSE calculations. To assess the accuracy of k_PX quantification introduced by denoising, errors in k_PX were calculated as $\frac{true{k}_{PX}-estimated{k}_{PX}}{true{k}_{PX}}\times 100\%$.

Figure 1. Spatiotemporal GL-HOSVD Numerical Simulation and Parameter Tuning.

(A) Abdominal numerical phantom featuring different organs initialized with different k_PX (s⁻¹) values. The numerical phantom was estimated from a healthy volunteer HP [1-¹³C]pyruvate abdominal study and previously found mean k_PX values from Lee et al.²², where k_PX values were initialized for each individual organ of interest as follows: liver = 0.02, pancreas = 0.015, spleen = 0.01, and kidney = 0.005 (s⁻¹). Influence of varying patch-based parameters k_global (B) and k_local (C) in the spatiotemporal GL-HOSVD algorithm on the mean RMSE of 250 denoised, simulated downstream metabolite images under varying Gaussian noise levels. (D) Assessment of the mean relative k_PX fit error as a function of non-denoised lactate SNR_AUC at different noise levels. The relative k_PX bias for lactate SNR_AUC > 8 is less than 15%.

Human HP-¹³C EPI data acquisition:

All human MRI experiments were performed on a 3T clinical scanner (MR750, GE Healthcare, Waukesha, WI) equipped with multi-nuclear imaging capability using a ¹³C volumetric transmitter along with an eight-channel flexible receiver array (QTAR, Clinical MR Solutions, Brookfield, WI) covering each subject’s abdomen. Human studies received approval from an institutional review board requiring informed consent as well as adhering to Food and Drug Administration investigation new drug application (FDA IND)-approved protocols. A 0.43 mL/kg dose of 250 mM HP [1-¹³C]pyruvate was injected into each patient, and a single-band spectral-spatial RF pulse was used to independently excite [1-¹³C]pyruvate, [1-¹³C]lactate, and [1-¹³C]alanine resonances with 20°, 30°, and 30° flip angles, respectively ^32,33. The datasets were acquired with an in-plane spatial resolution of 2 × 2 cm² for 7-14 20 mm thick slices with 20 timeframes with 3-second temporal resolution per metabolite (60-second total scan-time). The [1-¹³C]pyruvate was polarized for at least 2 hours using a 5T SPINlab instrument operating at 0.8 K (GE Healthcare). Proton T₂-weighted and T₁-weighted 3D images were acquired for anatomical reference. RF transmit power was calibrated on an unenriched dimethyl silicon phantom before each study utilizing a 2 ms hard pulse. The transmit gain for a 90° flip was then verified using a 30° flip which resulted in a spectral peak signal with half the maximum amplitude²². The spatiotemporal denoising method with the optimized parameters was applied to the HP [1-¹³C]pyruvate multi-slice EPI data from two abdominal human cohorts (n = 7 healthy volunteers and n = 8 PDA patients). The mean total SNR (area under the curve SNR, SNR_AUC) improvement was quantified. To determine SNR_AUC, the metabolite signal was summed through time and divided by the estimated standard deviation of a noise-only region. Voxel-wise kinetic modeling was performed on both the original non-denoised and denoised data to compare the number of voxels quantifiable based on the original SNR_AUC > 10 for pyruvate and original SNR_AUC > 5 [1-¹³C]lactate and [1-¹³C]alanine, along with fitting criteria (<30% relative error where σ_kPX < 0.3 · k_PX (X = lactate or alanine).

Results:

The patch-based spatiotemporal algorithm, leveraging sparseness in dynamic imaging to denoise the data, was optimized first in simulations and then applied to human HP [1-¹³C]pyruvate abdominal datasets. To select the appropriate parameters k_global and k_local for abdominal HP studies, denoising performance was investigated using simulated [1-¹³C]pyruvate dynamics at different noise levels in the abdominal numerical phantom. As shown in Figure 1A, the numerical simulation was initialized by varying k_PX (s⁻¹) between 0.005-0.03 (s⁻¹) representative of liver, pancreas, kidney, spleen, and tumor values. From 250 replicate simulations, Figure 1B-C demonstrates a broad range of k_global and k_local values which can denoise HP data while retaining image fidelity when evaluated by RMSE—k_global = 0.2-0.4 and k_local = 0.6-0.9 respectively. Similar to our previous study, varying the NxN patch-size from 3x3-7x7 and search window from 6-11 were not found to significantly alter SSIM, RMSE and relative k_PX fit error ⁹. At the parameterization of k_global = 0.2 and k_local = 0.9, simulation results demonstrated a 9.5 ± 0.1 fold increase in the relative mean apparent SNR of [1-¹³C]lactate and a SSIM near 1 for all twenty timepoints with σ_n = 0.1 added Gaussian noise (Figure S1). It is important to note that with σ_n = 0.1 added Gaussian noise, the [1-¹³C]lactate SNR_AUC (SNR of temporally summed metabolic signals) within the numerical phantom organs ranged approximately from 5 to 25 which is on the order of what has been previously observed from downstream metabolites [1-¹³C]lactate and [1-¹³C]alanine in human HP ¹³C studies. Using the parameterization of k_global = 0.2 and k_local = 0.9, Figure 1D illustrated the relative mean k_PX errors calculated from the abdominal numerical phantom k_PX maps obtained from 250 GL-HOSVD processed datasets simulated with 3, 5, 7, 10, 15, 20, 30 50, 60, 70, 80, 90 and 100% random Gaussian noise, and plotted over the original non-denoised SNR_AUC of [1-¹³C]lactate from the corresponding abdominal region. Here, the relative k_PX was underestimated by <15% when the non-denoised [1-¹³C]lactate or [1-¹³C]alanine SNR_AUC was greater than 8. This result showed that if the original metabolite signal has adequate SNR (as low as a SNR_AUC = 5) metabolite dynamic image features were retained and a less than −20% error on the quantification of the enzymatic conversion rates can be expected (Figure 1D and Figure S2). Denoised images improved qualitatively but were inaccurate when quantifying metabolism due to poor separation of signal and noise subspace at low SNR. Therefore, an original SNR_AUC threshold was implemented prior to kinetic fitting to address this potential error in quantification. The second criteria after kinetic fitting (k_PX fit with <30% relative errors) was chosen to further exclude voxels with high model uncertainty, which is the case for many low SNR voxels. Figure S3 shows correlation ratios of k_PX estimates from the denoised and noised added images compared to those of the corresponding voxels from the ground-truth k_PX numerical phantom at three different SNR_AUC ranges of noise-added [1-¹³C]lactate signals. When the noise-added [1-¹³C]lactate SNR_AUC > 5 the mean correlation ratio to the ground-truth k_PX values was 0.99 and 1.00 when [1-¹³C]lactate SNR_AUC > 20.

Fifteen HP [1-¹³C]pyruvate EPI scans of healthy volunteers and PDA patients illustrated the optimized spatiotemporal denoising method described above. The computation time for denoising EPI datasets (3 metabolites x 9 slices, matrix size of 16x16) using a patch size of 5x5 averaged 8.2 seconds. Figure 2 displays original non-denoised and denoised [1-¹³C]pyruvate, [1-¹³C]lactate, and [1-¹³C]alanine images from a healthy volunteer data set (HV 6). The top row for each metabolite shows original non-denoised dynamic images. The peak SNR_AUC was observed to be 370, 47, and 23 for [1-¹³C]pyruvate, [1-¹³C]lactate, and [1-¹³C]alanine, respectively. The second row displays the same metabolite images after denoising with the optimized parameters, here background noise was greatly reduced while preserving image fidelity. The lower SNR_AUC [1-¹³C]alanine images indicate most notable improvement after denoising. The [1-¹³C]alanine signals were not reliably distinguishable in the original images, especially at later time points, but are clearly observable after denoising with a 12.2-fold noise reduction. The [1-¹³C]pyruvate and [1-¹³C]lactate signal distributions were preserved after denoising images without noticeable distortion. As previously observed in human brains, there was close agreement between the denoised and original non-denoised pyruvate image ⁹. This indicates that the spatiotemporal denoising introduced minimal image bias to originally high SNR resonances. For this volunteer (HV 6), the apparent SNR_AUC gain for all 20 timeframes was calculated to be 8.2 for [1-¹³C]pyruvate, 11.1 for [1-¹³C]lactate, and 12.2 for [1-¹³C]alanine. Figure 3A illustrates the effective noise reduction seen in [1-¹³C]alanine dynamics in a PDA patient (PDA 1) after patch-based denoising. In this figure, the tumor is referenced in red and the normal appearing pancreas in white. Denoised images display more clearly pancreatic [1-¹³C]alanine distributions. Table S1 and Figure 4A-B summarize the improvement in metabolite SNR_AUC after denoising the dynamic HP-¹³C EPI images from both PDA patients and healthy volunteer datasets. For all 15 subjects, the mean increase in apparent SNR was 9.6 ± 3.3 for [1-¹³C]pyruvate, 8.7 ± 2.4 for [1-¹³C]lactate, and 11.4 ± 1.8 for [1-¹³C]alanine.

Figure 2. Dynamic Spatiotemporal Denoising Results from a Healthy Volunteer.

A. Dynamic HP-¹³C EPI data of pyruvate, lactate, and alanine signals from the abdomen of a healthy volunteer before (‘Orig.’) and after applying GL-HOSVD (‘DN’). B. Comparison between the ‘Orig.’ and ‘DN’ images of two individual lactate, and alanine timepoints at 12 (yellow) and 36 (green) seconds.

Figure 3. [1-¹³C]Alanine Spatiotemporal Denoising Results from a Pancreatic Cancer Patient.

(A) SNR improvement seen for [1-¹³C]alanine dynamic maps obtained before and after applying GL-HOSVD in a patient with pancreatic cancer. Tumor is outlined in red, pancreas in white. (B) In addition to increased SNR in the alanine dynamics, denoising increased the number of voxels that met the error criterion for k_PA quantification from 143 to 656.

Figure 4. Comparison of SNR_AUC and Number of Voxels Fit in k_PL and k_PA Maps from the Original and Denoised Dynamic HP-¹³C EPI Data.

(A-B) SNR_AUC and number of voxels fit in k_PL and k_PA maps from the original non-denoised (‘Orig.’) and denoised (‘DN’) dynamic HP-¹³C EPI images of the PDA patient and Healthy Volunteer datasets. For all 15 subjects, the mean increase in apparent SNR was 9.6 ± 3.3, 8.7 ± 2.4 and 11.4 ± 1.8 for pyruvate, lactate, and alanine, respectively. (C) After denoising there was an increase in the number of voxels fit that met the fitting criteria for k_PL and k_PA maps of 2.1 ± 0.4 and 4.8 ± 2.5-fold, respectively.

Spatial coverage improvement for k_PL and k_PA maps using GL- HOSVD was measured for denoised datasets from the healthy volunteers and PDA patient cohorts. The SNR_AUC cutoff and kinetic-modeling error criteria were applied to generate these maps based on numerical simulations as described in Methods Section and from Figure 1B-D. Figure 3B illustrates the increased number of k_PA values that met the SNR_AUC and error criteria after denoising, along with the distribution of k_PA values originally and after denoising. The data in this figure demonstrated a 4.9-fold gain in the number of voxels fit for this PDA patient resulting from denoising, and k_PA values were similar to the original. Figure 5A shows representative k_PL and k_PA values overlaid on anatomic images before and after denoising in a healthy volunteer. Apparent differences in coverage between the original non-denoised and denoised k_PL and k_PA maps can be understood by referencing the original SNR_AUC for each metabolite. Areas of low [1-¹³C]lactate and [1-¹³C]alanine SNR_AUC (near threshold value of SNR_AUC = 5) yielded high kinetic fit-values. Also, denoising increased the number of voxels fit for both k_PL and k_PA in the liver, kidney, pancreas and spleen, respectively (Figure 5B), while also reporting similar mean kinetic rates in each individual organ after denoising. The representative abdominal slice indicated similar mean k_PL and k_PA after denoising, when comparing k_PL and k_PA values at voxels that met the kinetic-fitting criteria and SNR_AUC criteria prior to denoising across the liver, pancreas, spleen and kidney (Figure 5C). Across all subjects as summarized in Table S1 and Figure 4C, after denoising there was an increase in the number of voxels fit for k_PL and k_PA maps of 2.1 ± 0.4 and 4.8 ± 2.5-fold, respectively.

Figure 5. Increase in the Number of Voxels Fit Across Abdominal FOVs for k_PL and k_PA Quantification Maps.

(A) A representative SSFSE axial slice from a healthy abdominal volunteer with corresponding SNR_AUC overlays for k_PL and k_PA quantification maps prior to denoising (‘Orig.’) and after denoising (‘DN’). (B) Visualization of increase in number of k_PL and k_PA voxels fit for the corresponding axial slice after denoising, suggesting denoising allows for greater spatial visualization of enzymatic ratio maps. (C) Mean k_PL and k_PA after before and after denoising, when comparing k_PL and k_PA values at voxels that met the SNR_AUC threshold for [1-¹³C]pyruvate > 10 and kinetic-fitting criteria of σ_kPX < 0.3 · k_PX across the liver, pancreas, spleen and kidney.

Discussion & Conclusion:

This study demonstrated that spatiotemporal denoising can greatly improve the detection of low SNR metabolites and quantification of pyruvate metabolism in dynamic HP-¹³C MR studies of the human abdomen. By denoising HP-¹³C EPI datasets using GL-HOSVD, low SNR metabolites, [1-¹³C]lactate and [1-¹³C]alanine, were better visualized, and more voxels were fit with lower relative error in k_PL and k_PA. A 9.6 ± 3.3, 8.7 ± 2.4 and 11.4 ± 1.8-fold gain in [1-¹³C]pyruvate, [1-¹³C]lactate and [1-¹³C]alanine peak SNR_AUC was observed, respectively. This improvement in SNR and image quality increased the coverage and utility of HP-¹³C MRI and supports the acquisition at finer spatial resolutions in future human abdominal studies, which could reduce partial volume effects leading to better detection of tumor heterogeneity. The abdominal numerical simulation results showed that the accuracy of kinetic rate quantification from the spatiotemporal GL-HOSVD denoised images depend on the original SNR_AUC of the noisiest metabolite and can be used to guide interpretation and analysis of denoised data. Based on these simulations, kinetic rates can be estimated with <20% error from voxels with metabolite SNR_AUC of 5 or higher. When metabolite SNR_AUC was <5_, we observed that the GL-HOSVD spatiotemporal denoising method was unable to reliably separate metabolite signal and noise. This manifests in the denoised images as a reduction in image sharpness for low-SNR objects especially in structures with sharp edges at object boundaries (Figure S2) and led to underestimation of downstream metabolic signal and errors in the calculated rate constants (Figure 1D). Other denoising techniques for multi-dimensional MR data include utilizing random matrix theory (RMT) to estimate pure noise and signal components ^34,35. These techniques may also be relevant to HP-¹³C MR by eliminating the need to tune the parameters to preserve image fidelity for different imaging contexts, but they have not been investigated for this application.

In this study, denoising resulted in a 2.1 ± 0.4-fold increase in the number of voxels adequately fit for mapping k_PL and a 4.8 ± 2.5-fold increase in the number of voxels for mapping k_PA. In cases of low SNR_AUC, spatiotemporal denoising may contribute to increased error in the calculations of kinetic-rate constants. B₁− and B₁+ are two additional variables to consider in these instances. In large-volume abdominal studies, signals decrease farther from the receive array. Also, B₁+ inhomogeneities could confound metabolic modeling as variations in flip angle due to inhomogeneous RF field can cause metabolite signal intensity variations that may lead to errors in the determination of k_PL and k_PA values in vivo. It is likely that the high k_PL and k_PA values observed at the periphery of the abdomen in this healthy volunteer are due to the B₁+ inhomogeneity of this flexible vest coil, which exhibits elevated B₁+ close to the coil. Previously, variations in B₁+ of up to ± 20% were shown to cause k_PL errors of up to 10% if uncorrected, and in the current HP experimental setup, flexible vest coils have more than ±50% variation in B₁+ throughout its volume ^8,22. The observed increase in abdominal spatial coverage of enzymatic rate maps after denoising could help empirical methods to better correct for B₁+ inhomogeneities. Additionally, in some organs where an increase in k_PX coverage was achieved after denoising compared to the original datasets, we observed a greater variation in standard deviation of k_PX estimations. This is likely due to additional voxels meeting fitting criteria after denoising that tend to have lower conversion rates, and if the analysis is restricted to only the voxels fit prior to denoising, we observe an overall reduction in k_PX standard deviation after denoising in each organ (Figure S4). Finally, this improvement in spatial coverage of [1-¹³C]pyruvate MRI may improve clinical research of patients with abdominal cancers including PDA, HCC, RCC and metastatic cancers.

Supplementary Material

Supinfo

Table S1. Patient Study Results. Summary of metabolite SNR_AUC and number of voxels fit in k_PL and k_PA maps values from the original non-denoised and denoised dynamic HP-¹³C EPI images of the PDA and Healthy Volunteer patient datasets. For all 15 subjects, the mean increase in apparent SNR was 9.6 ± 3.3, 8.7 ± 2.4 and 11.4 ± 1.8 for pyruvate, lactate, and alanine respectively. After denoising there was a 2.1 ± 0.4 and 4.8 ± 2.5-fold increase in the number of voxels fit for k_PL and k_PA maps, respectively.

Figure S1: Numerical simulations of metabolite dynamics. Left) SSIM of [1-¹³C]lactate dynamics with σ = 0.1 noise added (blue) and denoised (red) over 250 replicate simulations. Right) Relative mean values of the SNR of the noise-added and denoised [1-¹³C]lactate signals from 250 simulated data sets. The relative mean SNR gain here was observed to be 9.48 ± 0.14 The peak SNR_Lac of the noise-added data was normalized to 1.

Figure S2: Line-profiles of ground-truth, noise-added and denoised mean [1-¹³C]Lactate AUC images with random Gaussian noise. (A-C) The dynamic data was summed temporally to produce ground-truth, noise-added and denoised AUC plots. (D-E) Vertical and horizontal line intensity profiles were produced for each image (mean ± standard deviation across 250 noise iterations). The vertical line profile is drawn through high SNR_{AUC (}F) _and k_PL structures. The tumor (k_PL = 0.03 s⁻¹) represents a point source and there is good agreement between the denoised line profile and the ground truth. When SNR_AUC (< 5) is low, as in the blue horizontal line profile through the kidney, we do observe a systematic underestimation of signal at very low SNR_AUC (< 5) manifesting in the denoised images as a reduction in image sharpness for low-SNR objects when using a numerical phantom with sharp edges.

Figure S3: Correlation ratios of enzymatic rate constants in a numerical simulation regime. Box and whisker plots to visualize the k_PX rate-constant correlation ratio over three SNR_AUC ranges. Each box represents the upper and lower quartile range (25% - 75%), and the solid line within each box denotes the median value. The whisker bars extend to the lower and upper extremes of the data, defined as within 1.5 times the inter-quartile range from the upper or lower quartile, and only the outlier values, defined as values above and below 1.5 times the inter-quartile range, are explicitly plotted. The mean of the data was added to each boxplot denoted by the black ‘+’ mark. Here, after denoising for SNR_AUC < 5 there is a systematic underestimation and motivates our threshold for data analysis in the in-vivo datasets. When the noise-added [1-¹³C] downstream metabolite SNR_AUC > 5 the mean correlation ratio to the ground-truth k_PX values is 0.99 and 1.00 when [1-¹³C] downstream metabolite SNR_AUC > 20.

Figure S4: Mean k_PL and k_PA after before and after denoising in the liver, pancreas, spleen and kidney. As a companion to Fig. 5C, this figure restricts the Raw and DN k_PL and k_PA analysis to include only the voxels that were fit and passed the kinetic-fit criteria in both raw and denoised images. There is good agreement in the mean and a reduction in the standard deviations of the k_PL and k_PA values (s⁻¹) after denoising.