Deep Learning Robot Control

Learn to Accumulate Evidence from All Training Samples: Theory and Practice

Computationally Efficient Way To Turn A Determin-

istic neural network uncertainty-aware. The resul-

Uncertainty Using The Learned Evidence. To En-

sure theoretically sound evidential models, the ev-

Special Activation Functions For Model Training

and inference. This constraint often leads to infe-

Them To Many Large-Scale Datasets. To Unveil The

real cause of this undesired behavior, we theoreti- cally investigate evidential models and identify a

Into Such Regions. A Deeper Analysis Of Eviden-

tial activation functions based on our theoretical

Underpinning Inspires The Design Of A Novel Regu-

larizer that effectively alleviates this fundamental

Lenging Real-World Datasets And Settings Confirm

our theoretical findings and demonstrate the effec- tiveness of our proposed approach.

Ntroduction

Deep Learning (DL) models have found great success in many real-world applications such as speech recognition (Kamath et al., 2019), machine translation (Singh et al., 2017), and computer vision (Voulodimos et al., 2018). How- ever, these highly expressive models may easily fit the noise in the training data, which leads to overconfident predictions (Nguyen et al., 2015). The challenge is further compounded when learning from limited labeled data, which is common Proceedings of the 40 th International Conference on Machine Learning, Honolulu, Hawaii, USA. PMLR 202, 2023. Copyright 2023 by the author(s).

for applications from specialized domain (e.g., medicine, public safety, and military operations) where data collec- tion and annotation is highly costly. Accurate uncertainty quantification is essential for successful application of DL models in these domains. To this end, DL models have been augmented to become uncertainty-aware (Gal & Ghahra- mani, 2016; Blundell et al., 2015; Pearce et al., 2020). How- ever, commonly used extensions require expensive sampling operations (Gal & Ghahramani, 2016; Blundell et al., 2015), which significantly increase the computational costs (Lak- shminarayanan et al., 2017).

The recently developed evidential models bring together evidential theory (Shafer, 1976; Jøsang, 2016) and deep neural architectures that turn a deterministic neural network uncertainty-aware. By leveraging the learned evidence, evi- dential models are capable of quantifying fine-grained un- certainty that helps to identify the sources of ‘unknowns’.

Furthermore, since only lightweight modifications are intro- duced to existing DL architectures, additional computational costs remain minimum. Such evidential models have been successfully extended to classification (Sensoy et al., 2018), regression (Amini et al., 2020), meta-learning (Pandey & Yu, 2022a), and open-set recognition (Bao et al., 2021) settings.

Architectures In Rela-

nificant performance drop when facing large datasets with more complex features even in the common classification setting. As shown in Figure 1, an evidential model using ReLU activation and an evidential MSE loss (Sensoy et al., 2018) only achieves 36% test accuracy on Cifar100, which is almost 40% lower than a standard model trained using softmax. Additionally, most evidential models can easily break down with minor architecture changes and/or have a much stronger dependency on hyperparameter tuning to achieve reasonable predictive performance. The experiment section provides more details on these failure cases.

Arxiv:2306.11113V2 [Cs.Lg] 24 Jun 2023

Learn to Accumulate Evidence from All Training Samples: Theory and Practice Figure 2. Visualization of zero-evidence region for evidential mod- els with ReLU activation in a binary classification setting. Existing models fail to learn from samples that are mapped to such zero- evidence region (shared area at the bottom left quadrant).

To train uncertainty-aware evidential models that can also predict well, we perform a novel theoretical analysis with a focus on the standard classification setting to unveil the underlying cause of the performance gap. Our theoreti- cal results show that existing evidential models learn sub- optimally compared to corresponding softmax counterparts.

Such sub-optimal training is mainly attributed to the inher- ent learning deficiency of evidential models that prevents them from learning across all training samples. More specif- ically, they are incapable to acquire new knowledge from training samples mapped to “zero-evidence regions” in the evidence space, where the predicted evidence reduces to zero. The sub-optimal learning phenomenon is illustrated in Figure 2 (detailed discussion is presented in Section 4.2).

We analyze different variants of evidential models present in the existing literature and observe this limitation across all the models and settings. Our theoretical results inspire the design of a novel Regularized Evidential model (RED) that includes positive evidence regularization in its train- ing objective to battle the learning deficiency. Our major

Contributions Can Be Summarized As Follows:

• We identify a fundamental limitation of evidential models, i.e., lack the capability to learn from any data samples that lie in the “zero-evidence” region in the evidence space.

• We theoretically show the superiority of evidential models with exp activation over other activation functions. • We conduct novel evidence regularization that enables evidential models to avoid the “zero-evidence” region so that they can effectively learn from all training samples.

• We carry out experiments over multiple challenging real- world datasets to empirically validate the presented theory, and show the effectiveness of our proposed ideas.

Related Works

Uncertainty Quantification in Deep Learning.

Accu-

rate quantification of predictive uncertainty is essential for development of trustworthy Deep Learning (DL) models. Deep ensemble techniques (Pearce et al., 2020; Lakshmi- narayanan et al., 2017) have been developed for uncer- tainty quantification. An ensemble of neural networks is constructed and the agreement/disagreement across the en- semble components is used to quantify different uncertain- ties. Ensemble-based methods significantly increase the number of model parameters, which are computationally expensive at both training and test times. Alternatively, Bayesian neural networks (Gal & Ghahramani, 2016)(Blun- dell et al., 2015)(Mobiny et al., 2021) have been devel- oped that consider a Bayesian formalism to quantify dif- ferent uncertainties. For instance, (Blundell et al., 2015) use Bayes-by-backdrop to learn a distribution over neural network parameters, whereas (Gal & Ghahramani, 2016) enable dropout during inference phase to obtain predictive uncertainty. Bayesian methods resort to some form of ap- proximation to address the intractability issue in marginal- ization of latent variables. Moreover, these methods are also computationally expensive as they require sampling for uncertainty quantification.

Evidential Models Introduce A

conjugate higher-order evidential prior for the likelihood dis- tribution that enables the model to capture the fine-grained uncertainties. For instance, Dirichlet prior is introduced over the multinomial likelihood for evidential classification (Bao et al., 2021; Zhao et al., 2020), and NIG prior is in- troduced over the Gaussian likelihood (Amini et al., 2020; Pandey & Yu, 2022b) for the evidential regression models.

Adversarial robustness (Kopetzki et al., 2021) and calibra- tion (Tomani & Buettner, 2021) of evidential models have also been well studied. Usually, these models are trained with evidential losses in conjunction with heuristic evidence regularization to guide the uncertainty behavior (Pandey & Yu, 2022a; Shi et al., 2020) in addition to reasonable gen- eralization performance. Some evidential models assume access to out-of-distribution data during training (Malinin & Gales, 2019; 2018) and use the OOD data to guide the un- certainty behavior. A recent survey (Ulmer, 2021) provides a thorough review of the evidential deep learning field.

In this work, we focus on evidential classification models and consider settings where no OOD data is used during model training to make the proposed approach more broadly applicable to practical real-world situations.

Preliminaries And Problem Setup

Standard classification models use a softmax transformation on the output from the neural network FΘ for input x to ob- tain the class probabilities in K-class classification problem.

Such models are trained with the cross-entropy based loss. Learn to Accumulate Evidence from All Training Samples: Theory and Practice For a given training sample (x, y), the loss is given by

(1)

where smk is the softmax output.

These Models Have

achieved state-of-the-art performance on many benchmark effectively learn from all training data samples (see Ap- pendix A). Nevertheless, these models lack a systematic mechanism to quantify different sources of uncertainty, a Figure 3. Graphical model for Evidential Deep Learning Evidential classification models formulate training as an evidence acquisition process and consider a higher-order Dirichlet prior Dir(p|α) over the predictive Multino- mial distribution Mult(y|p). Different from a standard Bayesian formulation which optimizes Type II Maximum Likelihood to learn the Dirichlet hyperparameter (Bishop & Nasrabadi, 2006), evidential models directly predict α using data features x and then generate the prediction y by marginalizing the Multinomial parameter p. Figure 3 de- scribes this generative process. Such higher-order prior en- ables the model to systematically quantify different sources of uncertainty. In evidential models, the softmax layer of the standard neural networks is replaced by a non-negative

∀X ∈[−∞, ∞],

such that for input x, the neural network model FΘ with parameters Θ can output evidence e for different classes. Dirichlet prior α is evaluated as α = e+1 to ensure α ≥1.

The trained evidential model outputs Dirichlet parameters α for input x that can quantify fine-grained uncertainties in addition to the prediction y. Mathematically, for K−class

(4)

The activation function A(·) assumes three common forms to transform the neural network output into evidence: (1)

Relu(·) = Max(0, ·), (2) Softplus(·) = Log(1 +

exp(·)), and (3) exp(·). Evidential models assign input sample to that class for which the output evidence is greatest. Moreover, they quantify the confidence in the prediction for K class classification prob- lem through vacuity ν (i.e., measure of lack of confidence

(5)

For any training sample (x, y), the evidential models aim to maximize the evidence for the correct class, minimize the evidence for the incorrect classes, and output accurate confi- dence. To this end, three variants of evidential loss functions have been proposed (Sensoy et al., 2018): 1) Bayes risk with sum of squares loss, 2) Bayes risk with cross-entropy loss, and 3) Type II Maximum Likelihood loss. Please refer to equations (21), (22), and (23) in the Appendix for the spe- cific forms of these losses. Additionally, incorrect evidence regularization terms are introduced to guide the model to output low evidence for classes other than the ground truth class (See Appendix C for discussion on the regularization).

With evidential training, accurate evidential deep learning models are expected to output high evidence for the correct class, low evidence for all other classes, and output very high vacuity for unseen/out-of-distribution samples.

3.2. Theoretical Analysis of Learning Deficiency in

Evidential Learning

To identify the underlying reason that causes the perfor- mance gap of evidential models as described earlier, we consider a K class classification problem and a represen- tative evidential model trained using Bayes risk with sum of squares loss given in (21). We first provide an important definition that is critical for our theoretical analysis.

Definition 1 (Zero-Evidence Region). A Zero-evidence sample is a data sample for which the model outputs zero evidence for all classes. A region in the evidence space that contains zero-evidence samples is a zero-evidence region.

For a reasonable evidential model, novel data samples not yet seen during training, difficult data samples, and out-of- distribution samples should become zero-evidence samples.

Theorem 1. Given a training sample (x, y), if an evidential neural network outputs zero evidence e, then the gradients of the evidential loss evaluated on this training sample over the network parameters reduce to zero.

Proof. Consider an input x with one-hot ground truth label y. Let the ground truth class index be gt, i.e., ygt = 1, with corresponding Dirichlet parameter αgt, and y̸=gt = 0. Moreover, let o, e, and α represent the neural network output vector before applying the activation A, the evidence vector, and the Dirichlet parameters respectively.

(6)

Learn to Accumulate Evidence from All Training Samples: Theory and Practice Now, the gradient of the loss with respect to the neural network output can be computed using the chain rule:

(7)

Based on the actual form of A, we have three cases:

(8)

For a zero-evidence sample, the logits ok satisfy the rela-

= 0

Case II: SoftPlus(·) to transform logits to evidence

∂Ek

∂ok →0. Moreover, there is no term in the first part of the loss gradient in (7) to counterbalance these zero-approaching gradients.

So, for zero-evidence training samples, for any node k,

(11)

Since the gradient of the loss with respect to all the nodes is zero, there is no update to the model from such samples. This implies that the evidential models fail to learn from a zero-evidence data sample.

For completeness, we present the analysis of standard clas- sification models in Appendix A, detailed proof of the evi- dential models trained using Bayes risk with sum of squares error along with other evidential lossses in Appendix B, and impact of incorrect evidence regularization in Appendix C.

Evidential Models Can Not Learn From A Train-

ing sample that the model has never seen and for which the model accurately outputs “I don’t know”, i.e., ek = ∀k ∈[1, K]. Such samples are expected and likely to be present during model training. However, the supervised in- formation in such training data points is completely missed by evidential models so they fail to acquire any new knowl- edge from all such training data samples (i.e., data samples in zero-evidence region of the evidence space).

Corollary 1. Incorrect evidence regularization can not help evidential models learn from zero-evidence samples. Intuitively, the incorrect evidence regularization encourages the model to output zero evidence for all classes other than the ground truth class and the regularization does not have any impact on the evidence for the ground truth class. So, the regularization updates the model parameters such that the model is likely to map input samples closer to zero- evidence region in the evidence space. Thus, the regular- ization does not address the failure of evidential models to learn from zero evidence samples.

Theorem 2. For a data sample x, if an evidential model outputs logits ok ≤0 ∀k ∈[0, K], the exponential acti- vation function leads to a larger gradident update on the model parameters than softplus and ReLu.

Limited by space, we present the proof of Theorem 2 along with additional analysis in the Appendix D. The proof fol- lows the gradient analysis of the exponential, Softplus, and ReLU based models. It implies that the the training of evidential models is most effective with the exponential activation function. Intuitively, the ReLU based activation completely destroys all the information in the negative logits, and has largest region in evidence space in which training data have zero evidence. Softplus activation improves over the ReLU, and compared to ReLU, has smaller region in evidence space where training data have zero evidence.

However, Softplus based evidential models fail to cor- rect the acquired knowledge when the model has strong wrong evidence. Moreover, these models are likely to suf- fer from vanishing gradients problem when the number of classes increases (i.e., classification problem becomes more challenging). Finally, exponential activation has the smallest zero-evidence region in the evidence space without suffering from the issues of SoftPlus based evidential models.

Orrect Evidence Regularization

We now consider an evidential model with exponential func- tion to transform the logits into evidence. We propose a novel vacuity-guided correct evidence regularization term

S Represents The Regularization Term

whose value is given by the magnitude of the vacuity output by the evidential model and αgt −1 represents the predicted evidence for the ground truth class. The regularization term λcor determines the relative importance of the correct Learn to Accumulate Evidence from All Training Samples: Theory and Practice evidence regularization term compared to the evidential loss and incorrect evidence regularization and is treated as constant during model parameter update.

Theorem 3. Correct evidence regularization Lcor(x, y) can address the issue of learning from zero-evidence train- ing samples.

Proof. The proposed regularization term Lcor(x, y) does not contain any evidence terms other than the evidence for the ground truth node. So, the gradient of the regularization for nodes other than the ground truth node will be 0 i.e.

K̸=Gt = 0 And There Will Be No Update On These

nodes. For the ground truth node gt, ygt = 1, the gradient

(15)

The gradient value equals the magnitude of the vacuity. The vacuity is bounded in the range [0, 1], and zero-evidence sample, the vacuity is maximum, leading to the greatest

= −1. In Other Words, The Reg-

ularization encourages the model to update the parameters such that the correct evidence αgt −1 increases. As the model evidence increases, the vacuity decreases, and the contribution of the regularization Lcor(x, y) is minimized.

Thus, the proposed regularization enables the evidential model to learn from zero-evidence samples.

Evidential Model Training

We formulate an overall objective used to train the pro- posed Regularized evidential model (RED). Essentially, the evidential model is trained to maximize the correct evi- dence, minimize the incorrect evidence, and avoid the zero- evidence region during training. The overall loss is

(16)

where Levid(x, y) is the loss based on the evidential framework given by (21), (23), or (22) (See Appendix B), Linc(x, y) represents the incorrect evidence regularization (See Appendix Section C), Lcor(x, y) represents the pro- posed novel correct evidence regularization term in (12), and η1 = λ1 × min(1.0, epoch index/10) controls the impact of incorrect evidence regularization to the overall model training. In this work, we consider the forward-KL based incorrect evidence regularization given in (42) based on (Sensoy et al., 2018).

Evidence Space Visualization

Figure 4. Evidence space visualization to demonstrate the effec- tiveness of the proposed method. Figure 2 visualizes the evidence space in ReLU-based ev- idential models by considering the pre-ReLU output in a binary classification setting. Ideally, all samples that belong to Class 1 should be mapped to the blue region (region of high evidence for Class 1, low evidence for all other classes), all samples that belong to Class 2 should be mapped to the red region, and all out-of distribution samples should be mapped to the zero-evidence region (no evidence for all classes). To realize this goal, the models are trained using the evidential loss Levid with incorrect evidence regular- ization Linc. However, there is no update to the evidential model from such samples of zero-evidence region. Model’s prior belief of “I don’t know” for such samples does not get updated even after being exposed to the true label. For the samples with high incorrect evidence and low correct evidence, evidential model aims to correct itself. However, many such samples are likely to get mapped to the zero- evidence region (as shown by blue and orange arrows in Figure 2) after which there is no update to the model. Such fundamental limitation holds true for all evidential models.

The evidence space visualization for RED is shown in Figure 4 to illustrate how it addresses the above limitation. Cor- rect evidence regularization (indicated by green arrows) is weighted by the magnitude of the vacuity and is maximum in the zero-evidence region. In this problematic region, the proposed regularization fully dominates the model update as there is no update to the model from the two loss com- ponents (Levid and Linc) in (16). As the sample gets far away from the zero evidence region, the vacuity decreases proportionally, the impact of the proposed regularization to model update becomes insignificant, and the evidential losses (Levid & Linc) guide the model training. In this way, RED can effectively learn from all training samples irrespective of the model’s existing evidence.

We Consider The Standard Supervised

classification problem with MNIST (LeCun, 1998), Ci- far10, and Cifar100 datasets (Krizhevsky et al., 2009), and few-shot classification with mini-ImageNet dataset (Vinyals et al., 2016). We employ the LeNet model for MNIST, ResNet18 model (He et al., 2016) for Cifar10/Cifar100, and ResNet12 model (He et al., 2016) for mini-ImageNet.

We first conduct experiments to demonstrate the learning deficiency of existing evidential models to confirm our the- oretical findings. We then evaluate the proposed correct evidence regularization to show its effectiveness. We finally conduct ablation studies to investigate the impact of evi- dential losses on model generalization and the uncertainty quantification of the proposed evidential model. Limited by space, additional clarifications, experiment results includ- ing few-shot classification experiments, experiments over challenging tiny-Imagenet datasett with Swin Transformer, hyperparameter details, and discussions are presented in the

Earning Deficiency Of Evidential Models

Sensitivity to the change of the architecture.

We First

consider a toy illustrative experiment with two frameworks: 1) standard softmax, 2) evidential learning, and experiment with the LeNet (LeCun et al., 1999) model considered in EDL (Sensoy et al., 2018) with a minor modification to the architecture: no dropout in the model. To construct the toy dataset, we randomly select 4 labeled data points from the MNIST training dataset as shown in the Figure 5. For the evidential model, we use ReLU to transform the network outputs to evidence, and train the model with MSE-based evidential loss (Sensoy et al., 2018) given in (21) without incorrect evidence regularization. We train both models using only these 4 training data points.

Figure 6 compares the training accuracy and training loss trends of the evidential model with the standard softmax model (trained with the cross-entropy loss). Before any training, both models have 0% accuracy and the loss is high as expected. For the evidential model, in the first few iter- ations, the model learns from the training dataset, and the model’s accuracy increases to 50%. Afterward, the eviden- tial model fails to learn as the evidential model maps two of the training data samples to the zero-evidence region. Even in such a trivial setting, the evidential model fails to fit the 4 training data points showing their learning deficiency that empirically verifies the conclusion in Theorem 1. It is also worth noting that the range of the evidential model’s loss is significantly smaller than the standard model. This is mainly due to the bounded nature of the evidential MSE loss(i.e., it is bounded in the range [0, 2]) (a detailed theoretical analy- sis of the evidential losses is provided in the Appendix). In contrast, the standard model trained with cross-entropy loss

: 6

Figure 5. Toy dataset with 4 data points.

(B) Training Loss Trend

Figure 6. Training of standard and evidential models easily fits the trivial dataset, obtains near 0 loss, and perfect accuracy of 100% after a few iterations of training.

Evidence

Figure 7. Zero-evidence trend during model training Additionally, we visualize the zero-evidence data samples for the toy dataset setting. We plot the total evidence for each training sample as training progresses for the first 100 iterations. The total evidence trend as training progresses for the first 100 iterations is shown in Figure 7. The ev- idential model’s predictions are correct for data samples with ground truth labels of 3 and 6, and incorrect for the remaining two data samples. After few iterations of training, the remaining two samples have zero total evidence (i.e.

samples are mapped to zero evidence region), the model never learns from them, and the model only achieves overall 50% training accuracy even after 100 iterations. Clearly, the evidential model continues to output zero evidence for two of the training examples and fails to learn from them.

Such learning deficiency of evidential models limits their extension to challenging settings. In contrast, the standard model easily overfits the 4 training examples and achieves 100% accuracy.

Sensitivity to hyperparameter tuning.

N This Experi-

ment, evidential models are trained using evidential losses given in (21), (22), or (23) with incorrect evidence regular- ization to guide the model for accurate uncertainty quan- Learn to Accumulate Evidence from All Training Samples: Theory and Practice Figure 8. Impact of different incorrect evidence regularization strengths to the test set accuracy on Cifar100 dataset tification. We study the impact of the incorrect evidence regularization λ1 to the evidential model’s performance using Cifar100. The result shows that the generalization performance of evidential models is highly sensitive to λ1 values. To illustrate, we consider the Type II Maximum Likelihood loss in (23) with different λ1 to control KL reg- ularization (results on other loss functions are presented in the Appendix). As shown in Figure 8, when some regular- ization is introduced, evidential model’s test performance improves slightly. However, when strong regularization is used, the model focuses strongly on minimizing the incor- rect evidence. Such regularization causes the model to push many training samples into or close to the zero-evidence regions, which hurts the model’s learning capabilities. In contrast, the proposed model can continue to learn from samples in zero-evidence regions, which shows its robust- ness to incorrect evidence regularization. Moreover, our model has stable performance across all hyperparameter settings as it can effectively learn from all training samples.

Challenging datasets and settings.

We Next Consider

standard classification models for the Cifar100 dataset and 1-shot classification with the mini-ImageNet dataset. We develop evidential extensions of the classification models using Type II Maximum Likelihood loss given in (23) with- out any incorrect evidence regularization and use ReLU to transform logits to evidence. As shown in Figure 10, com- pared to the standard classification model, the evidential model’s predictive performance is sub-optimal (almost 20% lower for both classification problems). This is mainly due to the fact that evidential model maps many of the training data points to zero-evidence region, which is equivalent to the model saying “I don’t know to which class this sample belongs” and stopping to learn from them. Consequently, the model fails to acquire new knowledge (i.e., update itself), even after being exposed to correct supervision (the label information). In these cases, instead of learning, the eviden- tial model chooses to ignore the training data on which it does not have any evidence and remains to be ignorant.

Visualization of zero-evidence samples.

We Next Show

the 2-dimensional visualization of the latent representation for the randomly selected 500 training examples based on

(B) 1-Shot Results

Figure 10. Learning trends in complex classification problems the tSNE plot for ReLU based evidential model trained on the Cifar100 dataset with λ1 = 0.1. Figure 9 plot visualizes the latent embedding of zero evidence (Zero E) training sam- ples with non-zero evidence (Non-Zero E) training samples.

As can be seen, both zero and non-zero evidence samples ap- pear to be dispersed, overlap at different regions, and cover a large area in the embedding space. This further confirms the challenge of effectively learning from these samples

Effectiveness Of The Red

Evidential activation function.

We First Experiment With

different activation functions for the evidential models to show the superior predictive performance and generalization capability of exp activation validating our Theorem 2. We consider evidential models trained with evidential log loss given by (23) in Table 1 (Additional results along with hy- perparameter details are presented in Appendix Section F).

As can be seen, exp activation to transform network outputs into evidence leads to superior performance compared to ReLU and Softplus based transformations. Furthermore, our proposed model with correct evidence regularization further improves over the exp-based evidential models as it enables the evidential model to continue learning from zero-evidence samples.

±0.21

We next present the test set performance change as training Learn to Accumulate Evidence from All Training Samples: Theory and Practice progresses with MNIST dataset and two different evidential losses in Figure 11 where we observe similar results. The exp activation shows superior performance, as it has small- est zero-evidence region, and does not suffer from many learning issues present in other activation functions.

(B) Evidential Log Loss

Figure 11. Impact of evidential activation functions to the Test

Accuracy

Correct evidence regularization.

We Now Study The Im-

pact of the proposed correct evidence regularization using sider the evidential baseline model that uses exp activation to acquire evidence, and is trained with Type II Maximum Likelihood based loss with different incorrect evidence reg- ularization strengths. We introduce the proposed novel cor- rect evidence regularization to the model. As can be seen in Figure 12, the model with correct-evidence regularization has superior generalization performance compared to the baseline evidential model. This is mainly due to the fact that with proposed correct evidence regularization, the evi- dential model can also learn from the zero-evidence training samples to acquire new knowledge instead of ignoring them.

Our proposed model considers knowledge from all the train- ing data and aims to acquire new knowledge to improve its generalization instead of ignoring the samples on which it has no knowledge. Finally, even though strong incorrect evidence regularization hurts the model’s generalization, the proposed model is robust and generalizes better, empirically validating our Theorem 3. Limited by space, we present additional results in Appendix F.3.2.

Zero-evidence Sample Anaysis.

Similar To The Toy

MNIST zero-evidence analysis, we consider the Cifar100 dataset, and carry out the analysis for this complex dataset/setting. Instead of focusing on a few training ex- amples, we present the average statistics of the evidence (E) for the 50,000 training samples in the 100 class classi- fication problem for a model trained for 200 epochs using a log-based evidential loss in (23) with λ1 = 1.0. For ref- erence, the samples with less than 0.01 average evidence (i.e., E ≤0.01) are samples on which the model is not confident (i.e., having a high vacuity of ν ≥0.99), and are close to the ideal zero-evidence region. Our proposed RED model effectively avoids such zero evidence regions, and has the lowest number of samples (i.e. only 0.06% of total training dataset compared to 58.96% of SoftPlus based,

(D) Trend For Λ1 = 1.0

Figure 12. Impact of correct evidence regularization to test accu- racy: (a), (b) - MNIST Results; (c), (d) - Cifar100 Results and 100% of ReLU based evidential models) in very low evidence regions.

Table 2. Zero-Evidence Analysis for Complex Dataset-Setting

We Next Study The Impact Of

the evidential loss function on the model’s performance consider all three activations: ReLU, SoftPlus, and exp to transform neural network outputs to evidence and carry out experiments over CIFAR100 with identical model and settings. As seen in Table 3, the generalization performance of evidential model is consistently sub-optimal when trained with evidential MSE loss given by (21) compared to the two other evidential losses (22) & (23). This is consistent across all three evidence activation functions. This is mainly due to the bounded nature of the evidential MSE loss (21): for all training samples, evidential MSE loss is bounded in the range of [0, 2]. Type II Maximum Likelihood loss given in (23) and cross-entropy based evidential loss given in (22) show comparable empirical results.

Next, we consider exp activation and conduct experiments over the MNIST dataset for incorrect evidence regulariza- tion strengths of λ1 = 0&1. We again observe similar results where the training with the Evidential MSE loss in (21) leads to sub-optimal test performance. Additional re- sults, along with theoretical analysis are presented in the Appendix. In the subsequent experiments, we consider the Type II Maximum Likelihood loss (23) for evidential model training due to its simplicity and some theoretical advan- Learn to Accumulate Evidence from All Training Samples: Theory and Practice tages (see Appendix E). We leave a thorough investigation of these two evidential losses ((22) & (23)) as future work.

Table 3. Impact of evidential losses on classification performance

(B) Trend For Λ1 = 1.0

Figure 13. Impact of evidential losses on test set accuracy

Figure 14. Accuracy-Vacuity Curve

Study of uncertainty information.

We Now Investigate

the uncertainty behavior of the proposed evidential model with Cifar100 experiments.

We Present The Accuracy-

Vacuity curve for different incorrect evidence regulariza- tion strengths (λ1) in Figure 14. Vacuity reflects the lack of confidence in the predictions, and the accuracy of effec- tive evidential model should increase with lower vacuity threshold. Without any incorrect evidence regularization (i.e., λ1 = 0), the evidential model is highly confident on its predictions and all test samples are concentrated on the low vacuity region. As the incorrect evidence regularization strength is increased, the model outputs more accurate confi- dence in the predictions. Strong incorrect evidence regular- ization hurts the generalization over the test set as indicated by low accuracy when all test samples are considered (i.e., vacuity threshold of 1.0). In all cases, the evidential model shows reasonable uncertainty behavior: the model’s test set accuracy increases as the vacuity threshold is decreased.

Next, we look at the accuracy of the evidential models on their top-K % most confident predictions over the test set. Table 4 shows the accuracy trend of Top-K (%) confident samples. Consider the most confident 20% samples (cor- responding to 2000 test samples of Cifar100 dataset). The proposed model leads to highest accuracy (of 99.35%) com- pared to all the models. Similar trend is seen for different K values where the proposed model shows comparable to superior results demonstrating its accurate uncertainty quantification capability.

Red

We next consider out-of-distribution (OOD) detection ex- periments for the Cifar100-trained evidential model using SVHN dataset (as OOD) (Netzer et al., 2011). As seen in Table 5, the evidential models, on average, output very high vacuity for the OOD samples, showing the potential for OOD detection.

Red (Ours)

We present the AUROC score for Cifar100 trained models with SVHN dataset test set as the OOD samples in Table 6. In AUROC calculation, we use the maximum softmax score for the standard model, and predicted vacuity score for all the evidential models. As can be seen, the exp-based model outperforms all other activation functions, and the proposed model RED can learn from all the training samples that leads to the best performance.

Onclusion

In this paper, we theoretically investigate the evidential mod- els to identify their learning deficiency, which makes them fail to learn from zero-evidence regions. We then show the superiority of the evidential model with exp evidential activation over the ReLU and SoftPlus based models.

We further analyze the evidential losses, and introduce a novel correct evidence regularization over the exp-based ev- idential model. The proposed model effectively pushes the training samples out of the zero-evidence regions, leading to superior learning capabilities. We conduct extensive experi- ments that empirically validate all theoretical claims while demonstrating the effectiveness of the proposed approach.

IIS-1814450 and an ONR award N00014-18-1-2875. The views and conclusions contained in this paper are those of the authors and should not be interpreted as representing any funding agency.

Authors:

Peder EZ Larson 1, 2,* , Jenna ML Bernard1, James A Bankson 3, Nikolaj Bøgh 4, Robert A Bok1, Albert P. Chen 5, Charles H Cunningham 6,7, Jeremy Gordon1, Jan-Bernd Hövener 8, Christoffer Laustsen 4, Dirk Mayer 9,10, Mary A McLean11 12, Franz Schilling13, James Slater1, Jean-Luc Vanderheyden5, 14, Cornelius von Morze 15, Daniel B Vigneron1, 2, Duan Xu1, 2, and the HP 13C

94143, Usa.

Denmark. 5 GE Healthcare, Menlo Park, California, USA. 6 Physical Sciences, Sunnybrook Research Institute, Toronto, Ontario, Canada.

8 Section Biomedical Imaging, Molecular Imaging North Competence Center (MOIN CC), Medicine, Baltimore, MD, USA. Cambridge, United Kingdom.

14Jlvmi Consulting Llc, Dousman, Wi, Usa

#See Acknowledgements for a list of all HP 13C MRI Consensus Group Members This work was supported by the ISMRM Hyperpolarized Media MR Study Group, the ISMRM Hyperpolarization Methods & Equipment Study Group, and the Hyperpolarized MRI Technology Resource Center (NIH/NIBIB grant P41EB013598).

Abstract

MRI with hyperpolarized (HP) 13C agents, also known as HP 13C MRI, can measure processes such as localized metabolism that is altered in numerous cancers, liver, heart, kidney diseases, and more. It has been translated into human studies during the past 10 years, with recent rapid growth in studies largely based on increasing availability of hyperpolarized agent preparation methods suitable for use in humans. This paper aims to capture the current successful practices for HP MRI human studies with [1-13C]pyruvate - by far the most commonly used agent, which sits at a key metabolic junction in glycolysis. The paper is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification. In each area, we identified the key components for a successful study, summarized both published studies and current practices, and discuss evidence gaps, strengths, and limitations. This paper is the output of the “HP 13C MRI Consensus Group” as well as the ISMRM Hyperpolarized Media MR and Hyperpolarized Methods & Equipment study groups. It further aims to provide a comprehensive reference for future consensus building as the field continues to advance human studies with this metabolic imaging modality.

Keywords: Hyperpolarized MRI, metabolic imaging, carbon-13, pyruvate, dissolution dynamic

Introduction

MRI with hyperpolarized 13C agents, also known as hyperpolarized (HP) 13C MRI, has shown great potential as a novel imaging modality, particularly for its ability to probe metabolic processes in real time. The first human studies with HP [1-13C]pyruvate were performed in 2011 in prostate cancer patients (1).

Since then, there have been over 60 papers published with imaging results of human subjects from 13 different sites, with applications including prostate cancer, brain tumors, breast cancer, kidney cancer, pancreatic cancer, metastatic disease, liver disease, ischemic heart disease, diabetes and cardiomyopathies. The vast majority of these studies used [1-13C]pyruvate (1–63), where [2-13C]pyruvate (64) and 13C-urea (56) have been demonstrated too.

As clinical HP 13C MRI advances, there is a growing need to build consensus for best practices, which are critical for comparing data across sites, performing multi-site trials,deploying methods to new sites, partnering with vendors, and potentially for obtaining broader regulatory approvals.

In March 2022, we initiated an effort to build consensus within the HP 13C MRI community with this opportunity in mind, and it was greeted with strong enthusiasm. The “HP 13C MRI Consensus Group”, containing over 55 members from 27 sites, identified the area of greatest need and opportunity for consensus building to be HP [1-13C]pyruvate human

●

Pyruvate is the most mature and widely used HP agent and has the most significant translational evidence emphasizing the potential clinical impact.

●

Clinical trials, particularly multi-site trials, have the strongest need for consensus methods to ensure that data can be combined across sites. This work is a Position Paper for which the goal is to describe current successful practices and study methods for HP [1-13C]pyruvate human studies along with justification to support those practices. This is divided into four major topic areas: (1) HP 13C-pyruvate preparation, (2) MRI system setup and calibrations, (3) data acquisition and image reconstruction, and (4) data analysis and quantification (Fig. 1). The current successful practices and study methods include a literature review of published peer-reviewed journal papers showing human HP [1-13C]pyruvate study data, up to September 2022 (1–63), as well as new unpublished information from surveys of HP 13C study sites. Based on this information, we also highlight the evidence gaps, strengths, and limitations of current practices which are summarized at the end of each section.

Figure 1: Illustration of the HP 13C MRI human study process, including the 4 major areas covered in this paper: Hyperpolarized 13C-pyruvate preparation, MRI system setup and calibration, Acquisition and Reconstruction, and Data Analysis and Quantification.

Figure 2: Anatomical targets of HP [1-13C]pyruvate MRI human studies published up to September 2022.

Hyperpolarized 13C-Pyruvate Preparation

This section covers the processes for creating the HP agent, 13C pyruvate, and will include many aspects and considerations that are needed to safely and effectively prepare doses for metabolic imaging studies in human subjects. These include material, personnel, equipment and facility, fluid path preparation, quality control, and release.

It is helpful to understand that the specifications of a dose of 13C pyruvate suitable for in vivo MR HP metabolic imaging were shaped in part by early preclinical studies performed by GE HealthCare summarized in Ref. (65). In short, the safety of the two novel drug components, 13C pyruvate and the electron paramagnetic agent (EPA) AH111501, were demonstrated in those studies. The more precise formulation of the dose suitable for human use was then determined from clinical studies (66) that included two Phase 1 clinical trials in young and elderly healthy volunteers without hyperpolarization of the 13C nuclei and another Phase 1/2a dose escalation and imaging feasibility study with HP 13C pyruvate in 31 prostate cancer patients at the With the exception of the first HP 13C imaging clinical trial, which utilized a prototype device in a cleanroom (1), all HP 13C studies performed in humans to date have utilized the SPINlab polarizer (manufactured by GE HealthCare). Consequently all doses of the HP 13C pyruvate delivered by SPINlab have been produced using the “SPINlab Pharmacy Kit” that serves as the container-closure system for the various drug components (13C pyruvic acid and EPA mixture, dissolution medium, and neutralization and dilution medium) during sample polarization, dissolution and quality control (QC) processes. Thus many aspects of the HP sample preparation considerations discussed below are related to the SPINlab instrument and the consumables designed to be used with it (67).

General Considerations

While more than 860 patients or healthy subjects having been injected with HP 13C pyruvate as of January 2022 without reports of any serious adverse events (68), HP 13C pyruvate injection remains an investigational MR contrast agent and can only be administered by those with Investigational New Drug (IND) exemption from the Food and Drug Administration (FDA) in the USA, a Clinical Trial Application (CTA) in Canada, approval from National Research Ethics Committee Services in the UK, or approval from the relevant local regulatory body. Thus, methods and processes involved to produce a dose should have patient safety as the first priority. Since utilizing dissolution dynamic nuclear polarization (dissolution-DNP) for human use is still a relatively new development, there are no existing published regulatory guidelines specifically for this method.

There are two major production styles that determine how various sites approach the agent preparation. In the US, the most common approach is to rely on a sterilizing filter (“Terminal Sterilization”) to ensure sterility of the final product, akin to PET tracer production, where a starting molecule with a radioisotope is processed using various other ingredients to make the final, desired and injectable contrast agent within a necessarily short amount of time (69). For these sites, sterilization of the components and accessories upstream of this filter are not required, although many of them were manufactured and tested following Good Manufacturing Practice (GMP) or Good Laboratory Practice (GLP) requirements. The filling process is usually performed under an ISO 5 laminar flow hood, but a clean room or an isolator is not required.

This approach is typically accompanied by testing the integrity of the sterilizing filter prior to release of the dose for injection. Typically, post release endotoxin and sterility tests are performed using an aliquot reserved from each released dose.

In the UK and EU, the most common approach is to more-closely follow sterile pharmaceutical compounding guidelines (70), where all components and ingredients are required to be sterile or manufactured under GMP guidelines and are assembled and filled within a clean room environment or an isolator system (“Sterile Preparation”). Typically a batch of Pharmacy Kits for HP 13C pyruvate injection are prepared together. The sterility of the final dose is also ensured by batch validation testing, in addition to the sterility of the ingredients and the sterile compounding process. The endotoxin and sterility testing are performed for the process validation but are not performed for each injected dose.

Some institutions fill and assemble the Pharmacy Kit required for a specific study on the same day or the day prior to polarization, dissolution, and patient administration, but others have also demonstrated the feasibility of preparing a batch of kits, keeping them in a -20ºC freezer and using them over a period of a few months.

Beyond the obvious requirements that the process and the facility has to ultimately produce a dose that is safe to inject into a human, regulatory authorities will also focus on the question “Are you in control of your processes?”. To be in control of your process requires an in-depth and broad understanding of all processes involved in pre, post, and during the production process.

Personnel

It is typical and may be required to have licensed personnel involved in the production process depending on local regulations.Typically a pharmacist, radiopharmacist or other similarly qualified person (QP), in charge of the facility where the Pharmacy Kit filling and preparation is taking place, is responsible for the overall process and the release of the injectable dose.

Qualified cleanroom technicians are often involved in the Pharmacy Kit filling under the supervision of the pharmacist or QP. As is required for pharmaceutical compounding or PET tracer production, training requirements and training records for all personnel need to be maintained and available for audit by the FDA or equivalent.

Equipment And Facility

The facility and all equipment need to have standard operating procedures (SOPs) that describe how equipment is used, maintained, and calibrated to comply with relevant legislation. Currently, almost all the filling of the Pharmacy Kit takes place within a compounding laminar flow hood or isolator (typically ISO 5). At some sites, the filling is conducted within a cleanroom, while at others, it is conducted in a dedicated non-cleanroom space, reflecting differences in cleanroom approach and specifications between regulators worldwide (71). Some equipment or facilities, such as the compounding hood or cleanroom, may require external certified laboratories for testing.

Material Handling

Material handling guidelines (69,70) require SOPs detailing a system to track all of the materials involved in the HP production process for a particular patient dose, similar to current good manufacturing practice (cGMP) requirements for material handling for drug compounding. This includes acceptance standards, storage conditions, amount used in the patient dose for each ingredient and materials used in the assembly of the fluid path and Pharmacy Kit. Currently some users choose to open and inspect and sometimes modify the Pharmacy Kits upon arrival, but some users keep them in the sealed packaging until they are required for dose preparation.

Pharmacy Kit Filling And Assembling

As required by an IND or its equivalent, the preparation of the doses of HP 13C agent are detailed in the Chemistry, Manufacturing, and Control (CMC) section of an applicable regulatory submission; an example of this has been made available (72). It describes the processes of filling the Pharmacy Kit with the different components that make up the final drug product, and of assembling the final kit for either storage or immediate use in the polarizer. Special attention should be given to the laser welding process in order to satisfy installation qualification (IQ) and operational qualification (OQ). Typically, the final developed process is validated by process qualification (PQ) runs, during which 3 or more Pharmacy Kits are filled and used and the final HP 13C products are tested for endotoxin and sterility and to confirm that they meet the dose specifications for injections (usually including pyruvate concentration, residual EPA concentration, pH, liquid state polarization level and dose temperature). The data from 3 consecutive PQ runs are submitted as part of the IND submission (or its equivalent), and are often also reviewed by the Institutional Review Board (IRB) where the studies are conducted.

Quality Control And Dose Release

The quality control (QC) and dose release can be separated into two aspects: one is the QC and release of the filled Pharmacy Kit, and second is the QC and release of the HP 13C agent for injection, after polarization and dissolution. For institutions filling a batch of kits and storing them to use over a period of time, typically the batch can be released based on initial validation, environmental monitoring data from the day of kit production, and if filters are used during preparation of any of the components, filter integrity testing. But in some cases one or more kits are used for validation before the batch of kits are released for future use. For institutions that fill only the kits required for specific studies shortly before the experiment, the filled kits often do not go through separate release tests before they are used.

The quality control of the HP 13C pyruvate solution post dissolution is primarily performed to ensure that the agent meets the dose specifications (Table 1) before it is administered to the subject. These specifications target both safety (pH, residual EPA, temperature) and efficacy (pyruvate concentration, polarization, volume). Typically, the pyruvate concentration, residual EPA concentration, pH, dose temperature, dose volume, and liquid state polarization are measured by the QC accessory associated with the SPINlab polarizer. Some users perform a secondary measurement for one of the parameters, such as pH, using a different instrument or pH paper. For sites that do not go through a separate release testing process for batch filled kits, the integrity of the sterilization assurance filter, a part of the Pharmacy Kit, is typically tested as a part of the dose release. It is also common for these users to preserve an aliquot of the final HP 13C pyruvate solution for post-release endotoxin and sterility testing. This testing cannot be completed fast enough to test an individual dose prior to injection, but this is why other processes such as PQ runs and validation testing are done to minimize the chance a subject could be injected with a contaminated dose.

The Final Dose Release And Injection

should be done under the supervision of a licensed professional, based on local regulations.

Some Key Challenges

Many of the challenges associated with HP 13C pyruvate preparation can be attributed to the conditions required for the dissolution-DNP method of high magnetic field (~3-7 T) and very low temperature (~1 K) during polarization, with pressurized and superheated water necessary for the rapid dissolution event. These extreme conditions are quite challenging for the design of the container-closure and fluid path system. In particular, the cryogenic temperature in the polarizer requires special attention to any moisture or ambient (moist) air introduced into that portion of the fluid path, which can form an ice block at ~1 K. This ice can lead to flow restriction during the dissolution event and reduce the strength of the laser welded bond between the cryovial and its cap. This can ultimately produce failures in the dissolution step, including variations in final pyruvate concentration and pH that may fail to meet QC release criteria as well as fluid path ruptures that provide no available dose and result in polarizer down-time.

The polarization of the HP 13C pyruvate sample decays quickly over the span of a few minutes after dissolution, and thus the process of dissolution, QC for release, and injection should be completed as fast as possible to preserve the high polarization level achieved. Any delays in the preparation process, such as transportation time or equipment malfunction, can significantly reduce the final polarization and result in lower quality imaging data.

Current Practices

A summary of data collected from all sites performing clinical trials with HP 13C-pyruvate is shown in Fig. 3 and Table 1, including the specification of the final dose and how the quality control and release of the final dose are performed. There is a split in the Production Style, described in the General Considerations section above, with 8/13 sites using Sterile Preparation versus 5/13 using Terminal Sterilization. While many of the dose specifications show notable differences in acceptable ranges, all of these variations listed in tables have been successfully and safely been used to perform HP 13C pyruvate studies in humans. Their differences depend on the institutions’ preferences, resources and their particular regulatory situation. There is high similarity in pyruvate ranges, temperature ranges, EPA limits, and volume limits. There is modest variability in pH ranges and large variability in the endotoxin test limit. There is a 3-fold difference in acceptable polarization levels, which are measured to ensure a futile dose is not injected since the polarization is directly proportional to SNR. This reflects the decision by several sites to believe that useful data can be still be obtained with suboptimal polarizations.

Figure 3: Hyperpolarized agent preparation methods reported by sites currently performing HP

In House

Table 1: HP 13C-pyruvate preparation parameters, methods, and dose specifications used for quality control testing and release as well as validation. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. The parameters used for product release are noted in bold text, otherwise these parameters are measured for batch validation or other QC measurements. The endotoxin and sterility testing are performed during process validation of the batch and/or post-injection, and largely depends on the agent production approach.

Summary

The overall safety record of HP 13C-pyruvate has been very strong, and the SPINlab hyperpolarizer has proven to provide high polarizations at human sized doses while meeting numerous QC and release criteria. A weakness remains the failure modes of the SPINlab Phamacy Kits (e.g. ice blocks, path ruptures), which are placed under extreme requirements particularly during dissolution. The preparation process still requires a high degree of expertise.

Therefore, there is a significant need to improve the reliability, robustness, and ease of operation for generating HP 13C-pyruvate doses for human studies. Furthermore, there is a divide between manufacturing and sterile compounding style preparation as well as other site-specific practices, resulting in variations in SOPs and justification required to relevant regulatory bodies. There have also been no comparisons between these approaches. It is also unclear what release criteria and QC parameters are truly required to ensure patient safety.

However, all of the reported methods are acceptable and approved by the appropriate regulatory authorities, and have led to the rapid expansion of successful human studies in recent years.

Mri System Setup And Calibrations

This section covers the MRI system setup, including the imaging system, RF coils, phantoms, and prescan calibration methods.

Imaging System

The main prerequisite for a given MRI scanner to be capable of supporting studies with HP 13C is its “broadband” capability to transmit and receive radiofrequency (RF) signal at the frequency of 13C, which is around 4 times lower than 1H. This does not come as a default on clinical MR devices. The transmit power of the broadband amplifier should also be sufficient to support the intended flip angle and RF pulse shape with the employed transmission RF coil(s) for 13C. Most studies to date use relatively low flip angles (< 90 degrees) for HP 13C in order to preserve polarization for time-resolved imaging. The capability to receive 13C signal on multiple channels is also desirable to increase SNR, as discussed further in the “RF coils” section.

The choice of magnetic field strength is primarily dependent on the metabolites’ frequency separation due to chemical shift dispersion and 1H imaging. High field strengths do not enhance hyperpolarized 13C signal as they do for 1H because the signal strength in a HP experiment relies on manipulating the population of quantum energy states outside of the MRI scanner.

However, the injected HP 13C-pyruvate and its metabolic products have greater frequency separation at higher fields, and it may thus be easier to separate and quantify these resonances at higher fields. This comes at the cost of a reduction in the achievable T2* and often reduced T1. As the initial polarization is independent of the imaging field strength it has been proposed that the increased T2* at 1.5T can potentially be exploited to increase SNR by adapting the acquisition bandwidth or reduce off-resonance imaging effects in cases when the decay of the transverse magnetization is dominated by T2* (73). In practice, 3T has been used in all published human 13C-pyruvate studies surveyed (Supporting Table S1), and comprises the majority of scanners currently in use for human studies (Table 3). A field strength of 3T is well-suited for 1H MRI anatomical reference and correlative imaging.

Stronger and more rapidly slewing magnetic field gradients support more rapid spatial encoding, particularly for metabolite-specific single-shot imaging using echo-planar imaging (EPI) or spiral imaging (See “Acquisition and Reconstruction”). Although the spatial resolution acquired for HP 13C imaging is typically much coarser than for 1H MRI, the factor of ~4 in gyromagnetic ratio leads to the same reduction factor in performance of the gradient system, so 13C experiments are potentially more limited by gradient hardware performance. To date, all human studies have used the commercially-available integrated gradient systems provided in clinical MRI scanners.

Optimization of scanner design has understandably focused on minimization of artifacts in 1H MRI, where devices such as room lights, the gradient amplifiers, and the motors driving the patient bed are checked to ensure that they do not produce RF interference at the 1H frequency, but artifacts may arise at other frequencies. Eddy current compensation is also not always appropriately adjusted for nuclei at other frequencies (74). In order to optimize for 13C, many sites have performed checks on phantoms for RF interference, gradient artifacts, and eddy currents (74), including the use of post-hoc gradient impulse response function characterisation and correction, and some vendors have fixed these issues as well.

Rf Coils

For HP 13C imaging studies in humans, RF coils for both 1H and 13C nuclei are needed, with 1H MRI providing an anatomical reference for registration and optional additional multiparametric MRI readouts. At the Larmor frequency of 13C nuclei, the relative contributions from coil noise compared to sample noise increase compared to 1H (73,75), although sample noise still is likely the dominant contributor for human-sized coils at 32.1MHz - the resonance frequency of 13C nuclei at 3T.

The key requirement for human 13C-pyruvate RF coils are that the coil geometry and sensitive volume must cover the volume of interest in the subject. Table 2 and Figure 4 shows coil configurations that have been used and optimized for applications in different anatomic regions.

Volume resonators are most commonly used for transmit, as they surround the subject to

Provide B1 Transmit Across The Fov (B1

+). While 1H relies on a large birdcage (“body”) coil built into the scanner, 13C transmit coils must be placed inside the bore. This takes up valuable space within the magnet, and also has led to the use of designs with relatively inhomogeneous

B1

+. Many human studies have used Helmholz pair resonators for transmit, including the “clamshell coil”, which has a notably inhomogeneous B1

+ Profile But Has Been Used Because Of

relatively easy integration into the scanner bore. B1

+ Variation Results In Variations In The Flip

angles that control the use of the hyperpolarized magnetization and creates errors in common HP metrics (9,76). The exception are head coils, where birdcage designs with highly

Homogeneous B1

+ can be placed around the head while easily fitting inside the bore. As with 1H MRI, higher SNR can typically be achieved by smaller receive coil elements, such as surface coils or phased arrays, and the majority of 13C receive coils used have layouts similar to 1H phased arrays.

RF coil quality control is important to ensure proper functioning of the coils to provide consistent imaging quality, especially with limited natural abundance 13C signal in vivo. It typically involves 1) a physical integrity check of the coil cables and connectors and 2) phantom SNR tests to check the coil’s performance and to monitor it over time (see Phantoms below). An useful reference for RF coil quality control is outlined in the MRI accreditation program of the American College of Radiology (77) and can be adapted for 13C coils.

Notably, configurations for brain and prostate studies used dual-tuned 1H/13C coil designs, which greatly simplify workflow and registration of 1H and 13C images, as no switching of coils is needed.

(1)

Table 2: RF coil configurations reported for human HP [1-13C]pyruvate studies.

Tx = Transmit

coil, RX = receive coil. The commonly used “clamshell” TX coil is a Helmholz pair design. For 1H RF configurations, all used the Body coil for TX unless otherwise noted, and “repositioned” indicates the 13C coil was removed for 1H imaging. One representative reference is listed for each configuration. The RF coil configurations reported in the reviewed papers are shown in Supporting Table S1.

Figure 4: Examples of RF coil configurations used for human HP [1-13C]pyruvate brain studies. (A,B) 13C Clamshell TX (Helmholz pair) and 2× 4-channel paddle RX arrays. (C) 13C Birdcage volume TX and 32-channel RX array (RX array slides into TX coil). (D) 13C Birdcage volume TX and 24-channel RX array, combined with a 1H 8-channel RX array. Image reproduced with permission from Ref (16).

Phantoms

Since hyperpolarized magnetization is non-renewable, phantoms containing 13C nuclei are important to: 1) test the multi-nuclear capabilities of the imaging system, including all parts of the signal excitation and receive chain; 2) perform calibration measurements before a scan with hyperpolarized nuclei; and 3) perform necessary pre-scan adjustments (see “Prescan Calibration” section). The phantoms currently in use are listed in Table 3. Their composition must provide sufficient 13C signal, with additional considerations of conductivity, stability, chemical shift(s) present, potential for dynamic imaging, and cost. The phantom geometries are typically either compact, in order to be used alongside the subject during a HP scan, or large enough to mimic the inner volume of a RF coil for system testing.

One popular compact design contains enriched 13C-urea at high concentration, typically 8 M, which provides a single resonance, placed inside a small container ~1 mL. The most common recipe mixes 13C-urea in a 90% water/10% glycerol solution, with glycerol used to increase the urea solubility and doping with a Gd-based contrast agent to shorten T1 which increases the potential SNR per unit time. For example, when Dotarem is added at a 3:1000 volume ratio the 13C-urea T1 is around 500 ms and T2 is around 100 ms. However, when testing pulse sequences influenced by T1 and T2, doping should be used carefully. This phantom is suitable for frequency calibration, transmit gain calibration, sequence testing, and as a fiducial marker when placed next to a patient. However, enriched 13C-urea has a relatively high cost compared to natural abundance compounds.

For larger volumes (>100 ml), the phantoms most often used contain undiluted ethylene glycol, glycerol, or dimethyl silicone. These compounds have sufficiently high carbon concentrations to provide sufficient 13C signal even with the 1.1% natural abundance of 13C. These larger phantoms matching the inner volume of an RF coil are useful for coil testing, including transmit

+) And Receive (B1

-) coil profile mapping, as well as to mimic acquisitions using in vivo FOV requirements. In this case, size and conductivity should match the expected subject size in order to mimic coil loading and get a realistic estimation of B1+. Large-volume natural abundance urea phantoms have also been used by some sites, but suffer from higher conductivity compared to biological tissues. Typically, it is easier to increase the conductivity and hence coil loading of the non-conductive phantom by adding NaCl to match physiological loading (16,78).

Dynamic phantoms that aim to mimic metabolite kinetics have also been developed (79–81), and have the potential to more closely mimic the HP experiment, but so far these are not widely used.

Prescan Calibration

Prior to performing an MRI acquisition, the so-called prescan procedure is used to set the shim parameters to maximize B0 homogeneity over the field of view (FOV) or a specific region of interest (ROI), the scanner center frequency (CF), the RF transmit gain, and the receiver gain.

While this calibration procedure is usually automated for 1H, the lack of sufficient natural abundance 13C signal prevents use of automated methods. (Although natural abundance 13C lipid signal has been detected, there are so far no reports on using this signal for prescan.) Table 3 shows current practices across sites.

Maximizing B0 homogeneity is independent of the nucleus and is therefore performed prior to 13C imaging using the 1H water signal and existing shimming tools, such as by a standard automated process (“Auto Shimming”) or using high order shimming routines. Similarly, the 13C CF can be calculated from the 1H CF using a predetermined scaling factor that depends on the target chemical shift (82). Another common approach used is to have a small, high-concentration 13C phantom, e.g. 8M 13C-urea, integrated in the RF coil or placed next to the scan subject (1). The reference frequency can also be based on real-time measurements after the HP injection but prior to imaging (83). Both the CF and B0 shimming are critical when using spectrally-selective RF pulses, as inmetabolite-specific imaging methods, where the desired excitation bandwidths are typically very narrow and frequency offsets can lead to a failure mode that is only apparent after injection.

The calibration of the RF transmit power is typically performed on a small, high-concentration 13C phantom placed near the region of interest during the scan or on a large 13C phantom of similar size and coil loading as the subject, prior to the subject scan. Reference power is often done by sweeping the power in a pulse-acquire sequence (53,62), or the Bloch-Siegert method (52,84). When using a small phantom, the location of the phantom, B1

+ Inhomogeneity As Well

as any shielding effects, e.g., when the phantom is integrated into a coil (1), may degrade the accuracy. Other methods include real-time Bloch-Siegert method measurements after the HP injection (83), and using the stronger natural abundance 23Na signal that is close enough to the 13C resonance frequency to be detected by 13C coils (82).

The receiver gain is predetermined, either systematically based on independent phantom measurements and assuming the dose and polarization of the HP compound is known prior to injection, or based on past HP imaging studies.

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

8

13C-bicarbonate doped with dimethyl silicone, various

Power [Kw]

Phantom(s) - during study Phantom(s) - before study 13C Frequency

Maximum Values

Table 3: Summary of the imaging systems, phantoms, and prescan procedures used at sites currently performing HP 13C-pyruvate human studies. These were obtained from a survey of all sites performing clinical trials with HP [1-13C]pyruvate. *Previously performed studies with a Siemens 3T Tim Trio. The imaging systems, phantoms, and prescan procedures reported in the reviewed papers are shown in Supporting Table S1.

Summary

Commercially available 3T MRI systems are by far the most commonly used for human HP 13C-pyruvate studies, although a systematic investigation of the impact of B0 has only recently been investigated (73). The multi-nuclear RF transmit and receive chain has proven sufficient for current acquisition strategies, although many sites have observed artifacts due to RF interference, gradient interference, and residual eddy currents when operating at the 13C frequency. A variety of 13C RF coils, tailored for numerous anatomical targets, have been successfully demonstrated, with the main limitation that most transmit coils take up a lot of additional space inside the bore and provide relatively inhomogeneous B1

+ Profiles. The

phantoms used have converged into generally 2 categories - small phantoms containing 13C-enriched compounds that can be used during the study and human-sized phantoms containing compounds with high carbon concentrations but without 13C enrichment that are used to test and calibrate the coils. There are no standardized compositions or geometry, and dynamic phantoms that recapitulate in vivo kinetics would be desirable but are still an emerging area. Prescan calibration procedures were not well defined in most publications, so we surveyed individual sites to determine current practices. Calibration procedures for the B0 field (13C CF and shimming) for most sites take advantage of 1H signal and methods, while methods

For Calibration Of B1

+ is more variable across sites, likely a reflection of remaining challenges in how to perform this calibration. Standardization of both phantoms and calibration procedures would synergistically improve the robustness and reproducibility of HP 13C studies.

Acquisition And Reconstruction

Data acquisition strategies in human HP [1-13C]pyruvate MRI studies must account for multiple chemical shifts, efficiently utilize the non-renewable HP magnetization, and acquire data quickly relative to metabolism and relaxation decay processes. These studies require spectral encoding to separate metabolites, necessitating pulse sequences that efficiently encode up to 5D data (3 spatial + 1 spectral + 1 temporal dimension). RF pulses must efficiently sample without immediately saturating the non-renewable HP magnetization, and sequences must acquire data quickly and be robust to both experimental and physiologic variation (e.g. B1

+ Inhomogeneity,

variation in perfusion) to ensure reproducibility and minimize scan-to-scan variability. This section covers current successful practices for data acquisition in human [1-13C]pyruvate studies, and accompanying 1H imaging, from different anatomic regions, including scan parameters and image reconstruction.

Acquisition And Reconstruction Methods

The acquisition methods used in human [1-13C]pyruvate studies can be classified into 3 categories: 1) MR spectroscopy or MR spectroscopic imaging (“MRS/I”), 2) chemical shift encoding methods, and 3) metabolite-specific imaging (Fig. 5).

Mrs/I Methods Specifically

resolve a spectrum that can be analyzed to extract expected as well as unexpected resonances, making this approach very robust. It was used in many initial studies (1).

Chemical Shift

encoding methods, most commonly the Iterative Decomposition of water and fat with Echo Asymmetry and Least-squares estimation (IDEAL) method, use imaging sequences acquired with multiple TEs and rely on a model-based separation of expected chemical shifts (85).

Metabolite-specific imaging methods use specialized RF pulses that are spatially and spectrally selective to excite individual metabolites which are then typically imaged with fast k-space trajectories such as echo planar imaging (EPI) or spirals (86).

Their Application To Different

organ systems is described below. The image reconstruction methods used in human [1-13C]pyruvate studies have typically been conventional methods (e.g. FFT, non-uniform FFT, or equivalent). The incorporation of accelerated imaging and advanced reconstruction methods including parallel imaging (4,57,87) and compressed sensing (7) has also been applied in human studies for improved spatial resolution, temporal resolution and coverage, but have the potential for additional artifacts as well as SNR losses due to ill-conditioning of the reconstruction (e.g. g-factor).

The Majority Of

published studies do not use accelerated imaging indicating the resolution and coverage achievable without acceleration is currently adequate for successful data collection. Performing coil combination, even with fully sampled data has also been shown to have specific challenges for HP human images: using naive sum-of-squares methods suffer from high noise amplification in the relatively low SNR regime of HP [1-13C]pyruvate (compared to 1H), motivating several HP 13C-specific methods that include data-driven coil sensitivity estimation which have shown obvious improvements over sum-of-squares (11).

More recently denoising techniques have been applied as post-processing of human HP data(41,42,44). The techniques applied are based on spatial-temporal singular value decomposition for unsupervised estimation of signal and noise components. They have shown improvements in apparent SNR in the brain and liver, while care must be taken to choose parameters such as the rank threshold to avoid oversmoothing and overfitting to the estimated signal components.

Prostate Studies

Prostate cancer was the first human application of HP [1-13C]pyruvate (1), and data was acquired with MRS/I methods: 1D dynamic MRS, single-slice 2D dynamic echo-planar spectroscopic imaging (EPSI), and single time point 3D EPSI. Advances in imaging strategies led to the development and application of new acquisition schemes, including undersampled 3D EPSI with compressed-sensing (7), model-based chemical shift encoding methods that use a priori information (47,59), and metabolite-specific EPI (10), all of which can provide volumetric whole-organ coverage and dynamic acquisitions.

The pyruvate bolus arrival in the prostate can vary by ± 10 s between patients, necessitating dynamic imaging to reliably and consistently capture the pyruvate bolus (18). For this reason, all currently ongoing studies acquire dynamic data. While MRS/I, chemical shift encoding, and metabolite-specific imaging can all achieve dynamic imaging, chemical shift encoding and metabolite-specific imaging provide greater dynamic and volumetric coverage (85). For scan prescriptions, the FOV is designed to provide full prostate coverage and typically to match the orientation of the anatomic imaging used for registration. Flip angles used in current studies are constant through time, as quantification with a variable-through-time flip scheme is highly sensitive to bolus timing (8) and errors in the RF transmit (B1 +) field (76).

Heart Studies

Data acquisition methods for 13C imaging in the heart must be designed to meet the demands of significant cardiac motion and blood flow. To cope with the periodic cardiac motion, most human heart studies to date used gating to the diastolic window, the longest cardiac cycle interval, which has reduced motion (2,22,28,30,35,36,38,45,52). The duration of the diastolic window limits the available data sampling time, making cardiac acquisitions the most time-constrained of the HP 13C MRI applications. The most common acquisition approach is metabolite-specific imaging with spiral k-space trajectories (2). Their single-shot imaging capability makes these methods particularly robust to motion effects. Furthermore, spiral k-space trajectories provide rapid k-space coverage and relatively benign flow and motion artifacts. The majority of studies have used 2D multi-slice acquisitions, but 3D encoding has also been used successfully (35).

Brain Studies

For HP 13C MRI of the human brain, the majority of studies have also used 2D (slice selective) acquisitions (10–12,14,16,28,33,40,41,44,51,53,60), with a trend toward volumetric coverage using 2D multi-slice metabolite-specific imaging. 3D metabolite-specific imaging of the whole brain, with phase encoding of the slice direction (34,57), has been shown to provide similar SNR efficiency (88) compared with multislice imaging. A number of studies have employed MRS/I (5,6,29,31–33,50,55) resulting in a spectrum from each voxel, which has the advantage of not requiring a priori information about which peaks to encode. This was important in early brain studies when it was not known which peaks would be detectable. Chemical shift encoding, using a set of images with different echo times and an iterative reconstruction of the individual resonances (i.e. the IDEAL approach (85)), has also been used (12,49,54), with the drawback that coverage in the slice direction was limited due to the time required to acquire multiple echo time images.

Abdomen And Breast Studies

The fundamental approaches to data acquisition and reconstruction in the abdomen and breast are largely similar to the aforementioned applications, but demand attention to particular challenges associated with these anatomic regions, especially relating to respiratory motion.

Although it has been shown that a basic 2D MRSI approach based on phase encoding and FID readout can be successfully applied for HP 13C imaging in breast (15) and kidney (13), major advantages in terms of spatiotemporal resolution and coverage have been realized using tailored approaches based on metabolite-specific imaging (43,62) and chemical shift encoding (43), which have facilitated multi-slice or 3D dynamic acquisitions over large FOVs in the abdomen (4,37,46).

The significant respiratory motion encountered in these regions can directly blur 13C images, and has further favored these rapid acquisition strategies. Motion also degrades B0 homogeneity, which can shift frequency-selective excitation profiles and introduce artifacts into rapid imaging readouts. This makes accurate determination of the acquisition center frequency and shimming essential in these regions which often cover large FOVs. (See “Prescan Calibration” section for more information). In some studies, breath-holding was used to minimize motion effects and enforce frame-to-frame data consistency (42). A pragmatic and reasonably effective approach for dealing with respiratory motion during 13C data acquisition is an initial breath-hold (as long as can be tolerated), followed by free-breathing (46,62).

1H Imaging

Collection of 1H imaging data is essential both for prescribing the 13C acquisition and for interpretation of the resulting 13C data. Multi-planar 1H scouts are acquired prior to 13C acquisition to enable graphical prescription of the 13C imaging region. All human HP 13C-pyruvate imaging studies acquire conventional MRI scans (e.g. T1- and T2-weighted volumes) for anatomic reference, aiming to cover at least the full 13C FOV. Acquiring these anatomic scans as close as possible to the time of 13C imaging (immediately before or after) minimizes potential misregistration between the data sets. Depending on the application, other advanced 1H sequences are also acquired (e.g. diffusion-weighted imaging for cancer imaging).

When contrast-enhanced data is acquired, it is done after 13C imaging, as paramagnetic contrast agents will accelerate 13C relaxation.

Reported Study Parameters

Figures 5 and 6, and Supporting Table S2 shows the reported acquisition study parameters for human HP [1-13C]pyruvate studies published as of September 2022. Figure 5 shows a mixture of MRS/I, metabolite-specific imaging, and chemical shift encoding methods have been successfully used, where spectroscopy-based methods have become less prevalent in recent studies. Figure 6 shows the acquisition timing, including the important start time and interval/temporal resolution, is quite variable across studies.

Figure 5: Acquisition methods used in published HP [1-13C]pyruvate human studies published up to September 2022, classified into: MR spectroscopy and spectroscopy imaging (MRS/I); chemical shift encoding methods, such as IDEAL, that use multiple TEs and model-based reconstructions; and metabolite-specific imaging methods that use spectrally-selective excitation to image a single resonance at a time.

Figure 6: Temporal acquisition characteristics reported in HP [1-13C]pyruvate human studies published up to September 2022. (a) Reported referencing of acquisition start times.

(B)

Acquisition start times reported when using dynamic imaging and when timing was reported relative to the end of the injection. (c) Temporal resolutions. “Not Applicable” indicates dynamic imaging was not used.

Summary

Three general categories of acquisition strategies have been used successfully for human HP 13C-pyruvate studies: MRS/I, model-based chemical shift encoding (e.g. IDEAL) methods, and metabolite-specific imaging methods. These have enabled successful studies in the prostate, heart, brain, abdomen, and breast. Recent studies increasingly have used the imaging-based strategies of metabolite-specific imaging and chemical shift encoding which are the fastest methods, although a heads-to–head comparison between techniques has not been performed.

Metabolite-specific imaging is quite popular because of its speed and compatibility with single-shot imaging, but is sensitive to B0 field variations and thus requires careful calibrations. Nearly all studies surveyed acquired data dynamically, allowing measurement of the bolus and metabolite kinetics. The exact timings and associated flip angles vary quite widely across reported studies, with no consensus yet as to how to choose these parameters. Image reconstruction is typically done directly using Fourier Transform methods, and accelerated imaging strategies are uncommon.

Data Analysis And Quantification

This section covers the analysis of data from human HP [1-13C]pyruvate studies, including modeling and metrics, visualization, as well as considerations for how to store data and metadata. Depending on study design, the analysis may need to give quantitative or semi-quantitative output reflecting a biological process or may just reflect a contrast between different regions of interest for quantitative evaluation.

Metrics

Figure 7: HP [1-13C]pyruvate raw data (A) have typically been quantified using four categories of metrics depending on the acquisition. Data acquired as a single time point are often quantified using normalized metabolite images or metabolite ratios (B). Dynamic data can be quantified using normalized metabolite images or metabolite ratios (B), or with metabolite timings such as time-to-peak (TTP) or pharmacokinetic (PK) models (C). The latter two require the data to be time-resolved. [1-13C]alanine and 13C-bicarbonate are analyzed similarly to [1-13C]lactate but omitted here for display.

Metabolite images are commonly used as summary metrics for HP MRI data, often including some form of normalization as well as summed over time as an area under the time curve (AUC) (17). These are analogous to the visual evaluation that is most used for routine clinical work (89,90). In these metabolite images, we expect that the [1-13C]pyruvate AUC signal is predominantly weighted towards perfusion and uptake, while [1-13C]lactate, [1-13C]alanine and 13C-bicarbonate AUCs represent metabolic conversion. The strength of this approach lies in its simplicity and relatively few underlying assumptions. Limitations to the use of single-metabolite images or AUCs include sensitivity to inhomogeneous coil profiles (57,87,91), the acquisition strategy and acquisition parameters, pyruvate polarization and concentration level, and signal relaxation rates (92). Further, the reader must be careful to interpret all the images in conjunction to better understand the underlying biology; for example, increased [1-13C]lactate in the presence of decreased [1-13C]pyruvate delivery can have a very different meaning compared to increased [1-13C]lactate with increased [1-13C]pyruvate delivery.

In an attempt to address variations in coil sensitivity, polarization level, and pyruvate delivery, AUC images are often computed by normalizing to a specified parameter, such as the maximum pyruvate or average lactate signals, or presented as a ratio such as lactate/pyruvate or divided by “total Carbon” - the sum total of HP 13C signal observed across all metabolites. The AUC ratios between metabolites and pyruvate are proportional to the corresponding forward kinetic rates (81,93), but are not directly comparable to rate constants when magnetization loss rates (e.g. relaxation and losses due to signal excitation) differ between studies. Similarly, the ratios between the produced metabolites (e.g. bicarbonate/lactate) can reflect the balance between downstream metabolic pathways (12,55). Care must be taken to consider how AUC images are calculated and normalized before comparing values between studies.

To further quantify the interpretation, pharmacokinetic (PK) modeling approaches were developed to compute the apparent kinetics of pyruvate-to-metabolite exchange (92,94–99). These yield semi-quantitative to quantitative apparent rate constants, given in s-1. Some models require a vascular input function, while others avoid this requirement (95). PK models can explicitly account for acquisition-specific details such as excitation angle and repetition time, and thus may reduce the effects of these details on quantification. An input-less model, provided in the Hyperpolarized-MRI-Toolbox (https://github.com/LarsonLab/hyperpolarized-mri-toolbox) (100) and thus frequently employed for human data, has been shown to fit well and robustly to prostate and brain data (8,20). PK models are quantitative in nature, arguably provide more relevant biological information (8,20), and appear to be reproducible across sites (51). However, rate constants derived from PK models are still apparent rates, and likely do not reflect a single biological characteristic.

Some additional considerations include whether complex or magnitude data is used, as the noise behaviors will impact the analysis differently. Additionally, cut-off thresholds or other criteria may be used to identify and avoid voxels with insufficient SNR before analysis to improve robustness (20,41).

Regardless of the analysis approach, the underlying biology is not always clearly represented by the data; instead, the metrics may be influenced by perfusion, barrier permeability, intercellular shuttles, enzyme activities, co-substrate concentrations, or combinations thereof, depending on the organ and disease of interest (19,43,94,101–103). This may be addressed by incorporating complementary information. As an example, HP 13C pyruvate data is influenced by perfusion, and thus addition of perfusion MRI could be important for interpretation (98,104,105).

All the methods outlined above have been explored in clinical studies, described in Supporting Table 3 and summarized in Figure 8. As of September 2022, approximately 52% of studies involving human subjects report rate constants derived from a PK model with a few different models reported. A nearly equal fraction (51%) of the studies report AUC ratio values.

Approximately 66% of these studies report metabolite-specific images or AUC values. About 40% report SNR values; this metric is particularly frequent in manuscripts that describe technical developments for clinical HP MRI. Approximately 16% of these studies summarize model-free metrics, and 10% report measurements from a single timepoint. Most studies report a combination of quantities.

Figure 8: Reported metrics used for analysis in HP [1-13C]pyruvate human studies published up to September 2022.

Visualization

A wide variety of approaches have been used for visualizing data from human HP 13C-MRI studies. The challenges and practical considerations are: 1) choosing the appropriate metrics to display, 2) how to encode the parameters (e.g. the colormap), and 3) choosing how to provide anatomical context and other multi-parametric data. The choice of visualization also depends on the goal which could be for diagnostic interpretation, but also quality control, reproducibility among readers and publication.

Metrics

The choice of HP 13C metrics is described in detail above. At this stage in HP 13C development where there is no standardized metric, often a combination of metabolite images and ratios or PK model parameters are shown.

Parameter Encoding

The mapping function chosen should provide an adequate, often quantitative, impression of the parameter mapped. There is a consensus in the visualization field that perceptually uniform maps are best suited to visualize continuous parameters, like the greyscale typically used by radiologists as well as other monochrome (black to blue) and color ranges (fire-type, rainbow-type) (106,107). Multi-color heatmaps have been the most frequently employed method for HP 13C data, while greyscale has infrequently been used but it ensures there is no coloring-based bias as well as facilitating later reuse (Fig. 9a). Among the color schemes employed in the clinical HP 13C literature, fire-type scheme seems to be the most common [similar to “Plasma” or “Inferno” in matplotlib.org]. Next most commonly employed is the rainbow-type scheme [similar to “Rainbow” in matplotlib.org].

Anatomical Context

HP MRI faces the challenge that it does not necessarily depict the anatomical features, similar to PET, and thus requires an anatomical reference. Most often, a grayscale anatomical image is overlaid with a HP colormap (Fig. 9c,d). This approach is very intuitive, but can skew perception as the grey-scale anatomical reference may affect the brightness of the HP data (e.g. signal in the skull). This bias does not occur when showing adjacent maps (Fig. 9a, b). Here, anatomical outlines may help to provide reference (Fig. 9b).

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