# Anonymization of labeled TOF-MRA images for brain vessel segmentation using
generative adversarial networks

Tabea Kossen1,2, Pooja Subramaniam1,3, Vince I Madai1,4, Anja Hennemuth2,5,6,
Kristian Hildebrand7, Adam Hilbert1,2, Jan Sobesky8,9, Michelle Livne1, Ivana
Galinovic9, Ahmed A Khalil9,10,11,12, Jochen B Fiebach9 and Dietmar Frey1
1CLAIM - Charité Lab for AI in Medicine, Charité Universitätsmedizin Berlin,
Germany
2Department of Computer Engineering and Microelectronics, Computer Vision &
Remote Sensing, Technical University Berlin, Berlin, Germany
3Department of Electrical Engineering and Computer Science, Technical
University of Berlin, Berlin, Germany
4School of Computing and Digital Technology, Faculty of Computing, Engineering
and the Built Environment, Birmingham City University, Birmingham, UK
5Institute for Imaging Science and Computational Modelling in Cardiovascular
Medicine, Charité Universitätsmedizin Berlin, Berlin, Germany
6Fraunhofer MEVIS, Bremen, Germany
7Department VI Computer Science and Media, Beuth University of Applied
Sciences, Berlin, Germany
8Johanna-Etienne-Hospital, Neuss, Germany
9Centre for Stroke Research Berlin, Charité Universitätsmedizin Berlin,
Berlin, Germany
10Department of Neurology, Max Planck Institute for Human Cognitive and Brain
Sciences, Leipzig, Germany
11Mind, Brain, Body Institute, Berlin School of Mind and Brain, Humboldt
University Berlin, Berlin, Germany
12Berlin Institute of Health, Berlin, Germany

###### Abstract

Anonymization and data sharing are crucial for privacy protection and
acquisition of large datasets for robust medical image analysis. This
represents a major challenge, especially for brain imaging research. Here, the
unique structure of brain images allows for potential re-identification and
thus requires anonymization beyond conventional methods. Generative
adversarial networks (GANs) have the potential to provide anonymous images
while maintaining their predictive properties.

Analyzing brain vessel segmentation, we trained 3 GAN architectures on time-
of-flight (TOF) magnetic resonance angiography (MRA) patches of patients with
cerebrovascular disease for image-label pair generation: 1) Deep convolutional
GAN, 2) Wasserstein-GAN with gradient penalty (WGAN-GP) and 3) WGAN-GP with
spectral normalization (WGAN-GP-SN). First, the synthesized image-labels from
each GAN architecture were used to train a U-net for vessel segmentation. The
U-nets were then tested on real patient data. In total, 66 patients were used
for this analysis. In a second step, we simulated the application of our
synthetic patches in a transfer learning approach using a second, independent
dataset. Here, for an increasing number of up to 15 patients we evaluated
vessel segmentation model performance on real data with and without pre-
training on generated patches. Finally, performance for all models was
assessed by the Dice Similarity Coefficient (DSC) and the 95th percentile of
the Hausdorff Distance (95HD).

Comparing the 3 GAN architectures, the U-net model trained on synthetic data
generated by the WGAN-GP-SN showed the highest performance to predict brain
vessels (DSC/95HD 0.82/28.97) benchmarked by the U-net trained on real data
(0.89/26.61). The transfer learning approach showed superior performance for
the same GAN architecture compared to no pre-training, especially for one
labeled patient only (DSC/95HD 0.91/25.68 compared to DSC/95HD 0.85/27.36).

In a brain imaging segmentation paradigm, synthesized image-label pairs
preserved generalizable information and showed good performance for vessel
segmentation. Furthermore, we showed that synthetic patches can be used in a
transfer learning approach with an independent dataset. These results pave the
way to overcome the crucial challenges of scarce data and anonymization in the
medical imaging field. To facilitate further research, our synthetic image-
label pairs are being made available upon request.

###### keywords:

Anonymization, Generative Adversarial Networks, Image Segmentation

††pubyear: 2020††pagerange: Anonymization of labeled TOF-MRA images for brain
vessel segmentation using generative adversarial networks–Anonymization of
labeled TOF-MRA images for brain vessel segmentation using generative
adversarial networks

##  1 Introduction

Modern deep learning methods have revolutionized the field of natural image
analysis (Krizhevsky et al. (2017); Simonyan and Zisserman (2014)). These
methods are translated to medical image analysis with growing success (Litjens
et al. (2017); Ronneberger et al. (2015); Livne et al. (2019)). However, in
contrast to natural images, the number of data sets in medical image analysis
are usually orders of magnitude smaller since their availability is limited
owing to data privacy regulation. This poses a continuous challenge for deep
learning research in the medical imaging field. To meet this challenge,
anonymization of medical images is an essential method to ensure both data
privacy and data availability for research. However, current anonymization
methods in neuroimaging such as face blurring or face removal still allow re-
identification and thus cannot be applied (Abramian and Eklund (2019);
Ravindra and Grama (2019); Wachinger et al. (2015)). These results call for
new techniques to anonymize medical neuroimaging data to both protect patient
privacy and to facilitate research progress.

Generative adversarial networks (GANs) have the potential to fulfill this
need. GANs have already been applied successfully for medical imaging data
synthesis (Neff et al. (2017); Yi et al. (2019); Sorin et al. (2020)). Also,
first pilot studies have already made use of GANs for anonymization purposes
(Shin et al. (2018); Hukkelas et al. (2019)). However, applications for
neuroimages are scarce and synthesizing images often requires additional
patient information such as a segmentation label (Shin et al. (2018)). This
means that patient information is still fed into the model and the generated
images are then not properly anonymized. Thus, there is a need to investigate
the ability of GANs to create state-of-the-art anonymous synthetic
neuroimaging data maintaining the predictive properties of the original data.
Importantly, such an approach would have the most beneficial impact if the
corresponding labels would be created in the same process since many
supervised deep learning applications require time-consuming manual labeling
of the dataset by experienced physicians.

In this work, we utilize arterial brain vessel segmentation to test the
ability of GANs to create synthetic neuroimaging data and corresponding
labels. Moreover, we investigate the generalizability of the synthesized data
on a second, independent dataset. With respect to the generative
architectures, we train 3 different GAN architectures on time-of-flight (TOF)
magnetic resonance angiography (MRA) image patches of patients with
cerebrovascular disease: 1) Deep Convolutional GAN (DCGAN), 2) Wasserstein GAN
with gradient penalty (WGAN-GP) and 3) WGAN-GP using spectral normalization
(WGAN-GP-SN). With each GAN type, we synthesized both the image and the
corresponding label. We validate the generated synthetic patches using two
different approaches. In the first approach, we evaluate the quality of the
generated patches a) using the Fréchet inception distance (FID) and b) by
training a vessel segmentation U-net on the synthetic patches. The U-net’s
performance is then assessed on real test data. In total, 66 patients were
utilized for this analysis. In the second approach, we use the synthetic
patches to pre-train a vessel segmentation model and apply the network weights
in a transfer learning setting to pre-initialize the training of a U-net model
using up to 15 patients from a second, independent TOF-MRA dataset. The
performance of this model is then compared to a U-net model without any pre-
training. Finally, to facilitate and accelerate future research on arterial
vessel segmentation and to corroborate the usefulness of the effective
anonymization procedure, we make the synthesized image-label pairs generated
in our study available upon request.

Taken together, the contributions of the paper are: We present effectively
anonymized and labeled TOF-MRA patches for brain vessel segmentation for the
first time to our knowledge. Furthermore, we compare three different state-of-
the-art GAN architectures and evaluate our synthesized labeled data on an
independent, second dataset in a novel evaluation pipeline. We show that pre-
training a vessel segmentation network using our synthetic data yields
superior performance compared to no pre-training and can reduce the amount of
additional training data. Finally, we make our synthesized data available upon
request to facilitate further research.

##  2 Related Work

GANs have already been shown to be successful in many applications of data
augmentation in medical imaging (Frid-Adar et al. (2018), Sandfort et al.
(2019)) as well as in neuroimaging (Bowles et al. (2018), Foroozandeh and
Eklund (2020)). Here, real medical images together with synthesized images
were used to improve models that were trained on real data only. Whereas we
provide results on data augmentation, this study focussed on the models
trained on purely synthetic data and its generalizability to a new dataset.

Generating medical images with labels is not a new idea. Neff et al. (2017)
showed that lung x-rays with corresponding segmentation labels can be
generated using a GAN architecture. Guibas et al. (2018) demonstrated the
synthesization of labeled retina images using two GANs. While these studies
focused on 2D medical images, we use a 3D dataset and evaluate the performance
on an independent dataset. In the neuroimaging domain Foroozandeh and Eklund
(2020) recently showed that synthesized and labeled MR images can improve
tumor segmentation performance. However, the focus here was on augmentation
and models trained on synthesized data alone yielded comparably low
performance. Also, in contrast to the present study, only one dataset was used
for training and evaluation.

##  3 Methods

###  3.1 Network Architecture

The architecture of the proposed DCGAN was adapted from Radford et al. (2016)
and Neff et al. (2017). The WGAN-GP is an extension of the original
Wasserstein GAN (Arjovsky et al. (2017)) using gradient penalty for
regularization (Gulrajani et al. (2017)). For the third architecture WGAN-GP-
SN spectral normalization was used in the convolutional layers of the WGAN-GP
(Miyato et al. (2018)). Our code is available at
https://github.com/prediction2020/GANs-for-anonymized-labeled-TOF-MRA-patches.
The proposed methods and the structure of the GAN is shown in Fig. 1.

Figure 1:  Workflow of this study (A) and basic architecture of the generative
adversarial networks that were trained (B).

The generator $G$ of all architectures took a noise vector of length 100
sampled from a gaussian distribution as input. The noise vector was then fed
through 6 upsampling convolutional layers using a kernel size of 5 and stride
of 2. After each convolution layer, a batch normalization layer and a ReLU
activation layer were added, except for the last convolution layer. The
activation function used after the last convolution layer is the hyperbolic
tangent function. The network then outputs two 96 x 96 images that correspond
to one image-label pair $x_{\text{gen}}\sim p_{gen}$. The objective function
for the generators of all architectures were built upon:

$\mathcal{L}_{G}=\text{max}_{G}\mathbb{E}_{x_{\text{gen}}\sim
p_{\text{gen}}}[log(D(x_{\text{gen}}))]$ (1)

The discriminator $D$ for all architectures took two 96 x 96 images as input
which correspond to either a real image-label pair or generated image-label
pair. The pairs were again fed through 6 convolutional layers with a kernel
size of 5 and stride of 2. After each convolution layer, a batch normalization
layer and a leaky ReLU (with a slope of 0.2) were added, except for the last
convolution layer. The activation function used after the last convolution
layer in the DCGAN was a sigmoid function. The objective function of the
discriminator for the DCGAN was:

$\mathcal{L}_{D}=\text{max}_{D}\mathbb{E}_{x_{\text{real}}\sim
p_{\text{real}}}[logD(x_{\text{real}})]+\mathbb{E}_{x_{\text{gen}}\sim
p_{\text{gen}}}[log(1-D(x_{\text{gen}}))]$ (2)

where $x_{\text{real}}\sim p_{\text{real}}$ denoted the real image-label pair.

For the WGAN-GP and WGAN-GP-SN, a gradient penalty term for regularization was
added to the discriminator’s loss:

$\text{loss}_{D}=D(x_{\text{gen}})-D(x_{\text{real}})+\lambda(\lVert\nabla
D(\epsilon x_{\text{real}}+(1-\epsilon)x_{\text{gen}})\rVert-1)^{2},$ (3)

where $\epsilon\sim\textit{U}[0,1]$ and $\lambda=10$. Since the discriminator
acted as a critic, the sigmoid activation function in the last convolutional
layer was omitted. The batch normalization was replaced by instance
normalization to normalize across features and channels in the WGAN-GP. In the
WGAN-GP-SN architecture, spectral normalization was used instead of instance
normalization.

For training the DCGAN, the Adam optimizer (Kingma and Ba (2017)) with a
learning rate of 0.0003 with $\beta_{1}=0.5$ was used for both the generator
and the discriminator. The batch size was 512 and the model was trained for
178 epochs. To improve stability of the training, label smoothing (ranges
0.7-1.2/0-0.3) and feature matching between the last convolutional layer using
L1 norm were applied (Salimans et al. (2016)).

For WGAN-GP and WGAN-GP-SN, the Adam optimizer was utilized with a learning
rate of 0.0001 with $\beta_{1}=0$ and $\beta_{2}=0.9$ for both generator and
discriminator. The batch size was 300 and both models were trained for 180
epochs. In each epoch the discriminator was updated five times and the
generator once. All models were implemented in PyTorch and trained on two
GeForce GTX 1080Ti.

###  3.2 Patients

A total of 121 patient MRA data from two studies were used: PEGASUS (N=66) and
1000Plus (N=55). All patients were diagnosed with a cerebrovascular disease.
Details on both studies can be found in previous papers, for the PEGASUS study
see (Mutke et al. (2014)), for the 1000Plus study see (Hotter et al. (2009)).
All the patients gave their informed written consent. The studies have been
conducted in accordance with the authorized ethical review committee of
Charité - Universitätsmedizin Berlin.

Scans were performed on a clinical 3T whole-body system (Magnetom Trio,
Siemens Healthcare, Erlangen, Germany; using a 12-channel receive
radiofrequency coil (Siemens Healthcare) tailored for head imaging.

Parameters PEGASUS: voxel size = (0.5x0.5x0.7) mm3; matrix size: 312x384x127;
TR/TE = 22ms/3.86ms; acquisition time: 3:50 min, flip angle = 18 degrees.

Parameters 1000Plus: voxel size = (0.5x0.7x0.7) mm3; matrix size: 384x268x127;
TR/TE = 22ms/3.86ms; acquisition time: 3:50 min, flip angle = 18 degrees.

For both datasets, skull-stripping was applied. The segmentation labels were
produced semi-manually using a standardized pipeline along with 4 raters
correcting the labels as described in Livne et al. (2019).

###  3.3 Data Splitting and Patch Extraction

For the anonymization, 41 out of the 66 PEGASUS patients were used as a
training set, 11 were used for validation and 14 for testing. For the transfer
learning approach, one to 15 patients in increments of two of the 1000Plus
data were utilized for training. The 1000Plus validation set consisted of 10
and the test set of 40 patients.

Due to memory considerations, 2D patches of size 96x96 were extracted from
each patient instead of using the whole volume. The data contained 1% vessels
and 99% background. To compensate for this imbalance, 500 patches per patient
with a brain vessel in the center were extracted. Then, 500 random patches per
patient were added. The input patches were normalized to a range between -1
and 1 for the GAN used for anonymization. For the U-net segmentation model,
the input was normalized patch-wise to zero-mean and unit-variance.

###  3.4 Performance Evaluation

The generated images were first visually inspected and then quantitatively
compared to the real data using the Fréchet inception distance (FID) (Heusel
et al. (2018)). The FID measures the similarity of the real and generated
images by feeding both into an Inception-v3 network. The difference between
the activations in the pool3 layer inside the Inception-v3 network is then
calculated as follows:

$\text{FID}=\parallel\mu_{real}-\mu_{gen}\parallel^{2}+\text{Tr}(\sigma_{real}+\sigma_{gen}-2(\sigma_{real}\sigma_{gen})^{1/2}),$
(4)

where $x_{real}\sim\mathcal{N}(\mu_{real},\sigma_{real})$ and
$x_{gen}\sim\mathcal{N}(\mu_{gen},\sigma_{gen})$ are the distributions of the
features in the pool3 layer of the real and generated data respectively.

The FID was calculated for 41,000 generated patches of all three architectures
with the respective 41,000 real patches. The lower the FID, the higher the
similarity of the generated data to the original data.

As a second evaluation, the state-of-the-art "half U-net" used in Livne et al.
(2019) was trained with generated data as well as both real and generated
data. The parameters learning rate and dropout rate were tuned with respect to
the validation set. Additionally, classical augmentation was used as described
in Livne et al. (2019) if this led to an improved performance on the
validation set. Each segmentation network was trained for 15 epochs. Then, the
performance was evaluated on the binary segmentation maps of the test set by
the DSC and the 95th percentile of the Hausdorff distance (95HD):

$\text{DSC}=\frac{2\text{TP}}{2\text{TP}+\text{FP}+\text{FN}},$ (5)

where TP are the true positives, FP the false positives and FN the false
negatives. The Hausdorff distance is defined as:

$\text{HD}=\text{max}(\text{max}_{i\in[0,N-1]}d(i,P,G),\text{max}_{i\in[0,M-1]}d(i,G,P)),$
(6)

where $N$ and $M$ denote the number of voxels on the vessel tree of the ground
truth $G$ and the prediction $P$ respectively. $d(i,P,G)$ is defined as the
distance from vessel voxel $i$ in $G$ to the closest vessel voxel in $P$. The
95HD was then the 95th percentile Hausdorff distance for each voxel, averaged
over each voxel and each patient. It was measured in millimeters.

In the second part of the analysis, the performance of the U-net trained on
generated patches was evaluated on the 1000Plus dataset. For an increasing
number of training patients (1, 3,…, 15) the U-net was trained from scratch
and using the weights from the model trained on the generated image-label
pairs (transfer learning). The performance of using real data only and
transfer learning was then compared by assessing the DSC and 95HD on the
validation (10 patients) and test set (40 patients).

##  4 Results

Overall, generated synthetic patches showed high similarities to the training
set patches, in particular those that were synthesized by the WGAN-GP-SN. The
patches generated by the DCGAN showed a lower resolution with slight
checkerboard artifacts compared to the original patches. The generated
corresponding labels fit well to the patches for all models. A subset of the
synthesized image-label pairs for all GAN architectures as well as original
image-label pairs are shown in Fig. 2A to D. In the quantitative assessment,
the data generated by the WGAN-GP-SN architecture showed the highest
similarity to the real data with a FID of 38.05 compared to 105.96 for the
worst performing DCGAN. All FID values for real and synthesized data can be
found in Table 1.

Figure 2: Real and synthesized image patches with corresponding labels. (A) to (C) show image-label pairs generated by DCGAN (A), WGAN-GP (B) and WGAN-GP-SN (C) respectively. (D) show real patches and corresponding labels. The synthesized patches resemble real vessel patches and the labels fit well to the patches, especially those generated by WGAN-GP-SN (C). Table 1: Fréchet inception distance (FID) as a quantitative measurement of the generated image’s similarity compared to the real images for each of the three GAN architectures. WGAN-GP-SN showed the highest similarity to the real data in terms of FID. GAN architecture | FID
---|---
DCGAN | 105.96
WGAN-GP | 52.53
WGAN-GP-SN | 38.05

In the first validation approach, The U-net trained on data generated by the
WGAN-GP-SN showed the highest performance of all GAN models with a
segmentation performance of 0.82 DSC/28.97 95HD. The U-net trained on real
PEGASUS data showed a performance of 0.89 DSC/26.57 95HD. The same model
showed a similarly high performance in the external validation on the 1000Plus
data with 0.88 DSC/25.68 95HD. Quantitative results for all models trained on
generated and/or real data can be found in Table 2.

Table 2: Summary of the Dice similarity coefficient (DSC) and the 95th-percentile Hausdorff distance (95HD) of the U-net on validation and test set. Each metric is averaged over patients. The artificial patches were generated by Generative Adversarial Networks (GANs) trained on the PEGASUS dataset. For data augmentation, both real and generated patches have been used for training. For anonymization the U-net was trained on generated patches only. Models trained on anonymized, synthetic data only show performances close to the model trained on real data. | mean DSC | mean 95HD [mm]
---|---|---
val | test | val | test
U-net on real PEGASUS data (Livne et al.) | 0.88 | 0.89 | 29.50 | 26.57
Data augmentation (real data (PEGASUS) and generated data) |  |  |  |
DCGAN | 0.89 | 0.90 | 30.22 | 25.61
WGAN-GP | 0.89 | 0.89 | 28.03 | 30.01
WGAN-GP-SN | 0.89 | 0.89 | 29.91 | 26.51
Anonymization (trained on generated data only) |  |  |  |
PEGASUS anonymization models: validated and evaluated on PEGASUS data |  |  |  |
DCGAN | 0.82 | 0.79 | 34.58 | 31.25
WGAN-GP | 0.82 | 0.78 | 34.64 | 33.70
WGAN-GP-SN | 0.85 | 0.82 | 30.88 | 28.97
PEGASUS anonymization models evaluated on real 1000Plus |  |  |  |
DCGAN | - | 0.76 | - | 26.79
WGAN-GP | - | 0.85 | - | 26.98
WGAN-GP-SN | - | 0.88 | - | 25.68

In the second validation approach applying transfer learning, the U-net pre-
initialized with the weights from training on synthesized patches exhibited a
higher performance compared to the model trained from scratch on real data
only could be observed. Particularly when training on patches from one patient
only (n=1000), transfer learning using patch-label pairs generated by the
WGAN-GP-SN led to a higher performance in terms of DSC and 95HD (DSC/95HD
0.91/25.68 compared to 0.85/27.36). This observed performance difference
between pre-initialized models and models trained from scratch became smaller
when more patients were used for training. Results of the transfer learning
approach are visualized in Fig. 3. Fig. 4 shows the error maps for both
approaches on one example patient in large vessels (Fig. 4A and C) and small
vessels (Fig. 4B and D).

Figure 3: Performance evaluation for segmentation for an increasing number of
patients on the 1000Plus dataset when trained from scratch (green) and using
transfer learning (blue). The black dotted lines indicate the performance of
the Unet on the real PEGASUS dataset. The error bars show the standard
deviation over the patients. Especially for up to 5000 data samples the pre-
trained WGAN-GP-SN outperform the models without any pre-training. Figure 4:
Error maps for one example patient from the 1000Plus study using one patient
when training from scratch (A, B) and using transfer learning from WGAN-GP-SN
generated patches (C, D). True positives are shown in red, false positives in
green and false negatives in yellow. Transfer learning led to less errors,
especially on small vessels (B, D).

##  5 Discussion

We present a Wasserstein-GAN based model for the generation of synthetic TOF-
MRA imaging data and corresponding labels. The model generated synthetic data
of high quality, as evidenced visually and through the FID measure, and
retained much of the predictive properties of the original images. Here, a
predictive model for vessel segmentation trained on synthetic data alone
showed a good performance on one dataset and excellent performance on an
external validation set. The synthetic data were also successfully applied in
a transfer learning approach where training was pre-initialized with weights
from a model trained on synthetic data. It outperformed the models trained on
real data. Our results mark a significant step towards the use of GAN-based
models to generate synthetic and effectively anonymous data. Consequently,
this approach has the potential to significantly accelerate research in the
field of neuroimaging.

While the image-label pairs synthesized by the DCGAN showed some artifacts,
the more recent GAN architectures (WGAN-GP and WGAN-GP-SN) produced higher
resolution data that looked similar to the real data (Fig. 2). The superiority
of the WGAN-approaches was confirmed by lower FID values as well as the
improved performance of the U-net segmentation models trained on synthetic
data. This can be explained by the inherent differences between Wasserstein-
GANs and the DCGAN. In contrast to the DCGAN, the loss function of the WGAN-GP
architectures utilizes the Earth Mover’s distance and is bounded by a
Lipschitz constraint (Arjovsky et al. (2017); Gulrajani et al. (2017)). This
works as a robust regularization and enhances training stability while
diminishing mode collapse at the same time. This explains why the WGAN-GP
produced more realistic looking image-label pairs. Other studies confirm the
superiority of Wasserstein GAN architectures over the DCGAN (Arjovsky et al.
(2017); Gulrajani et al. (2017)). A recent addition to GAN architectures was
the introduction of spectral normalization. This method additionally restricts
the discriminator’s weights for each layer in order to stabilize training even
for high learning rates (Miyato et al. (2018)). As evidenced in our work,
spectral normalization is also beneficial for the application of Wasserstein
GANs, and the combination of both regularization techniques (WGAN-GP-SN)
yielded the best image quality both by visual inspection as well as in terms
of FID. These techniques have thus supported the preservation of the
predictive properties for vessel segmentation within the synthetic patches.
Therefore, it is likely that more sophisticated (future) GAN architectures
will further improve the generation of synthetic data. Here, potential current
candidate methods are progressive growing GAN (PG-GAN) or stacked GAN
architectures (Karras et al. (2018); Huang et al. (2017)).

Whereas the data generated by WGAN-GP-SN consistently yielded the highest DSC
in the transfer learning approach, this is not as apparent in other parts of
the results. First, the 95HD did not show a consistent trend. Since the
Hausdorff distance is vulnerable to outliers, we argue that it might not be as
reliable as the DSC. This is also corroborated by the high standard deviation
over the patients. Secondly, when training the U-net with real data and
additional synthesized data (data augmentation), the performance only slightly
increased for the WGAN-GP-SN. In addition, the DCGAN seems to perform slightly
better. Here, the real training data used for the U-net was the same as for
GAN training. Hence, the generated data contains information from the same
underlying distribution and did not add much value. Further, the best
performance by adding DCGAN generated data needs to be taken with caution and
does not necessarily mean that this was the overall best performing generative
model.

GAN architectures have the potential to generate anonymized data since the
generator does not have direct access to the training data. This also holds
true for this study: the generator synthesizes patch-label pairs from a noise
vector. However, a recent study by Hayes et al. (2019) shows that DCGANs might
be vulnerable to so-called membership inference attacks (Shokri et al.
(2017)). Such attacks aim to identify whether a given data sample was part of
the original training set or not. To prevent this, differentially private GANs
(DPGANs) have been introduced (Xie et al. (2018)). Here, carefully adjusted
noise is introduced in the gradients during the discriminator’s training.
While these GANs have the potential to ensure a certain level of privacy, they
show poorer performance to date (Mukherjee et al. (2020)) and have only been
trained on natural image datasets yet. Training a DPGAN on sparse medical
imaging datasets remains a major challenge. While DPGANs might provide even
further advantages in anonymization, we argue that our synthesized patch-label
pairs are effectively anonymized. For one, in the WGAN-GP-SN approach, we
apply Lipschitz regularization techniques such as gradient penalty and
spectral normalization. Wu et al. (2019) found that these techniques might
reduce information leakage and might even make the trained models resistant to
membership inference attacks. Furthermore, we use randomly sampled 2D patches
in this study. Thus, for a successful membership inference attack two events
must coincide: First, the real training data that is protected by state-of-
the-art hospital security systems has to be leaked. Second, the patches need
to be extracted in the exact same way as in the GAN-training process to allow
re-identification. The minuscule probability of these events to happen is
comparable to other theoretical scenarios of state-of-the-art anonymization.
For example, any tabular data anonymized using state-of-the-art techniques
could be re-identified when compared with the leaked original data. Thus, we
consider our generated patches anonymous and hence make them available for
researchers upon request.

Our results are also promising for AI in healthcare product development
(Higgins and Madai (2020)). In the medical AI research setting, a strong focus
on performance in homogeneous samples can be observed. This is in stark
contrast to the requirements for a medical imaging product. A product is
supposed to be used in a real world setting confronted with highly
heterogeneous data reflecting different settings and multiple hardware
options. Thus, product development should focus as much on training on
heterogeneous data as on keeping the necessary performance (Higgins and Madai
(2020)). This, however, is currently highly challenging as data is a scarce
resource due to limited availability. Our results show that a relatively small
amount of data is sufficient to generate robust results. Thus, a GAN-based
anonymization approach could allow the generation of high quality data from a
smaller number of patients from multiple locations that - in total - reflect
the full distribution of soft- and hardware settings in the clinical setting.
Here, the possibility to generate high-quality labels as evidenced by our
study is also a great advantage. Notably, a GAN model also learns the quality
of the labels provided during training. Thus, the final performance of any
model trained on synthetic data will also be dependent on the quality of the
real labels. Providing high-quality labels is no simple task and requires
usually hours of manual labor by highly qualified medical staff. Thus, a novel
GAN-based approach to product development could entail the high-quality
labeling of relatively small data-sets from multiple data providers that are
then anonymized and pooled for training. This would on one hand keep
development costs relatively low which is a prerequisite for startup success.
On the other hand, such an approach would ensure both high performance and low
bias as the chance for out-of-sample data in the clinical setting would be
significantly lowered.

Our study has several limitations. The DCGAN is 2D due to computational
restrictions. 3D approaches could help extracting information about the 3D
vessel tree structure and in this way improve the performance of the
segmentation task. The computational restrictions also did not allow to try
out more advanced GAN architectures such as PG-GAN. Another limitation is the
calculation of the FID. Due to computational restrictions it was only
calculated to confirm the quality of visually inspected images and not for
every epoch in an end-to-end solution. Secondly, the FID for assessing the
image quality might not be ideal. Although it is used as a quality measurement
in the medical field (Haarburger et al. (2019); Cao et al. (2020)), it was
originally designed for natural images and hence might not entirely capture
relevant features for medical imaging. Thus, further research on assessing
image quality specific to medical images should be undertaken.

##  6 Conclusion

This study marks a crucial step towards true anonymization of medical imaging
data while maintaining essential predictive features within the image patch.
We show that these features might be generalizable to another, independent
dataset. Our initial performance for vessel segmentation on the PEGASUS
dataset already is relatively high. We show that training more advanced GAN
architectures can further increase the quality of synthesized image-label
pairs. By using only one patient from a different cohort, we can achieve a
high comparable performance on an independent dataset. Our synthesized image-
label pairs allow other researchers to build models that only require few
labeled patient data and will significantly facilitate research in this
domain. It may be the case that our framework achieves similar results on
other medical segmentation tasks. This could lead to a lower demand of labeled
patient data and allow more data sharing of anonymized data. Nevertheless,
further studies should assess the generalizability of this analysis to other
(more complex) segmentation problems.

## Disclosures

Tabea Kossen reported receiving personal fees from ai4medicine outside the
submitted work. Dr Madai reported receiving personal fees from ai4medicine
outside the submitted work. Adam Hilbert reported receiving personal fees from
ai4medicine outside the submitted work.

While not related to this work, Dr Sobesky reports receipt of speakers
honoraria from Pfizer, Boehringer Ingelheim, and Daiichi Sankyo. Furthermore,
Dr Fiebach has received consulting and advisory board fees from BioClinica,
Cerevast, Artemida, Brainomix, Biogen, BMS, EISAI, and Guerbet. Dr Frey
reported receiving grants from the European Commission, reported receiving
personal fees from and holding an equity interest in ai4medicine outside the
submitted work.

## Acknowledgements

This work has received funding by the German Federal Ministry of Education and
Research through (1) the grant Centre for Stroke Research Berlin and (2) a Go-
Bio grant for the research group PREDICTioN2020 (lead: DF).

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# Reaction of deuteron with a heavy nucleus at low energies

Pankaj<EMAIL_ADDRESS>1 Department of Space, Planetary and
Astronomical Sciences & Engineering, Indian Institute of Technology, Kanpur,
208016, India
2 Physics Department, Indian Institute of Technology, Kanpur, 208016, India
Harishyam<EMAIL_ADDRESS>1 Department of Space, Planetary and
Astronomical Sciences & Engineering, Indian Institute of Technology, Kanpur,
208016, India
2 Physics Department, Indian Institute of Technology, Kanpur, 208016, India
K<EMAIL_ADDRESS>1 Department of Space, Planetary
and Astronomical Sciences & Engineering, Indian Institute of Technology,
Kanpur, 208016, India
2 Physics Department, Indian Institute of Technology, Kanpur, 208016, India

###### Abstract

We extend the recently proposed mechanism for inducing low energy nuclear
reactions (LENR) to compute the reaction rate of deuteron with a heavy
nucleus. The process gets dominant contribution at second order in the time
dependent perturbation theory and is assisted by a resonance. The reaction
proceeds by breakdown of deuteron into a proton and a neutron due to the
action of the first perturbation. In the second, nuclear perturbation, the
neutron gets captured by the heavy nucleus. Both perturbations are assumed to
be electromagnetic and lead to emission of two photons, one at each vertex.
The heavy nucleus is taken to be 58Ni although many other may be considered.
The reaction rate is found to be very small unless assisted by some special
conditions. In the present case we assume the presence of a nuclear resonant
state. In the presence of such a state we find that the reaction rate is
sufficiently large to be observable in laboratory even at low energies.

##  1 Introduction

Nuclear fusion are expected to be strongly suppressed at low energies due to
the Coulomb barrier [1]. However, there exists considerable experimental
evidence that such reactions may occur at observable rates even at very low
energies [2, 3, 4, 5, 6, 7]. The current status is reviewed in [8, 9, 10, 11].
Theoretically, there have been many attempts to explain these processes in
terms of electron screening [12, 13, 14], correlated states [15, 16],
electroweak interactions [17], formation of clusters of nuclear particles
[18], relativistic electrons in deep orbits [19] and phonon induced reactions
[20]. A critical review of many claims in this field is provided in [21].

It has been proposed that there exist additional processes which open up at
second order in the perturbation theory for which the rate may be significant
even at low energies. [22, 23, 24, 25]. These reactions are different from the
standard first order fusion process which is heavily suppressed [1]. The basic
idea is that the reaction proceeds due to two interactions with widely
different distance scales. The first perturbation causes the system to go into
a state which is a linear superposition of all eigenstates of the unperturbed
Hamiltonian. Due to the presence of eigenstates of relatively high energy, it
is possible that the Coulomb barrier may not be a very serious issue. Although
the amplitude for such high energy eigenstates is suppressed, the suppression
may not be as strong as that due to the Coulomb barrier. The resulting
amplitude gets contribution from relatively large atomic scale distances and
we refer to it as the molecular matrix element. The second perturbation leads
to the nuclear transition and gets contribution from small distances. We refer
to it as the nuclear matrix element.

We applied this formalism explicitly to the process involving fusion of proton
with deuteron to form helium nucleus with $A=3$ [23, 24]. The perturbation was
assumed to be electromagnetic leading to either emission or absorption of
photons. The dominant process was found to be the one in which two photons are
spontaneously emitted. The rate was found to be very small in free space due
to cancellation among different eigenstates contributing to the intermediate
state. Similar phenomenon is seen for other low energy nuclear processes [26,
27, 28]. However it was argued that in a medium, under special conditions, the
rate may be significant and observable [23, 24, 25, 26, 28]. This mechanism
has also been used to provide the theoretical explanation for the
experimentally observed low energy reaction of Palladium with Deuteron in
medium to produce Silver and neutron [29].

It has recently been shown that the rate may be substantial, even in free
space, in the presence of a resonance [27]. In this paper we considered the
reaction in which an incident proton converts into a neutron through weak
interactions and the neutron gets absorbed by a heavy nucleus emitting a
photon. We applied this explicitly to the case of nickel nucleus with A=58
which has a resonance at 7 keV with width of approximately 7 eV. The rate for
this process was found to be relatively large and observable. Here we extend
this mechanism to another process involving fusion of deuteron 2H and a heavy
nucleus AX of atomic number $Z$ and atomic mass $A$. The process we shall
consider is the following:

${}^{2}{\rm H}+{}^{A}{\rm X}\rightarrow{}^{A+1}{\rm X}+{}^{1}{\rm H}+Y$ (1)

where $Y$ may consist of a wide range of final states. In this paper we shall
confine ourselves to a particular process in which the final state involves
two photons. In this case,

$Y=\gamma(\omega_{1})+\gamma(\omega_{2})\,.$ (2)

and both perturbations are electromagnetic.

Within the framework of the second order perturbation theory, the reaction
proceeds in two steps. Due to the action of the first perturbation, the
deuteron forms an intermediate state with emission of a photon and a proton.
We refer to this as the first vertex, as shown in Fig. 1. The second
perturbation leads to capture of neutron by the nucleus AX with emission of
another photon. We refer to this as the second vertex. We point out that the
photon with frequency $\omega_{1}$ can also be emitted from the second vertex
and that with frequency $\omega_{2}$ from the first vertex. The intermediate
state, formed after the action of the first perturbation, involves
superposition of all eigenstates of the unperturbed Hamiltonian. The 2H is
initially in the ground state. The next state available is the one in which
the neutron and the proton are barely free. This state corresponds to zero
kinetic energies for both of these particles. In the second order perturbation
theory we need to sum over all intermediate eigenstates, up to infinite
energy.

Figure 1: Schematic illustration of the reaction described in Eq. 1.

##  2 Deuteron-X fusion at second order in perturbation theory

The Hamiltonian of the system is given by

$H=H_{0}+H_{I}$ (3)

where $H_{0}$ denote the unperturbed Hamiltonian and $H_{I}$ is a time
dependent perturbation. The unperturbed part contains the kinetic energy terms
and screened Coulomb and nuclear potentials. We assume a simple spherical well
model for the neutron-proton nuclear potential. Hence for small separations,
the potential experienced by the neutron-proton system is given by

$V_{pn}=-V_{0}$ (4)

for $r<r_{0}$ and $V_{pn}=0$ for larger distances. Here we set $V_{0}=34.6$
MeV and $r_{0}=2.066$ fm which leads to the deuteron bound state of correct
binding energy. For the nuclear potential of the $X$ nucleus we use the shell
model potential, same as that used in [27],

$V_{X}=-{V_{0}\over 1+\exp[(r-R_{X})/a_{X}]}$ (5)

where, $V_{0}=50$ MeV, $a_{X}=0.524$ fm and $R_{X}=1.25A^{1/3}$ [30]. This
leads to the nickel 2p state at energy of -9.2 MeV, in good agreement with the
observed value of -9 MeV for 59Ni.

We denote the coordinates of proton, neutron and the X nucleus by
$\vec{r}_{1}$, $\vec{r}_{2}$ and $\vec{r}_{X}$ respectively. We define the
relative coordinate $\vec{r}=\vec{r}_{2}-\vec{r}_{1}$ and denote the center of
mass coordinate of the proton-neutron system by $\vec{R}_{cm}$. The overall
center of mass coordinate of the three particle system is denoted by
$R^{\prime}_{cm}$. The initial state wave function can be expressed as

$\Psi_{i}=\Psi_{cm}(\vec{R}^{\prime}_{cm})\psi_{D}(\vec{r},\vec{R}_{cm})$ (6)

where $\Psi_{cm}(\vec{R}^{\prime}_{cm})$ is the overall center of mass wave
function and $\psi_{D}$ is the deuteron wave function. We assume that in the
initial state the deuteron and X atoms form a molecular bound state. We may
express the deuteron wave function as

$\psi_{D}(\vec{R}_{cm},\vec{r})=\psi_{i}(\vec{R}_{cm})\chi_{D}(r)=\psi_{i}(\vec{R}_{cm}){u_{D}(r)\over\sqrt{4\pi}\,r}$
(7)

where $\chi_{D}(r)$ is the deuteron nuclear ground state wave function
corresponding to $l=0$ and $\psi_{i}$ is the molecular bound state wave
function of the deuteron-X system. We shall assume that the molecular wave
function $\psi_{i}$ is spherically symmetric. The nucleus $X$ is assumed to be
very heavy and we ignore its recoil, which is expected to be a good
approximation. We point out that the overall center of mass motion does not
add any essential detail to the calculation and only impose overall momentum
conservation.

The interaction Hamiltonian [31, 32] is given by

$H_{I}(t)=\sum_{i}{Z_{i}e\over cm_{i}}\vec{A}(\vec{r}_{i},t)\cdot\vec{p}_{i}$
(8)

where $Z_{i}$, $m_{i}$, $\vec{r}_{i}$ and $\vec{p}_{i}$ are respectively the
charge, mass, position vector and momentum vector of the particle $i$. The
electromagnetic field operator $\vec{A}(\vec{r},t)$ is given by

$\vec{A}(\vec{r},t)={1\over\sqrt{V}}\sum_{\vec{k}}\sum_{\beta}c\sqrt{\hbar\over
2\omega}\left[a_{\vec{k},\beta}(t)\vec{\epsilon}_{\beta}e^{i\vec{k}\cdot\vec{r}}+a^{\dagger}_{\vec{k},\beta}(t)\vec{\epsilon}^{\,*}_{\beta}e^{-i\vec{k}\cdot\vec{r}}\right]$
(9)

The leading order contribution to the process in Eq. 1 with $Y$ given by Eq. 2
is obtained at second order in the time dependent perturbation theory. Let the
wave vectors of the two emitted photons be $\vec{k}_{1}$ and $\vec{k}_{2}$ and
their frequencies $\omega_{1}$ and $\omega_{2}$ respectively. The transition
amplitude at this order can be expressed as,

$\langle
f|T(t_{0},t)|i\rangle=\left(-{i\over\hbar}\right)^{2}\sum_{E_{n}}\int_{t_{0}}^{t}dt^{\prime}e^{i(E_{f}-E_{n})t^{\prime}/\hbar}\langle
f|H_{I}(t^{\prime})|n\rangle\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}e^{i(E_{p}+E_{n}-E_{i})t^{\prime\prime}/\hbar}\langle
pn|H_{I}(t^{\prime\prime})|i\rangle\ .$ (10)

where the sum is over the intermediate neutron states. Here $|pn\rangle$
denotes the intermediate state proton and neutron wave function with $E_{p}$,
$E_{n}$ being the energies of the proton and neutron states respectively
produced at the first vertex.

###  2.1 Molecular Matrix Element

Let us first consider the transition from initial to intermediate state which
we refer to as the molecular matrix element. We note that in the sum over $i$
in Eq. 8 only the proton contributes while computing the matrix element in the
integral over $t^{\prime\prime}$. Either of the two photons can emerge from
this interaction leading to two amplitudes. Let us first write to the
amplitude corresponding to emission of photon with frequency $\omega_{1}$. The
$t^{\prime\prime}$ integral then gives

$\displaystyle\int_{t_{0}}^{t^{\prime}}dt^{\prime\prime}e^{i(E_{p}+E_{n}-E_{i})t^{\prime\prime}/\hbar}\langle
pn|H_{I}(t^{\prime\prime})|i\rangle$ $\displaystyle=$
$\displaystyle-{ie\hbar\over m_{p}}\sqrt{\hbar\over 2\omega_{1}V}\langle
pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{p}_{1}\ |i\rangle$ (11)
$\displaystyle\times$
$\displaystyle{e^{i(E_{p}+E_{n}-E_{i}+\hbar\omega_{1})t^{\prime}/\hbar}\over
E_{p}+E_{n}-E_{i}+\hbar\omega_{1}}$

where we have dropped the term depending on the arbitrary initial time $t_{0}$
which is expected to give negligible contribution [31, 32]. The matrix element
can be written as [32]

$\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{p}_{1}\
|i\rangle={im_{1}\over\hbar}(E_{p}+E_{n}-E_{i})\,\langle
pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{r}_{1}\ |i\rangle$ (12)

We next express $\vec{r}_{1}$ as

$\vec{r}_{1}=\vec{R}_{cm}-{m_{2}\over M}\ \vec{r}$ (13)

where $M=m_{1}+m_{2}$, with $m_{1}=m_{p}$ and $m_{2}=m_{n}$. Let us assume
that the $R_{cm}$ dependence of the wave function is given by,

$\psi_{i}={N_{cm}\over\sqrt{4\pi}}\ e^{-(R_{cm}-R_{0})^{2}/\Delta^{2}}$ (14)

where $N_{cm}$ is the normalization factor. This is essentially the molecular
bound state wave function of the D-X system and is controlled by the molecular
potential. Here we directly assume the form of the wave function, applicable
at $R_{cm}$ larger than roughly 1 atomic units. For smaller $R_{cm}$ values
the form will be modified by the Coulomb repulsion and at nuclear distances by
the nuclear potential. However, this region contributes negligibly to the
molecular matrix element and hence the difference can be ignored. The assumed
form is convenient since it allows analytic calculations. A more realistic
form is expected to have a similar behavior and hence will not lead to
substantially different results. We consider values of $R_{0}$ and $\Delta$
roughly equal to 3.5 and 0.3 in atomic units, which are reasonable for a
molecular bound state. Note that with these values the wave function is
heavily suppressed for distances smaller than about 1 atomic unit.

The matrix element on the right hand side of Eq. 12 can now be computed by
solving the Schrodinger equation to obtain all eigenstates of the proton
neutron system in terms of the relative coordinate $\vec{r}$. Here the initial
state is the ground state, i.e. the deuteron. We ignore the $l=2$ component of
deuteron and set $l=0$, where $l$ is the orbital quantum number. Hence the
initial state has quantum numbers $S=1$, $l=0$ and $j=1$. The dominant
contribution from the intermediate state is obtained with orbital quantum
number $l=1$. Hence, with $S=1$ we can have $j=0,1,2$.

Let us consider the initial state with $m=1$ where $\langle j_{z}\rangle=\hbar
m$. The spin part of the wave function is not affected in the entire process.
Hence both the proton and neutron have spin up for the entire process and here
we focus entirely on the space part of the wave function. Let us choose the
photon momentum $\vec{k}_{1}$ such that,

$\vec{k}_{\gamma 1}=k_{\gamma 1}\left(\cos\theta_{\gamma
1}\hat{z}+\sin\theta_{\gamma 1}(\cos\phi_{\gamma 1}\hat{x}+\sin\phi_{\gamma
1}\hat{y})\right]$ (15)

The two polarization vectors of this photon can be expressed as

$\displaystyle\vec{\epsilon}_{1a}$ $\displaystyle=$
$\displaystyle-\sin\theta_{\gamma 1}\hat{z}+\cos\theta_{\gamma
1}(\cos\phi_{\gamma 1}\hat{x}+\sin\phi_{\gamma 1}\hat{y})$
$\displaystyle\vec{\epsilon}_{1b}$ $\displaystyle=$
$\displaystyle-\sin\phi_{\gamma 1}\hat{x}+\cos\phi_{\gamma 1}\hat{y}$ (16)

Here we consider only the polarization vector $\vec{\epsilon}_{1a}$. The other
polarization will add incoherently and can only enhance the computed result.
We can express the matrix element as

$\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{r}_{1}|i\rangle=\int\int
d^{3}rd^{3}R_{cm}\Psi^{*}_{pn}(\vec{r},\vec{R})\vec{\epsilon}^{\,*}_{1a}\cdot\vec{r}_{1}e^{-i\vec{k}_{\gamma
1}\cdot\vec{r}_{1}}\psi_{i}(R_{cm})\chi(r)$ (17)

Here $\Psi_{pn}$ is the intermediate state wave function of the proton-neutron
system. It is a wave function of neutron and proton in the potential due to
the $X$ particle.

The wave function $\Psi_{pn}$ is somewhat complicated and we make some
simplifying assumptions. Let us first consider this wave function in the
absence of the particle $X$. In this case the $\vec{R}_{cm}$ dependence is
expected to be a plane wave and we can express it as,

$\Psi_{pn}={1\over\sqrt{V}}e^{i\vec{k}_{+}\cdot\vec{R}_{cm}}\chi_{pn}(\vec{r})$
(18)

where $\vec{k}_{+}=\vec{k}_{1}+\vec{k}_{2}$ and $\vec{k}_{1}$ and
$\vec{k}_{2}$ are the proton and neutron wave vectors respectively. The wave
function $\chi_{pn}$ is the $l=1$ proton-neutron wave function in the nuclear
potential corresponding to $E>0$. It depends on
$\vec{k}_{-}=(\vec{k}_{2}-\vec{k}_{1})/2$ and we express it as,

$\chi_{pn}={u_{pn}(r)\over r}\zeta(\hat{r})$ (19)

where the angular dependence is contained in $\zeta(\hat{r})$ and can be read
off from $l=1$ part of the plane wave expansion

$e^{i\vec{k}_{-}\cdot\vec{r}}=4\pi\sum_{l=0}^{\infty}\sum_{m=-l}^{l}i^{l}j_{l}(k_{-}r)\
Y_{l}^{m}(\hat{r})Y_{l}^{m*}(\hat{k}_{-})$ (20)

We point out that we are not explicitly displaying the spin part of the wave
function since it does not play any role in our calculation. The wave function
$\Psi_{pn}$ gets modified due to the presence of the particle X and the
factorized form assumed in Eq. 18 is not applicable. We are interested in the
final state in which proton is free and neutron gets captured by X. Hence a
reasonable approximation is to assume a plane wave form for the proton and
determine the neutron wave function by using Schrodinger equation. For the
molecular matrix element we require this wave function only at large distances
from X, i.e. large $R_{cm}$. This is because the initial state wave function
is strongly suppressed at small $R_{cm}$. Furthermore, the dominant
contribution is obtained for very small values of the relative coordinate
$r=|\vec{r}|$. Hence it is reasonable to assume following form of the wave
function,

$\Psi_{pn}={1\over\sqrt{V}}\psi_{n}(R_{cm})e^{i\vec{k}_{1}\cdot\vec{R}_{cm}}\chi_{pn}(\vec{r})$
(21)

Here, in the intermediate state $\psi_{n}$ refers to the neutron wave function
in the presence of the particle X. It depends on the variable $\vec{r}_{2}$.
However since only very small values of $r$ contribute to molecular matrix
element, it is reasonable to assume $\vec{r}_{2}\approx\vec{R}_{cm}$.
Similarly we have assumed $\vec{r}_{1}\approx\vec{R}_{cm}$. The wave function
$\psi_{n}$ may be obtained by solving the Schrodinger equation for neutron in
the presence of particle X. As we shall have seen in [27], a significant
contribution is obtained only in the presence of a resonance in the nuclear
matrix element. This is applicable as long as we assume free space wave
functions at large distances. In order to include the contribution of
resonance, here we directly model the phase shift in the wave function [27],
as explained later in this section.

The matrix element can be written as

$\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{r}_{1}\ |i\rangle=\langle
pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{R}_{cm}\ |i\rangle-{m_{2}\over
M}\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{r}\ |i\rangle$ (22)

Let us first consider the matrix element corresponding to $\vec{R}_{cm}$. We
have

$\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{R}_{cm}|i\rangle=\int\int
d^{3}rd^{3}R_{cm}\Psi^{*}_{pn}(\vec{r},\vec{R})\vec{\epsilon}^{\,*}_{1a}\cdot\vec{R}_{cm}e^{-i\vec{k}_{\gamma
1}\cdot\vec{R}_{cm}}e^{i\vec{k}_{\gamma
1}\cdot\vec{r}m_{2}/M}\psi_{i}(R_{cm})\chi_{D}(r)$ (23)

We get dominant contributions only from very small values of $r$, of the order
of the deuteron radius. Hence the exponential factor involving $r$ can be set
to unity at leading order. The integral over $r$ then involves a direct
overlap between the deuteron bound state wave function and a free proton-
neutron wave function corresponding to deuteron potential. This clearly
vanishes and hence, to leading order, this matrix element is zero. We
therefore focus on the second matrix element on the right hand side of Eq. 22
which involves $\vec{r}$. This matrix element can be written as,

$\displaystyle\langle pn|\vec{\epsilon}^{\,*}_{\beta}\cdot\vec{r}|i\rangle$
$\displaystyle=$ $\displaystyle{1\over\sqrt{V}}\int\int
d^{3}rd^{3}R_{cm}\chi_{pn}^{*}(\vec{r})e^{-i\vec{k}_{+}\cdot\vec{R}_{cm}}\vec{\epsilon}^{\,*}_{1a}\cdot\vec{r}e^{-i\vec{k}_{\gamma
1}\cdot\vec{R}_{cm}}e^{i\vec{k}_{\gamma
1}\cdot\vec{r}m_{2}/M}\psi_{i}(R_{cm})\chi_{D}(r)$ (24) $\displaystyle=$
$\displaystyle I_{a}I_{b}$

where

$I_{a}=\int
d^{3}r\chi_{pn}^{*}(\vec{r})\vec{\epsilon}^{\,*}_{1a}\cdot\vec{r}e^{i\vec{k}_{\gamma
1}\cdot\vec{r}m_{2}/M}\chi_{D}(r)$ (25)

and

$I_{b}={1\over\sqrt{V}}\int
d^{3}R_{cm}\psi^{*}_{n}(R_{cm})e^{-i\vec{k}_{1}\cdot\vec{R}_{cm}}e^{-i\vec{k}_{\gamma
1}\cdot\vec{R}_{cm}}\psi_{i}(R_{cm})$ (26)

The wave function $\chi_{pn}$ depends on the wave vector $\vec{k}_{-}$ and is
normalized to a plane wave. We denote the angular coordinates of $\vec{k}_{-}$
as $(\theta_{-},\phi_{-})$. We get contributions from all the three components
$l_{z}=-1,0,1$. Performing the angular integrals in $I_{a}$, we find that

$I_{a}={1\over 2}I_{\Omega}I^{\prime}_{a}$ (27)

where

$I^{\prime}_{a}=\int dru^{*}_{pn}(r)ru_{D}(r)$ (28)

and

$I_{\Omega}=-\cos\theta_{-}\sin\theta_{\gamma
1}+\sin\theta_{-}\cos\theta_{\gamma 1}\cos(\phi_{-}-\phi_{\gamma 1})$ (29)

The radial integral in Eq. 27 gets dominant contribution from the leading
order term proportional to $j_{1}$ and is approximately proportional $k_{-}$
at low energies relevant to our calculation. The integral $I_{b}$ can be
expressed as,

$I_{b}={4\pi\over\sqrt{V}K}\int_{0}^{\infty}dR_{cm}u^{*}_{n}(R_{cm})\sin(KR_{cm})e^{-(R_{cm}-R_{0}^{2})^{2}/\Delta^{2}}$
(30)

where $K=|\vec{k}_{1}+\vec{k}_{\gamma 1}|$ and we have expressed
$\psi_{n}=u_{n}/R_{cm}$.

At large distances, the intermediate state neutron wave function can be
written as

$u_{n}(r_{2})=\sin(k_{2}r_{2}+\delta(k_{2}))$ (31)

where $\delta(k_{2})$ is the phase shift. Here we directly model the phase
shift for the neutron-nickel system by taking guidance from the observed data.
This system shows the first resonance at 7 keV which has a width of about 7
eV. We denote the value of $k_{n}$ corresponding to the resonance as $k_{R}$.
We are interested in a phase shift which accurately models this resonance. The
standard expansion of the phase shift for $l=0$ can be written as,

$k\cot\delta=-a_{1}+a_{2}k^{2}+...$ (32)

We use the expansion up to $k^{6}$ and model the phase shift as,

$\displaystyle\sin\delta$ $\displaystyle=$
$\displaystyle{-k_{n}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}(1+b_{1}k_{n}^{2}+b_{2}k_{n}^{4})^{2}}}$
$\displaystyle\cos{\delta}$ $\displaystyle=$
$\displaystyle{(a_{1}-a_{2}k_{n}^{2})(1+b_{1}k_{n}^{2}+b_{2}k_{n}^{4})\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}(1+b_{1}k_{n}^{2}+b_{2}k_{n}^{4})^{2}}}$
(33)

Here we have isolated the factor $a_{1}-a_{2}k_{n}^{2}$ which leads to a
resonance at $k_{n}=k_{R}=\sqrt{a_{1}/a_{2}}$. This is a generalization of the
model used in [27] and allows us to properly model the height of the
resonance. As expected, $\delta=0$ in the limit $k_{n}$ goes to zero. We can
now directly choose the parameters $a_{1}$, $a_{2}$, $b_{1}$ and $b_{2}$ so
that the phase shift is properly modelled at low energies, as explained later.
The main assumption in this model is that the $k_{n}$ dependence of the
nuclear matrix element is dominated by the resonant term which is given by the
denominator in Eq. 33.

Here, as well as [27], we are directly modelling the resonance in terms of the
phase shift and not using a potential model. We may also consider some simple
potential models to determine if they lead to a result which is not
negligible. Here we consider a square well potential, with $V=-V_{0}$ for
$r<r_{0}$ and $V=0$ for $r\geq r_{0}$. In general this potential produces
resonances with rather wide width which will lead to an amplitude close to
zero. However, it is possible to find a resonance close to $E=0$, with energy
slightly larger than zero and with a relatively narrow width. Such a resonance
exists, for example, for parameters, $V_{0}=30$ MeV and $r_{0}=2.19529\times
10^{-4}$ atomic units (approximately 10 fm). The nuclear bound state energy
for these parameters is found to be $-22.2$ MeV. We find many such parameter
choices which allow presence of a low energy resonance. For such cases, we
find a non-negligible contribution for small $K$ of order unity. In our
calculation we did not include the neutron bound states in the sum over
intermediate states. However, these states are highly suppressed at large
distances and will lead to extremely small molecular matrix elements. Hence,
it is reasonable to neglect the contribution from such states. A more detailed
investigation of potential models is postponed to future research.

###  2.2 Nuclear Matrix Element

We next consider the nuclear matrix element $\langle
f|H_{I}(t^{\prime})|n\rangle$. In this case, the intermediate state neutron
undergoes fusion with the heavy nucleus. As for the case of free neutron, a
wide range of processes associated with standard neutron capture can take
place. Here we shall focus on the process with photon emission. The process is
complicated since as the neutron interacts with the nucleus, it can interact
with all the nucleons, leading to a change in the multiparticle wave function.
Furthermore, the photon can be emitted by any one of the proton or neutron in
the system. Here we assume that we can treat the nucleus $X$ as a single
particle of charge $Z_{X}$. We take the emitted photon momentum to be
$\vec{k}_{\gamma 2}$. Let the angular coordinates of this photon momentum be
$(\theta_{\gamma 2},\phi_{\gamma 2})$. We point out that we also need to add
the amplitude for which photon one and two are interchanged, i.e. photons with
frequencies $\omega_{1}$ and $\omega_{2}$ are emitted from second and first
vertex in Fig. 1 respectively.

In order to proceed further we need to specify the final state eigenfunction.
We take this state to be $l=1$, $j=3/2$ shell model state with one unpaired
neutron in the outer most shell. As stated above we shall consider the state
for which neutron has spin up and set $j_{z}=3/2$. We point out that 58Ni
(spin 0) satisfies our requirements. It has 30 neutrons which means that it
has two neutrons in the outermost $2p_{3/2}$ level. Adding one more neutron in
the $l=1$ level will lead to a total spin $3/2$ with $j_{z}=3/2$. Using the
square well potential (Eq. 5), we find that the energy eigenvalue of this
state is 9.1 MeV in good agreement with the observed value. We point out that
we can get transitions into other states also but this choice is convenient,
as explained below. Furthermore the heavy nucleus need not be Ni. It would be
interesting to consider other possibilities.

We perform the calculation by specializing to a particular polarization vector
of the emitted photon with frequency $\omega_{2}$. This is taken to be

$\vec{\epsilon}_{2b}=-\sin\phi_{\gamma 2}\hat{x}+\cos\phi_{\gamma 2}\hat{y}$
(34)

The photon is emitted only by the particle $X$. We denote its position vector
by $\vec{r}_{X}$. Let $\vec{R}^{\prime}_{cm}$ and $\vec{r}^{\prime}$ be
respectively the center of mass and relative coordinate of the neutron-$X$
system. The matrix element can be expressed as,

$\langle f|H_{I}(t^{\prime})|n\rangle={iZ_{X}e\over\hbar}\sqrt{\hbar\over
2V\omega_{2}}e^{iE_{\gamma 2}t^{\prime}/\bar{h}}(E_{f}-E_{n})\langle
f|\vec{\epsilon}^{*}_{2b}\cdot\vec{r}_{X}|n\rangle$ (35)

where $|n\rangle$ represents the neutron intermediate state of energy $E_{n}$.
We express $\vec{r}_{X}$ in terms of the center of mass coordinate
$\vec{R}^{\prime}_{cm}$ and relative coordinate $\vec{r}^{\prime}$ of the
neutron-$X$ system,

$\vec{r}_{X}=\vec{R}^{\prime}_{cm}-{m_{2}\over M^{\prime}}\vec{r}^{\prime}$
(36)

where $M^{\prime}=m_{X}+m_{2}$ is the mass of the final state nucleus.

We first consider

$\langle f|\vec{\epsilon}^{*}_{2b}\cdot\vec{R}^{\prime}_{cm}|n\rangle=\int
d^{3}R^{\prime}_{cm}d^{3}r^{\prime}\psi^{*}_{f}(\vec{r}^{\prime})\psi^{*}_{fc}(\vec{R}^{\prime})\vec{\epsilon}^{*}_{2b}\cdot\vec{R}^{\prime}_{cm}e^{-ik_{\gamma
2}[\vec{R}^{\prime}_{cm}-{m_{n}\over
M^{\prime}}\vec{r}^{\prime}]}\psi_{nX}(\vec{r}^{\prime})\psi_{nXc}(\vec{R}^{\prime})$
(37)

Here $\psi_{nX}$ and $\psi_{nXc}$ are the intermediate state wave functions of
the neutron-$X$. Similarly $\psi_{f}$ and $\psi_{fc}$ are the final state wave
functions. Let us consider the integral over $r^{\prime}$. Since the final
state wave function is significant only for very small values of $r^{\prime}$,
we can drop the $\vec{r}^{\prime}$ dependent term in the exponential to
leading order. The integral then vanishes since the two $\vec{r}^{\prime}$
dependent wave functions are orthogonal to one another. We next consider the
$\vec{r}^{\prime}$ dependent term. We obtain

$\langle f|\vec{\epsilon}^{*}_{2b}\cdot\vec{r}^{\prime}|n\rangle=\int
d^{3}R^{\prime}_{cm}d^{3}r^{\prime}\psi^{*}_{f}(\vec{r}^{\prime})\psi^{*}_{fc}(\vec{R}^{\prime})\vec{\epsilon}^{*}_{2b}\cdot\vec{r}^{\prime}e^{-ik_{\gamma
2}[\vec{R}^{\prime}_{cm}-{m_{n}\over
M^{\prime}}\vec{r}^{\prime}]}\psi_{nX}(\vec{r}^{\prime})\psi_{nXc}(\vec{R}^{\prime})$
(38)

Integration over the center of mass variable just imposes the overall momentum
conservation and we focus on the relative variable. Again we can drop the
$\vec{r}^{\prime}$ dependence in the exponential function. The state
$\psi_{f}(\vec{r}^{\prime})$ is the $l=1$ state of the shell model nuclear
wave function, as discussed above. The state $\psi_{nX}$ is taken to the $l=0$
state which gives dominant contribution for energies under consideration. Note
that the neutron-proton system has $l=1$. The neutron intermediate state can
take all $l$ values which can lead to $l=1$ for the neutron-proton system.
However the values $l\neq 0$ for the neutron state lead to very small
amplitudes and can be neglected. The matrix element can be written as,

$\langle
f|\vec{\epsilon}^{*}_{2b}\cdot\vec{r}^{\prime}|n\rangle=I^{\prime}_{\Omega}I_{f}$
(39)

where

$I_{f}=\int dr^{\prime}u^{*}_{f}(r^{\prime})r^{\prime}u_{n}(r^{\prime})$ (40)

and

$I^{\prime}_{\Omega}=i\,\sqrt{2\pi\over 3}e^{-i\phi_{\gamma 2}}$ (41)

Here we have expressed
$\psi_{f}=Y^{1}_{1}(\hat{r}^{\prime})u_{f}(r^{\prime})/r^{\prime}$ and
$\psi_{nX}=Y^{0}_{0}u_{n}(r^{\prime})/r^{\prime}$.

We estimate the nuclear matrix element $I_{f}$ at $k_{n}=0$ using the shell
model potential for the wave functions given in Eq. 5. However to determine
its dependence on $k_{n}$ we directly use the model for the phase shift given
in Eq. 33. We note that $I_{f}$ is expected to be almost independent of
$k_{n}$ at low energies. At medium energies it has a mild dependence and shows
dramatic change close to resonance. This is captured by the following overall
factor in the wave function $u_{n}$ at very small $r^{\prime}$, corresponding
to nuclear distances,

${a_{1}\over\sqrt{k_{n}^{2}+(a_{1}-a_{2}k_{n}^{2})^{2}(1+b_{1}k_{n}^{2}+b_{2}k_{n}^{4})^{2}}}$
(42)

which goes to unity in the limit $k_{n}\rightarrow 0$. This will lead to the
proper behaviour of the standard neutron absorption cross section cross
section both at low energy and in the vicinity of the resonance. The
parameters are taken to be $a_{1}=1.6\times 10^{4}$, $a_{2}=1.2\times
10^{-2}$, $b_{1}=0$ and $b_{2}=8.0\times 10^{-11}$, which correctly model the
observed resonance in neutron-nickel system and energy of 7 keV. Here we have
set $b_{1}=0$ for simplicity. This parameter can be adjusted by making a more
detailed fit to the observed neutron absorption cross section. Here we are
only interested in the contribution from resonance and it is reasonable to set
it to zero. We point out that the estimated value of $I_{f}$, using the shell
model potential, leads to a good agreement of the standard neutron absorption
cross section on 58Ni with photon emission, which is found to be about 200
barns at incident energy of $10^{-5}$ eV.

##  3 Reaction Rate

We can express the transition matrix element as,

$\displaystyle\langle f|T(t,t_{0})|i\rangle$ $\displaystyle=$ $\displaystyle
i{e^{2}Z_{X}m_{2}\over
2\hbar^{2}M^{\prime}V}\sqrt{1\over\omega_{1}\omega_{2}}I^{\prime}_{\Omega}\sum_{n}\int_{t_{0}}^{t}dt^{\prime}e^{i(E_{f}-E_{i}+E_{\gamma
1}+E_{\gamma 2}+E_{p})t^{\prime}/\hbar}$ (43) $\displaystyle\times$
$\displaystyle\sum_{n}I_{f}I_{a}I_{b}{(E_{f}-E_{n})(E_{p}+E_{n}-E_{i})\over
E_{p}+E_{n}-E_{i}+E_{\gamma 1}}$

where $E_{\gamma 1}=\hbar\omega_{1}$ and $E_{\gamma 2}=\hbar\omega_{2}$. As
mentioned earlier, we also need to add another amplitude in which the two
photons are reversed. However, in most of the kinematic regime only one of the
two amplitudes dominate and we can neglect the other. This is because we
obtain dominant contribution from the neutron momentum $k_{n}$, and hence $K$,
close to the resonant value. Since $\vec{K}=\vec{k}_{1}+\vec{k}_{\gamma 1}$,
for a fixed final state proton momentum, it is clear that we obtain dominant
contribution from a narrow range of photon momentum $\vec{k}_{\gamma 1}$
emerging from the first vertex. In other words, for any event, given the
observed proton momentum, the photon momentum $\vec{k}_{\gamma 1}$ is fixed
within a narrow range, with the corresponding value of $\vec{k}_{\gamma 2}$
being very different. Hence the second amplitude in which the photon with
momentum $\vec{k}_{\gamma 1}$ emerges from the second vertex will give
negligible contribution and, for brevity, we neglect it. The sum over the
intermediate neutron states can be computed by converting it into an integral
$\int Vd^{3}k_{2}/(2\pi)^{3}$.

The transition rate can now be expressed as

${dP\over dt}={1\over\Delta
T}\int{Vd^{3}k_{p}Vd^{3}k_{1}Vd^{3}k_{2}\over(2\pi)^{9}}|\langle
f|T(t_{0},t)|i\rangle|^{2}$ (44)

where $\Delta T$ is the total time. The time integral in the transition matrix
element is proportional to $\Delta T\delta(E_{f}-E_{i}+E_{\gamma 1}+E_{\gamma
2}+E_{p})$. The reaction rate can now be expressed as

$\displaystyle{dP\over dt}$ $\displaystyle=$ $\displaystyle{\alpha^{2}\over
64\pi^{5}\hbar^{3}c^{2}}\left({Z_{X}m_{2}\over
M^{\prime}}\right)^{2}V\int{k_{p}^{2}dk_{p}}d\Omega_{p}dk_{\gamma 1}k_{\gamma
1}d\Omega_{\gamma 1}E_{\gamma 2}|I^{\prime}_{\Omega}|^{2}$ (45)
$\displaystyle\times$ $\displaystyle\Bigg{|}{V\over(2\pi)^{3}}\int
dk_{n}k_{n}^{2}d\Omega_{n}I_{f}I^{\prime}_{a}I_{\Omega}I_{b}{(E_{f}-E_{n})(E_{n}+E_{p}-E_{i})\over
E_{p}+E_{n}-E_{i}+E_{1}}\Bigg{|}^{2}$

where $d\Omega_{\gamma 1}$ and $d\Omega_{\gamma 2}$ are the measures for the
angular integrations over the two photon momenta and $d\Omega_{2}$ corresponds
to the intermediate state neutron. The integration over the proton angular
variables is trivial and just gives $4\pi$.

The reaction rate for the deuteron-nickel system is found to be approximately
100 per sec. This is a fairly large rate and easily observable experimentally.
The photon energy $E_{2}$ shows a broad peak centered at approximately 2 MeV.
As mentioned earlier, the energy of the first photon $E_{1}$ for any event is
fixed by the requirement that $K=|\vec{k}_{1}+\vec{k}_{\gamma 1}|$ is close to
the resonant value, which in the present case is found to be approximately
1140 in atomic units.

In our calculation we have focussed on the contribution from a single isolated
resonance. It is important to determine whether the contribution from other
resonances may change our result significantly. We have tested this
possibility by assuming a resonance at a higher value of $k_{n}$ with the
value of $K$ fixed to approximately 1140, applicable for the resonance under
consideration. We find that the amplitude falls exponentially if the value of
$k_{n}$ is widely different from $K$ and hence the contribution from other
resonances is expected to be negligible.

##  4 Conclusions

In this paper we have computed the rate for the nuclear reaction given in Eq.
1 at second order in time dependent perturbation theory. The initial state
consists of a deuteron and a heavy nucleus AX at very low energies, of order
eV or less. The reaction proceeds by formation of an intermediate state which
is composed of a free proton and neutron. Despite the low energy of the
initial state, such a state can exist for a small time interval. Due to the
small time interval, this leads to a suppression factor in the rate of this
process. However this suppression is not as prohibitive as the very strong
suppression arising due to Coulomb barrier. Hence the rate of this process may
be substantially higher in comparison to expectations based on the standard
Coulomb barrier. The neutron gets captured by the heavy nucleus which is
assumed to happen with emission of a photon, although other reactions may also
be possible. Overall the final state consist of a proton, two photons and the
nucleus A+1X.

We have computed the rate assuming that the initial state deuteron is bound to
the heavy nucleus with the formation of a molecule, with the wave function
given by Eq. 14. Furthermore we have considered, as an example, the heavy
nucleus to be 58Ni which absorbs a neutron into the $2p_{3/2}$ level. We find
that the rate for such a process is, in general, very small. However, in the
presence of a resonance the rate may be substantial and observable [27]. We
find that there exists an isolated resonance in the neutron-nickel system at
energy of 7 keV, which has a width of approximately 7 eV. Assuming that this
resonance dominates the amplitude, we find that the rate for the process is
approximately 100 per second, which is easily observable. This rate turns out
to be much larger than that found for the weak interaction process considered
in [27]. The process leads to emission of two photons and a proton, with one
of the photons showing a broad peak at approximately 2 MeV. Furthermore, the
total momentum of the second photon and proton should show a peak at 1140 in
atomic units, i.e. close to the resonant value $k_{R}$.

We have pointed out that besides the process considered, several other
transitions are possible in which only one or no photon may be emitted.
Alternatively, the transition may not be to the ground state of 59Ni. In this
case, the two primary photons would have lower total energy and will be
followed by a cascade of lower energy photons. These lower energy photons have
a much higher probability of being captured by the medium and hence will lead
to excess heat in the medium. Excess heat will also be generated by the
nuclear recoil, which will lead to transfer of energy to the medium. Hence we
also expect to see lower energy primary photons along with a cascade of lower
energy photons and excess heat in the medium. A detailed study of this process
is postponed to future research.

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