Specialized computational methods for denoising, B₁ correction, and kinetic modeling in hyperpolarized ¹³C MR EPSI studies of liver tumors

Abstract

Purpose:

To develop a novel post-processing pipeline for hyperpolarized (HP) ¹³C MRSI that integrates tensor denoising and ${\text{B}}_1^{+}$ correction to measure pyruvate-to-lactate conversion rates (k_PL) in patients with liver tumors.

Methods:

Seven HP ¹³C MR scans of progressing liver tumors were acquired using a custom ¹³C surface transmit/receive coil and the echo-planar spectroscopic imaging (EPSI) data analysis included B₀ correction, tensor rank truncation, and zero-and first-order phase corrections to recover metabolite signals that would otherwise be obscured by spectral noise as well as a correction for inhomogeneous transmit (${\text{B}}_1^{+}$) using a ${\text{B}}_1^{+}$ map aligned to the coil position for each patient scan. Processed HP data and corrected flip angles were analyzed with an inputless two-site exchange model to calculate k_PL.

Results:

Denoising averages SNR increases of pyruvate, lactate, and alanine were 37.4-, 34.0-, and 20.1-fold, respectively, with lactate and alanine dynamics most noticeably recovered and better defined. In agreement with Monte Carlo simulations, over-flipped regions underestimated k_PL and under-flipped regions overestimated k_PL. ${\text{B}}_1^{+}$ correction addressed this issue.

Conclusion:

The new HP ¹³C EPSI post-processing pipeline integrated tensor denoising and ${\text{B}}_1^{+}$ correction to measure k_PL in patients with liver tumors. These technical developments not only recovered metabolite signals in voxels that did not receive the prescribed flip angle, but also increased the extent and accuracy of k_PL estimations throughout the tumor and adjacent regions including normal-appearing tissue and additional lesions.

1 |. INTRODUCTION

Hyperpolarized (HP) ¹³C MRSI using dissolution dynamic nuclear polarization techniques¹ enables quantitative imaging of enzyme-catalyzed metabolism in humans and has been applied to study in vivo metabolism of cardiac diseases,^2,3 breast cancer,^4,5 brain cancer,^6–9 and prostate cancer.^10–12 In this metabolic imaging approach, HP ¹³C-pyruvate can be safely injected into human subjects while retaining an observable signal that is 10,000+-fold higher than thermal values at clinical field strengths.¹³ Analysis and kinetic modeling can quantify enzymatic conversion rates in key metabolic pathways, most notably the pyruvate-to-lactate conversion through lactate dehydrogenase (LDH).^13–15 The measured rate constant, k_PL, is a potential biomarker that is greatly increased in cancer^13,16 and has been shown to reflect treatment response in animal models^17–20 and in humans.^10,12

Although prior studies demonstrated the ability to detect increased pyruvate-to-lactate conversion in high grade cancers and reduced conversion following successful therapy,^{5,7,10,12,17–20} current methods are often sub-optimal with several technical challenges yet to be fully overcome.¹³ Background noise in HP spectra can obscure downstream metabolite signals limiting the achievable spatial resolution and reducing the accuracy of k_PL estimations in these cases. Surface transmit/receive (T/R) coils can improve SNR limitations increasing the sensitivity in these experiments, but they produce an inhomogeneous transmit (${\text{B}}_1^{+}$) profile. This results in spatially varying flip angles that could bias quantification.^14,21

The goal of this project was to develop a novel post-processing pipeline to improve spectral SNR using tensor rank truncation denoising and to correct k_PL data for ${\text{B}}_1^{+}$ variations in human studies of metastatic and primary cancers in the liver. This was designed to enable quantitative HP ¹³C-pyruvate MR to address an important unmet clinical need for measuring cancer metabolism and response to therapy in liver tumors.

2 |. METHODS

2.1 |. Patients

Seven MRI exams including a HP ¹³C MR acquisition were performed on a clinical 3T MRI scanner with multi-nuclear capability from three patients enrolled in a HP MRI study of metastatic cancer. These seven exams are tabulated in Table 1. The human studies were approved by an Institutional Review Board (IRB) requiring informed consent and followed Food and Drug Administration Investigational New Drug application (FDA IND) approved protocols.

TABLE 1.

Selected voxels from seven patient scans showing the optimized tensor ranks (with dimensions: frequency, x, y, and time), improvements in SNR after denoising, the flip correction scaling factor (over-flipped: >1, under-flipped: <1), the k_PL values before and after ${\text{B}}_1^{+}$ correction, and the % difference

Scan no.	Cancer origin and notes	Optimized tensor ranks [freq, x, y, t]	Denoising SNR multiplicative increases	Flip angle scaling	kPL before correction (s−1)	kPL after correction (s−1)	% Difference
1	Rectal adenocarcinoma	[18, 9, 10, 5]	Pyr: 17.5	1.20	0.031	0.042	−27.96
			Lac: 9.4	1.03	0.038	0.040	−4.60
			Ala: 4.0	1.10	0.044	0.052	−15.24
2	Rectal adenocarcinoma	[12, 8, 11, 8]	Pyr: 15.9	0.97	0.044	0.042	+5.77
			Lac: 20.2	1.03	0.037	0.039	−5.58
			Ala: 9.1	1.13	0.033	0.042	−19.49
3	Intrahepatic cholangiocarcinoma	[7, 5, 11, 18]	Pyr: 26.5	1.13	0.051	0.063	−19.01
			Lac: 27.6	0.87	0.080	0.066	+20.22
			Ala: 35.1	0.97	0.051	0.048	+5.31
4	Intrahepatic cholangiocarcinoma	[5, 6, 8, 6]	Pyr: 82.4	1.37	0.030	0.053	−43.62
			Lac: 101.8	1.27	0.048	0.073	−34.79
			Ala: 17.7	1.17	0.043	0.057	−23.64
5	Recurrent grade 1 metastatic pancreatic neuroendocrine	[14, 8, 5, 7]	Pyr: 58.7	0.97	0.043	0.040	+5.64
			Lac: 26.2	0.97	0.029	0.027	+5.60
			Ala: 28.9	1.03	0.024	0.026	−5.38
6	Recurrent grade 1 metastatic pancreatic neuroendocrine	[10, 7, 14, 3]	Pyr: 47.9	1.27	0.043	0.069	−37.65
			Lac: 42.3	0.87	0.059	0.047	+26.96
			Ala: 37.5	0.97	0.049	0.046	+6.21
7	Recurrent grade 1 metastatic pancreatic neuroendocrine	[20, 7, 12, 7]	Pyr: 12.6	1.27	0.041	0.066	−37.86
			Lac: 10.4	0.97	0.038	0.036	+5.90
			Ala: 8.7	0.97	0.032	0.031	+5.84

There were variations between the SNR gains of the metabolite peaks, where larger SNR gains were shown in those with lower SNR dynamic spectra before denoising.

2.2 |. HP ¹³C patient MRI scans

The studies were conducted on a clinical 3T MRI (GE Healthcare, Milwaukee, WI) with a custom figure-8 surface coil (shown in Figure 1) for ¹³C transmit and receive and either a commercial 32-channel torso array or a 16-channel flex array for ¹H receive. As seen in Figure 1A, the figure-8 coil measured 17 cm × 10 cm. Vitamin E capsules were embedded in the coil at its corners and along one central axis, and therefore, facilitated coil positioning because they were visible on T₁-weighted images. A ¹³C-enriched urea phantom was also visible on T₁-weighted images and was used for transmit power calibration. As seen in Figure 1B, both vitamin E capsules and the urea phantom, highlighted by yellow dots, were used during the initial ${\text{B}}_1^{+}$ map acquisition and subsequent alignment. Details for the ${\text{B}}_1^{+}$ map and correction are presented in section 2.5. Figure 1C–D show an example coil-placement for a patient scan, here with the figure-8 coil and a 32-channel ¹H torso array.

FIGURE 1.

(A) The figure-8 T/R coil opened with a ¹³C-enriched urea phantom used for power calibration highlighted by a purple circle and vitamin E capsule fiducial markers highlighted by light blue rectangles. (B) The coil’s ${\text{B}}_1^{+}$ profile acquired using the double angle method. The contour map indicates the actual flip angle delivered relative to the prescribed flip angle. (C and D) Photos of an example coil-placement for a patient scan. Exterior markers were used for verifying coil placement following localizer scans, but were not used for ${\text{B}}_1^{+}$ correction.

Good manufacturing practice (GMP) grade [1-¹³C]pyruvic acid (ISOTEC Stable Isotope Division, MilliporeSigma, Miamisburg, OH) with trityl radical (GE Healthcare) was prepared and loaded in pharmacy kits (GE Healthcare) according to IRB- and FDA IND-approved stable-isotope manufacturing processes. Following polarization in a 5T SPINLab clinical-research polarizer (GE Healthcare) and dissolution, 240 ± 5.2 mM sterile pyruvate with 7.8 ± 0.2 pH, at 33.9°C ± 1.6°C, and a polarization of 37.7% ± 1.7%, was injected following pharmacist approval at 60.1 ± 9.8 s after dissolution at a rate of 5 mL/s followed by a 20 mL saline flush. The injection volume was determined using the weight-based dose of 0.43 mL/kg (total body weight) with a maximum of 40 mL.

2.3 |. Data acquisition

HP ¹³C data were acquired using a 2D dynamic MR echo-planar spectroscopic imaging (EPSI) pulse sequence consisting of a slice-selective spectral-spatial pulse followed by phase-encode and EPSI readout similar to prior prostate cancer studies.^12,22,23 The scan parameters were as follows: 130 ms/3.5–5.2 ms TR/TE, 1.2- to 3-centimeter slice thickness, 1.2 × 1.2 cm in-plane spatial resolution and 3-second temporal resolution, 60-second acquisition window, 545 Hz spectral bandwidth, and constant flip angles of 10° for pyruvate, 20° for lactate, and 15° for alanine. Acquisition was started 5 s after the injection completed. Each patient was asked to breath-hold for as long as comfortable once the acquisition began. T₁-weighted spoiled gradient-echo ¹H images (with 4.0 ms/1.8 ms TR/TE) and T₂-weighted single-shot fast spin echo ¹H images (with 545.24 ms/81.662 TR/TE) images were acquired for anatomic references.

2.4 |. Denoising

The analysis pipeline began with a B₀ inhomogeneity correction that shifted peaks in the spectral dimension.^15,24 For each voxel, the cross correlation between a standard spectrum and the voxel’s spectrum was computed and the index of the maximum value was used to obtain the new pyruvate peak index. The pipeline continued with tensor rank truncation for denoising the HP data.^25,26 The spectral, spatial, and temporal dimensions of HP ¹³C MRSI data were broken down into principal component fibers using high-order singular value decomposition. Because the chemical shift spectrum, geometric space, and metabolic dynamics were inherently low rank, the dimensions of the HP ¹³C MRSI data could be represented by a truncated number of fibers. The optimal number of tensor ranks to preserve in each dimension was computed using a bias-variance tradeoff that balances between spurious artifacts and noise removal.²⁶ Finally, tensor rank truncation was followed by zero-order and first-order phase corrections and a baseline correction. Metabolite signals were calculated by integrating each peak without any overlap because of the sparsity of the HP ¹³C spectra. Images were sinc interpolated to a matrix size of 1024 × 1024. Pyruvate and lactate SNR thresholds (defined as the mean of the metabolite’s signal divided by the standard deviation of the spectral noise) were implemented to mask out noise. Voxels outside of the coil’s sensitive region, determined by the acquired ${\text{B}}_1^{+}$ map described below, were removed.

2.5 |. ${\text{B}}_1^{+}$ transmit field inhomogeneity correction

Prior to the patient studies, a transmit inhomogeneity ${\text{B}}_1^{+}$ field map was acquired from a phantom (Figure 1B) using the double angle method (DAM).^27–29 This method calculates a flip-angle map as an indirect measurement of the B₁ excitation field. Two images are acquired, I₁ with the prescribed flip angle α₁ and I₁ with the prescribed flip angle α₂ = 2α₁. Other acquisition parameters were kept constant. Using a TR of 3 s, T₁ effects were negligible and the equation below was used to compute the ${\text{B}}_1^{+}$ map, α(r):

$$\alpha (r)=\text{arccos\hspace{0.17em}}\left(\frac{I_2(r)}{2I_1(r)}\right).$$ (1)

The surface coil was strapped to a 25-cm diameter cylindrical ethylene glycol phantom.³⁰ The ¹³C center frequency was placed on the central ethylene glycol resonance. The ${\text{B}}_1^{+}$ data were acquired using a metabolite-selective sequence with a single-shot echo planar readout, with α₁ = 30° and α₂ = 60°, a 20 × 20 matrix zero-filled to 32 × 32, with voxel sizes of 0.875 × 0.875 × 4.0 cm. With two flip angles, a NEX of 100, and a TR of 3 s, the total scan time was 600 s. The ${\text{B}}_1^{+}$ map was computed using Eq. (1) and thresholded to remove noise voxels.

Because the fiducial markers were present in T₁-weighted images from the initial DAM acquisition and from subsequent patient scans, each marker had corresponding left, posterior, superior (LPS) coordinates in DICOM images. A least squares estimation method^31,32 was used to compute rotation and translation matrices that would transform the fiducial markers’ DAM acquisition LPS coordinates to patient scan coordinates. The ${\text{B}}_1^{+}$ map and the ¹³C images were correspondingly converted from pixels to physical dimensions, matching the LPS coordinate system. The ${\text{B}}_1^{+}$ map was transformed as above and interpolated to the resolution of the ¹³C scans, resulting in a registered ${\text{B}}_1^{+}$ map where each ¹³C voxel corresponded to a ${\text{B}}_1^{+}$ scaling factor. This scaling factor, the relative transmit power per voxel, was used to scale the nominal flip angle accordingly. The corrected flip angle was subsequently provided as a parameter for an inputless two-site exchange model¹⁴ to estimate the forward reaction rate of pyruvate to lactate, or k_PL. k_PL maps were sinc interpolated to a matrix size of 1024 × 1024. Monte Carlo simulations were used to compute expected k_PL variations when varying ${\text{B}}_1^{+}$ in 10% increments from −20 to +100% relative to its original value.

3 |. RESULTS

3.1 |. Patient scans

Seven MR exams including HP ¹³C were acquired from patients with progressing liver tumors from multiple primary cancers as summarized in Table 1. Representative data from scan 5 are shown in Figures 2 and 3 and were acquired with the coil set-up shown in Figure 1. The conversion of pyruvate-to-lactate in the liver metastases was observed to be upregulated, consistent with the metabolic reprogramming known to occur in cancers.^13,33,34

FIGURE 2.

The left panel (orange box) shows the acquired HP ¹³C EPSI spectrum and metabolite dynamics before and after denoising for the tumor voxel. Likewise, the right panel (green box) shows the acquired EPSI spectrum and metabolite dynamics before and after denoising for a normal-appearing voxel. The adjacent T₁-weighted anatomic images highlight locations of the metastases and the selected voxel. Both are representative voxels taken from scan 5.

FIGURE 3.

From scan 5, the metabolite peaks from pyruvate, lactate, and alanine were integrated and the resulting time courses are shown on the left side of the figure. For each metabolite, the top row of images is from the original spectra and the bottom row of images shows the post-denoised metabolite maps. Every entire row of images is independently window-leveled relative to the other rows. On the right are displayed zoomed-in images before and after denoising for a selected time point (18 s from the start of acquisition) are highlighted on the right side of the figure.

3.2 |. Denoising and ${\text{B}}_1^{+}$ transmit field inhomogeneity correction

The denoising pipeline consisting of B₀ correction, tensor rank truncation, and zero- and first-order phase corrections was applied before ${\text{B}}_1^{+}$ correction and k_PL analysis for all seven scans with results summarized in Table 1. On average for all the voxels, pyruvate SNR increased by 37.4-fold, lactate SNR increased by 34.0-fold, and alanine SNR increased by 20.1-fold. Figure 2 shows the spectral data before and after denoising for two selected voxels from a patient with metastatic pancreatic cancer to the liver (scan 5). From left to right, the top row shows the acquired EPSI spectra before and after denoising for a tumor voxel and a normal-appearing voxel, respectively. The adjacent T₁-weighted anatomic images highlight the tumors and selected voxels.

Before denoising, there was a high noise floor obscuring the alanine peak. After denoising, both the lactate and alanine peaks were more clearly resolved both spectrally and dynamically through time with SNR increases of 26.2- and 28.9-fold, respectively. The lactate dynamic profile was likewise recovered and smoothed. Because the acquired pyruvate SNR was high, no significant changes in kinetics were observed after denoising. However, the mean pyruvate SNR increase was 58.7-fold. The dynamic curve for pyruvate remained similar pre- and post-denoising, however, the depiction of lactate and alanine dynamics were greatly improved with the denoising pipeline, shown in the bottom row of Figure 2.

For the same scan, the metabolite peaks of interest were integrated to create images of their spatiotemporal distributions shown in Figure 3. For each metabolite, the top row corresponds to pre-denoised spectra and the bottom row to post-denoised spectra, both rows independently normalized for display. Shown on the right are zoomed-in images for each metabolite pre-and post-denoising for the selected time point 18 s from the start of acquisition, corresponding to the peak of the lactate signal. The background noise was most apparent in the lactate and alanine pre-denoised images. In the denoised lactate images, bright voxels at t = 18 s correspond to the increased conversion to lactate in the metastatic tumors. However, the shape and location of these hot spots were blurred before denoising. These differences are clearly shown in the images on the right of the figure. The alanine signal is also noticeably recovered from a noisy background at t = 21 s. The improvement for pyruvate detection is not as significant because of its high SNR before denoising.

In addition to denoising, this analysis pipeline provided flip angle corrections using a ${\text{B}}_1^{+}$ map acquired from a phantom scan to improve k_PL estimation (Figure 1B). The ${\text{B}}_1^{+}$ map was acquired and calculated using DAM and is shown as a contour map in Figure 1B showing variations in the ${\text{B}}_1^{+}$ field. The corresponding color bar indicates the relative power delivered to the region; 100% indicates the region received the prescribed flip angle.

Representative ${\text{B}}_1^{+}$ correction data are shown in Figure 4 from a patient with cholangiocarcinoma (scan 3). Figure 4A shows an axial T₁-weighted spoiled gradient-echo anatomic scan with arrows highlighting large tumors. Two tumor voxels were selected, labeled voxel 1 and 2. Figure 4B shows the previously acquired ${\text{B}}_1^{+}$ map aligned using the coil’s fiducial markers, highlighted in yellow. Figure 4C–D depict the k_PL values for voxel 1 before and after ${\text{B}}_1^{+}$ correction, respectively. Before ${\text{B}}_1^{+}$ correction, the estimated k_PL for this voxel was 0.051 s⁻¹. After ${\text{B}}_1^{+}$ correction, the estimated k_PL was 0.063 s⁻¹. This voxel was over-flipped by 13% resulting in a −19.01% underestimation of k_PL without ${\text{B}}_1^{+}$ correction. Figure 4E–F depict the k_PL values for voxel 2 before and after ${\text{B}}_1^{+}$ correction, respectively. Before ${\text{B}}_1^{+}$ correction, the estimated k_PL for this voxel was 0.080 s⁻¹. After ${\text{B}}_1^{+}$ correction, the estimated k_PL was 0.066 s⁻¹. This voxel was under-flipped by 13% resulting in a 20.22% overestimation of k_PL without ${\text{B}}_1^{+}$ correction. These voxels corresponded with the Monte Carlo simulations in that over-flipping resulted in an underestimation of k_PL and under-flipping resulted in an overestimation of k_PL (Supporting Information Figure S1).

FIGURE 4.

From scan 3, (A) an axial T₁-weighted spoiled gradient-echo anatomic scan with an over-flipped tumor voxel highlighted in green (voxel 1) and an under-flipped tumor voxel highlighted in orange (voxel 2). (B) The ${\text{B}}_1^{+}$ profile aligned to its position during the scan using the coil’s fiducial markers. (C) The k_PL map before ${\text{B}}_1^{+}$ correction. (D) The k_PL map after ${\text{B}}_1^{+}$ correction. For voxel 1, over-flipped by +13%, there was a −19.01% underestimation of k_PL without ${\text{B}}_1^{+}$ correction (0.051 s⁻¹ before denoising and 0.063 s⁻¹ after denoising). For voxel 2, under-flipped by −13%, there was a +20.22% overestimation of k_PL without ${\text{B}}_1^{+}$ correction (0.080 s⁻¹ before denoising and 0.066 s⁻¹ after denoising).

4 |. DISCUSSION

HP ¹³C MRI has been used to quantitatively and dynamically image in vivo metabolic activity in a variety of human studies including cardiac diseases,^2,3 breast cancer,^4,5 brain cancer,^6–9 and prostate cancer.^10–12 Due to metabolic reprogramming, cancer cells demonstrate upregulated expression and activity of LDH,^35,36 the enzyme catalyzing the conversion of pyruvate to lactate, even in the presence of adequate oxygen (the “Warburg effect”).³³ Therefore, metabolic conversion from pyruvate to lactate (or k_PL) within the cancer cell can serve as a biomarker of tumor characterization,³⁷ tumor aggressiveness and grading,^11,38–40 metabolic pathway inhibition, and response to drug therapies.¹⁰ In mouse models, HP lactate signals correlated significantly with histologic prostate cancer grade as expected with increased LDH activity.^38,40 Furthermore, the metabolic flux of pyruvate to lactate intratumorally has been shown to be a biomarker of response to therapy.^17–19 Notably, changes in cancer HP [1-¹³C]pyruvate metabolism has been shown to be a more sensitive biomarker of early response to therapy compared to [¹⁸F]FDG-PET in preclinical models.²⁰ In human studies, HP [1-¹³C]pyruvate MR has been used to interrogate metabolism in healthy volunteers^2,41–43 and cancer patients, investigating metabolic reprogramming in primary tumors and metastases.^{5–7,10,12,22,44–47} These studies demonstrated that k_PL can serve as a biomarker and provide insight into the reprogrammed metabolism of tumors, elucidate response to therapy, and afford the ability to look at cancer progression and aggressiveness.

Differing from prior studies, this new project was designed to develop specialized methods to investigate liver tumor metabolism in patients using HP [1-¹³C]pyruvate. The goal of the project was to characterize these liver metastases, requiring a dedicated surface coil for acquisition. However, surface coils have an inherently inhomogeneous ${\text{B}}_1^{+}$ profile resulting in flip angles that vary through space. ${\text{B}}_1^{+}$ correction is especially important for studying inter- and intra-tumor heterogeneity and/or comparing cancerous tissue to normal-appearing tissue. Hence, for the first time, we implemented a ${\text{B}}_1^{+}$ correction method within a denoising pipeline for improved spectral SNR using a figure-8 T/R ¹³C surface coil.

Monte Carlo simulations were computed to elucidate the relationships between ${\text{B}}_1^{+}$ error and k_PL estimations in the inputless model. All excitations of each time point were consolidated into a single effective flip angle.¹⁴ As seen from the Monte Carlo simulations presented in Supporting Information Figure S1, inaccuracies in k_PL estimations arise when it is assumed that all voxels receive the same prescribed nominal flip angle when in reality, there is non-negligible variation of power as a function of space. Hence, to more accurately estimate rates of conversion, namely the first-order conversion rate of pyruvate to lactate (k_PL), one must use a ${\text{B}}_1^{+}$ map to scale each voxel’s flip angle correspondingly. In this work, we developed a novel EPSI processing pipeline that appends ${\text{B}}_1^{+}$ correction after a series of spectral processing steps: B₀ inhomogeneity corrections, tensor rank truncation to improve metabolite SNR, and zero- and first-order phase corrections.

In processing the HP ¹³C EPSI data, the spectral data were denoised before ${\text{B}}_1^{+}$ correction. The spectra pre- and post-denoising of a representative voxel are shown in Figure 2. On average, pyruvate, lactate, and alanine SNRs were improved by 37.4-, 34.0-, and 20.1-fold in the spectral dimension, respectively, therefore, substantially recovering metabolite dynamics. This is most noticeable with the lactate and alanine peaks, which are either slightly above or entirely shrouded by the noise floor. By removing extraneous ranks primarily in the frequency dimension of the HP ¹³C MRSI tensor, low signal metabolites can be extracted for improved subsequent kinetic modeling. The denoised lactate dynamics shown in Figure 2 also more accurately reflected physiological expectations. Compared to the pre-denoised dynamics, the post-denoised dynamics have an elongated lactate plateau beginning from around 18 s until 40 s. With the liver’s dual input blood supply from the hepatic artery and the portal vein, we should expect a sustained lactate peak as the portal vein, which delivers two-thirds of the liver’s blood supply, brings HP ¹³C lactate produced by the gastrointestinal tract to the liver. This physiological phenomenon is not as clearly perceptible before denoising and is reflected in the denoised dynamics. The denoised spectra were also corrected for ${\text{B}}_1^{+}$ field inhomogeneities before modeling metabolite dynamics in each voxel. A previously measured ${\text{B}}_1^{+}$ map using DAM was used to scale the nominal flip angle for each voxel. The corrected flip angles with the denoised spectra were provided as parameters for an inputless two-site model to estimate k_PL. The corrected k_PL values were compared against the expected changes derived from Monte Carlo simulations in which ${\text{B}}_1^{+}$ was varied. As shown in Supporting Information Figure S1, the corrected k_PL values fit closely to the ±1 SD bounds and match the inverse relationship between relative ${\text{B}}_1^{+}$ error and fractional k_PL error: positive errors in ${\text{B}}_1^{+}$ resulted in underestimations of k_PL and negative errors resulted in overestimations.

The developed methods here not only restored quantitative accuracy in regions receiving excess or insufficient RF power because of transmit field inhomogeneity, but more importantly increased the number of available voxels outside of the B₁-calibrated region of interest for k_PL estimations both in other lesions and in normal-appearing tissue. Limitations of the methods developed in this project include the need for prior mapping of the ${\text{B}}_1^{+}$ inhomogeneity profile specific to the coil. Recent real-time power calibration and ${\text{B}}_1^{+}$ mapping⁴⁸ could improve this approach for the future.

5 |. CONCLUSIONS

In this project, a specialized HP ¹³C EPSI post-processing pipeline was developed that integrated tensor denoising and ${\text{B}}_1^{+}$ correction to measure k_PL conversion rates in patients with liver tumors, in order to improve the quantitative accuracy of surface coil HP ¹³C metabolic MR in cancer patients. This approach could benefit future clinical trials using HP ¹³C MRSI to measure cancer metabolism and response to targeted drug therapies.

Supplementary Material

Supporting Info

FIGURE S1 (A) The simulated pyruvate and lactate signals. Following similar approaches,¹ the inputless 2-site exchange model was used, an approach that only fits the lactate magnetization and not the pyruvate magnetization while the measured pyruvate magnetization is used as the input at each time point. The nominal simulation values were k_PL = 0.03 s⁻¹, T₁ of pyruvate = 30 s, and T₁ of lactate = 25 s. The noise standard deviation was set at 0.004 relative to a fixed total magnetization input normalized to 1. This value was chosen to approximately match the SNR of the simulated data with typical empirical data. The γ-distribution function to simulate the pyruvate and lactate signals had an arrival time of 4 s and a full width at half-maximum of 8 s. All excitations of each time point were consolidated into a single effective flip angle.¹ (B) Monte Carlo simulations of the 2-site inputless model demonstrated an inverse relationship between the relative ${\text{B}}_1^{+}$ error and the fractional k_PL error, the simulated k_PL values (k_PL,fit) against the input k_PL of 0.03 s⁻¹ (k_PL,truth). The solid and dashed lines indicate the mean and ±1 standard deviation, respectively. Positive errors in ${\text{B}}_1^{+}$ resulted in underestimations of k_PL and negative errors resulted in over-estimations, agreeing with acquired data. This relationship corresponded with the computed k_PL maps before and after ${\text{B}}_1^{+}$ correction. The selected voxels presented in Table 1 are indicated with blue circles and all but 1 are within the 1 SD bounds