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[fn1]Authors contribute equally to the paper. [cor1]Corresponding authors:
Yizhou Yu<EMAIL_ADDRESS>and Yuqing Zhang<EMAIL_ADDRESS>[cor2]Corresponding author at: Department of Computer Science, The University
of Hong Kong, Pokfulam, Hong Kong

# SDR-Former: A Siamese Dual-Resolution Transformer for Liver Lesion
Classification Using 3D Multi-Phase Imaging

Meng Lou  Hanning Ying  Xiaoqing Liu  Hong-Yu Zhou  Yuqing Zhang  Yizhou Yu
Artificial Intelligence Laboratory, Deepwise Healthcare, Beijing, China
Department of General Surgery, Sir Run Run Shaw Hospital, Zhejiang University
School of Medicine, Hangzhou, Zhejiang, China  Department of Radiology, Ningbo
Medical Center Lihuili Hospital, Ningbo, Zhejiang, China  Department of
Computer Science, The University of Hong Kong, Pokfulam, Hong Kong

###### Abstract

Automated classification of liver lesions in multi-phase CT and MR scans is of
clinical significance but challenging. This study proposes a novel Siamese
Dual-Resolution Transformer (SDR-Former) framework, specifically designed for
liver lesion classification in 3D multi-phase CT and MR imaging with varying
phase counts. The proposed SDR-Former utilizes a streamlined Siamese Neural
Network (SNN) to process multi-phase imaging inputs, possessing robust feature
representations while maintaining computational efficiency. The weight-sharing
feature of the SNN is further enriched by a hybrid Dual-Resolution Transformer
(DR-Former), comprising a 3D Convolutional Neural Network (CNN) and a tailored
3D Transformer for processing high- and low-resolution images, respectively.
This hybrid sub-architecture excels in capturing detailed local features and
understanding global contextual information, thereby, boosting the SNN’s
feature extraction capabilities. Additionally, a novel Adaptive Phase
Selection Module (APSM) is introduced, promoting phase-specific
intercommunication and dynamically adjusting each phase’s influence on the
diagnostic outcome. The proposed SDR-Former framework has been validated
through comprehensive experiments on two clinical datasets: a three-phase CT
dataset and an eight-phase MR dataset. The experimental results affirm the
efficacy of the proposed framework. To support the scientific community, we
are releasing our extensive multi-phase MR dataset for liver lesion analysis
to the public. This pioneering dataset, being the first publicly available
multi-phase MR dataset in this field, also underpins the MICCAI LLD-MMRI
Challenge. The dataset is accessible at: https://bit.ly/3IyYlgN.

###### keywords:

Liver Lesion Classification Multi-phase Imaging Siamese Neural Networks Dual-
Resolution Transformer

##  1 Introduction

Combating liver cancer remains a significant challenge in global health,
demanding immediate progress in both diagnosis and treatment methods[1].
Presently, contrast-enhanced multi-phase computed tomography (CT) and magnetic
resonance (MR) imaging are the primary techniques for diagnosing liver lesions
[2]. These imaging methods provide extensive anatomical and functional
insights, from vascular patterns to lesion features, enhancing their accuracy
and reliability in lesion identification [3]. Nevertheless, analyzing 3D
multi-phase images manually is a demanding and challenging task, even for
experienced radiologists. This difficulty is further exacerbated by the
variability in lesion appearances, indistinct lesion margins, and the unique
characteristics presented in each imaging phase, all of which make lesion
classification more complex [4, 5]. These challenges highlight the urgent need
to develop a precise and resilient computer-aided diagnostic (CAD) system,
aiming at assisting radiologists in providing more accurate and rapid
diagnostic assessments.

Deep learning has rapidly risen to prominence in medical image analysis,
primarily due to its exceptional capacity to encapsulate intricate
representations of medical images [6]. This advancement is particularly
evident in the domain of multi-phase medical image analysis, where state-of-
the-art deep learning techniques have been instrumental in achieving
groundbreaking results [7, 8, 9, 10, 11, 12, 13]. Presently, the field of
multi-phase medical image analysis is predominantly guided by two influential
approaches: image-level fusion and feature-level fusion. The image-level
fusion approach is notable for its scalability and computational efficiency.
It treats each phase as a separate channel in a unified input image [10],
allowing for seamless adaptation to datasets with different numbers of phases
while conserving computational resources. Additionally, this approach paves
the way for transfer learning across datasets with different phase counts,
facilitating the application of insights gained from one dataset to enhance
outcomes on another. Conversely, by setting up a dedicated model for each
phase, the feature-level fusion approach is distinguished by its ability to
thoroughly harness multi-phase information, promoting detailed interaction
among features from various phases [11, 12, 13]. This approach enhances the
model’s capacity to represent multi-phase characteristics, resulting in
improved performance.

However, both approaches come with intrinsic limitations. The image-level
fusion approach has a risk of diluting the richness of feature representation
within individual phases. This often stems from the preliminary fusion of
multi-phase images at early stages, potentially limiting the model’s
representational capabilities and leading to sub-optimal performance. In
contrast, feature-level fusion faces distinct hurdles. Notably, assigning a
separate model for each phase results in considerable computational demands,
which may surpass hardware capacities as the number of phases increases. This
high computational load can hinder the practicality of this approach, both in
model training and in actual clinical settings. Additionally, the adaptability
of the model is hindered when dealing with datasets having different phase
counts. This limitation affects the smooth implementation of transfer learning
and necessitates extra modifications to ensure model compatibility with
varying phase counts.

To overcome the inherent drawbacks of both image-level and feature-level
fusion approaches, we propose the Siamese Dual-Resolution Transformer (SDR-
Former), a novel framework designed for liver lesion classification in 3D
multi-phase CT and MR imaging with varying phase counts. Our framework
utilizes a Siamese Neural Network (SNN) for handling multi-phase inputs,
offering two major benefits: (1) By allocating an individual neural network to
each phase, our method surpasses the image-level fusion approach in feature
representation capabilities; (2) Through the use of a weight-sharing network
for encoding multi-phase images, our proposed framework achieves exceptional
scalability and computational efficiency, effectively addressing the
limitations of the feature-level fusion approach. However, our experiments
revealed limitations when employing a CNN-based SNN, attributed to the static
weight generation of CNN models, limiting their representation capabilities.
To enhance the feature representation power of the SNN, we design a hybrid
CNN-Transformer network named the Dual-Resolution Transformer (DR-Former),
which combines the strengths of CNNs and Transformers: it utilizes a 3D CNN
for detailed local feature extraction from high-resolution images and an
efficient 3D Transformer for global context analysis from low-resolution
images. We also introduce a Bilateral Cross-resolution Integration Module
(BCIM) to enable semantic interactions and extract complementary features from
the dual-resolution feature maps produced by the DR-Former. Additionally, we
propose an Adaptive Phase Selection Module (APSM) to enhance inter-phase
contextual communication and dynamically adjust the influence of each phase in
the final diagnostic classification.

In summary, our contributions are multifaceted:

  * •

We introduce the SDR-Former, a versatile framework tailored for liver lesion
classification in 3D multi-phase CT/MR imaging. This framework stands out for
its effective multi-phase feature extraction, exceptional scalability, and
computational efficiency, effectively overcoming the constraints of
traditional fusion techniques.

  * •

To augment the feature representation capabilities of the SNN, we design a
unique hybrid CNN-Transformer network, namely DR-Former. Additionally, we
introduce the APSM, designed to dynamically adjust the relevance of each
imaging phase, thereby enhancing diagnostic accuracy and contextual
sensitivity.

  * •

We thoroughly validate our SDR-Former framework using two clinical datasets: a
three-phase CT dataset with two lesion types and an eight-phase MR dataset
with seven different lesion categories. Our experimental results confirm the
superiority of the SDR-Former, establishing it as a cutting-edge solution in
liver lesion classification.

  * •

As a contribution to the research community, we are making our comprehensive
3D multi-phase MR dataset available publicly. The complete dataset will be
accessible post-manuscript approval. To the best of our knowledge, this is the
first public MR dataset that encompasses up to eight imaging phases with seven
lesion categories, dedicated to liver lesion diagnosis.

The rest of the paper is organized as follows: Section 2 reviews relevant
prior studies. Section 3 details the proposed methodology. Section 4 outlines
the experimental design, and Section 5 analyzes the experimental results. The
paper concludes in Section 6, summarizing our primary findings and
contributions.

##  2 Related Work

###  2.1 Information Exploitation in Multi-phase Imaging

The abundant information contained in multi-phase images has led to
significant strides in their analysis, thanks to the application of diverse
deep learning techniques. Liang et al. [11] adapted ResNet into both global
and local processing pathways, incorporating a bidirectional long short-term
memory (LSTM) for efficient classification of focal liver lesions in multi-
phase CT images. Jiang et al. [12] developed an advanced network utilizing
multi-phase and multi-level feature integration, enabled by dual feature
encoders and an elaborate adaptive cross-refinement module, thereby enhancing
pancreas segmentation accuracy in CT imaging. Wang et al. [10] successfully
integrated a Transformer with a 3D CNN for multi-modal MR image analysis,
achieving significant improvements in brain tumor segmentation by fusing
multi-modal data at the image level. Recently, Qu et al. [13] introduced
M3Net, a multi-scale, multi-view approach for CT-based multi-phase pancreas
segmentation. The authors employed a non-local attention mechanism [14] to
improve inter-phase feature connections, resulting in effective segmentation
of the pancreas in varied pathological conditions.

###  2.2 Incorporating Multi-resolution Data in Image Analysis

Leveraging multi-resolution inputs is pivotal for capturing both coarse- and
fine-grained information, a principle that has led to various advancements in
medical image analysis. Xu et al. developed a multi-scale, cost-sensitive
neural network (MSCSNN), employing three parallel ResNets to process images at
different resolutions, thereby generating multi-scale representations for
accurately detecting malignant lung nodules [15]. HookNet utilizes dual CNNs
to separately process low- and high-resolution images, combining detailed and
broader features for semantic segmentation in histopathology images [16]. In
the semi-supervised reverse adversarial network by Fu et al., a multi-
resolution CNN is used to enhance the interaction between inputs of varying
resolutions, improving the diagnostic precision of pulmonary nodules [17]. In
this work, our hybrid CNN-Transformer network (i.e., DR-Former) capitalizes on
this multi-resolution concept. It utilizes a 3D CNN for analyzing high-
resolution images and a 3D Transformer to handle low-resolution images, based
on the idea that convolutional and self-attention mechanisms serve as
respective high- and low-pass filters and can capture unique spatial details
crucial [18].

###  2.3 Siamese Neural Networks

Siamese Neural Networks (SNNs), a distinct category of neural network designs,
are tailored for analyzing paired inputs through a shared-weights mechanism
[19]. Their adaptability has led to diverse applications in fields like object
tracking [20], few-shot image classification [21], and the development of
invariant visual representations using contrastive learning [22, 23, 24]. In
the realm of multi-phase medical imaging, we propose that SNNs hold
considerable promise for the effective extraction of features from multi-phase
imagery, a prospect that our study seeks to investigate for liver lesion
classification in multi-phase CT and MR scans. By utilizing an SNN framework,
we endeavor to not only extract robust features from each imaging phase but
also develop a model that is flexible and scalable across datasets with
varying phase counts.

###  2.4 Advancements in Vision Transformer Models

Drawing inspiration from Transformer models in natural language processing
(NLP) [25], Vision Transformer (ViT) models have ushered in a new era in
computer vision. Similar to NLP tokens, ViTs dissect images into patch
sequences and use self-attention mechanisms to understand the global
relationships between these patches [26]. Although traditional ViTs have
achieved impressive results, they tend to be resource-intensive, particularly
with high-resolution images. To address these computational demands, various
refined Transformer models have been developed. The Swin Transformer
introduces a method of creating hierarchical image structures and implementing
self-attention in shifting local windows across layers [27]. Pyramid Vision
Transformer (PVT) manages computational efficiency by reducing the spatial
dimensions of key-value pairs at different stages [28]. Pyramid Pooling
Transformer (P2T) streamlines self-attention by condensing key-value data
through multi-scale pooling [29]. Furthermore, hybrid models that merge CNNs
with Transformers, such as BoTNet [30], UniFormer [31], and MaxViT [32], have
achieved success by combining the advantages of both designs. In the field of
medical image analysis, customized ViT models have shown great promise. For
example, the nnFormer integrates convolutional processes with self-attention
for effective 3D medical image segmentation [33]. The H2Former, fuses CNNs and
Transformers for medical image segmentation, highlighting the efficacy of
these models in specific applications [34]. Our study extends these
innovations, emphasizing the collaborative benefits of CNNs and Transformers
to improve feature representation in multi-phase medical image analysis.

##  3 Methodology

###  3.1 Overview

Figure 1:  This diagram illustrates the SDR-Former framework, which utilizes a
Siamese Neural Network (SNN) for processing each phase in the multi-phase
imaging series independently yet simultaneously. Central to this framework is
the hybrid Dual-Resolution Transformer (DR-Former), a fusion network that
effectively merges CNN and Transformer approaches to handle high- and low-
resolution inputs using shared weights. The Bilateral Cross-Resolution
Integration Module (BCIM) is strategically incorporated into this architecture
to facilitate semantic interactions and enhance the coalescence of the
distinct high- and low-resolution streams. Furthermore, the Adaptive Phase
Selection Module (APSM) plays a key role in dynamically merging these feature
representations, ensuring a versatile and perceptive amalgamation of multi-
phase features.

As illustrated in Figure 1, our SDR-Former framework employs a dual-stage
strategy to effectively utilize multi-phase image data, consisting of an SNN
for feature extraction and an APSM for integrating phase-specific features.

Unlike traditional multi-phase image analysis approaches (i.e., image-level
and feature-level), the SDR-Former uniquely processes multi-phase images
through a weight-sharing network of the SNN. Note that the number of weight-
sharing networks employed in SNN within this study aligns with the number of
input phases. This straightforward architecture ensures the model’s
scalability and efficiency, as well as its seamless adaptation to datasets
with different phase counts, thus enabling transfer learning across diverse
multi-phase datasets. Nonetheless, the simple use of the SNN to extract multi-
phase features may fall short in isolating distinctive phase features,
potentially leading to weak feature representation.

To address this, we propose the novel DR-Former, a dual-branch network
designed for the simultaneous processing of images at varying resolutions,
serving as the weight-sharing network of the SNN. This configuration processes
high-resolution images with a 3D CNN, adept at capturing spatial details, and
low-resolution images with a 3D Transformer, aimed at recognizing global low-
frequency contextual information. As a result, the weight-sharing mechanism
intrinsic to the SNN underpins scalability, effectively accommodating datasets
with diverse phase counts, thereby enabling transfer learning between such
datasets. Moreover, incorporating the DR-Former within the SNN framework
substantially enhances the model’s capability in representing features,
particularly in multi-phase imaging scenarios as this dual approach allows the
DR-Former to leverage the complementary strengths of CNNs and Transformers.
Concurrently, the BCIM is introduced to bridge the CNN and Transformer
pathways, facilitating a smooth exchange of features across different
resolutions.

Regarding post feature extraction, the APSM dynamically merges these features
in a phase-sensitive fashion, highlighting the most diagnostically valuable
features for each phase. The combined feature maps are then merged through a
Global Average Pooling (GAP) layer and a Fully-Connected (FC) layer for final
diagnostic classification.

###  3.2 Dual-Resolution Transformer (DR-Former)

Considering that high-resolution images naturally contain abundant high-
frequency details, while low-resolution images predominantly focus on low-
frequency elements [35, 36], our proposed DR-Former employs a CNN for high-
resolution imagery and a Transformer for low-resolution imagery to optimize
feature representation across different frequency ranges.

As depicted in Figure 2, the DR-Former comprises two separate processing
paths: a 3D CNN for analyzing high-resolution images and a 3D Transformer for
low-resolution image encoding. The CNN pathway incorporates multiple residual
blocks [37], with each block containing two $3\times 3\times 3$ convolutional
layers linked by a shortcut connection. In parallel, the Transformer pathway
consists of a sequence of attention blocks, each featuring a self-attention
layer followed by a multi-layer perceptron (MLP).

Given the significant computational demands of 3D imaging and multi-phase data
handling—particularly the quadratic complexity of the Multi-Head Self-
Attention (MHSA) in the ViT [26], we have evaluated four distinct self-
attention schemes for computational efficiency: Shifted Window (Swin) [27],
Spatial Reduction Attention (SRA) [28], Pooling-based Self-attention (PSA)
[29], and Grid Self-attention (GSA) [32]. Our comprehensive experimental
analyses, which will be detailed in Section 5.2, demonstrate GSA’s superior
ability to strike a balance between computational efficiency and performance.

Figure 2:  A diagrammatic depiction outlining the structure of the Hybrid
Dual-Resolution Transformer (DR-Former).

Bilateral Cross-resolution Integration Module (BCIM) specifically designed to
foster semantic consistency and elevate the feature interaction between high-
and low-resolution data streams, as an effective and streamlined component,
the BCIM facilitates the flexible information exchange between features
processed by the CNN and Transformer paths. It enables both pathways to absorb
relevant information from each other, thereby intensifying semantic links and
enriching feature complementarity. This integration is crucial for the
effective combination of high- and low-resolution features, thus boosting the
overall efficacy of the DR-Former.

The mechanism of the BCIM is graphically illustrated in Figure 3,
demonstrating its role in relation to a high-resolution feature map
$\textbf{F}{c}\in\mathbb{R}^{C\times D\times H\times W}$ and a lower-
resolution feature map $\textbf{F}{v}\in\mathbb{R}^{C\times
D\times\frac{H}{2}\times\frac{W}{2}}$, as encoded by their respective
branches. Initially, GAP is applied to each feature map, denoted as
$\left\\{{\textbf{F}{c},\textbf{F}{v}}\right\\}$. This step is followed by a
two-layer Multi-Layer Perceptron (MLP) with a sigmoid activation function,
yielding two vectors of attention coefficients
$\left\\{\textbf{\emph{c}},\textbf{\emph{v}}\right\\}\in\mathbb{R}^{C\times
1\times 1\times 1}$. Subsequently, $\textbf{F}{c}$ is down-scaled via average
pooling to $\textbf{D}{c}\in\mathbb{R}^{C\times
D\times\frac{H}{2}\times\frac{W}{2}}$, while $\textbf{F}{v}$ is up-scaled to
$\textbf{U}{v}\in\mathbb{R}^{C\times D\times H\times W}$. The subsequent step
involves the element-wise multiplication of $\textbf{U}{v}$ with _c_ and
$\textbf{D}{c}$ with _v_ , producing modified feature maps
$\left\\{{\mathbf{U}{v}^{\prime}\in\mathbb{R}^{C\times D\times H\times
W},\mathbf{D}{c}^{\prime}\in\mathbb{R}^{C\times
D\times\frac{H}{2}\times\frac{W}{2}}}\right\\}$ aimed at enabling semantic
integration. These adjusted maps
$\left\\{{\mathbf{U}{v}^{\prime},\mathbf{D}{c}^{\prime}}\right\\}$ are then
merged with their original feature maps
$\left\\{{\textbf{F}{c},\textbf{F}{v}}\right\\}$. The combined maps are then
processed through two $3\times 3\times 3$ convolutional layers, each followed
by Batch Normalization (BN) and ReLU activation, resulting in the finalized
output feature maps
$\left\\{{\textbf{F}{c}^{\prime},\textbf{F}{v}^{\prime}}\right\\}$. The BCIM
can be formally described as follows:

$\begin{array}[]{c}c=\sigma(MLP(\sum_{i=1}^{D}\sum_{j=1}^{H}\sum_{k=1}^{W}\mathbf{F}_{c}(i,j,k)))\\\
v=\sigma(MLP(\sum_{i=1}^{D}\sum_{j=1}^{H}\sum_{k=1}^{W}\mathbf{F}_{v}(i,j,k)))\\\
\mathbf{D}_{c}=Down(\mathbf{F}_{c});\mathbf{U}_{v}=Up(\mathbf{F}_{v})\\\
\mathbf{D}_{c}^{\prime}=v\times\mathbf{D}_{c};\mathbf{U}_{v}^{\prime}=c\times\mathbf{U}_{v}\\\
\mathbf{F}_{c}^{\prime}=W_{3}^{2}(\hat{\mathcal{C}}(\mathbf{F}_{c},\mathbf{U}_{v}^{\prime}));\mathbf{F}_{v}^{\prime}=W_{3}^{2}(\hat{\mathcal{C}}(\mathbf{F}_{v},\mathbf{D}_{c}^{\prime}))\\\
\end{array}$ (1)

Here, the terms $\sigma(\cdot)$, $\hat{\mathcal{C}}(\cdot)$, and
$W_{3}^{2}(\cdot)$ denote the sigmoid activation function, the operation of
channel-wise concatenation, and a sequence of two $3\times 3\times 3$
convolutional layers, each followed by BN and ReLU activation, respectively.

Figure 3: A schematic illustration of the Bilateral Cross-resolution
Integration Module (BCIM).

###  3.3 Adaptive Phase Selection Module (APSM)

Inspired by the critical role of phase-specific imaging attributes in clinical
diagnostics [38, 39], we have developed the APSM, tailored to facilitate
detailed interaction among phases and dynamically modulate the focus on
features from each phase in line with their clinical significance.

As illustrated in Figure 4, the APSM processes feature maps for each phase,
$\left\\{\mathbf{P}_{1},\mathbf{P}_{2},\cdots,\mathbf{P}_{N}\right\\}\in\mathbb{R}^{C\times
D\times H\times W}$, generated by the DR-Former, where _N_ indicates the total
number of phases. The process initiates with a channel-wise concatenation of
the multi-phase feature maps, followed by refinement using a $1\times 1\times
1$ convolution and a GAP, leading to a unified global descriptor
$\mathbf{M}\in\mathbb{R}^{C\times 1\times 1\times 1}$. Following this, _N_
parallel $1\times 1\times 1$ convolutions are applied to create individual
global descriptors
$\left\\{\mathbf{M}_{1},\mathbf{M}_{2},\cdots,\mathbf{M}_{N}\right\\}$. These
descriptors are then used to calculate phase-specific attention coefficients
$\left\\{p{1},p{2},\cdots,p{N}\right\\}$ via a channel-wise softmax function,
ensuring that the impact of each phase is proportionally weighted according to
its diagnostic relevance, mathematically expressed as:

$p_{k}=\frac{{e^{\mathbf{M}_{k}^{j}}}}{e^{\sum_{i=1}^{N}\mathbf{M}_{i}^{j}}}$
(2)

Here, _j_ = $\left\\{1,2,3,\cdots,C\right\\}$ and _k_ =
$\left\\{1,2,3,\cdots,N\right\\}$. The attention-weighted feature maps are
concatenated and further refined by a $3\times 3\times 3$ convolution with BN
and ReLU activation to produce the final feature map
$\mathbf{V}\in\mathbb{R}^{C\times D\times H\times W}$ as expressed by the
following equation:

$\mathbf{V}=W_{3}(\hat{\mathcal{C}}(p_{1}\times\mathbf{P}{1},p{2}\times\mathbf{P}{2},\cdots,p{N}\times\mathbf{P}_{N}))$
(3)

In the architecture of the SDR-Former, the APSM is tactically situated
following the encoding of features from both high- and low-resolution images
across multiple phases. This placement empowers the model to adaptively
regulate the significance of features across different resolutions. As a
result, the diagnostic accuracy of the SDR-Former is notably improved, owing
to its proficient integration of phase-specific and inter-phase contextual
information.

Figure 4:  A schematic illustration of the Adaptive Phase Selection Module
(APSM).

##  4 Experimental Configuration

###  4.1 Datasets

In this study, the performance of the SDR-Former is assessed using two
distinct multi-phase imaging datasets derived from clinical sources: one
dataset comprises CT images, while the other includes MR images.

CT Image Dataset: This dataset encompasses 4,304 multi-phase liver lesions
from 10 different hospitals in China, involving over 2,000 patients. Each
lesion is documented in three separate phases: venous, arterial, and non-
contrast, forming distinct volumetric data. We implement patient-level
segregation to split the dataset into training, validation, and test groups,
ensuring no data overlap patient-wise. The training group contains 3,248
lesions (1,625 benign, 1,623 malignant), the validation group includes 434
lesions (253 benign, 181 malignant), and the testing group comprises 622
lesions (339 benign, 283 malignant), all of which are from an independent
hospital for testing purpose.

MR Image Dataset: Sourced from Ningbo Medical Center Lihuili Hospital, this
dataset includes 498 annotated multi-phase liver lesions from an equal number
of patients. Each lesion is represented across eight phases: non-contrast,
arterial, venous, delay, T2-weighted imaging, diffusion-weighted imaging, T1
in-phase, and T1 out-of-phase, each contributing an individual volume. The
lesions are classified into seven categories: hepatocellular carcinoma (HCC),
intrahepatic cholangiocarcinoma (ICC), liver metastases (HM), hepatic cysts
(HC), hepatic hemangioma (HH), focal nodular hyperplasia (FNH), and hepatic
abscess (HA). Pre-partitioned for research purposes, the dataset is divided
into training (316 lesions), validation (78 lesions), and testing sets (104
lesions). In line with our commitment to advancing multi-phase medical imaging
analysis, this dataset has been publicly released. Importantly, the MICCAI
LLD-MMRI Challenge111https://github.com/LMMMEng/LLD-MMRI2023 utilizes this
dataset as its core resource, aiming to promote the development and refinement
of computer-aided diagnosis (CAD) systems within this field. Furthermore,
within the scope of this paper, we provide annotations for an additional 104
cases, constituting a testing set that is not incorporated within the
challenge.

###  4.2 Implementation Details

Our SDR-Former framework is developed using the PyTorch framework and all
experiments are conducted on a NVIDIA RTX 3090 GPU. For training, we employ
the AdamW optimizer, starting with an initial learning rate of 0.0001, which
is dynamically adjusted using a cosine annealing schedule to fine-tune
performance. A weight decay of 0.05 is applied to prevent overfitting. The
training process spans over 200 epochs, with the first 5 epochs designated as
a warm-up phase where the learning rate gradually escalates. The standard
cross-entropy loss function is employed to compare the model’s output with the
ground-truth. Due to hardware limitations, our batch size is set to 8, and
each lesion volume is uniformly resized to $16\times 128\times 128$. To
further mitigate overfitting, a variety of data augmentation techniques are
employed, including random rotations, erasing, and flips across different
anatomical axes. In the training stage, we randomly crop the lesion volumes to
$14\times 112\times 112$, while for evaluation, a central crop of the same
size is extracted. The model’s effectiveness is rigorously assessed using a
range of established metrics: Accuracy (ACC), Area Under the Curve (AUC),
F1-score (F1), and Cohen’s Kappa (Kappa). We also conduct a paired t-test to
statistically ascertain and differentiate the performance of our model against
benchmark methods, ensuring an in-depth and statistically validated
performance analysis.

##  5 Results and Discussion

This section delves into a thorough assessment of the SDR-Former. First, an
ablation study is conducted to determine the structural effectiveness of the
hybrid DR-Former architecture via comparing it with traditional CNN and
Transformer models. Following this, the specific roles of the BCIM and the
APSM in the model are meticulously examined, emphasizing their influence on
the model’s performance. The analysis further encompasses various self-
attention mechanisms embedded in the model’s Transformer branch, assessing
their role in augmenting feature extraction efficiency. Additionally, the
importance of utilizing multi-phase imaging is investigated. A detailed
comparison with the current state-of-the-art methods is presented,
establishing the proposed SDR-Former as a new benchmark in the field. The
final part of the evaluation explores the model’s ability for cross-modality
transfer learning and its adaptability to different datasets, affirming the
SDR-Former’s flexibility and scalability in medical image analysis.

###  5.1 Comparison with State-of-the-Art (SOTA) Methods

In this work, the performance of the SDR-Former model is benchmarked against a
suite of SOTA models. This comparison includes two renowned CNN architectures,
ResNet-50 [37] and DenseNet-121 [40], along with two advanced Transformer-
based models, BoTNet-50 [30] and UniFormer-S [31]. To highlight the advantages
of our hybrid design, we also included the MSCSNN [15], designed for multi-
resolution input, and the H2Former [34], another hybrid model. Note that
H2Former is principally intended for segmentation, so we employed only its
encoder in our experiments. Additionally, we explored the resilience of our
multi-phase analysis approach by integrating these models into the SDR-
Former’s weight-sharing network framework, resulting in variants like SNN-
ResNet-50, SNN-DenseNet-121, SNN-BoTNet-50, SNN-UniFormer-S, SNN-MSCSNN, and
SNN-H2Former, all under the umbrella of an SNN architecture. For a thorough
and unbiased evaluation, we also considered STIC [41], a publicly well-known
method for multi-phase image classification. The outcomes of this
comprehensive comparative study are summarized in Table 1 and depicted in the
ROC curves shown in Figure 5.

Table 1: Comparative Performance Metrics for Multi-phase Lesion Classification Across CT and MR Datasets.  Method | ACC | AUC | F1 | Kappa
---|---|---|---|---
CT (3-phase) |  |  |  |
STIC | 0.6913 | 0.7394 | 0.6933 | 0.3883
ResNet-50 | 0.7814 | 0.8553 | 0.7834 | 0.5670
SNN-ResNet-50 | 0.7974 | 0.8747 | 0.7872 | 0.5946
DenseNet-121 | 0.7588 | 0.8423 | 0.7611 | 0.5224
SNN-DenseNet-121 | 0.7669 | 0.8555 | 0.7745 | 0.5403
MCSCNN | 0.7749 | 0.8480 | 0.7603 | 0.5485
SNN-MCSCNN | 0.7621 | 0.8683 | 0.7716 | 0.5314
BoTNet-50 | 0.7669 | 0.8589 | 0.7724 | 0.5395
SNN-BoTNet-50 | 0.8087 | 0.8747 | 0.8072 | 0.6143
UniFormer-S | 0.7862 | 0.8784 | 0.7932 | 0.5783
SNN-UniFormer-S | 0.8312 | 0.8991 | 0.8305 | 0.6611
H2Former | 0.8103 | 0.8662 | 0.8097 | 0.6200
SNN-H2Former | 0.8344 | 0.8896 | 0.8335 | 0.6681
SDR-Former | 0.8537 | 0.9168 | 0.8529 | 0.7058
p-value | <0.01 | <0.01 | <0.01 | <0.01
MR (8-phase) |  |  |  |
STIC | 0.6346 | 0.8969 | 0.6254 | 0.5572
ResNet-50 | 0.6923 | 0.9298 | 0.6898 | 0.6244
SNN-ResNet-50 | 0.7115 | 0.9339 | 0.7168 | 0.6541
DenseNet-121 | 0.7404 | 0.9346 | 0.7171 | 0.6797
SNN-DenseNet-121 | 0.7500 | 0.9370 | 0.7394 | 0.6961
MCSCNN | 0.7115 | 0.9379 | 0.7089 | 0.6536
SNN-MCSCNN | 0.7308 | 0.9220 | 0.7409 | 0.6770
BoTNet-50 | 0.7212 | 0.9314 | 0.7139 | 0.6628
SNN-BoTNet-50 | 0.7692 | 0.9445 | 0.7572 | 0.7135
UniFormer-S | 0.7308 | 0.9254 | 0.7123 | 0.6662
SNN-UniFormer-S | 0.7692 | 0.9207 | 0.7639 | 0.7250
H2Former | 0.7212 | 0.9311 | 0.7342 | 0.6660
SNN-H2Former | 0.7788 | 0.9478 | 0.7745 | 0.7263
SDR-Former | 0.7885 | 0.9536 | 0.7910 | 0.7467
p-value | <0.05 | <0.01 | <0.01 | <0.01
|  |  |  |

Firstly, the STIC method exhibits the least favorable performance among the
methods evaluated, potentially due to its naive fusion of CNN and RNN, which
may not be adequate for fully capturing the intricate relationships present in
multi-phase images. Secondly, our hybrid CNN-Transformer model significantly
outperforms the MSCSNN, which utilizes three parallel ResNets for multi-
resolution data processing. This highlights the superior ability of our model
to handle such complex inputs effectively. Thirdly, our model exceeds the
performance of the H2Former, probably due to our deliberate allocation of
different resolutions to the specific CNN and Transformer branches. This
method fosters a more nuanced interaction of features compared to H2Former’s
simpler aggregate fusion approach. Notably, traditional CNN-based models, such
as SNN-DenseNet, SNN-ResNet, and SNN-MCSCNN, exhibit only modest improvements
within our multi-phase framework, indicating limited benefits over their
single-phase versions. On the contrary, Transformer-based models demonstrate
more substantial performance enhancements in the multi-phase context, owing to
their inherent dynamic self-attention capabilities. These capabilities enable
Transformer models to generate phase-sensitive attention maps, a functionality
not inherent in CNN-exclusive models. Ultimately, the integration of
Transformers within our multi-phase framework substantially elevates
performance, underscoring the importance of Transformer-based self-attention
in multi-phase lesion classification tasks.

Figure 5:  Receiver Operating Characteristic (ROC) curves for various methods.

Overall, the proposed SDR-Former model exhibits outstanding performance
compared to other SOTA models, as evidenced by its robust performance metrics.
In a CT dataset with three phases and two lesion types, the SDR-Former model
achieves an ACC of 0.8537, an AUC of 0.9168, an F1 of 0.8529, and a Kappa
0.7058. In a more complex MR dataset featuring eight phases and seven lesion
categories, the model obtains an ACC of 0.7885, an AUC of 0.9536, an F1 of
0.7910, and a Kappa of 0.7467. Furthermore, to establish statistical
significance, we conducted independent two-sample t-tests comparing the
performance of our SDR-Former with the best-performing method in each metric.
The null hypothesis assumes that our method doesn’t surpass the best-
performing method. Our analysis reveals p-values below 0.01 (or 0.05) for SDR-
Former across both datasets, indicating compelling evidence against the null
hypothesis. Consequently, our approach demonstrates significant advancements
over previous methods in two distinct datasets, highlighting its superiority.

###  5.2 Ablation Study

####  5.2.1 Effectiveness of the DR-Former

The DR-Former, central to the feature extraction process of the SDR-Former
framework, is constructed as a hybrid architecture combining CNN and
Transformer mechanisms. This design’s foundation is the exploitation of the
collaborative strengths of convolutional operations and self-attention
mechanisms [18].

In our implementation, high-resolution images are processed through the DR-
Former’s convolutional branch, while its self-attention branch handles the
low-resolution images. This strategic allocation is designed to fully utilize
the unique benefits of each processing approach. To confirm the DR-Former’s
hybrid design’s efficiency, we conducted a performance analysis of its
individual CNN and Transformer branches, applying each separately for feature
extraction. The results of this analysis are detailed in Table 2.

Table 2: Performance Assessment of Individual Sub-Networks in the Hybrid Dual-
Resolution Transformer (DR-Former).

Method | ACC | AUC | F1 | Kappa
---|---|---|---|---
CT (3-phase) |  |  |  |
Only Transformer | 0.7621 | 0.8552 | 0.7549 | 0.5131
Only CNN | 0.7765 | 0.8733 | 0.7705 | 0.5434
SDR-Former | 0.8537 | 0.9168 | 0.8529 | 0.7058
MR (8-phase) |  |  |  |
Only Transformer | 0.6731 | 0.9360 | 0.6721 | 0.6056
Only CNN | 0.6827 | 0.9228 | 0.7078 | 0.6282
SDR-Former | 0.7885 | 0.9536 | 0.7910 | 0.7467
|  |  |  |

The experimental results robustly support the notion that exclusive reliance
on either CNN or Transformer architectures separately results in less than
optimal performance. Notably, the standalone usages of CNNs demonstrate
superior outcomes compared to Transformers used alone. This underscores
potential challenges in optimizing pure Transformer models, as suggested in
recent research [42]. The DR-Former’s design, which merges CNN and Transformer
and seamlessly integrates them through the BCIM, achieves notably enhanced
performance over the singular use of each technique. This dual-branch,
synergistic method excels in extracting multi-resolution features. The CNN’s
inherent biases aid in more efficient training of the Transformer component,
leading to a notable increase in the precision of lesion classification.

####  5.2.2 Influence of Core Integral Modules

The BCIM and APSM are crucial components of our SDR-Former architecture, each
bringing unique enhancements to the processing of multi-phase features. The
BCIM boosts the feature representation by facilitating interactions between
convolutional and self-attention-derived features within each imaging phase.
Simultaneously, the APSM is designed for dynamic inter-phase communication,
enhancing the significance of features across phases based on the particular
characteristics of the input lesions. To thoroughly assess the individual and
collective contributions of these components, a set of ablation studies was
performed on both CT and MR datasets. Initially, we established a Baseline
model devoid of the contextual interaction offered by the BCIM and the APSM.
This Baseline served as a comparative foundation for further experiments.
Subsequently, the Baseline was enhanced with the APSM module, termed ’w/
APSM,’ to gauge its effect on the adaptive prioritization of phase-specific
features. In a separate experiment, the BCIM module was integrated into the
Baseline, labeled ’w/ BCIM,’ to evaluate the benefit of improved feature
interaction. The final configuration, encompassing both the APSM and BCIM, was
analyzed to determine their combined influence within the SDR-Former
framework. The results from these analyses are concisely depicted in Table 3.

Table 3: Evaluation of the Effects of Integrating Various Network Components.

Method | ACC | AUC | F1 | Kappa
---|---|---|---|---
CT (3-phase) |  |  |  |
Baseline | 0.7974 | 0.8776 | 0.7942 | 0.5889
Baseline+BCIM | 0.8376 | 0.9134 | 0.8359 | 0.6720
Baseline+APSM | 0.8055 | 0.8995 | 0.8018 | 0.6044
SDR-Former | 0.8537 | 0.9168 | 0.8529 | 0.7058
MR (8-phase) |  |  |  |
Baseline | 0.7404 | 0.9410 | 0.7508 | 0.6908
Baseline+BCIM | 0.7692 | 0.9418 | 0.7730 | 0.7196
Baseline+APSM | 0.7596 | 0.9520 | 0.7636 | 0.7051
SDR-Former | 0.7885 | 0.9536 | 0.7910 | 0.7467
|  |  |  |

The findings offer compelling evidence of the performance enhancements
attributed to the BCIM and APSM modules within the SDR-Former model, as
evidenced by various quantitative metrics. Notably, the APSM module
demonstrates a more pronounced impact on the MR dataset compared to the CT
dataset. This likely stems from the MR dataset’s greater phase variety,
encompassing eight phases versus the CT dataset’s three, enabling the APSM to
more effectively perform adaptive feature selection across a broader range of
phase scenarios. Regarding the BCIM, an observable boost in model performance
is noted on both datasets, underscoring the module’s vital role in mediating
contextual information exchange between the CNN and Transformer branches, thus
improving feature representation. The simultaneous incorporation of BCIM and
APSM leads to even more substantial improvements, verifying the advantages of
their synergistic integration as reflected in the outcomes. Furthermore, Table
5.2.2 provides insights into the computational load associated with BCIM and
APSM. The results indicate that despite their relatively low computational
requirements, these modules make a significant contribution to overall
performance, confirming their effectiveness and efficiency.

Table 4: Evaluation of Computational Complexity for Various Models: Benchmarks
Performed Using a Single NVIDIA GeForce RTX 3090 GPU with Multi-Phase Inputs.
Dimensions: 14 $\times$ 112 $\times$ 112.

N | Method | FLOPs (G) | Params (M)
---|---|---|---
3 Phases | Baseline | 102.07 | 27.87
| w/ BCIM | 102.29 | 28.43
| w/ APSM | 102.07 | 27.96
| SDR-Former | 102.30 | 28.52
8 Phases | Baseline | 40.17 | 19.02
| w/ BCIM | 40.26 | 19.25
| w/ APSM | 40.18 | 19.11
| SDR-Former | 40.26 | 19.34
|  |  |

Visualization of BCIM’s Role: The Grad-CAM technique [43] was employed to
visualize the impact of the BCIM on activation patterns within the SDR-Former
model. This visualization targets intermediate output feature maps from both
CNN and Transformer branches.

Figure 6 illustrates that, in configurations without the BCIM, the CNN branch
exhibits relatively limited receptive fields, while the Transformer branch
shows numerous non-specific activations. However, with the inclusion of the
BCIM, which encourages interaction between the branches, a notable improvement
in feature representation quality is observed. This enhanced representation
leads to more precise lesion localization in various imaging phases. The CNN
branch becomes more efficient in pinpointing localized features without excess
noise, and the Transformer branch, utilizing its capacity for global context
comprehension, accurately outlines lesion contours. To conclude, the BCIM acts
as a connective mechanism, effectively reducing the disparity between the two
branches and enabling them to work together more effectively, thus enriching
the quality of multi-phase feature representations.

Figure 6: Feature Activation Visualization Using Grad-CAM: Comparative
Analysis for (a) CT and (b) MR Datasets. Green overlays indicate the model
without BCIM, whereas red overlays depict the model with BCIM integrated. The
abbreviations ‘CNN’ and ‘Trans.’ refer to the CNN and Transformer segments of
the SDR-Former model, respectively. For enhanced detail, zooming in is
advisable.

Visualization of APSM’s Role: The adaptive phase selection capability of the
APSM is also visually illustrated through its ‘Attention Coefficients’
mapping. This visualization aims to exhibit how APSM variably allocates
attention weights across phases, based on the relative importance of their
feature activations. As shown in Figure 7, the APSM is adept at
differentiating between phases, assigning greater attention to those with more
pronounced feature activations that suggest the presence of lesions.
Additionally, the attention variation among phases validates the module’s
effectiveness in identifying the most informative phases for precise lesion
localization. This confirms APSM’s dynamic phase selection capacity and its
significant contribution to the enhanced performance of the SDR-Former model.

Figure 7: Illustration of Significance-Oriented Coefficients Produced by the
Adaptive Phase Selection Module (APSM) for (a) CT and (b) MR Datasets. The
intensity of the blue color indicates the strength of the coefficients,
reflecting the impact of the APSM on the selection of phase-specific features.
For enhanced detail, zooming in is advisable.

####  5.2.3 Selection of Self-attention Modules

In our investigation to determine the most suitable self-attention
architecture for the Transformer branch of our framework, we conducted an
analytical comparison of four contemporary self-attention designs: Swin [27],
SRA [28], PSA [29], and GSA [32]. Originally developed for 2D imagery, these
architectures underwent necessary adaptations for their application in 3D
imaging scenarios. These modifications were meticulously implemented following
the principles of the original methodologies, ensuring a smooth transition
from 2D to 3D image processing capabilities within our model.

Table 5: Performance Analysis with Various Self-Attention Mechanisms.

Method | ACC | AUC | F1 | Kappa
---|---|---|---|---
CT (3-phase) |  |  |  |
Swin | 0.8087 | 0.8989 | 0.8086 | 0.6221
SRA | 0.8264 | 0.9127 | 0.8230 | 0.6468
PSA | 0.8441 | 0.9075 | 0.8434 | 0.6871
GSA | 0.8537 | 0.9168 | 0.8529 | 0.7058
MR (8-phase) |  |  |  |
Swin | 0.7787 | 0.9508 | 0.7738 | 0.7511
SRA | 0.7788 | 0.9530 | 0.7760 | 0.7705
PSA | 0.7884 | 0.9507 | 0.7830 | 0.7700
GSA | 0.7885 | 0.9536 | 0.7910 | 0.7467
|  |  |  |

The comparative analysis of self-attention modules, as outlined in Table 5,
indicates that Swin’s efficacy is somewhat lower compared to other models. The
architecture of Swin, with its emphasis on localized window-based attention,
may not be as effective in capturing expansive relationships, a key element in
Transformer design. Conversely, SRA and PSA maintain a more comprehensive view
of self-attention by evaluating token-to-region relationships, though they do
reduce the dimensions of key and value matrices. While this approach betters
Swin’s output, it comes with the potential shortcoming of information
diminution due to feature downsampling. GSA, however, stands out as the most
effective among the evaluated modules. It retains the full architecture of the
query, key, and value matrices, preventing any unintended loss of information.
Additionally, GSA employs a sparse token-linking strategy that broadens its
range to a global scale, significantly boosting its utility in the evaluated
tasks.

####  5.2.4 Benefits of Multi-Phase Imaging in Diagnostic Accuracy

To evaluate the impact of multi-phase imaging on enhancing diagnostic
accuracy, we performed an ablation study using the CT dataset. It is important
to note that with single-phase input, the APSM is not utilized. Table 6
demonstrates that relying solely on non-contrast phases leads to the least
favorable outcomes, aligning with clinical observations. The contrast-enhanced
phases significantly improve diagnostic performance as arterial and/or venous
phases enhance lesion visibility. Especially noteworthy is the venous phase,
which excels in clearly differentiating lesions from adjacent tissues. Our
analysis also highlights the SDR-Former’s capability to effectively utilize
multi-phase data, evidently outperforming single-phase approaches. This
substantiates the premise that multi-phase data is vital in enriching feature
depiction for more precise lesion identification. The model achieves optimal
performance when incorporating all available phases, thereby emphasizing the
critical importance of a comprehensive multi-phase imaging approach for
accurate liver lesion classification.

Table 6: Influence of Multi-Phase Imaging in Diagnostic Accuracy.

Non-contrast | Arterial | Venous | ACC | AUC | F1 | Kappa
---|---|---|---|---|---|---
|  |  | 0.7788 | 0.8594 | 0.7787 | 0.5606
|  |  | 0.7885 | 0.8516 | 0.7865 | 0.5736
|  |  | 0.7981 | 0.8241 | 0.7938 | 0.5907
|  |  | 0.8077 | 0.8627 | 0.8051 | 0.6115
|  |  | 0.8173 | 0.8579 | 0.8159 | 0.6322
|  |  | 0.8269 | 0.8675 | 0.8246 | 0.6504
|  |  | 0.8537 | 0.9168 | 0.8529 | 0.7058
|  |  |  |  |  |

###  5.3 Efficacy in Cross-Modality Transfer Learning

Our proposed SDR-Former model is inherently designed for effective transfer
learning across datasets with varied phase counts, showcasing its scalability.
To validate this capability, we carried out a set of empirical tests.
Initially, the SDR-Former was trained using our three-phase CT dataset, then
its learned weights were fine-tuned for application to the eight-phase MR
dataset. Conversely, a similar experiment was conducted where the model,
initially trained on our eight-phase MR dataset, had its weights fine-tuned
for the three-phase CT dataset. The experimental results outlined in Table 7,
demonstrate a notable performance improvement with the application of cross-
modality transfer learning, underscoring the SDR-Former model’s adaptability
and scalability. Remarkably, utilizing pre-trained weights from the more
complex, eight-phase MR dataset yielded more pronounced improvements than
those from the simpler three-phase dataset, despite its smaller size. This
suggests that the complex patterns gleaned from a dataset with a greater
number of phases can enrich the model’s knowledge base, thereby enhancing its
performance even on datasets with fewer phases.

Table 7: Efficacy of Transfer Learning.

SDR-Former | ACC | AUC | F1 | Kappa
---|---|---|---|---
CT (3-phase) |  |  |  |
From Scratch | 0.8537 | 0.9168 | 0.8529 | 0.7058
MR pre-training | 0.8923 | 0.9352 | 0.8915 | 0.7830
MR (8-phase) |  |  |  |
From Scratch | 0.7885 | 0.9536 | 0.7910 | 0.7467
CT pre-training | 0.7981 | 0.9350 | 0.8003 | 0.7581
|  |  |  |

###  5.4 Versatility Across Various Medical Datasets

The DR-Former, the cornerstone of our framework, is designed as a universal
vision backbone. In this section, we broaden the assessment of our model via
extending its primary usage of multi-phase imaging. To be specific, we
assessed the feature extraction capability of the DR-Former using MedMNIST v2,
a recognized benchmark in medical image classification [44], specifically
focusing on subsets from MedMNIST3D, including OrganMNIST3D and NoduleMNIST3D.
A distinctive feature of the DR-Former is its dual-resolution feature map
generation. In our experiments, we down-scaled the higher-resolution maps to
match the lower-resolution ones, then merged them. This combination was
followed by processing through a GAP layer and a FC layer for classification.
The comparative results presented in Table 8 consistently demonstrate the DR-
Former’s enhanced performance in comparison to other top-performing methods
across multiple metrics. These findings highlight the DR-Former’s robustness
and establish it as an effective tool for medical image analysis, confirming
its broad effectiveness beyond multi-phase imaging applications.

Table 8: Comparative analysis of model performance on MedMNIST v2 dataset.

Method | ACC | AUC | F1 | Kappa
---|---|---|---|---
OrganMNIST3D |  |  |  |
ResNet-50 | 0.9541 | 0.9988 | 0.9599 | 0.9489
DenseNet-121 | 0.9459 | 0.9978 | 0.9551 | 0.9397
MCSCNN | 0.9426 | 0.9983 | 0.9521 | 0.9361
BoTNet-50 | 0.9475 | 0.9987 | 0.9553 | 0.9415
UniFormer-S | 0.9279 | 0.9977 | 0.9391 | 0.9196
H2Former | 0.9410 | 0.9980 | 0.9495 | 0.9342
DR-Former | 0.9623 | 0.9990 | 0.9681 | 0.9580
NoduleMNIST3D |  |  |  |
ResNet-50 | 0.8871 | 0.8939 | 0.8267 | 0.6534
DenseNet-121 | 0.8806 | 0.9057 | 0.8168 | 0.6336
MCSCNN | 0.8806 | 0.9177 | 0.8229 | 0.6460
BoTNet-50 | 0.8742 | 0.8889 | 0.8022 | 0.5595
UniFormer-S | 0.8548 | 0.8521 | 0.7798 | 0.5602
H2Former | 0.8710 | 0.9190 | 0.8053 | 0.6107
DR-Former | 0.9032 | 0.9216 | 0.8430 | 0.6865
|  |  |  |

##  6 Conclusion

This study introduces the SDR-Former, an innovative architecture specifically
developed for classifying liver lesions in 3D multi-phase CT and MR images.
The model capitalizes on an SNN to concurrently process multiple phases,
thereby enhancing the robustness of feature representation while maintaining
scalability and computational efficiency. To enhance the feature
representation power of the SNN, a hybrid DR-Former is incorporated. This DR-
Former combines the capabilities of CNNs and Transformers to optimize both
local details and global context understanding. In addition, a BCIM is
incorporated to enable interaction between features from the CNN and
Transformer branches. Furthermore, an APSM is utilized to strategically
highlight key multi-phase features for accurate classification. Extensive
testing on two clinical 3D multi-phase CT and MR datasets demonstrates our
model’s flexibility and effectiveness under various phase conditions. To
contribute to research in multi-phase liver lesion analysis, we plan to
release our MR dataset publicly.

Despite its strengths, the SDR-Former does have limitations, particularly its
dependence on an SNN for multi-phase input processing. While the model
exhibits remarkable scalability, its effectiveness could be less pronounced in
scenarios with fewer phases, such as dual-phase cases, where feature-level
phase fusion might be more effective. However, the model’s ability to adapt to
a range of phase conditions is a notable advantage, typically unachievable by
feature-level fusion methods due to their high computational demands. Future
developments will focus on refining the feature extraction process by
integrating dynamic convolutions into the CNN branch [45]. This enhancement
aims to produce phase-specific features within a weight-sharing framework,
potentially leading to unique feature extractors for each imaging phase.

## Ethical Approval

The ethical approval for the utilization of CT dataset in this study has been
granted by the Institutional Review Board (IRB) at Sir Run Run Shaw Hospital,
Zhejiang University School of Medicine, Hangzhou, Zhejiang, China (IRB
Approval No: 20200903-43). Similarly, the ethical approval for the utilization
of MR dataset has obtained from the IRB at Ningbo Medical Center Lihuili
Hospital, Ningbo, Zhejiang, China (IRB Approval No: KY2022SL446-01).

## Data Availability

The MR dataset is available at: https://bit.ly/3IyYlgN.

## Acknowledgement

This work was supported in part by the National Key Research and Development
Program of China (No. 2021ZD0113302).

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[c]Judah Unmuth-Yockey

# Effective ${\mathbb{Z}}_{3}$ model for finite-density QCD with tensor
networks

Jacques Bloch  Robert Lohmayer  Sophia Schweiß

###### Abstract

The tensor renormalization group is a promising numerical method used to study
lattice statistical field theories. However, this approach is computationally
expensive in 2+1 and 3+1 dimensions. Here we use tensor renormalization group
methods to study an effective three-dimensional ${\mathbb{Z}}_{3}$ model for
the heavy-quark, high-temperature, strong-coupling limit of single-flavor 3+1
dimensional quantum chromodynamics. Our results are cross-checked using the
worm Monte Carlo algorithm. We present the phase diagram of the model through
the measurement of the Polyakov loop, the nearest-neighbor Polyakov loop
correlator, and their susceptibilities. The tensor renormalization group
results are in good agreement with the literature.

##  1 Introduction

Tensor-network methods provide a tool to approximately coarse-grain a
classical partition function. Moreover, in certain cases, the method is immune
to the sign problem, since there is no probabilistic sampling involved. Here
we apply two tensor methods—the higher-order tensor renormalization group
(HOTRG) [1], and the the triad tensor renormalization group (TTRG) [2]—in
conjunction with the worm Monte Carlo algorithm [3], to study an effective
theory for one-flavor, finite-density, 3+1-dimensional quantum chromodynamics
(QCD). The effective theory has a sign problem in the original formulation,
however this can be overcome with a change of variables, making the model
ideal to test the tensor methods against the Monte Carlo results. We first
derive the effective theory, then provide a tensor representation of the
theory, and finally report our results.

##  2 Single-flavor, finite-density QCD

The starting action is the standard lattice Wilson gauge action, and Wilson
single-flavor fermion action. The action has two parts, $S=S_{g}+S_{f}$, with
the gauge action given by

$\displaystyle S_{g}$
$\displaystyle=-\frac{\beta}{3}\sum_{x=1}^{N}\left[\frac{a}{a_{t}}\sum_{i=1}^{3}\text{Re}[\text{Tr}\left[U_{x,i4}\right]]+\frac{a_{t}}{a}\sum_{i<j=1}^{3}\text{Re}[\text{Tr}\left[U_{x,ij}\right]]\right]$
(1)

and the fermion action by

$\displaystyle S_{f}=\sum_{x,y}\bar{\psi}_{x}M_{xy}\psi_{y},\quad M$
$\displaystyle=\mathbf{1}-\kappa\frac{a_{t}}{a}H_{s}-\kappa H_{t}$ (2)

with the spatial and temporal hoppings

$\displaystyle H_{s}=\sum_{i=1}^{3}\left(T_{i}^{+}+T_{i}^{-}\right),\quad
H_{t}=e^{\tilde{\mu}a_{t}}T_{4}^{+}+e^{-\tilde{\mu}a_{t}}T_{4}^{-}$ (3)

and

$\displaystyle(T_{\nu}^{\pm})_{xy}$
$\displaystyle=(1\pm\gamma_{\nu})U_{x,\pm\nu}\delta_{y,x\pm\hat{\nu}}.$ (4)

Here, the gauge coupling enters through $\beta=6/g^{2}$ and we identify
$\kappa$ as the fermion hopping parameter. The chemical potential is
$\tilde{\mu}$, and $a$ and $a_{t}$ are the spatial and temporal lattice
spacings, respectively. In anticipation of deriving the effective theory, we
define the temperature as given by the inverse physical extent of the lattice,

$T=\frac{1}{N_{t}a_{t}},$ (5)

with $N_{t}$ the number of lattice sites in the temporal direction. We work on
a finite lattice with anti-periodic boundary conditions in the temporal
direction, and periodic boundary conditions in the spatial directions.

##  3 The effective action

The effective action we study corresponds to a particular limiting case. To
derive it we follow Ref. [4] closely. We consider the strong-coupling, high-
temperature limit, where we take the temporal extent of the lattice very
short—in this case we set $N_{t}=1$—and shrink the temporal lattice spacing.
This corresponds to

$\displaystyle\beta\ll 1,\quad a_{t}\ll 1.$ (6)

In addition we take the large-chemical-potential, large-quark-mass limit,
which in the action corresponds to

$\displaystyle\tilde{\mu}\gg 1,\quad\kappa\ll 1.$ (7)

In $S_{g}$ spatial plaquettes are suppressed by a factor of $a_{t}^{2}$
relative to the temporal plaquettes, and in $S_{f}$ spatial hopping is
suppressed. Since the confinement-deconfinement transition is believed to be
related to center-symmetry breaking, we replace the $SU(3)$ group elements by
elements of the center of the group, ${\mathbb{Z}}_{3}$, which is an Abelian
group. For $S_{g}$ we find

$\displaystyle
S_{g}\rightarrow-\frac{\beta}{2a_{t}}\sum_{x}\sum_{\nu=1}^{3}P^{*}_{x}P_{x+\hat{\nu}}+\text{c.c.}$
(8)

and for $S_{f}$

$\displaystyle
S_{f}\rightarrow-\kappa\sum_{x}\left[e^{\mu}P_{x}+e^{-\mu}P^{*}_{x}\right]$
(9)

with $P_{x}\in{\mathbb{Z}}_{3}$, which can be interpreted as a Polyakov loop
in the short, compactified temporal direction, and $\mu=a_{t}\tilde{\mu}$.
This results in an effective action

$S_{\text{eff}}=-\sum_{x}\left[\tau\sum_{\nu=1}^{3}\left(P^{*}_{x}P_{x+\hat{\nu}}+\text{c.c.}\right)+\left(\eta
P_{x}+\bar{\eta}P^{*}_{x}\right)\right]$ (10)

with $\tau\equiv\beta/2a_{t}$ and $\eta\equiv\kappa e^{\mu}$ and
$\bar{\eta}=\kappa e^{-\mu}$. A graphical illustration of the Polyakov loop
variables on the lattice can be seen on the left-hand side of Fig. 1. The
effective theory lives in three Euclidean dimensions, and is to be interpreted
as a statistical theory, in equilibrium, in three spatial dimensions.

##  4 The tensor network

Figure 1: (left) An illustration of the original degrees of freedom. At each
site there is a ${\mathbb{Z}}_{3}$ element (a Polyakov loop) which interacts
with its nearest neighbors. (right) In the Fourier basis, there are “charges”
and “currents” on the sites, and links, respectively, which must obey a
conservation-law constraint at each site.

The partition function for the effective action describing this theory of
interacting ${\mathbb{Z}}_{3}$ “spins” is given by,

$\displaystyle Z=\sum_{\\{P\\}}e^{-S_{\text{eff}}},$ (11)

where the sum, $\sum_{\\{P\\}}$, means a sum over all possible configurations
of the ${\mathbb{Z}}_{3}$ variables. This can be recast to a theory of
“currents” and “charges” by Fourier expansion of the elementary Boltzmann
weights [5],

$\displaystyle\exp(\tau[P_{x}P_{x+\hat{\nu}}^{*}+\text{c.c.}])$
$\displaystyle=C\sum_{n_{x,\nu}\in\\{-1,0,1\\}}B^{|n_{x,\nu}|}(P_{x}P_{x+\hat{\nu}}^{*})^{n_{x,\nu}}$
(12)

and

$\displaystyle\exp(\eta P_{x}+\bar{\eta}P_{x}^{*})$
$\displaystyle=\sum_{m_{x}\in\\{-1,0,1\\}}V_{m_{x}}P_{x}^{m_{x}}$ (13)

where $B$ and $V$ can be found by inverting the above. These expansions allow
the original ${\mathbb{Z}}_{3}$ variables to be summed over, leaving only the
$n$ and $m$ integer variables as the new degrees of freedom. The resulting
partition function is written as,

$\displaystyle
Z=\sum_{\\{n\\}}\sum_{\\{m\\}}\left(\prod_{x,\nu}B_{n_{x,\nu}}(\tau)\right)\left(\prod_{x}V_{m_{x}}(\eta,\bar{\eta})\Delta^{(3)}(j_{x})\right),$
(14)

with $j_{x}=\sum_{\nu=1}^{3}(n_{x,\nu}-n_{x-\hat{\nu},\nu})+m_{x}$, and
$\Delta^{(3)}(j_{x})=\delta_{j_{x}\text{mod}3,0}$. A graphical illustration of
the new degrees of freedom on the lattice can be seen in Fig. 1 on the right-
hand side.

A tensor network can be built from the tensor,

$\displaystyle
T^{(x)}_{n_{x-\hat{1},1}n_{x,1}n_{x-\hat{2},2}n_{x,2}n_{x-\hat{3},3}n_{x,3}}=\sqrt{B_{n_{x-\hat{1},1}}B_{n_{x,1}}B_{n_{x-\hat{2},2}}B_{n_{x,2}}B_{n_{x-\hat{3},3}}B_{n_{x,3}}(\tau)}V_{m_{x}}(\eta,\bar{\eta})\Delta^{(3)}(j_{x})$
(15)

and the partition function written as a tensor trace

$\displaystyle Z=\text{Tr}\left[\prod_{x}T^{(x)}\right].$ (16)

We calculate the average Polyakov loop and average nearest-neighbor
correlator, and their susceptibilities,

$\displaystyle\langle P\rangle=\frac{1}{V}\frac{\partial\ln
Z}{\partial\eta},\quad\langle E\rangle=-\frac{1}{V}\frac{\partial\ln
Z}{\partial\tau}$ (17) $\displaystyle\chi_{P}=V(\langle P^{2}\rangle-\langle
P\rangle^{2}),\quad\chi_{E}=V(\langle E^{2}\rangle-\langle E\rangle^{2}).$
(18)

We do this using “impure” tensors [6, 7], and finite differences of the
logarithm of the partition function.

##  5 Numerical methods

Figure 2: A graphical illustration of the decomposition of the fundamental
six-indexed tensor into four triad tensors.

Using $T$ from Eq. (15) we contract the tensor network using the HOTRG. This
can be done in the original HOTRG prescription [1], or with additional
approximations using the TTRG [2]. Within the TTRG, The $T$ tensor can be
decomposed into four, three-indexed tensors called triads,

$\displaystyle T_{ijklmn}=\sum_{\alpha,\beta,\gamma}A_{ij\alpha}B_{\alpha
k\beta}C_{\beta l\gamma}D_{\gamma mn}.$ (19)

A graphical representation of the triad decomposition can be seen in Fig. 2.
The tensor network contraction with these can be done efficiently. In addition
to the two tensor methods, we also use the worm Monte Carlo method [3, 5]. We
report results using the original HOTRG, triad HOTRG, and worm Monte Carlo.

##  6 Results

We calculate the observables mentioned in Sec. 4 using three algorithms: the
HOTRG, the TTRG and worm Monte Carlo, and compare them in various
circumstances. In Fig. 3 we calculate the average nearest-neighbor Polyakov
loop correlator at $\kappa=0$. We do this using the TTRG with a bond dimension
($D$) equal to 40 for several volumes to demonstrate convergence. On the
right-hand side of Fig. 3 we show the average Polyakov loop at a small value
of $\kappa=1\times 10^{-5}$, calculated using the TTRG using $D=40$. In both
figures we see evidence of a phase transition.

Figure 3: (left) The average nearest-neighbor Polyakov loop correlator
evaluated at $\kappa=0$ as a function of $\tau$ using the TTRG with $D=40$
states. Here multiple volumes are shown to demonstrate convergence. (right)
The average Polakov loop as a function of $\tau$, evaluated at $\kappa=1\times
10^{-5}$ using the TTRG with $D=40$ states, at a volume of $V=1024^{3}$.

In Fig. 4 we calculated the average Polyakov loop at finite $\tau$ and
$\kappa$, with $\tau=0.1$, and $\kappa=0.05$, for a range of $\mu$ values. On
the left-hand side we see a comparison between the three methods, with all
three showing good agreement. On the right-hand side we plot a heat-map
visualization of the average Polyakov loop in the $\tau$-$\mu$ plane, at
$\kappa=0.05$. This heat-map data was generated using the HOTRG with $D=9$.

Figure 4: (left) The average Polyakov loop evaluated at fixed $\tau=0.1$, and
$\kappa=0.05$ as a function of $\mu$. Three algorithms are compared: the
HOTRG, the TTRG, and the worm Monte Carlo. We find good agreement between all
three methods, and the number of states used in each tensor method is in the
legend. (right) A heat map of the average Polyakov loop in the $\mu$-$\tau$
plane, evaluated at $\kappa=0.05$, using the HOTRG method with $D=9$ states.

Figure 5 shows the Polyakov loop susceptibility. On the left-hand side we
again have a comparison between the three numerical methods, and the three
showing good agreement, except at small values of $\mu$ where all three
methods are somewhat noisy. On the right-hand side we plot a heat-map of the
susceptibility in the $\tau$-$\mu$ plane at $\kappa=0.05$. This data was
generated using the HOTRG with $D=9$.

Figure 5: (left) The Polyakov loop susceptibility evaluated at $\tau=0.1$,
$\kappa=0.05$ as a function of $\mu$. Here three algorithms are compared: the
HOTRG, the TTRG, and the worm Monte Carlo. We find good agreement between the
three methods, with noise among all three at small $\mu$. The number of states
for the two tensor methods is found in the legend. (right) A heat-map of the
Polyakov loop susceptibility in the $\mu$-$\tau$ plane evaluate at
$\kappa=0.05$ using the HOTRG method with $D=9$ states.

##  7 Conclusion

We have studied 3+1-dimensional, single-flavor, finite-density QCD in the
strong-coupling, high-temperature limit where the model undergoes dimensional
reduction to a three-dimensional Abelian spin model. We studied this effective
theory using three different methods: the triad tensor renormalization group,
the higher-order tensor renormalization group, and the worm Monte Carlo
sampling method. Where calculations were clean, the three methods agree with
each other well. We found the tensor methods were able to reproduce known
features of the phase diagram well; however, calculations at very small values
of $\kappa$, and small values of $\mu$ were noisy and unreliable in all three
methods.

## References

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  * [2] D. Kadoh and K. Nakayama, _Renormalization group on a triad network_ , _arXiv_ [hep-lat] (2019) 1912.02414.
  * [3] N. Prokof’ev and B. Svistunov, _Worm algorithms for classical statistical models_ , _Phys. Rev. Lett._ 87 (2001) 160601.
  * [4] T. Degrand and C. Detar, _Phase structure of qcd at high temperature with massive quarks and finite quark density: A z(3) paradigm_ , _Nuclear Physics B_ 225 (1983) 590.
  * [5] Y.D. Mercado, H.G. Evertz and C. Gattringer, _Qcd phase diagram according to the center group_ , _Phys. Rev. Lett._ 106 (2011) 222001.
  * [6] Z.-C. Gu, M. Levin and X.-G. Wen, _Tensor-entanglement renormalization group approach as a unified method for symmetry breaking and topological phase transitions_ , _Phys. Rev. B_ 78 (2008) 205116.
  * [7] S. Morita and N. Kawashima, _Calculation of higher-order moments by higher-order tensor renormalization group_ , _Computer Physics Communications_ 236 (2019) 65.


[a]Diksha Garg

# Atmospheric muon fluxes at sub-orbital neutrino detectors

Mary Hall Reno

###### Abstract

Very-high-energy and ultra-high-energy neutrinos are messengers of energetic
sources in the universe. Sub-orbital and satellite-based neutrino telescopes
employ detectors of the atmospheric Cherenkov emission from extensive air
showers (EASs) generated by charged particles. These Cherenkov detectors can
be pointed below or above the Earth’s limb. Cherenkov emissions produced from
directions below the limb are from upward-going EASs produced in the
atmosphere sourced by Earth-skimming neutrinos. When the Cherenkov telescope
is pointed slightly above the Earth’s limb, signals from EASs are initiated by
cosmic ray interactions in the atmosphere. For sub-orbital detectors, muons
produced from cosmic rays in the atmosphere can directly hit the Cherenkov
telescope. Using a semi-analytic technique with cascade equations for
atmospheric particle fluxes, we quantify the atmospheric muon flux that
reaches sub-orbital telescopes like Extreme Universe Space Observatory Super
Pressure Balloon 2 (EUSO-SPB2). We assess this potential background to the EAS
signals. The calculation technique may also provide an understanding of the
evolution of the muon content in individual EAS.

##  1 Introduction

Neutrinos, neutral and weakly interacting particles, can travel astronomical
distances unhindered and act as messengers from the distant universe. To
detect the very-high-energy (VHE) ($E>10^{15}$ eV) neutrino flux, large volume
neutrino targets/detectors are needed. One approach is to use Earth as a
neutrino converter, where neutrinos propagating through the Earth interact to
produce charged leptons (electrons, muons, and tau-leptons). The charged
leptons can exit the Earth and create extensive-air-showers (EASs) in the
atmosphere. The EASs have optical Cherenkov, radio emission and fluorescent
radiation associated to them. The radiation can be detected by ground-based,
sub-orbital, and orbital neutrino telescopes such as IceCube [1], EUSO-SPB2
[2, 3], and a future POEMMA [4]. Another feature of cosmic ray interactions in
the atmosphere is the generation of atmospheric neutrino and muon fluxes. The
atmospheric muon flux incident on EUSO-SPB2 is the subject of this study.

EUSO-SPB2 had a fluorescence telescope (FT) (nadir pointing) and a Cherenkov
telescope (CT) that pointed above the limb to detect EAS from cosmic rays. It
flew at an altitude of 33 km from the surface of the Earth. The muons produced
from cosmic rays interactions in the atmosphere produce charged pions and
kaons that can decay to muons. These muons can directly hit the CT and the FT.
This is depicted in fig. 1, where $\alpha$ describes the incident direction of
the cosmic rays, which can be below or above the telescope’s horizon. For the
telescopes at an altitude of 33 km, $\alpha>84.2^{\circ}$ for cosmic ray
trajectories to be above the Earth’s limb.

Direct muon hits from the atmospheric muon flux can act as a potential
background to the EAS signals measured by the CT and the FT. The atmospheric
lepton fluxes are well-studied for ground-based instruments (see, e.g., refs.
[5, 6, 7, 8, 9]), but less so for sub-orbital instruments like EUSO-SPB2.
Here, we make a first estimate of the rate of atmospheric muons hitting the
EUSO-SPB2 telescopes.

Figure 1: Sub-orbital telescope (like, EUSO-SPB2) at $33$ km altitude. In red
is the particle trajectory to the telescope surface at different $\alpha$
angles ($\alpha=0^{\circ}$ at nadir). The cosmic rays ($p$) interact in the
atmosphere to create pions and kaons which decay to muons. At a given point in
the particle’s trajectory, $h$ represents the altitude of that point and
$\ell$ represents the remaining trajectory distance from that point to the
telescope. The figure is not to scale.

##  2 Atmospheric lepton fluxes with the $Z$-moment approximation

The atmospheric particle flux $\phi_{j}(E,X)$ for particle $j$, as a function
of energy $E$ and column depth $X$ can be written as

$\displaystyle\frac{d\phi_{j}(E,X)}{dX}$ $\displaystyle=$
$\displaystyle-\frac{\phi_{j}(E,X)}{\lambda_{j}(E)}-\frac{\phi_{j}(E,X)}{\lambda_{j}^{\rm
dec}(E)}+\sum S(k\to j)\,,$ (1) $\displaystyle S(k\to j)$ $\displaystyle=$
$\displaystyle\int_{E}^{\infty}dE^{\prime}\frac{\phi_{k}(E^{\prime},X)}{\lambda_{k}(E^{\prime})}\frac{dn(k\to
j;E^{\prime},E)}{dE}\,,$ (2)

for all particles except for muons. The symbol $\lambda_{j}(E)$ and
$\lambda_{j}^{\rm dec}(E)=E\tau_{j}\rho/m_{j}$ are the interaction length and
the decay length, respectively, of the particle $j$ in the atmosphere. The
quantities $E$, $\tau_{j}$, $m_{j}$ are the energy, lifetime and mass of the
particle $j$. We have suppressed the angular ($\alpha$) dependence of
$\phi_{j}$ and $X$. The column depth is given as:

$X(\ell,\alpha)=\int_{\ell}^{\infty}d\ell^{\prime}\rho(h(\ell^{\prime},\alpha)),$
(3)

where $h(\ell,\alpha)$ is the altitude, $\ell$ is the trajectory distance and
$\alpha$ is the nadir angle, also shown in fig. 1. The atmospheric density
($\rho$) considered here is given by an exponential distribution,

$\rho=\rho_{0}\exp(-h/h_{0})\ ,$ (4)

where $h_{0}=6.4$ km and $\rho_{0}h_{0}=1300$ g/cm2. We approximate particle
trajectories as one-dimensional, a good approximation above 10 GeV [9]. The
interaction and decay distributions in the source term $S(k\to j)$ (eq. (2))
in terms of cross-section ($\sigma$) and decay width ($\Gamma$) are

$\displaystyle\frac{dn(k\to j;E^{\prime},E)}{dE}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{kA}(E^{\prime})}\frac{d\sigma(kA\to
jY;E^{\prime},E)}{dE}\quad{\rm(interaction)\,,}$ (5)
$\displaystyle\frac{dn(k\to j;E^{\prime},E)}{dE}$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma_{k}(E^{\prime})}\frac{d\Gamma(k\to
jY;E^{\prime},E)}{dE}\quad{\rm(decay)}\ .$ (6)

For muons, electromagnetic energy loss is incorporated in the continuous-
energy-loss approximation, where

$\Biggl{\langle}\frac{dE}{dX}\Biggr{\rangle}\simeq\frac{dE}{dX}\simeq-(a+bE)\equiv-\beta(E)\,.$
(7)

Here, $a$ and $b$ are the energy loss parameters accounting for ionization,
and bremsstrahlung, pair production and photo-nuclear energy loss,
respectively. In this approximation

$\frac{d\phi_{\mu}(E,X)}{dX}=-\frac{\phi_{\mu}(E,X)}{\lambda_{j}^{\rm
dec}(E)}+\frac{\partial}{\partial
E}\Bigl{[}\beta(E)\phi_{\mu}(E,X)\Bigr{]}+\sum S(k\to\mu)\,.$ (8)

We use the same analytic approximation method involving spectrum-weighted
$Z$-moments that is successful for calculating the atmospheric lepton fluxes
at the Earth’s surface [5, 6]. The analytic approximation to determine
atmospheric lepton fluxes relies on flux-weighted integrals of differential
distributions for particle production and decay. The source of particle $j$
from initial particle $k$, denoted $S(k\to j)$, is given by eq. (2) and can be
re-written in terms of $Z$-moment as

$\displaystyle S(k\to j)$ $\displaystyle\simeq$
$\displaystyle\Biggl{[}\int_{E}^{\infty}dE^{\prime}\frac{\phi_{k}^{0}(E^{\prime})}{\phi_{k}^{0}(E)}\frac{\lambda_{k}(E)}{\lambda_{k}(E^{\prime})}\frac{dn(k\to
j;E^{\prime},E)}{dE}\Biggr{]}\frac{\phi_{k}(E,X)}{\lambda_{k}(E)}$ (9)
$\displaystyle\equiv$ $\displaystyle Z_{kj}(E)\frac{dn(k\to
j;E^{\prime},E)}{dE}\,,$

where $\phi_{k}(E,X)=\phi_{k}^{0}(E)f(X)$ and
$\Lambda_{k}=\lambda_{k}/(1-Z_{kk})$ so that $f(X)=\exp{(-X/\Lambda_{k})}$.

Figure 2: Muon energy (left) and the survival probability of muon (right)
produced $\ell$ distance away from the EUSO-SPB2 balloon. The balloon is at
$\ell=0$ km. It is shown for three final muon energies at the balloon. The
dashed line is for $\alpha=87^{\circ}$ and solid line is for
$\alpha=90^{\circ}$. The plot on the left shows the muon reaching the balloon
(at $\ell=0$ km) with a final energy of $E_{\mu,f}=1,5,10$ GeV starts its
trajectory at a higher energy.

To evaluate the flux of muons, we step through the column depth of the
atmosphere with steps $\Delta X$ that are small relative to
$\lambda_{\mu}^{\rm dec}$ and $\lambda_{j}$ for interactions and decays. For
cosmic ray nucleons $N$, and for the pions and kaons they produce in the
atmosphere, this translates for each step to

$\displaystyle\phi_{N}(E,X+\Delta X)$ $\displaystyle=$
$\displaystyle\phi_{N}(E,X)\Biggl{(}1-\frac{\Delta X}{\Lambda_{N}}\Biggr{)}$
(10) $\displaystyle\phi_{j}(E,X+\Delta X)$ $\displaystyle=$
$\displaystyle\phi_{j}(E,X)\Biggl{(}1-\frac{\Delta
X}{\Lambda_{j}}-\frac{\Delta X}{\lambda_{j}^{\rm dec}}\Biggr{)}+Z_{N\to
j}\phi_{N}(E,X)\frac{\Delta X}{\lambda_{N}}$ (11)

where $j=\pi,K$. For muons

$\displaystyle\phi_{\mu}(E,X+\Delta X)$ $\displaystyle=$
$\displaystyle\Biggl{[}\phi_{\mu}(E^{\prime},X)\Biggl{(}1-\frac{\Delta
X}{\lambda_{\mu}^{\rm
dec}}\Biggr{)}+\sum_{j=\pi,K}Z_{j\to\mu}\phi_{j}(E^{\prime},X)\frac{\Delta
X}{\lambda_{j}^{\rm dec}}\Biggr{]}\exp(b\Delta X)$ (12) $\displaystyle
E^{\prime}$ $\displaystyle=$ $\displaystyle(E+a/b)\exp(-b\Delta X)-a/b\,.$
(13)

To simplify the calculation further, we make several approximations. The
energy loss parameters $a$, $b$, and the $Z$-moments are taken to be energy
independent with their respective values are taken from ref. [5], including
pion and kaon decays as sources of muons in the atmosphere. We don’t account
for the Earth’s magnetic field, an effect most important for muons with
energies below 10 GeV [9]. We approximate the cosmic ray flux ($\phi_{N}^{0}$)
as a function of energy per nucleon $E$ by

$\phi_{N}^{0}(E)\Biggl{[}\frac{\rm nucleons}{\rm
cm^{2}\,s\,sr\,GeV}\Biggr{]}=1.7\ (E/{\rm GeV})^{-2.7},\quad E<5\cdot 10^{6}\
{\rm GeV}\ .$

The muon flux is evaluated according to eq. (12). Our evaluation of the
atmospheric muon flux on the ground is in agreement with the results in ref.
[5]. For total atmospheric column depth $X(\alpha)$, the muon energy at the
beginning of the trajectory has initial energy $E_{i}=(E+a/b)\exp(bX)-a/b$,
and for each of the $n$ steps in column depth, the muon energy is decreased
according to eq. (13). Therefore, the muons reaching the telescope with a
final energy $E_{\mu,f}$ ($E_{\mu}=E_{\mu,f}$ at $\ell=0$) will have started
their journey with a higher initial energy, as shown in the left panel of fig.
2 for two different column depths corresponding to $\alpha=87^{\circ}$ and
$\alpha=90^{\circ}$. Muon energy loss also impacts the survival probability of
the muons, as shown in the left panel of fig. 2 which shows the muon survival
probability as a function of $\ell$ for three final muon energies and two
column depths.

##  3 Results

Figure 3: Left: Altitude $h$ (left) and trajectory distance $\ell$ (right) as
a function of column depth $X$ starting from very high altitude, for
trajectories below the horizon, at the horizon and above the horizon
(different $\alpha$ values) for an instrument at an altitude of 33 km. The
atmospheric density is approximated by an exponential function.

Figure 4: Muon flux (scaled by $E_{\mu,f}^{2.7}$) as a function of final muon
energy reaching the detector. It is shown for different trajectories above and
below the horizon. Right: Muon flux (scaled by $E_{\mu,f}^{2.7}$) as a
function of $\alpha$. It is shown for different final muon energies reaching
the detector. Figure 5: The ratio of muon flux produced from charged pions to
charged kaons, as a function of final muon energies. It is shown for different
trajectories below and above the horizon.  Figure 6: Muon flux reaching per
cm2 of area of the detector per second scaled by final muon energies, as a
function of final muon energy. Solid line is for the detector pointing to the
horizon, and dashed line is for the detector pointing in nadir direction.

The cosmic rays interact after traversing $X\simeq 120$ g/cm2 in the
atmosphere to form charged mesons, like pions and kaons, which can decay to
muons. The column depth depends on the density of the air in the atmosphere,
and is given by eq. (3). The atmosphere near the surface of the Earth is
denser and its density decreases as we go farther up from the surface.
Therefore, understanding the column depth variation along a particle’s
trajectory as a function of $h$ and $\ell$ is important. Figure 3 shows the
column depth variation in the atmosphere as a function of altitude (left) and
particle’s trajectory distance (right). It is shown for different trajectories
above and below the horizon.

Our main results for the muon flux with energy $E_{\mu,f}$ at a detector at an
altitude of 33 km are shown in fig. 4. Figure 4 (left) shows the muon flux
scaled by the $E_{\mu,f}^{2.7}$, as a function of $E_{\mu,f}$, for several
values of $\alpha$, from both below and above the horizon of the telescope.
Figure 4 (right) shows the muon flux scaled by $E_{\mu,f}^{2.7}$ as a function
of $\alpha$ for fixed final muon energies at the balloon.

The particle trajectories with smaller $\alpha$, have to travel longer
distances to reach the balloon as compared to the particle trajectories with
higher $\alpha$. For higher $\alpha$, the density of the atmosphere is lower,
and it leads to the cosmic rays interacting to produce charged pions and kaons
closer to the detector. This means the production of muons is also closer to
the detector, increasing the muon flux reaching the telescopes on the balloon
for higher $\alpha$ trajectories. This is why the muon flux reaching the
balloon with a final muon energy of 1 GeV, for example, is lower for smaller
$\alpha$ but increases with higher $\alpha$ (shown in fig. 4 right).

We can see in fig. 4 (left), the muon flux increases with increasing muon
final energies as more muons are able to reach the balloon. But the flux
starts falling for $E_{\mu,f}\gtrsim 10^{3}$ GeV. This is because fewer muons
at higher energies are being produced as the probability of interaction for
pions and kaons is higher than their decay probability.

For reference, we show the ratio of the muon flux reaching the balloon
produced from charged pion decays to the flux from charged kaon decays in fig.
5 for different values of $\alpha$. The muon flux from pion decays is a factor
of $\sim 8-11$ higher than the flux from kaon decays at lower final muon
energies. For higher final muon energies, the ratio of the muon flux produced
from pions to kaons is nearly constant as a function of energy, with a value
of $\sim 1.5$.

To calculate the rate of muons incident on the CT pointed horizontally, and
from the upward muon flux incident on the FT pointing to the nadir, we
integrate the flux of muons over the appropriate solid angle, starting with
$\alpha=84.2^{\circ}$ at the Earth’s limb. For these two cases, the flux
integrated over solid angle and energy for energy $E_{\mu,f}$, denoted by
$\Phi_{\mu}(E_{\mu,f})$ are

$\displaystyle\Phi_{\mu}(E_{\mu,f})$ $\displaystyle\equiv$
$\displaystyle\int_{E_{\mu,f}}dE_{\mu}\int_{\varphi=-\pi/2}^{\varphi=\pi/2}\int_{\alpha=84.2^{\circ}}^{\alpha=180^{\circ}}\phi_{\mu}(E_{\mu},\alpha)\sin^{2}\alpha\cos\varphi\,d\alpha
d\varphi,\quad{\rm horizontal}$ (14) $\displaystyle\Phi_{\mu}(E_{\mu,f})$
$\displaystyle\equiv$
$\displaystyle\int_{E_{\mu,f}}dE_{\mu}\int_{\varphi=0}^{\varphi=2\pi}\int_{\alpha=84.2^{\circ}}^{\alpha=90^{\circ}}\phi_{\mu}(E_{\mu},\alpha)\cos\alpha\sin\alpha\,d\alpha
d\varphi,\quad{\rm nadir}\,.$ (15)

The results for $E_{\mu,f}\times d\Phi_{\mu}/dE_{\mu,f}$ are shown in fig. 6
as a function of final muon energy are shown for each orientation. The rate of
up-going muons in the atmospheric flux that reach the FT, parallel to the
Earth (pointing to the nadir) is much smaller than the rate incident on the CT
pointing to the horizon. Integrating over energy with a minimum energy of
$E_{\mu,f}=1$ GeV yields $6.6\times 10^{-3}$/cm2/s on the horizontally-
pointing CT, and $2.1\times 10^{-6}$/cm2/s on the nadir-pointing FT. The flux
of muons on the ground is approximately 1/cm2/min$\,\simeq 1.7\times
10^{-2}$/cm2/s [10].

While our approximation that for a given $\alpha$ we can use the cascade
equations in a 1-dimensional approximation is good for $E_{\mu,f}\gtrsim 10$
GeV, is not as good an approximation for lower energy muons where the Earth’s
magnetic field bends the muon trajectory (and makes it longer) [11]. For
reference, $\Phi_{\mu}(E_{\mu,f}=10$ GeV) is $3.6\times 10^{-4}$/cm2/s for the
horizontal orientation of the telescope, and $1.3\times 10^{-6}$/cm2/s for the
nadir orientation.

##  4 Discussion

Our muon flux calculation is an approximate result, but for the orientation of
CT pointing to its horizon, or near to it, is interesting enough to pursue
with a more detailed evaluation. As muons pass through the plane of the
detector, they are ionizing and deposit energy. More detailed modeling of the
detector is required to understand their impact.

Given the areas of the CT, if we assume that 100% of the muon rate triggers
the detector, the atmospheric muon rate for muons above 1 GeV is 1.21 Hz and
above 10 GeV is 0.066 Hz, on the CT of area 184 cm2 with horizontal pointing.
There may be an opportunity to measure the atmospheric muon flux with EUSO-
SPB2 using data collected to determine the air glow background.

Acknowledgements We thank J. Krizmanic and J. Szabelski for illuminating
discussions. This work was supported in part the the US Department of Energy
grant DE-SC-0010113.

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