Spatially Constrained Hyperpolarized 13C MRI Pharmacokinetic Rate Constant Map Estimation using a Digital Brain Phantom and a U-Net

Abstract

Fitting rate constants to Hyperpolarized [1-¹³C]Pyruvate (HP C13) MRI data is a promising approach for quantifying metabolism in vivo. Current methods typically fit each voxel of the dataset using a least-squares objective. With these methods, each voxel is considered independently, and the spatial relationships are not considered during fitting.

In this work, we use a convolutional neural network, a U-Net, with convolutions across the 2D spatial dimensions to estimate pyruvate-to-lactate conversion rate, k_PL, maps from dynamic HP C13 datasets. We designed a framework for creating simulated anatomically accurate brain data that matches typical HP C13 characteristics to provide large amounts of data for training with ground truth results. The U-Net is initially trained with the digital phantom data and then further trained with in vivo datasets for regularization.

In simulation where ground-truth k_PL maps are available, the U-Net outperforms voxel-wise fitting with and without spatiotemporal denoising, particularly for low SNR data. In vivo data was evaluated qualitatively, as no ground truth is available, and before regularization the U-Net predicted k_PL maps appear oversmoothed. After further training with in vivo data, the resulting k_PL maps appear more realistic.

This study demonstrates how to use a U-Net to estimate rate constant maps for HP C13 data, including a comprehensive framework for generating a large amount of anatomically realistic simulated data and an approach for regularization. This simulation and architecture provide a foundation that can be built upon in the future for improved performance.

Graphical Abstract

Introduction:

Hyperpolarized [1-¹³C]Pyruvate (HP C13) MRI is a powerful and robust minimally-invasive method to dynamically image metabolism within the body^1–3. To fully leverage HP C13’s power, acquired dynamic metabolite images must be translated to clinically relevant results. The metabolite dynamics can be modelled using pharmacokinetic (PK) modelling, where the apparent rate constants between metabolites describe the rate of exchange. The pyruvate to lactate PK apparent rate constant, k_PL, has previously shown to be an important clinical biomarker to quantify metabolic reprogramming^3–5.

In previous work, PK models have been fit to the dynamics of each voxel to obtain k_PL maps^6–11. With these methods, each voxel is considered independently of one another regardless of spatial proximity. However, spatial relationships are particularly relevant to consider when estimating k_PL maps from HP C13 MRI which can suffer from low SNR. With noisy data, spatial constraints may minimize anomalies during k_PL fitting and smooth k_PL maps.

Limited previous work has explored this challenge within the field of HP C13. Maidens et al.¹² proposed a spatio-temporally constrained PK model fitting using L2 and total variation regularizations. This method resulted in improved appearance of in vivo parameter maps particularly in low SNR cases.

As an alternative to this optimization-based method, we propose a deep-learning method: to use a convolutional neural network, a U-Net, to estimate k_PL maps from dynamic HP C13 data. The convolutional layers will inherently spatially constrain the k_PL maps. Additionally, a deep learning solution could speed up the k_PL quantification task compared to least-squares fitting or other iterative methods. In the field of DCE MRI, several others have found promise in using convolutional neural networks to estimate pharmacokinetic parameters^13–17.

A challenge to training a U-Net to estimate k_PL maps is the limited number of HP C13 training data available and the lack of k_PL ground truths. As a solution, we propose to develop an anatomically accurate digital HP C13 brain phantom that matches typical HP C13 characteristics and use this synthetic phantom data for training. The pretrained model will then be further trained using in vivo HP C13 brain datasets for regularization. The objective of this work is to explore the feasibility of using a convolutional neural network to estimate PK parameter maps for HP C13 data.

Methods:

Anatomical Digital Phantom Data Creation:

Segmented proton MR data from the BrainWeb^18–24 database from 19 different brains was used as a base of the anatomical digital phantom (Figure 1). For each brain, the segmented grey matter, white matter and vasculature masks were each assigned model parameters of k_PL, k_PB, k_TRANS and M_z0 values. These maps were down sampled, then, per voxel a single physical compartment pharmacokinetic model was used to generate dynamics (Eqs 1–3). The dynamic images were multiplied by coil sensitivity maps (for “coil combined” images), Rician noise was added and then the dynamic images were up sampled to the output size. The data was simulated as a “multi-resolution” or “variable resolution” dataset²⁵ by varying the sample matrix size when generating the dynamics. The input function was modeled as a gamma variate function where T_bolus was set to 8s and T_arrival was varied between −4s and 4s.

Figure 1:

Overview of the creation of anatomical BrainWeb-based digital phantom datasets. Briefly, k_TRANS, k_PL, M_z,0 maps are derived using BrainWeb masks (A). The maps are down sampled and randomly spatially altered (B). Dynamics of each voxel are then modeled using pharmacokinetic equations (Eqs 1–3, C). The dynamics are multiplied by coil sensitivity maps and noise is added to the dynamics (C,D).

The dynamics were modeled including the effect of RF excitation pulses using the sim_Nsite_model() function from the Hyperpolarized MRI Toolbox^6,26 which can be described with the following equations:

For $1<t\le N_t$ :

$$\begin{gathered} M_z^{+}[t]=\exp (A\cdot \text{TR})\left(M_z[t-1]+\left[\begin{array}{c}{k}_{TRANS}\ast u[t-1]\\ 0\end{array}\right]\right) \\ {M}_z[t]=\left[\begin{array}{c}{P}_z[t]\\ L_z[t]\end{array}\right]=M_z^{+}[t]\left[\begin{array}{c}\cos {\alpha }_P\\ \cos {\alpha }_L\end{array}\right] \\ {M}_{xy}[t]=\left[\begin{array}{c}{P}_{xy}[t]\\ L_{xy}[t]\end{array}\right]=M_z^{+}[t]\left[\begin{array}{c}\sin {\alpha }_P\\ \sin {\alpha }_L\end{array}\right] \end{gathered}$$ 1

For $t=1$ :

$$\begin{gathered} M_z[t]=\left[\begin{array}{l}{P}_z[t]\hfill \\ L_z[t]\hfill \end{array}\right]=M_{z,0}\left[\begin{array}{l}\cos {\alpha }_P\hfill \\ \cos {\alpha }_L\hfill \end{array}\right] \\ {M}_{xy}[t]=\left[\begin{array}{l}{P}_{xy}[t]\hfill \\ L_{xy}[t]\hfill \end{array}\right]=M_{z,0}\left[\begin{array}{c}\sin {\alpha }_P\\ \sin {\alpha }_L\end{array}\right] \end{gathered}$$ 2

where:

$$A=\left[\begin{array}{cc}-R_{1P}-k_{PL}& 0\\ k_{PL}& -R_{1L}\end{array}\right]$$ 3

In these equations, N_t is the number of time points, TR is the temporal resolution or time between subsequent pyruvate image acquisitions, u is the input function modeled as a gamma variate function and α_P and α_L are the flip angles for pyruvate and lactate, respectively. M_z,0 is the initial longitudinal magnetization of pyruvate and lactate. M_xy[t] is used as the signal dynamics for each voxel of the anatomical digital brain phantom.

To create a diverse training, validation and test set, many parameters for the anatomical digital brain phantom were randomly varied. Parameter ranges were chosen based on previous literature^8,9,27–29 and in vivo data characteristics. The flip angles and temporal resolution were kept consistent and matched the in vivo data acquisition. The full set of parameters and parameter ranges for the anatomical digital phantom used for training data can be found in Table 1. The parameters were randomly sampled from a uniform distribution such that the model could learn across the full range. The final MATLAB function used to generate the anatomical digital brain phantom data is available in the Hyperpolarized MRI Toolbox²⁶.

Table 1:

Parameter values or ranges for various simulation parameters for the anatomical digital phantom data creation.

Parameter	Vasculature (Vasc)	Grey Matter (GM)	White Matter (WM)
kTRANS lower limit (s−1) [min, max]	[1, 1]	[0.15, 0.25]	[0.12, 0.22]
kTRANS upper limit (s−1) [min, max]	[5, 11]	[0.35, 0.45]	[0.32, 0.42]
kPL (s−1) [min, max]	[0, 0]	[0.007, 0.035]	[0.005, 0.03]
Pyruvate Mz,0 Scale [min, max]	[1, 1]	[0.5, 1]	[0.5, 1]
Lactate Mz,0 Scale [min, max]	[0, 0]	[0, 0.01]	[0, 0.01]
	Pyruvate	Lactate
Flip Angle (deg)	20	30
T1 (s)	30	25
SNR [min, max]	[70, 320]	[15, 75]
Sample Matrix Size [Nx, Ny, Nz]	[32, 32, 8]	[16, 16, 8]
	Value
Temporal Resolution (s)	3
Tarrival (s) [min, max]*	[−4, 4]
Tbolus (s)	8
Coil Sensity Dropoff Scale [min, max]	[0.2, 0.6]
Output Matrix Size [Nx, Ny, Nz]	[64, 64, 8]
X Translation (# voxels) * [min, max]	[−2, 2]
Y Translation (# voxels) * [min, max]	[−2, 2]
X Reflection *	True/False
Scale [min, max]	[0.95, 1.2]
Rotation (deg) [min, max]	[−5, 5]
Brain Index * [min, max]	Train/Val: [1, 19] exc. 5, 16 Test: 5 or 16

(*= sampled from a discretized distribution, otherwise ranges are sampled from a continuous distribution)

Using the parameters and ranges from Table 1, a training/validation dataset of 450 3D volumes was created with 8 slices each for a total of 3600 2D images. The 3600 2D images were randomly separated into training (3200 2D images) and validation (400 2D images). A separate test set (referred to as the “simulated test set”) of 50 3D volumes (400 2D images) was created. The test set used two BrainWeb templates that were not used in the training/validation set.

Out of distribution (OOD) simulated datasets were created with the following parameters above or below the parameters used for the training set: high SNR: 350 and 90 for pyruvate and lactate, low SNR: 50 and 10 for pyruvate and lactate, high k_PL: 0.05, 0.045, 0 s⁻¹ for grey matter, white matter and vasculature, low k_PL: 0.0015, 0.001, 0 s⁻¹ for grey matter, white matter and vasculature. The two BrainWeb templates used for the simulated test set were also used here.

In Vivo Dataset:

To test generalization to in vivo data and as training data for the regularization step, a dataset of 21 healthy volunteer in vivo HP C13 brain images was used^30–32. All of these datasets were variable resolution acquisitions^25,28 where lactate was acquired at 2x coarser resolution than pyruvate. The data was acquired with a metabolite-selective EPI sequence with a spectrally and spatially selective RF excitation. The flip angles used for acquisition were 20°, 30°, 30° for pyruvate, lactate and bicarbonate. Metabolite data was acquired dynamically for 20 time points with a temporal resolution of 3s. All datasets were acquired with a 24 channel RAPID Biomedical receive coil. Briefly, EPI reconstruction was performed using MATLAB and the GE Orchestra Toolbox. Nyquist ghost artifact correction was performed, the data was prewhitened and coil combination was performed using pyruvate to estimate the coil weights³³.

The in vivo dataset (21 3D volumes or 168 2D images) was split into a training/validation set (16 3D volumes or 128 2D images) for regularization and an in vivo test set (5 3D volumes or 40 2D images) randomly. The training/validation set was further split so that 12 2D images were used for validation and the remaining 116 2D images were used for training in the regularization step. All patient studies were conducted under a University of California San Francisco Institutional Review Board approved protocol.

U-Net Model & Training:

A 2D Basic U-Net model³⁴ from MONAI³⁵ was used with feature layers of (32, 32, 64, 128, 256, 32) and an encoder-decoder depth of 4. The input into the model was the simulated dynamics of pyruvate and lactate concatenated in the time dimension (Figure 2). The time dimension of the data was input into the model as the channels. The output was a single channel k_PL map³⁶. The model was trained for 1500 epochs and the weights that resulted in the lowest validation loss were used (epoch 213). A voxel-wise L1 loss was used and summed across the k_PL map. ADAM optimizer was used with a learning rate of 1e-3. For training and validation, the batch size was 2 and dropout probability was set to 0.3. The model was trained in Python with PyTorch and PyTorch Lightning. A Nvidia RTX A6000 GPU was used to speed up training.

Figure 2:

A graphical representation of the U-Net architecture used. This model is taken from MONAI’s³⁵ Basic U-Net implementation which was inspired by Falk et al³⁴. The dynamics are input into the model with pyruvate and lactate concatenated in the time dimension. The output is a single channel k_PL map.

After initial training with the simulated data, the in vivo training and validation dataset was used for further training for regularization. During this regularization experiment, all of the layers of the network were updated. The learning rate was decreased to 1e-5 and the model was trained for 3000 epochs with model weights saved every 300 epochs.

Pharmacokinetic Modeling & Denoising:

As comparisons to the U-Net results, pharmacokinetic fitting was performed with and without data denoising to obtain k_PL maps. The data was denoised using the global-local higher order SVD (GL-HOSVD) denoising method³⁷ with the following parameters: kglobal=0.4, klocal = 0.8, patchsize =5, step=2 and search window radius=11 as recommended in prior work³⁷. The data with or without denoising was fit to k_PL maps using an inputless one physical compartment pharmacokinetic model^6,8,26. For fitting, initial k_PL = 0.02, pyruvate T₁=30s, and lactate T₁=25s. Pharmacokinetic model-derived k_PL maps were masked with a brain mask. For simulated data, the k_PL map was thresholded to derive a brain mask. For in vivo data, the lactate SNR was thresholded to obtain a brain mask.

As another comparison to the U-Net results, a spatio-temporally constrained modeling technique proposed by Maidens et al.¹² was used to derive k_PL maps. The parameters used were pyruvate T₁=30s, lactate T₁=25s, ρ = 1e3, λ_TV = 10 and λ_l2 = 500. The ρ and λ values were chosen such that the model converged in less than 100 iterations and the resulting k_PL values were similar to the inputless pharmacokinetic model. The spatio-temporally constrained model maps were only compared with in vivo data maps as a new set of parameters would have to be optimized for simulated datasets.

Monte Carlo Dropout:

To estimate the uncertainty of the U-Net, a Monte Carlo dropout technique³⁸ was used after 3000 regularization epochs. During inference, the dropout layers of the network were kept enabled such that for each iteration of inference the dropout mask changed. The dropout probability was kept the same value it was for training, 0.3. For each in vivo slice, 1000 Monte Carlo iterations were run with randomly changing dropout masks to generate a distribution of possible k_PL maps. The mean and variance of each voxel was calculated. The mean map should match the model’s prediction without Monte Carlo dropout enabled. The variance map represents the spatial uncertainty of the model, where the voxels with highest variance represent where the model is least certain about the estimated k_PL. The Monte Carlo dropout variance maps were normalized by the pyruvate area-under-the-curve (AUC).

Results:

Simulated Anatomical Digital Brain Phantom Data:

Examples from the simulated digital brain phantom data are shown in Figure 3. The simulated data examples largely resemble in vivo datasets. The sagittal sinus signal dominates the pyruvate signal. The typical signal drop-off in the center of the brain is also captured using the simulated coil sensitivity maps. The simulated examples show some differences from the in vivo data. In simulation, the pyruvate signal is higher for more time points whereas in the in vivo examples the signal decays faster. The reverse is true for lactate, where in the simulated examples lactate decays faster than the in vivo examples. These differences can likely be addressed by modifying or increasing the complexity of the perfusion and PK model.

Figure 3:

Three examples of simulated anatomical digital brain phantom datasets and two examples of in vivo datasets. Each simulated example is a different dataset and a different image slice (5, 7 and 2, top to bottom). On the left are the dynamics of pyruvate (PYR) and lactate (LAC) and on the right is pyruvate & lactate AUC, k_PL, k_TRANS, pyruvate & lactate M_z0 maps. AUC=Area Under the Curve

Base U-Net:

A 2D U-Net was trained with the anatomical digital brain phantom data. The validation loss reached a minimum at epoch 213. Figure 4 shows examples of U-Net predicted k_PL maps on the simulated test set. These are compared with using pharmacokinetic model fitting and using pharmacokinetic model fitting after HOSVD denoising.

Figure 4:

U-Net k_PL map predictions for three examples of slices from the simulated test set are shown. The U-Net predictions are compared to using a Voxelwise PK model without denoising data and a Voxelwise PK model with HOSVD denoised data. Error maps for each k_PL map estimation method are below and are normalized to the ground truth maps. The mean absolute error is listed above each error map. The PK model maps are thresholded. AUC=Area Under the Curve, PK = pharmacokinetic, HOSVD = higher-order singular value decomposition, Abs Err = absolute error

The U-Net predicted results were more robust to added noise compared with the PK model results even with HOSVD denoising (Figure 4). The U-Net inherently ignores low SNR regions outside the brain, whereas the PK model maps had to be manually thresholded. Using denoised data for the PK model only resulted in small improvements to using raw data as input into the model. Compared to the ground truth maps, the U-Net predictions fail to recreate the higher spatial frequency edges and details of the brain. For example, in the second example (rows 3-4) in Figure 4, the U-Net prediction does not capture the sulci of the brain apparent in the ground truth map. Quantitatively, the U-Net predictions performed better on the simulated test data than the PK model (Table 2).

Table 2:

Simulated Test Metrics for U-Net, PK Model and PK Model with HOSVD denoising.

Metrics	U-Net Prediction	Voxelwise PK Model	HOSVD Denoising + Voxelwise PK Model
Sum of Abs Error	0.207	3.572	3.634
Sum of Sq Error	0.0004	0.129	0.129
Avg SSIM	0.864	0.325	0.330

The U-Net model was tested further with out of distribution (OOD) simulated data examples, where the datasets were created with SNR or k_PL values outside of the training range (Figure 5). In the low SNR case, the U-Net performed significantly better than PK modeling, but still suffered in recreating the ground truth k_PL map, specifically in the locations of the ventricles. The PK model overestimated the k_PL when there was very low SNR. In the high SNR case, the U-Net predicted maps performed better than the PK model. For both low k_PL and high k_PL, the U-Net either overestimated or underestimated the k_PL, resulting in large bias in the k_PL map. Interestingly, in the low k_PL case, the U-Net predicted map did not properly mask the brain and the map included regions outside of the brain. This can be attributed to the vasculature structure of the BrainWeb templates used in the simulated training data.

Figure 5:

U-Net and PK model k_PL map estimations shown for four out-of-distribution cases, where the data had either k_PL values or SNR above or below the training data. The U-Net performed better than the PK model in the low SNR case but did not generalize well to k_PL values outside of the training data’s k_PL values. The error maps are normalized to the ground truth maps and the mean absolute error is listed above each error map. The PK model maps are thresholded. AUC=Area Under the Curve, PK = pharmacokinetic, HOSVD = higher-order singular value decomposition, Abs Err = absolute error

The U-Net’s performance was next tested with in vivo data (Figure 6). The U-Net predicted k_PL maps were compared with the PK model derived k_PL. The U-Net successfully SNR thresholded the resulting k_PL maps, ignoring regions outside the brain with no signal. Quantitatively, the U-Net predicted maps reflected k_PL values within the same range as the PK model maps. The U-Net predicted maps were more uniform across the brain in k_PL value compared to the voxelwise PK model and spatio-temporally constrained results. The spatio-temporally constrained maps showed some evidence of denoising compared to the voxelwise maps, with similar performance to the voxelwise+HOSVD denoising maps.

Figure 6:

Four examples of the U-Net’s performance on the in vivo test set. For each example, the U-Net predicted k_PL map is compared with the PK model estimated maps with or without HOSVD denoising and spatio-temporally constrained maps. The voxelwise PK model and maps are manually SNR thresholded whereas the U-Net internally SNR thresholds the estimated k_PL maps. The spatio-temporally constrained maps are also not thresholded, showing some evidence of internal SNR thresholding as well. AUC=Area Under the Curve, PK = pharmacokinetic, HOSVD = higher-order singular value decomposition

Regularization:

All of the layers of the previously simulated data-trained U-Net were trained further using the in vivo dataset with the voxelwise PK model derived k_PL maps as the ground truth. To investigate the effect of these regularization training steps, the U-Net predicted k_PL maps were compared before regularization and after regularization training for 300, 1200 and 3000 epochs (Figure 7). As training proceeded, the U-Net predictions gained more similarities to the voxelwise PK model, and similar regions had the same k_PL values. Compared with the prior results using the base U-Net, the U-Net predictions no longer fully SNR threshold regions outside of the brain and fit k_PL values greater than 0 to the background. The regularization results appear most similar to the spatio-temporally constrained and voxelwise+HOSVD denoising k_PL maps.

Figure 7:

U-Net predicted k_PL maps for four images from the in vivo test set before regularization and after 300, 1200 and 3000 epochs of regularization. The voxelwise PK model maps are SNR thresholded. As regularization training continues, the U-Net predicted maps show more contrast that is similar to the PK model maps, but also exhibit higher noise levels. AUC=Area Under the Curve, PK = pharmacokinetic, HOSVD = higher-order singular value decomposition, Reg = regularization

Monte Carlo Dropout:

To measure model uncertainty, the U-Net model after 3000 epochs of regularization was used for Monte Carlo dropout. A total of 1000 Monte Carlo iterations of inference were run to generate a distribution of k_PL maps. The variance maps represent the U-Net model’s uncertainty (Figure 8). Across the in vivo dataset, the highest regions of uncertainty were in the posterior regions of the brain and interestingly, a region outside of the brain. This region outside of the brain is similar in location to the vasculature regions apparent in the BrainWeb templates.

Figure 8:

Monte Carlo dropout mean and variance maps using the U-Net model after 3000 epochs of regularization. Monte Carlo dropout was run for 1000 iterations. The variance maps were normalized with the pyruvate AUC. MC = Monte Carlo, AUC=Area Under the Curve

Discussion:

To our knowledge, we show here the first approach to estimate PK model parameters for HP C13 data using a convolutional neural network. In this work, to overcome the limited HP C13 data available for supervised learning, an anatomically accurate digital HP brain phantom was developed. Using digital phantom data for training, a base U-Net was trained. Although this U-Net performed well on the simulated test data (Figure 4, Table 2), it resulted in over-smoothed k_PL maps in the in vivo datasets that lacked high spatial frequency features and expected gray/white matter contrast (Figure 6). To improve performance on in vivo data, the model was further trained using a small in vivo dataset for regularization. The regularization models were able to generate more gray/white matter contrast while apparently maintaining denoising properties (Figure 7). The level of regularization (number of epochs for a given learning rate) must be optimized in future work to obtain the correct tradeoff between spatial blurring and k_PL accuracy.

One of the biggest disadvantages of the most common least-squares based PK models to estimate k_PL is evaluating each voxel independently rather than taking advantage of the relationships between voxels. Convolutional layers within the U-Net naturally spatially constrain and can improve k_PL estimation for noisy HP C13 data. The PK model in the digital phantom datasets also resulted in k_PL estimation errors in the center of the brain due to lower SNR of pyruvate and lactate from coil geometry. This led to blooms of error near the ventricles (Figure 4, last row) which were avoided when using the U-Net. In simulation, the U-Net approach showed strongly improved denoising performance compared to the use of HOSVD denoising prior to PK fitting. In vivo, the U-Net results after regularization were more similar to these other approaches. Based on the simulations, we believe that with better training data the U-Net performance will exceed these prior methods. The U-Net approach also has the advantage that at inference time the U-Net estimates a full k_PL map in less than a second which is a large time improvement compared with the iterative, least-squares constrained methods which can take on the order of 1-2 minutes with similar hardware.

Anatomical Digital Phantom:

We present an open-source framework to generate HP C13 anatomical digital phantoms of the healthy brain²⁶. In this work, the digital phantom is utilized to train a convolutional neural to estimate k_PL maps but there are plenty of further use cases for the digital phantom. The digital phantom may be used to evaluate and validate new sampling, acquisition or quantification methods without requiring an in vivo acquisition. Prior work has previously used anatomically-accurate digital phantoms^28,39,40. Here, a digital phantom of the healthy brain is described but this framework may be extended to other anatomy as well.

The anatomical digital phantom is a good approximation of in vivo data, including multiple subjects and slices to capture potential variations, yet it makes some assumptions and has limitations. The metabolite kinetics are somewhat varied from typical in vivo data (Figure 3). This could be a result of the input function, including arrival time, being kept constant across the brain. Additionally, a single physical compartment PK model is used to simulate the dynamics whereas more complicated models may be more reflective of the biochemistry¹⁰. Using multiple compartment models for the digital phantom could further minimize model-mismatch errors when training with digital phantom data and generalizing to in vivo datasets. The digital phantom framework could easily be adapted for other PK models when generating the dynamics. Another limitation of the digital phantom was that the simulated coil sensitivity maps were simple and didn’t include much variation.

Another challenge of the anatomical digital phantom was properly modelling the high sagittal sinus signal that typically dominates in vivo pyruvate data. For the simulated datasets, the vasculature signal was weighted higher in the posterior brain. However, in some BrainWeb templates, there was other vasculature signal present in the posterior brain which was also enhanced in the simulated data. This manifested as regions of signal outside of the brain which were learned by the model and at times translated into the predicted maps or resulted in model uncertainty (Figure 5, Figure 8). A more targeted approach to weight the sagittal sinus signal should be used in the future to avoid these regions of k_PL outside of the brain.

Monte Carlo Dropout:

In this work, we provide the Monte Carlo dropout results as a potential method of exploring uncertainty when using a convolutional neural network to estimate k_PL (Figure 8). For pharmacokinetic modelling, metrics such as lactate R squared are used to evaluate the model fit and express model uncertainty on a voxelwise basis. When using a neural network, a typical pipeline does not measure uncertainty. Monte Carlo dropout is one method in which we can generate a distribution of k_PL values for each voxel³⁸. The variance of this distribution can give us an approximation of the uncertainty of the model. Although these variance values may not be able to be compared quantitatively to PK model metrics, the uncertainty of the respective models within the regions of the brain can be compared. The dropout probability rate was chosen arbitrarily in this work as 0.3. The choice of this probability rate and methods of evaluating uncertainty is an open question and an active field of research.

Extensions & Future Work:

This work was specifically concerned with estimating k_PL maps using a U-Net. However, the same pipeline can be extended to estimate other rate constant maps of interest, such as k_PB or k_PA. To estimate more than one rate constant map using the U-Net, one could output multiple channels from the U-Net and calculate a multi-channel loss. The low lactate SNR example (Figure 5) may reflect the U-Net’s performance in estimating k_PB or k_PA as bicarbonate and alanine typically have lower SNR than lactate.

The in vivo dataset used here was uniform and had the same acquisition parameters using the same scanner. We did not answer the question as to how well the U-Net would generalize to changes in the HP C13 acquisition. We expect it may not generalize well if there are changes to the signal dynamics due to variations in flip angles and acquisition timing, as these are typically necessary to account for in PK modeling. It may also not generalize well to changes in the coil, like coil geometry or number of coil channels. The model may also not generalize well to cases with any pathological variation as the training and evaluation cases were all healthy brains. The simulation framework is very flexible and would support generating new training datasets for these changes. HP C13 data of anatomical regions outside of the brain would likely require a different simulation framework.

To overcome the limited number of in vivo HP C13 datasets, the simulated anatomical digital brain phantom data was used for training. A different and perhaps more direct workaround to this challenge could instead be considering unsupervised deep learning methods. Taking inspiration from DCE parameter estimation, Oh et al.¹⁷ and Ottens et al.¹⁴ both used unpaired methods with a physics-informed loss. A second challenge is the lack of ground truth in vivo parameter maps. A solution here could be to use a physics-informed loss. The output estimated k_PL maps can be converted back into dynamic images using a PK model. Then, the loss can be calculated across dynamic images which would not require ground truth maps.

Other neural network architectures may also be considered. Again in the field of DCE MRI, there has been promising work using recurrent neural networks (GRU, LSTM)¹⁴ or attention-based⁴¹ networks for PK parameter estimation which could be explored in addition to convolution-only networks.

Conclusions:

In this work, we have provided a demonstration of convolutional neural networks to estimate k_PL maps as well as a framework to generate anatomically accurate dynamic brain HP C13 datasets. The simulated anatomical digital phantom data resembled in vivo data and constituted a diverse training dataset. In simulation, the U-Net outperforms voxel-wise fitting with and without spatiotemporal denoising, particularly for low SNR data. The base U-Net trained with the simulated data resulted in over-smoothed appearing k_PL maps for in vivo datasets. After regularization, the U-Net predicted parameter maps more similar to PK model derived maps generated after denoising or with spatio-temporally constrained optimization, but the U-Net has the advantage of much faster reconstruction times.

• A flexible anatomical digital Hyperpolarized C-13 brain phantom framework was developed

• The digital phantom data mimicked in vivo datasets well

• A U-Net was trained to estimate k_PL maps for Hyperpolarized C-13 datasets

• The U-Net resulted in spatially-constrained and smoothed k_PL maps