# A robust multi-domain network for short-scanning amyloid PET reconstruction

Hyoung Suk Park 111These authors contributed equally to this work. Young Jin
Jeong 222These authors contributed equally to this work. Kiwan Jeon  for the
Alzheimer’s Disease Neuroimagin Initiative 333Data used in preparation of this
article were obtained from the Alzheimer’s Disease Neuroimaging Initiative
(ADNI) database (adni.loni.usc.edu). As such, the investigators within the
ADNI contributed to the design and implementation of ADNI and/or provided data
but did not participate in analysis or writing of this report. A complete
listing of ADNI investigators can be found at: to the design and
implementation of ADNI and/or provided data but did not participate in
analysis or writing of this report. A complete listing of ADNI investigators
can be found at:http://adni.loni.usc.edu/wp-
content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf National Institute
for Mathematical Sciences, Daejeon, 34047, Republic of Korea  Department of
Nuclear Medicine, Dong-A University Hospital and College of Medicine, Dong-A
University, Busan, 49201, Republic of Korea

###### Abstract

This paper presents a robust multi-domain network designed to restore low-
quality amyloid PET images acquired in a short period of time. The proposed
method is trained on pairs of PET images from short (2 minutes) and standard
(20 minutes) scanning times, sourced from multiple domains. Learning relevant
image features between these domains with a single network is challenging. Our
key contribution is the introduction of a mapping label, which enables
effective learning of specific representations between different domains. The
network, trained with various mapping labels, can efficiently correct amyloid
PET datasets in multiple training domains and unseen domains, such as those
obtained with new radiotracers, acquisition protocols, or PET scanners.
Internal, temporal, and external validations demonstrate the effectiveness of
the proposed method. Notably, for external validation datasets from unseen
domains, the proposed method achieved comparable or superior results relative
to methods trained with these datasets, in terms of quantitative metrics such
as normalized root mean-square error and structure similarity index measure.
Two nuclear medicine physicians evaluated the amyloid status as positive or
negative for the external validation datasets, with accuracies of 0.970 and
0.930 for readers 1 and 2, respectively.

###### keywords:

Amyloid positron emission tomography , multi-domain learning , short scanning
time , noise reduction , mapping label

††journal: Arxiv

##  1 Introduction

Amyloid positron emission tomography (PET) is a widely utilized nuclear
medicine imaging modality for detecting amyloid plaques in patients with
memory disorders, such as Alzheimer’s disease. Various radiotracers, including
F-18 florbetaben (FBB), F-18 florapronol (FPN), F-18 flutemetamol (FMM), and
F-18 florbetapir (FBP), are commercially available for amyloid PET imaging,
typically requiring a scanning time of 20-30 minutes. However, elderly
patients with memory disorders often struggle to remain still for such an
extended period, resulting in head movements that can cause motion artifacts
in PET images and degrade their diagnostic value. In some instances, patients
may require re-scanning or additional radiation exposure due to poor image
quality caused by movement. Consequently, there is an increasing demand for
reducing scan time while maintaining diagnostic reliability. Nonetheless, PET
images acquired from shorter scanning times can suffer from low signal-to-
noise ratios and diminished diagnostic reliability, potentially limiting their
clinical utility.

In recent years, deep learning techniques have demonstrated significant
potential for improving PET image quality by reducing noise [2, 16, 23]. Deep
learning models learn the relationship between low-quality and high-quality
PET images obtained from source and target domains. Several studies have
focused on minimizing noise caused by reduced radiotracer injection using
various deep learning techniques, such as convolutional neural network (CNN)
[4, 5, 17] and generative adversarial networks (GAN) [13, 18, 22]. Some
research has attempted to decrease noise resulting from shortened PET
acquisition times and enhance image quality. For example, Jeong et al . [7]
and Sanaat et al . [14] employed deep learning for short-scanning PET image
restoration. They used multiple pairs of PET images acquired at short-scanning
times of 2 and 3 minutes and standard scanning times of 20 and 27 minutes,
respectively, for training purposes.

Nonetheless, most existing methods have been trained and validated using PET
images from a single source domain. PET image domains can exhibit different
characteristics depending on various factors, such as the type of radiotracer,
PET scanner, reconstruction method, and scanning protocol. Due to these domain
discrepancies, the correction performance of deep learning methods may
significantly degrade when applied to PET images from an unseen source domain.

In this paper, we propose a deep learning-based method for short-scanning
amyloid PET image restoration that offers enhanced generalization across
various image domains. To improve generalization, we simultaneously learn a
correction map using amyloid PET images sampled from multiple source and
target domains. However, capturing all relevant image features between source
and target domains with a single correction network is challenging [9, 12]. To
address this issue, we introduce a mapping label, which enables the single
network to effectively learn specific representations between multiple source
and target domains. Importantly, the trained network with various mapping
labels can effectively correct not only amyloid PET datasets in multiple
training domains but also unseen PET datasets obtained with new radiotracers,
acquisition protocols, and PET scanners.

We performed quantitative and qualitative analyses to demonstrate that the
proposed method for amyloid PET imaging can reduce scanning time while
maintaining high diagnostic reliability.

Figure 1: Schematic diagram of proposed method for restoring short-scanning
amyloid PET images. The mapping label $\boldsymbol{c}$ allows single network
$G$ to effectively learn specific representations between $N$ different source
and target domains $\mathcal{Z}_{i}$ and $\mathcal{X}_{i},i=1,2,\ldots,N$. The
network, trained with various mapping labels, has the capability to
effectively correct amyloid PET datasets from an unseen domain
$\mathcal{Z}_{T}$.

##  2 Method

Let $\mathbf{z}$ and $\mathbf{x}$ be amyloid PET images acquired from short
and normal scan times, respectively. We assume that
$\mathbf{z}=\mathbf{x}+\mathbf{n}$, where $\mathbf{n}$ is the noise artifact
in $\mathbf{z}$ caused by the short-scanning time. The goal of this study is
to learn a correction map $G:\mathbf{z}\to\mathbf{x}$ using $N$ different
training datasets
${\mathcal{S}}_{i}=\\{(\mathbf{z},\mathbf{x})~{}:~{}\mathbf{z}\in\mathcal{Z}_{i},~{}\mathbf{x}\in\mathcal{X}_{i}\\},~{}i=1,2\ldots,N$.
Here, $\mathcal{Z}_{i}$ and $\mathcal{X}_{i}$ are source and target domains
that respectively exhibit different image characteristics depending on the
acquisition environments, including the type of radiotracers used, the type of
PET scanner, the type of reconstruction methods employed, and the scanning
protocol.

Learning the map $G$ between two domains $\mathcal{Z}_{i}$ and
$\mathcal{X}_{i}$ from the training datasets
$\cup_{i=1}^{N}{\mathcal{S}}_{i}$, which suffer from diverse perturbations, is
challenging. To address this difficulty, we introduce mapping labels
$\boldsymbol{c}_{i},~{}i=1,2,\ldots,N$, which enable the
$G:(\mathbf{z};\boldsymbol{c}_{i})\to\mathbf{x}$ maps
$\mathbf{z}\in\mathcal{Z}_{i}$ to $\mathbf{x}\in\mathcal{X}_{i}$. We assigned
the mapping label $\boldsymbol{c}_{i}$ as a one-hot vector of length $N$,
where the $i$-th element was set to 1 and the other elements are set to 0.
Interestingly, the trained $G$ with these mapping labels possesses the
capability to effectively map $\mathbf{z}$ to $\mathbf{x}$ sampled from the
unseen source and target domains $\mathcal{Z}_{T}$ and $\mathcal{X}_{T}$,
respectively. Specifically, we assumed that $\mathcal{Z}_{T}$ and
$\mathcal{X}_{T}$ are sufficiently close to the source and target domains
$\mathcal{Z}_{i}$ and $\mathcal{X}_{i},~{}i=1,2,\ldots,N$, respectively. Then,
for $\mathbf{z}\in\mathcal{Z}_{T}$, and $\mathbf{x}\in\mathcal{X}_{T}$, there
exists $\boldsymbol{c}_{T}\in{\mathbb{R}}^{N}$ such that
$G(\mathbf{z};\boldsymbol{c}_{T})\approx\mathbf{x}$. Based on this
observation, for the trained $G$ using $\cup_{i=1}^{N}{\mathcal{S}}_{i}$, we
estimated $\boldsymbol{c}_{T}$ by solving the following minimization problem:

$\displaystyle\arg\min_{\boldsymbol{c}\in{\mathbb{R}}^{N}}E_{(\mathbf{z},\mathbf{x})\sim{\mathcal{S}}_{T}}~{}\|G(\mathbf{z};\boldsymbol{c})-\mathbf{x}\|_{2}^{2},$
(1)

where $E[\cdot]$ denotes the expectation and
${\mathcal{S}}_{T}=\\{(\mathbf{z},\mathbf{x})~{}:~{}\mathbf{z}\in\mathcal{Z}_{T},~{}\mathbf{x}\in\mathcal{X}_{T}\\}$.
The schematic diagram of the proposed approach is illustrated in Fig. 1.

###  2.1 How to train network $G$ with the domain label $\boldsymbol{c}$

In this study, we incorporated the adaptive instance normalization (AdaIN)
layer [6] into the network $G$ to facilitate the conversion of feature maps
from the $\mathcal{Z}_{i}$ to $\mathcal{X}_{i}$. We utilized the label
$\mathbf{c}$ to generate the mean $\mu$ and variance $\sigma$ for the AdaIN
layer, which are achieved through a mapping network $f$. Specifically, let
${\mathbf{h}}^{(k)}_{j}$ be a vectorized feature map of $G$ at the $j$-th
channel of the $k$-th stage. Then, the AdaIN operation $\Gamma$ is applied as
follows:

$\displaystyle\Gamma({\mathbf{h}}^{(k)}_{j},f^{(k)}(\boldsymbol{c}))=\frac{\sigma_{j}^{(k)}}{\sigma({\mathbf{h}}^{(k)}_{j})}({\mathbf{h}}^{(k)}_{j}-\mu({\mathbf{h}}^{(k)}_{j})\mathbf{1})+\mu_{j}^{(k)}\mathbf{1},$
(2)

where $\sigma(\mathbf{x})$ and $\mu(\mathbf{x})$ respectively denote variance
and mean of $\mathbf{x}$, and $\mathbf{1}$ denotes a vector of ones whose
every element is equal to one, and its size is same as that of
${\mathbf{h}}^{(k)}_{j}$. Here, $f^{(k)}$ is a mapping network at $k$-th stage
that maps

$\displaystyle
f^{(k)}:\boldsymbol{c}\to\left[\begin{array}[]{cc}\mu_{1}^{(k)}&\sigma_{1}^{(k)}\\\
\vdots&\vdots\\\ \mu_{J_{k}}^{(k)}&\sigma_{J_{k}}^{(k)}\\\
\end{array}\right].$ (6)

Here, $J_{k}$ is the number of channels of feature map at $k$-th stage.

In clinical practice, it is common to encounter data imbalance during
training, where the number of collected samples in each domain is not equal.
This can have a negative impact on the training performance of correction
functions $G$. To address this issue, we used a weighted least squared loss
that considers training samples $\cup_{i=1}^{N}{\mathcal{S}}_{i}$ from $N$
different domains, as shown in eqn. (11). However, using least-squares loss
alone can lead to over-smoothed images with loss of details [7, 20]. To
overcome this limitation, we employed CycleGAN loss, which includes
adversarial and cycle-consistency losses, in addition to the weighted least
squared loss. The incorporation of CycleGAN loss in paired learning could
significantly enhance the overall performance and quality of the generated
images [19]. In the proposed CycleGAN framework, the correction function $G$
is trained simultaneously with an additional backward function
$F:(\mathbf{x};\boldsymbol{c})\to\mathbf{z}$, which maps the generated image
back to the original domain $\mathcal{Z}$. Moreover, two discriminators
$D_{\mathcal{X}}$ and $D_{\mathcal{Z}}$ are introduced to distinguish between
the generated images and real images in their respective domains $\mathcal{X}$
and $\mathcal{Z}$.

In summary, the proposed correction function $G$ is trained along with the
backward function $F$ and the discriminators $D_{\mathcal{X}}$ and
$D_{\mathcal{Z}}$ by solving the following min-max problem:

$\displaystyle\operatorname*{\arg\min}_{G,F}\operatorname*{\arg\max}_{D_{\mathcal{X}},D_{\mathcal{Z}}}~{}$
$\displaystyle{\mathcal{L}}_{adv}(G,F,D_{\mathcal{X}},D_{\mathcal{Z}})+\lambda_{1}{\mathcal{L}}_{cyc}(G,F)$
(7) $\displaystyle+\lambda_{2}{\mathcal{L}}_{wls}(G,F),$ (8)

where

  * 1.

${\mathcal{L}}_{adv}$ denotes the adversarial loss. To improve the training
stability, we use the least-squares loss [11]:

${\mathcal{L}}_{adv}(G,F,D_{\mathcal{X}},D_{\mathcal{Z}})=\\\
E_{(\mathbf{x},\mathbf{z})\sim\cup_{i=1}^{N}{\mathcal{S}}_{i}}\left[(D_{\mathcal{X}}(\mathbf{x}))^{2}+(1-D_{\mathcal{X}}(G(\mathbf{z};\boldsymbol{c})))^{2}\right.\\\
\left.(D_{\mathcal{Z}}(\mathbf{z}))^{2}+(1-D_{\mathcal{Z}}(F(\mathbf{x};\boldsymbol{c})))^{2}\right].$
(9)

  * 2.

${\mathcal{L}}_{cyc}$ denotes the cycle-consistency loss given by

${\mathcal{L}}_{cyc}(G,F)=\\\
E_{(\mathbf{x},\mathbf{z})\sim\cup_{i=1}^{N}{\mathcal{S}}_{i}}\left[\|F(G(\mathbf{z};\boldsymbol{c});\boldsymbol{c})-\mathbf{z}\|_{2}^{2}+\|G(F(\mathbf{x};\boldsymbol{c});\boldsymbol{c})-\mathbf{x}\|_{2}^{2}\right].$
(10)

  * 3.

${\mathcal{L}}_{wls}$ denotes the weighted least squared loss given by

${\mathcal{L}}_{wls}(G,F)=\\\
E_{(\mathbf{x},\mathbf{z})\sim\cup_{i=1}^{N}{\mathcal{S}}_{i}}\left[\|{\boldsymbol{w}}(G(\mathbf{z};\boldsymbol{c})-\mathbf{x})\|_{2}^{2}+\|{\boldsymbol{w}}(F(\mathbf{x};\boldsymbol{c})-\mathbf{z})\|_{2}^{2}\right],$
(11)

where the weight is implemented as for each paired sample
$(\mathbf{z},\mathbf{x})\in{\mathcal{S}}_{i}$,

$\displaystyle{\boldsymbol{w}}_{i}=\frac{\frac{1}{|{\mathcal{S}}_{i}|}}{\sum_{i=1}^{N}\frac{1}{|{\mathcal{S}}_{i}|}}.$
(12)

Here, $|{\mathcal{S}}|$ denotes the number of training samples in the set
${\mathcal{S}}$.

  * 4.

$\lambda_{1}>0$ and $\lambda_{2}>0$ denote regularization parameters.

###  2.2 Network architectures

Figure 2: Network architectures of the proposed method for restoring short-
scanning amyloid PET images consist of generators and discriminators.
Generators include two repeated $4\times 4$ convolutions, each followed by
AdaIN and leaky rectified linear unit. Mapping label
$\boldsymbol{c}\in{\mathbb{R}}^{N}$ is passed through mapping networks to
generate mean and variance for AdaIN layer. Discriminators were designed as a
standard CNN without a fully connected layer. Numbers around the rectangular
boxes indicate the number of channels.

Network architectures of the proposed method are illustrated in Fig. 2. The
networks $G$ and $F$ were designed based on deep convolutional framelets [21],
which were constructed with encoder-decoder structures and skipped
connections. The networks consisted of two repeated $4\times 4$ convolutions,
each followed by the AdaIN and a leaky rectified linear unit (LReLU). In this
setup, the mapping networks $f^{(k)},k=1,\ldots,K$ that generate $J_{k}$
different $(\mu,\sigma)$ pairs were designed with two fully connected layers,
each consisting of 64 nodes, and an output layer with $2J_{k}$ nodes. The
generators employed 2-D Haar wavelet decomposition and reconstruction for
down-sampling and up-sampling, respectively. In the encoder path, three high-
pass filters after wavelet decomposition were directly skipped to the decoder
path, while one low-pass filter was concatenated with the features in the
decoder path at the same step. A convolution layer with a $1\times 1$ window
was added at the end to match the input and output image dimensions.

The discriminators $D_{\mathcal{Z}}$ and $D_{\mathcal{X}}$ were designed using
a standard CNN without a fully connected layer. They consisted of three
convolution layers, each with a $4\times 4$ window and strides of two in each
direction of the domain, followed by batch normalization (BN) and a LReLU with
a slope of 0.2. At the end of the architecture, a $1\times 1$ convolution was
added to generate a single-channel image.

###  2.3 Implementation details

We utilized the Adam optimization [10] to update the parameters of the
networks for $G$, $F$, $D_{\mathcal{X}}$, and $D_{\mathcal{Z}}$. The learning
rate for the optimization process was set to 0.0002. The initial weights of
the networks were randomly initialized from a Gaussian distribution with a
mean of 0 and a standard deviation of 0.01. We ran the training process for a
total of 200 epochs. Training was implemented using TensorFlow [1] with
Horovod [15] on a 4-way GPU system (NVIDIA RTX 3090 24GB) with dual CPUs
(Intel(R) Xeon Gold 6226R, 2.9GHz). With a mini-batch size of 32, the training
took approximately 206 hours. In our study, the regularization parameters
$\lambda_{1}$ and $\lambda_{2}$ were set to $10$ and $10$, respectively.

Table 1:  Quantitative results of the correction methods for the internal
validation.

Training datasets Methods FBB (n=119) FMM (n=30) FPN (n=15) NRMSE SSIM NRMSE
SSIM NRMSE SSIM 2 min image 15.613 0.839 20.494 0.807 31.360 0.798 Single
(${\mathcal{S}}_{i}~{},i=1,2,3$) [M1] adv+ls 11.983 0.876 14.633 0.851 12.736
0.865 [M2] adv+ls+cycle 10.921 0.893 13.736 0.869 12.607 0.877 Multiple
($\cup_{i=1}^{3}{\mathcal{S}}_{i}$) [M3] adv+wls 12.846 0.885 14.038 0.859
13.226 0.858 [M4] adv+wls+cycle 11.956 0.879 13.558 0.855 14.386 0.854 [ours]
adv+wls+cycle+$\boldsymbol{c}$ 10.837 0.894 13.589 0.867 12.131 0.880

Table 2:  Quantitative results of the correction methods for the temporal
validation.

Training datasets Methods FBB (n=20) FMM (n=20) FPN (n=20) NRMSE SSIM NRMSE
SSIM NRMSE SSIM 2 min image 15.179 0.823 19.882 0.811 34.934 0.815 Single
(${\mathcal{S}}_{i}~{},i=1,2,3$) [M1] adv+ls 11.626 0.866 13.615 0.859 12.367
0.865 [M2] adv+ls+cycle 10.648 0.886 13.058 0.873 12.493 0.882 Multiple
($\cup_{i=1}^{3}{\mathcal{S}}_{i}$) [M3] adv+wls+cycle 11.984 0.886 12.844
0.876 13.866 0.881 [ours] adv+wls+cycle+$\boldsymbol{c}$ 10.557 0.888 12.820
0.873 11.763 0.886

##  3 Experiments

The retrospective study protocol (DAUHIRB-22-053) was reviewed and approved by
the Institutional Review Board (IRB) of Dong-A University Hospital (DAUH).
Informed consentsince only anonymized data was used for research purposes. All
methods used in this study were in accordance with relevant guidelines and
regulations. Some of the data used in this article were obtained from the
Alzheimer’s Disease Neuroimaging Initiative (ADNI) database
(adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private
partnership, led by Principal Investigator Michael W. Weiner, MD. The primary
goal of ADNI has been to test whether serial magnetic resonance imaging (MRI),
positron emission tomography (PET), other biological markers, and clinical and
neuropsychological assessment can be combined to measure the progression of
mild cognitive impairment and early Alzheimer’s disease. For up-to-date
information, see www.adni-info.org.

###  3.1 Datasets

In this study, a total of 814 FBB-PET, 415 FMM-PET, and 208 FPN-PET scans were
collected from the three different domains. These scans were divided into
training, internal validation, and temporal validation sets. Specifically,
734, 60, and 20 scans were used for training, internal validation, and
temporal validation for FBB-PET, respectively. Similarly, 365, 30, and 20
scans were used for FMM-PET, and 173, 15, and 20 scans were used for FPN-PET.
Each scan consisted of 110 images of size $400\times 400$. The training and
internal validation datasets were obtained between December 2015 and December
2021, while the temporal validation dataset was obtained between May and
September 2022. All FBB-, FMM-, and FPN-PET scans were acquired using a
Biograph mCT flow scanner (Siemens Healthcare, Knoxville, TN, USA). The PET/CT
imaging was conducted in accordance with the standard protocol of DAUH, which
is consistent with the method used in previous studies [8, 7]. In each scan
protocol, the patients were intravenously injected with 300 MBq of FBB, 185
MBq of FMM, and 370 MBq of FPN. PET images were acquired 90 minutes after
injection for FBB- and FMM-PET, and 60 minutes for FPN-PET. The PET scans
covered the skull vertex to the skull base and were acquired for 20 minutes
using 3-dimensional and list modes. The ground-truth $\mathbf{x}$ and short-
scanning static PET image $\mathbf{z}$ were reconstructed from the 20-minute
and first 2-minute scan data, respectively, with the same parameters. The
paired training samples for FBB-, FMM-, and FPN-PET were denoted as
${\mathcal{S}}_{1}$, ${\mathcal{S}}_{2}$, and ${\mathcal{S}}_{3}$,
respectively, and the mapping labels were assigned as
$\boldsymbol{c}_{1}=[1,~{}0,~{}0]$ for FBB-PET,
$\boldsymbol{c}_{2}=[0,~{}1,~{}0]$ for FMM-PET, and
$\boldsymbol{c}_{3}=[0,~{}0,~{}1]$ for FPN-PET.

For external validations, we collected 54 FBB-PET and 30 FBP-PET scans from
the ADNI database. These scans contained a series of $4\times 5$ min PET
images. Note that the proposed model, trained on the dataset of paired
2-minute and 20-minute images, was tested on the first 5-minute PET images.
Here, a Gaussian filter with 4 mm FWHM was applied to all FBB-PET images of
the ADNI datasets. Among the collected ADNI scans, 8 FBB-PET and 6 FPN-PET
scans were randomly selected for estimation of unknown label
$\boldsymbol{c}_{T}$, while the remaining scans were employed for external
validations.

In our study, we determined the value of $\boldsymbol{c}_{T}$ as follows:
first, we restricted the domain ${\mathbb{R}}^{3}$ to
$[-\varepsilon,1+\varepsilon]^{3}$, where $\varepsilon$ is a small positive
value. Next, we discretized the restricted domain with a linear spacing and
computed the objective function defined in equation (1) for each point within
the discrete domain. Finally, we selected the point with the minimum objective
function value as our estimate for $\boldsymbol{c}_{T}$. The computed
objective function values on the domain $[-\varepsilon,1+\varepsilon]^{3}$ are
visually presented in Fig. 3.

Figure 3: Illustration of mapping labels $\boldsymbol{c}_{T}$ for ADNI-FBB
(left figure) and ADNI-FBP datasets (right figure). Color indicates value of
the objective function in (1). Table 3:  Quantitative results of the
correction methods for external validation.

Training datasets Methods ADNI-FBB (n=46) ADNI-FBP (n=24) NRMSE SSIM NRMSE
SSIM 5 min images 11.963 0.861 16.117 0.808 FBB $({\mathcal{S}}_{1})$ [M1]
adv+ls 10.775 0.867 11.386 0.861 Multiple $(\sum_{i=1}^{3}{\mathcal{S}}_{i})$
[M2] adv+wls+cycle 9.597 0.893 11.974 0.891 [ours]
adv+wls+cycle+$\boldsymbol{c}_{T}$ $\boldsymbol{c}_{T}=[0.94,0.06,0]$ for FBB
$\boldsymbol{c}_{T}=[0.9,0.1,0.5]$ for FBP 8.218 0.899 8.251 0.897 Training
datasets Methods ADNI-FBB (n=46) ADNI-FBP (n=24) NRMSE SSIM NRMSE SSIM ADNI-
FBB (n=54) [M3] adv+wls+cycle 8.490 0.897 - - ADNI-FBP (n=30) [M4]
adv+wls+cycle - - 8.910 0.899

###  3.2 Quantitative, statistical, and qualitative analyses

Image quality was evaluated using normalized root mean-square error (NRMSE)
and structure similarity index measure (SSIM) metrics. Precisely, for two
vectors $\mathbf{z}$ and $\mathbf{x}$, the NRMSE is computed as follows:

$\displaystyle
NRMSE(\mathbf{z},\mathbf{x})=\sqrt{\frac{\|\mathbf{z}-\mathbf{x}\|_{2}^{2}}{\|\mathbf{x}\|_{2}^{2}}}\times
100,$ (13)

where $\|\cdot\|_{2}$ denotes the standard $l_{2}$ norm.

The SSIM is computed as follows:

$\displaystyle
SSIM(\mathbf{z},\mathbf{x})=\frac{(2\mu_{\mathbf{z}}\mu_{\mathbf{x}}+\alpha_{1})(2\sigma_{\mathbf{z}\mathbf{x}}+\alpha_{2})}{(\mu_{\mathbf{z}}^{2}+\mu_{\mathbf{x}}^{2}+\alpha_{1})(\sigma_{\mathbf{x}}^{2}+\sigma_{\mathbf{x}}^{2}+\alpha_{2})},$
(14)

where $\alpha_{1}$ and $\alpha_{2}$ are parameters. We chose
$\alpha_{1}=(0.0002\times 65535)^{2}$ and $\alpha_{2}=(0.0007\times
65535)^{2}$ based on the experimental results reported in our previous study
[7].

We also computed the standardized uptake value ratio (SUVR) using PMOD 3.6
software (PMOD Technologies, Zurich, Switzerland) [3]. PET images were
normalized spatially using the transformation matrix of each participant and
were applied to an automated anatomical labeling template of PMOD. We
normalized spatially and applied all pairs of corrected and ground-truth PET
images (i.e., 20-min PET images) to the amyloid cortical composite atlas in
the PMOD program. The cortical composite was set up as the frontal, temporal,
and parietal lobes and cingulate of the brain, and we reconstructed the
volume-of-interests of the atlas to calculate the SUVRs of the representative
areas. The reference region used for SUVR calculation was the whole
cerebellum. Finally, we compared the SUVRs of the identical areas between the
corrected and ground-truth PET images of the external validation dataset
(ADNI-FBB and ADNI-FBP).

In addition, we performed visual interpretation to evaluate the clinical
applicability of the corrected PET images. Following the conventional
interpretation protocol, two nuclear medicine physicians certified and
experienced in amyloid PET reading independently determined the amyloid status
(positive or negative) for both corrected and ground-truth PET images. They
were blinded to the clinical data and read all PET images of the external
validation dataset. We measured the accuracy of the visual interpretation for
corrected PET images for each physician and analyzed the intra-observer
agreements of binary classification of corrected and ground-truth PET images
for each physician.

Figure 4: Correction results for external validation (ADNI-FBB). Each column
(from left to right) shows the input (5 min), adv+ls [M1], adv+wls+cycle [M2],
proposed and reference (20 min) images. The second and fourth columns show
difference images from the reference (20 min) image. WW/WC = [7000, 3500] for
PET images, and WW/WC = [1200, 600] for difference images.

###  3.3 Correction results

####  3.3.1 Internal validations

Table 1 illustrates the quantitative results of correction methods for
internal validation datasets. For quantitative evaluation, we calculated NRMSE
and SSIM between corrected images and the corresponding ground-truth images.
In Table 1, the ‘M1’ to ‘M4’ denote the methods trained using the different
combination of the losses such as
${\mathcal{L}}_{adv},~{}{\mathcal{L}}_{ls},~{}{\mathcal{L}}_{wls},~{}{\mathcal{L}}_{cycle}$,
and the different training datasets such as single and multiple training
datasets. We note that Method 1 is the method used in our pervious study [7].

The results demonstrate that Method 2 outperformed Method 1 for the FBB, FMM,
and FPN datasets, respectively, suggesting that the cycle-consistency loss led
to improved performance for each individual training dataset. However, Methods
3 and 4 performed worse than Methods 1 and 2, likely due to the added
complexity of dealing with multiple source and target domains
$\mathcal{Z}_{i}$ and $\mathcal{X}_{i}$, where $i=1,2,3$. The proposed method
using mapping labels achieved the best performance for the FBB and FPN
datasets and was close to the best performance for the FMM dataset. These
findings suggest that the proposed network using mapping labels stably learned
the relationships between multiple source and target domains. Furthermore, the
proposed method achieved comparable or better performance compared to methods
trained using only a single source and target domain (Method 2).

Table 2 summarizes the quantitative results of the correction methods for
temporal validation dataset. The results were consistent with the those
obtained from the internal validation dataset. Overall, the proposed method
achieved the best performance, except for the SSIM value of the FMM dataset.

Figure 5: Correction results for external validation (ADNI-FBP). Each column
(from left to right) shows the input (5 min), adv+ls [M1], adv+wls+cycle [M2],
proposed and reference (20 min) images. The second and fourth columns show
difference images from the reference (20 min) image. WW/WC = [7000, 3500] for
PET images, and WW/WC = [1200, 600] for difference images.

####  3.3.2 External validations

We performed an external validation of our proposed method by applying it to
the ADNI-FBB and ADNI-FBP datasets, which were obtained from unseen domains.
The estimated $\boldsymbol{c}_{T}$ for ADNI-FBB was found to be
$[0.94,0.06,0]$, which is very close to the $\boldsymbol{c}_{1}=[1,0,0]$
assigned for the FBB dataset. On the other hand, the estimated
$\boldsymbol{c}_{T}$ for ADNI-FBP was $[0.9,0.1,0.5]$. Since the FBP dataset
was not used in training, it was represented as a suitable linear combination
of $\boldsymbol{c}_{1}$, $\boldsymbol{c}_{2}$, and $\boldsymbol{c}_{3}$. We
presented the quantitative results in Table 3. Method 1 was trained using the
FBB dataset (${\mathcal{S}}_{1}$), while Method 2 and the proposed method were
trained using all FBB (${\mathcal{S}}_{1}$), FMM (${\mathcal{S}}_{2}$), and
FPN (${\mathcal{S}}_{3}$) datasets. The proposed method, with the mapping
label $\boldsymbol{c}_{T}$, outperformed Methods 1 and 2 for both ADNI-FBB and
ADNI-FBP datasets. To further evaluate our proposed method, we compared its
performance with Methods 3 and 4, which were trained using the 54 ADNI-FBB and
30 ADNI-FBP datasets used for external validation, respectively. Due to the
limited number of datasets, we calculated the NRMSE and SSIM on the training
datasets, and the results are presented in Table 3. Even when compared with
the training results of Methods 3 and 4, the proposed method achieved better
or comparable results. We showed the visual comparisons in Figs. 4 and 5.

Figure 6: Comparison of mean SUVRs in ground-truth (GT) and corrected PET
images of external validation. Figure 7: The Bland-Altman graphs for ADNI-FBB
(left figure) and ADNI-FBP (right figure) show the differences between ground-
truth and corrected PET images. Red line represents average difference between
the ground-truth and corrected PET images. Blue lines represent the upper and
lower 95 % confidence interval limits for the average difference.

We compared the range of SUVRs in the identical cortical composite area
between the corrected image and the corresponding ground-truth image for both
ADNI-FBB and ADNI-FBP datasets. As shown in Fig. 6, no statistically
significant difference was revealed in the SUVRs of the cortical composite
areas between ground-truth and corrected images for both cases. There was a
very strong positive correlation in the ADNI-FBB dataset ($r=0.998$,
$p<0.001$) and the ADNI-FBP dataset ($r=0.994$, $p<0.001$). We also evaluated
the difference in the SUVRs between the two PET images. In the Bland-Altman
analysis, the mean difference of SUVR between ground-truth and corrected
images was 0.030 (95% confidence interval (CI) -0.024, 0.036) in the ADNI-FBB
dataset and 0.009 (95% CI -0.000, 0.018) in the ADNI-FBP dataset. Upper and
lower limits of agreement were 0.070 (95% CI 0.060, 0.081) and -0.010 (95% CI
-0.021, 0.000) in the ADNI-FBB dataset, and 0.051 (95% CI 0.035, 0.067) and
-0.033 (95% CI -0.049, -0.018) in the ADNI-FBP dataset, respectively. The
Bland-Altman graphs for ADNI-FBB and ADNI-FBP were shown in Fig. 7.

The accuracy of visual binary classification for corrected PET images of the
external validation set was 0.960 (reader 1), 0.920 (reader 2) for ADNI-FBP-
PET images, and 0.980 (reader 1), 0.940 (reader 2) for ADNI-FBB-PET images.
Additionally, we evaluated the intra-observer agreement using Cohen’s weighted
kappa by comparing visual interpretations between ground-truth and corrected
PET images. The computed Cohen’s weighted kappa was 0.911 (reader 1) and 0.826
(reader 2) for ADNI-FBP-PET images, and 0.959 (reader 1) and 0.874 (reader 2)
for ADNI-FBB-PET images.

##  4 Discussion and conclusion

In this paper, we proposed a deep learning-based method for short-scanning
amyloid PET image restoration that can be generalized across various image
domains. The proposed method learns a correction map using amyloid PET images
sampled from multiple source and target domains. Our main contribution is the
introduction of a mapping label that facilitates the effective learning of
specific representations between these domains. Furthermore, the mapping label
enables the effective correction of not only amyloid PET datasets in multiple
training domains but also PET datasets from unseen domains.

Our experiments demonstrate that the proposed method has the potential to
reduce scanning time while maintaining high diagnostic reliability. The
proposed method effectively restores short scan amyloid PET images from
multiple training source domains, as evidenced by quantitative metrics such as
NRMSE and SSIM. For the external validation datasets (ADNI-FBB and ADNI-FBP),
the proposed method achieved comparable or even superior results compared to
methods trained using the datasets used for external validations.
Additionally, there was no statistically significant difference in the SUVRs
of the cortical composite areas between the ground-truth and corrected images
$(p<0.001)$. The accuracy for determining amyloid status on the corrected PET
images was high, with values of 0.970 for reader 1 and 0.930 for reader 2.

In our study, although the proposed method was trained using FBB, FMM, and FPN
datasets from three different domains $(N=3)$, it can be easily generalized
with the mapping labels $\boldsymbol{c}_{i},~{}i=1,2,\ldots,N$ for large $N$.
For external validation, various optimization methods can be applied to find
the optimal $\boldsymbol{c}_{T}$ in the problem of equation (1). The proposed
method, which utilizes domain labels for short-scanning amyloid PET image
restoration, can be a promising approach to overcoming the challenges of
multi-domain learning in PET imaging.

## Acknowledgments

This work was supported by the National Institute for Mathematical
Sciences(NIMS) grant funded by the Korean government (No. NIMS-B23910000).
Y.J.J was supported by the National Research Foundation of Korea(NRF) grant
funded by the Korea government(MSIT) (No. 2022R1A2C1005016). Data collection
and sharing for this project was funded by the Alzheimer’s Disease
Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01
AG024904) and DOD ADNI (Department of Defense award number W81XWH-12-2-0012).
ADNI is funded by the National Institute on Aging, the National Institute of
Biomedical Imaging and Bioengineering, and through generous contributions from
the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery
Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb
Company; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli
Lilly and Company; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated
company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd.; Janssen
Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson
Pharmaceutical Research & Development LLC.; Lumosity; Lundbeck; Merck & Co.,
Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Neurotrack Technologies;
Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier;
Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian
Institutes of Health Research is providing funds to support ADNI clinical
sites in Canada. Private sector contributions are facilitated by the
Foundation for the National Institutes of Health (www.fnih.org). The grantee
organization is the Northern California Institute for Research and Education,
and the study is coordinated by the Alzheimer’s Therapeutic Research Institute
at the University of Southern California. ADNI data are disseminated by the
Laboratory for Neuro Imaging at the University of Southern California.

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