#  Deep Learning-based Diffusion Tensor Cardiac Magnetic Resonance
Reconstruction: A Comparison Study

Jiahao Huang  National Heart and Lung Institute, Imperial College London,
London, United Kingdom  Cardiovascular Research Centre, Royal Brompton
Hospital, London, United Kingdom  Pedro F. Ferreira  National Heart and Lung
Institute, Imperial College London, London, United Kingdom  Cardiovascular
Research Centre, Royal Brompton Hospital, London, United Kingdom  Lichao Wang
National Heart and Lung Institute, Imperial College London, London, United
Kingdom  Department of Computing, Imperial College London, London, United
Kingdom  Yinzhe Wu  National Heart and Lung Institute, Imperial College
London, London, United Kingdom  Cardiovascular Research Centre, Royal Brompton
Hospital, London, United Kingdom  Angelica I. Aviles-Rivero  Department of
Applied Mathematics and Theoretical Physics, University of Cambridge,
Cambridge, United Kingdom  Carola-Bibiane Schönlieb  Department of Applied
Mathematics and Theoretical Physics, University of Cambridge, Cambridge,
United Kingdom  Andrew D. Scott  National Heart and Lung Institute, Imperial
College London, London, United Kingdom  Cardiovascular Research Centre, Royal
Brompton Hospital, London, United Kingdom  Zohya Khalique  National Heart and
Lung Institute, Imperial College London, London, United Kingdom
Cardiovascular Research Centre, Royal Brompton Hospital, London, United
Kingdom  Maria Dwornik  National Heart and Lung Institute, Imperial College
London, London, United Kingdom  Cardiovascular Research Centre, Royal Brompton
Hospital, London, United Kingdom  Ramyah Rajakulasingam  National Heart and
Lung Institute, Imperial College London, London, United Kingdom
Cardiovascular Research Centre, Royal Brompton Hospital, London, United
Kingdom  Ranil De Silva  National Heart and Lung Institute, Imperial College
London, London, United Kingdom  Cardiovascular Research Centre, Royal Brompton
Hospital, London, United Kingdom  Dudley J. Pennell  National Heart and Lung
Institute, Imperial College London, London, United Kingdom  Cardiovascular
Research Centre, Royal Brompton Hospital, London, United Kingdom  Sonia
Nielles-Vallespin  National Heart and Lung Institute, Imperial College London,
London, United Kingdom  Cardiovascular Research Centre, Royal Brompton
Hospital, London, United Kingdom  Co-last senior authors. Send correspondence
to<EMAIL_ADDRESS>Guang Yang  National Heart and Lung
Institute, Imperial College London, London, United Kingdom  Cardiovascular
Research Centre, Royal Brompton Hospital, London, United Kingdom  Co-last
senior authors. Send correspondence to<EMAIL_ADDRESS>
###### Abstract

In vivo cardiac diffusion tensor imaging (cDTI) is a promising Magnetic
Resonance Imaging (MRI) technique for evaluating the micro-structure of
myocardial tissue in the living heart, providing insights into cardiac
function and enabling the development of innovative therapeutic strategies.
However, the integration of cDTI into routine clinical practice is challenging
due to the technical obstacles involved in the acquisition, such as low
signal-to-noise ratio and long scanning times. In this paper, we investigate
and implement three different types of deep learning-based MRI reconstruction
models for cDTI reconstruction. We evaluate the performance of these models
based on reconstruction quality assessment and diffusion tensor parameter
assessment. Our results indicate that the models we discussed in this study
can be applied for clinical use at an acceleration factor (AF) of $\times 2$
and $\times 4$, with the D5C5 model showing superior fidelity for
reconstruction and the SwinMR model providing higher perceptual scores. There
is no statistical difference with the reference for all diffusion tensor
parameters at AF $\times 2$ or most DT parameters at AF $\times 4$, and the
quality of most diffusion tensor parameter maps are visually acceptable.
SwinMR is recommended as the optimal approach for reconstruction at AF $\times
2$ and AF $\times 4$. However, we believed the models discussed in this
studies are not prepared for clinical use at a higher AF. At AF $\times 8$,
the performance of all models discussed remains limited, with only half of the
diffusion tensor parameters being recovered to a level with no statistical
difference from the reference. Some diffusion tensor parameter maps even
provide wrong and misleading information.

###### keywords:

Deep Learning, CNN, Transformer, Cardiac Diffusion Tensor, MRI Reconstruction

## Introduction

In vivo cardiac diffusion tensor (DT) imaging (cDTI) is an emerging Magnetic
Resonance Imaging (MRI) technique that has the potential to describe the
micro-structure of myocardial tissue in the living heart. The diffusion of
water molecules occurs anisotropically due to the restrictions imposed by the
micro-structure of the myocardium, which can be approximated by fitting three-
dimensional (3D) tensors with a specific shape and orientation in cDTI.
Various parameters can be derived from the DT, including mean diffusivity (MD)
and fractional anisotropy (FA), which are crucial indices that can indicate
the structural integrity of myocardial tissues. The helix angle (HA) signifies
local cell orientations, while the second eigenvector (E2A) represents the
average sheetlet orientation [1]. The development of cDTI provides insights
into the myocardial micro-structure and offers new perspectives on the elusive
connection between cellular contraction and macroscopic cardiac function [1,
2]. Furthermore, it presents opportunities for novel assessments of the
myocardial micro-structure and cardiac function, as well as the development
and evaluation of innovative therapeutic strategies [3].

Despite the numerous advantages, there are still significant technical
obstacles that must be overcome to integrate cDTI into routine clinical
practice. For the calculation of the DT, diffusion-weighted images (DWIs) with
diffusion encoding in at least six distinct directions need to be collected.
Due to the movement derived from the heart beat and human breath, in vivo cDTI
exploits single-shot encoding acquisition for repetitive fast scanning, e.g.,
single-shot echo planar imaging (SS-EPI) or spiral diffusion-weighted imaging
[4]. The utilisation of these single-shot encoding acquisitions, which lead to
low signal-to-noise (SNR) images, usually requires multiple repetitions to
enhance the accuracy of the DT estimation[5, 6]. Each repetition necessitates
an additional breath-hold for the patient when using breath-hold acquisitions,
which significantly increases the total scanning time and leads to
uncomfortable patient experience.

Numerous studies have been proposed to accelerate cDTI technique, which can be
mainly categorised as 1) reducing the total amount of DWIs used for the
calculation of the DT; 2) general fast DWIs by k-space undersampling and
reconstruction using compressed sensing (CS) or deep learning techniques. This
study focuses on the second strategy.

Deep learning has emerged as a powerful technique for image analysis,
capitalising on the non-linear and complex nature of networks through
supervised or unsupervised learning, and has found widespread applications in
medical image researches [7]. Deep learning-based MRI reconstruction [8, 9]
has gained significant attention, leveraging its capable of learning complex
and hierarchical representations from large MRI datasets [10].

In this work, we investigate the application of deep learning-based methods
for cDTI reconstruction. We explore and implement three different types of
deep learning-based models on cDTI datasets with acceleration factor (AF) of
$\times 2$, $\times 4$ and $\times 8$. These models include a Convolutional
Neural Network (CNN)-based unrolling method, i.e., D5C5 [11], a CNN-based and
conditional Generative Adversarial Network (GAN)-based non-unrolling method,
i.e., DAGAN [12], and a Transformer-based non-unrolling methods, i.e., SwinMR
[13]. The performance of these three models are evaluated by the
reconstruction quality assessment and the DT parameters assessment.

Our experiments demonstrate that the models discussed in this paper can be
applied for clinical use at AF $\times 2$ and AF $\times 4$, since the both
the reconstruction of DWIs and DT parameters reach satisfactory levels. Among
these models, D5C5 shows superior fidelity for the reconstruction, while
SwinMR provides results with higher perceptual scores. There is no statistical
difference with the reference for all the DT parameters at AF $\times 2$ or
most of the DT parameters at AF $\times 4$. The quality of most the DT
parameter maps we considered are visually acceptable. Considering various
factors, SwinMR is recommended as the optimal approach for the reconstruction
with AF $\times 2$ and AF $\times 4$.

However at AF $\times 8$, the performance of these three models, including the
best-performing SwinMR, is still limited. The reconstruction quality is not
unsatisfactory due to the artefact remaining and the noisy (DAGAN) or ‘fake’
(SwinMR) estimation. Only half of the DT parameters can be recovered to a
level that is no statistical difference with the reference. Some DT parameter
maps even provide wrong and misleading information, which is unacceptable and
dangerous for clinical use.

## Related Works

### Diffusion Tensor MRI Acceleration

A major drawback of DTI is its extended scanning time, as it requires multiple
DWIs with varying b-value and diffusion gradient directions to calculate the
DT. In theory, the estimation of the DT requires only six DWIs with different
diffusion gradient directions and one reference image. Practically for cDTI, a
considerable number of cardiac DWIs and multiple averages are typically
required to enhance the accuracy of DT estimation, due to the inherently low
SNR of single-shot acquisitions.

Strategies to accelerate the DTI technique have been explored. One technical
route aims to reduce the number of DWIs required for the DT estimation [14,
15, 16, 17, 18, 19, 20], which can be further categorised into three sub-
class.

1) Learn a direct mapping from reduced repetition (or gradient direction) of
DWIs, to the DT or DT parameter maps. Ferreira et al. [14] proposed a U-Net-
based method for cDTI acceleration, which directly estimates the DT, using
DWIs collected within one breath-hold, instead of solving a conventional
linear-least-square (LLS) tensor fitting. Karimi et al. [15] introduced a
Transformer-based model with coarse-and-fine strategy to provide accuracy
estimation of the DT, using only six diffusion-weighted measurements. Aliotta
et al. [21] proposed a neural network for brain DTI, namely DiffNet, which
estimated MD and FA maps directly from diffusion-weighted acquisitions with as
few as three diffusion-encoding directions. They further improved their method
by combining a parallel U-Net for slice-to-slice mapping and a multi-layer
perceptron for pixel-to-pixel mapping [16]. Li et al. [17] developed a CNN-
based model for brain DTI, i.e., SuperDTI, to generate FA, MD and
directionally encoded color maps with as few as six diffusion-weighted
acquisitions.

2) Enhance DWIs (denoising). This kind of methods usually apply only a small
amount of enhanced images to achieve comparable estimation results with the
results via standard protocol. Tian et al. [18] developed a novel DTI
processing framework, entitled DeepDTI, that minimised the required data for
DTI to six diffusion-weighted images. The core idea of this framework was to
use a CNN, which takes a non-diffusion-weighted (b0) image, six DWIs as well
as a anatomical (T1- or T2-weighted) image as input, to produce high-quality
b0 images and six DWIs. Phipps et al. [19] applied a denoising CNN to enhance
the quality of b0 images and corresponding DWIs for cDTI.

3) Refine the DT quality. Tänzer et al. [20] proposed a GAN-based Transformer,
to directly enhance the quality of the DT that was calculated with reduce
amount of DWIs in a end-to-end manner.

Another technical route follows the general DWIs acceleration by k-space
undersampling and reconstruction [22, 23, 24, 25]. Zhu et al. [22] directly
estimated the DT from highly undersampled k-space data. Chen et al. [23]
incorporated the joint sparsity prior of different DWIs with the L1-L2 norm
and the DT’s smoothness using the total variational (TV) semi-norm to
efficiently expedite DWI reconstruction. Huang et al. [24] utilised a local
low-rank model and 3D TV constraints to reconstruct the DWIs from undersampled
k-space measurements. Teh et al. [25] introduced a directed TV-based method
for DWI images reconstruction, applying the information on the position and
orientation of edges in the reference image.

In addition to these major technical routes, Liu et al. [26] explored the deep
learning-based image synthetics for the inter-directional DWIs generation. The
true b0 and 6 DWIs were concatenated with the generated data and passed to the
CNN-based tensor fitting network.

### Deep Learning-Based Reconstruction

The aim of MRI reconstruction is to recover the ground truth image $x$ from
the undersampled k-space measurement $y$, which is mathematically described as
an inverse problem:

$\displaystyle
x=\operatorname{arg}\min_{x}\frac{1}{2}||\mathcal{A}x-y||_{2}^{2}+\lambda\mathcal{R}(x),$
(1)

in which the degradation matrix $\mathcal{A}$ can be further presented as the
combination of the undersampling trajectory $\mathcal{M}$, Fourier transform
$\mathcal{F}$ and coil sensitivity maps $\mathcal{S}$. $\lambda$ is the
coefficient that balances regularisation term $\mathcal{R}(x)$.

Deep learning technique has been widely used for MRI reconstruction. Based on
the association with traditional iterative CS algorithms, deep learning-based
MRI reconstruction methods can be categorised into 1) unrolling-based models
[27, 11, 28] and 2) non-unrolling-based models [12, 13].

Unrolling-based models usually integrate neural networks with traditional CS
algorithms, simulating the iterative reconstruction algorithms through
learnable iterative blocks [9]. Yang et al. [27] reformulated an Alternating
Direction Method of Multipliers (ADMM) algorithm to a multi-stage deep
architecture, namely Deep-ADMM-Net, for MRI reconstruction, of which each
stage corresponds to an iteration in traditional ADMM algorithm. Some
unrolling-based models improved Eq (1) with a deep learning-based regulariser
[11, 28], which can be formulated as:

$\displaystyle
x=\operatorname{arg}\min_{x}\frac{1}{2}||\mathcal{A}x-y||_{2}^{2}+\lambda||x-f_{\theta}(x_{u})||_{2}^{2},\quad\text{s.t.
}x_{u}=\mathcal{F}^{-1}y,$ (2)

in which $f_{\theta}(\cdot)$ is a deep neural network and $x_{u}$ is the
undersampled zero-filled images (ZF). Schlemper et al. [11] designed a deep
cascade of CNNs for cardiac cine reconstruction, in which a spatio-temporal
correlations can be also efficiently learned via the data sharing approach.
Aggarwal et al. [28] proposed a model-based deep learning methods, namely
MoDL, which exploited a CNN-based regularisation prior for MRI reconstruction.

Non-unrolling-based models usually train an deep learning-based function
$f_{\theta}(\cdot)$ that maps the undersampled k-space measurement $y$ or
zero-filled images $x_{u}$ to estimate fully-sampled images $\hat{x}_{u}$ or
its residual in an end-to-end manner, which can be formulated as
$\hat{x}_{u}=f_{\theta}(x_{u})$ or $\hat{x}_{u}=f_{\theta}(x_{u})+x_{u}$. Yang
et al. [12] proposed a de-alising Generative Adversarial Networks for MRI
reconstruction, in which the U-Net-based generator produced the estimated
fully-sampled MRI images in an end-to-end manner. Feng et al. [29] exploited
the task-specific novel cross-attention and designed a end-to-end Transformer-
based model for jointly MRI reconstruction and super-resolution. Huang et al.
[13] proposed a Swin Transformer-based model, namely SwinMR, for end-to-end
MRI reconstruction, and they further explored the combination of Swin
Transformer and GAN for the edge and texture preservation in MRI
reconstruction [30].

Deep learning community constantly provides a wide range of novel and powerful
network structures for both kinds of MRI reconstruction methods, including
CNNs [11, 12], Recurrent Neural Networks [31, 32], Graph Neural Networks [33],
recently thriving Transformers [29, 13, 30, 34, 35], etc. These rapidly
evolving deep learning-based networks enable advances in for MRI
reconstruction.

## Methodology

In this study, we implement three deep learning-based MRI reconstruction
methods, namely, DAGAN [12], D5C5 [11] and SwinMR [13], and assess their
performance on cDTI dataset. The overall data flow is depicted in Figure 1.

Figure 1:  The data flow of our implementation for cardiac diffusion tensor
imaging data. The whole procedure consists (A) data acquisition, (B) data pre-
processing, (C) deep learning-based reconstruction and (D) data post-
processing. It is noted that D5C5 does not require the cropping and pasting
step and additionally takes the undersampled k-space data and the
corresponding undersampling mask as input.

### Data Acquisition

All data used in this study were approved by the National Research Ethics
Service. Written informed consent was obtained from all subjects.

Retrospectively acquired cDTI data were acquired using Siemens Skyra 3T MRI
scanner and Siemens Vida 3T MRI scanner (Siemens AG, Erlangen, Germany). A
diffusion-weighted stimulated echo acquisition mode (STEAM) SS-EPI sequence
with reduced phase field-of-view and fat saturation. Some MR sequence
parameters are listed: $\text{TR}=2\ \text{RR intervals}$; $\text{TE}=23\
\text{ms}$; SENSE or GRAPPA with $\text{AF}=2$; echo train duration $=13\
\text{ms}$; spatial resolution $=2.8\times 2.8\times 8.0\
\text{mm}^{\text{3}}$. Diffusion-weighted images were encoded in six
directions with diffusion-weighted of $\text{b}=150\ \text{and}\ 600\
\text{sec/mm}^{\text{2}}$ (namely b150 and b600) in a short-axis mid-
ventricular slice. Reference images, namely b0, were also acquired with with a
minor diffusion weighting.

We used 481 cDTI cases including 2 cardiac phases, i.e., diastole ($n=232$)
and systole ($n=249$), for the experiments section. The dataset contains 241
healthy cases, 31 amyloidosis (AMYLOID) cases, 47 dilated cardiomyopathy (DCM)
cases, 35 in-recovery DCM (rDCM) cases, 39 hypertrophic cardiomyopathy (HCM)
cases, 48 HCM genotype-positive–phenotype-negative (HCM G+P-) cases, and 40
acute myocardial infarction (MI) cases. The overall data distribution of our
dataset is shown in Table 1. The detailed data distribution per cohort and
cardiac phase can be found in Table S1 in Supplementary.

This work separately discussed the reconstruction of systole and diastole
cases. For each deep learning-based methods, two network weights were trained
for either systole or diastole reconstruction. In the training stage, we
applied 5-fold-cross-validation strategy, using 169 diastole cases
(TrainVal-D) or 183 systole cases (TrainVal-S). In the testing stage, four
testing sets were utilised, including mixed ordinary testing set with diastole
cases (Test-D) or systole cases (Test-S) and out-of-distribution MI testing
set with diastole cases (Test-MI-D) or systole cases (Test-MI-S). According to
Table S1, Test-D and Test-S includes the data of Health, AMYLOID, rDCM, DCM,
HCM and HCM G+P-, which are also included in the TrainVal. For further
examining the model robustness and ability to handle out-of-the-distribution
data, Test-MI dataset includes only MI cases, which are ‘invisible’ for models
during the training stage.

Table 1: The overview of the dataset. Cardaic Phase |  Total |  TrainVal |  Test |  Test-MI
---|---|---|---|---
Case |  Slice |  Case |  Slice |  Case |  Slice |  Case |  Slice
Diastole |  272 |  22509 |  169 |  14182 |  43 |  3630 |  20 |  1470
Systole |  290 |  23654 |  183 |  15054 |  46 |  3938 |  20 |  1470

### Data Pre-Processing

In the data pre-processing stage, all DWIs (b0, b150 and b600) were processed
following the same protocol.

The pixel intensity ranges of DWIs vary considerably across different
b-values. To address this, We normalised all DWIs in the dataset to a pixel
intensity range of $0\sim 1$ using the max-min method, while the maximum and
minimum pixel values of all DWIs were recorded for the pixel intensity range
recovery at the beginning of the data post-processing stage.

In our dataset, the majority of DWIs have a resolution of $256\times 96$,
while a small subset of 2D slices exhibit a resolution of $256\times 88$. In
order to standardise the resolution, we zero-padded the edges of the images
with a resolution of $256\times 88$ to achieve a resolution of $256\times 96$.

In this study, GRAPPA-like Cartesian k-space undersampling masks with AF
$\times 2$, $\times 4$ and $\times 8$, generated by the official protocol of
fastMRI dataset [10], were applied to simulate the k-space undersampling
process. Since all the 2D slices have been reconstructed with zero-padding
factor of two, the phase encoding (PE) of our undersampling masks was set to
48 instead of 96, for a more realistic simulation. The undersampling masks
were then zero-padded from $128\times 48$ to $256\times 96$ as shown in Figure
1. More details regarding the undersampling masks can be found in Figure S1 in
Supplementary.

For DAGAN and SwinMR, DWIs were further cropped to $96\times 96$, as both
models only support square-shaped input images.

### Deep Learning-Based Cardiac Diffusion Tensor Imaging Reconstruction

In this stage, deep learning-based models were utilised to took the k-space
undersampled data as the input and produced the reconstructed MR images. We
implemented and evaluated three deep learning-based models, namely DAGAN [12],
D5C5 [11] and SwinMR [13] in this stage.

#### DAGAN

DAGAN [12] is a conditional GAN-based and CNN-based model designed for general
MRI reconstruction, of which the model structure is presented in Figure. 2.
DAGAN comprises two components: a generator and a discriminator, which are
trained in an adversarial manner as a two-player game.

The generator is a modified CNN-based U-Net [36] with a residual connection
[37], which takes the k-space zero-filled MR images as input and aims to
produce reconstructed MR images as close as possible to the ground truth
images. The discriminator is a standard CNN-based classifier that attempts to
distinguish the ‘fake’ reconstructed MR images generated by the generator,
from the ground truth MR images.

During the inference stage, only the generator is applied, which takes the ZF
MR images as input and outputs the reconstructed images.

DAGAN is trained with a hybrid loss function including an image space $l2$
loss, a frequency space $l2$ loss, a perceptual $l2$ loss based on a pre-
trained VGG [38], as well as an adversarial loss [39]. More implement details
can be found in the original paper [12].

Figure 2:  The model architecture of DAGAN. (A) the generator of DAGAN is a
modified Convolutional Neural Network (CNN)-based U-Net with a residual
connection; (B) the discriminator of DAGAN is a standard CNN-based classifier.
Conv2D: 2D convolution layer; Recon: reconstructed MR images; GT: groung truth
MR images.

#### D5C5

D5C5 [11] is a CNN-based model for MRI reconstruction, with its model
structure presented in Figure. 3. Originally proposed for cine MRI
reconstruction, D5C5 also supports general MRI reconstruction.

D5C5 takes the undersampled k-space measurement as well as ZF MR images as the
input and outputs the reconstructed MR images. It is composed of multiple
stages, each comprising a CNN block and a data consistency (DC) layer. The CNN
block contains a cascade of convolutional layers with Rectifier Linear Units
(ReLU) for feature extraction, an optional data sharing (DS) layer for
learning spatio-temporal features, as well as a residual connection [37]. The
DC layer takes a linear combination between the output of the CNN block and
the undersampled k-space data, enforcing the consistency between the
prediction of CNNs and the original k-space measurements. D5C5 has five
stages, with five convolution layers in each CNN block, and no DS layer is
applied for our 2D MRI reconstruction task.

D5C5 is trained end-to-end using an image space $l2$ loss function. Further
implementation details can be found in the original paper [11].

Figure 3:  (A) The model architecture of D5C5. D5C5 has five stages, each
comprising a Convolutional Neural Network block (CNN Block) and a data
consistency layer (DC). (B) The structure of the CNN Block. One optional data
sharing module (DS) and five convolutional layers (Conv Layers) are included
in the CNN Block. (C) The structure of the DC. $\mathcal{M}$ denotes the
undersampling mask, and $\overline{\mathcal{M}}=\mathcal{I}-\mathcal{M}$.
$\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier and inverse Fourier
transform. $\lambda$ is an adjustable coefficient controlling the level of DC.

#### SwinMR

SwinMR [13] is a Swin Transformer-based model for MRI reconstruction, with its
model structure shown in Figure 4. SwinMR takes the ZF MR images as the input
and directly outputs the reconstructed images.

SwinMR is composed of a CNN-based input module and output module for
projecting between the image space and the latent space, a cascade of residual
Swin Trasformer blocks (RSTBs), and a convolution layer with a residual
connection for feature extraction. A patch embedding and a patch unembedding
layer are placed at the beginning and end of each RSTB, facilitating the
inter-conversion of feature maps and sequences, since the computation of
Transformers is based on sequences. Multiple standard Swin Transformer layers
(STLs) [40] and a single convolutional layer are applied between the patch
embedding and unembedding layer.

SwinMR is trained end-to-end with a hybrid loss function consisting of an
image space $l1$ loss, a frequency space $l1$ loss, a perceptual $l1$ loss
based on a pre-trained VGG [38]. More implementation details can be found in
the original paper [13].

Figure 4:  (A) The model architecture of SwinMR. (B) The structure of the
residual Swin Transformer block (RSTB). (C) The structure of the Swin
Transformer layer (STL). Conv2D: 2D convolutional layer. MLP: multi-layer
perceptron; LN: layer normalisation; Q: query; K: key; V: value. MSA: multi-
head self-attention.

### Data Post-Processing

We applied our in-house developed software (MATLAB 2021b, MathWorks, Natick,
MA) for cDTI post-processing, following the protocol described in [1, 14]. The
post-process procedure for reference data includes: 1) manual removal of low-
quality DWIs; 2) DWI registration; 3) semi-manual segmentation for left
ventricle (LV) myocardium; 4) DT calculation via the LLS fit; 5) DT parameter
calculation including FA, MD, HA and E2A. The initial post-processing of
reference data was performed by either Z.K. (7 years of experience), R.R. (3
years of experience) or M.D. (2 years of experience), and subsequently
reviewed by P.F. (10 years of experience).

For the post-processing of deep leanring-based reconstruction results, the
output ($96\times 96$) of DAGAN and SwinMR were ‘pasted’ back to the
corresponding zero-filled images ($256\times 96$) at their original position.
(This process does not affect the final post-processing results since the ROI
region is set in the central $96\times 96$ area.)

All the DWIs were ‘anti-normalised’ (pixel value range recovery) to their
original pixel intensity range using the maximum and minimum values recorded
in the pre-processing stage.

The reconstruction results were arranged to construct new reconstruction
dataset with the same structure as the reference dataset. The reconstructed
dataset was then automatically post-processed following the configuration of
reference data (e.g., low-quality removal information, registration shifting,
segmentation masks) for a fair comparison.

## Experiments and Results

In this section, the experimental results are presented from the perspective:
1) the quality of DWIs reconstruction and 2) the quality of DT parameter
estimations.

### Reconstruction Quality Assessment

Table 2:  The quantitative reconstruction results on the testing sets Test-S
and Test-D with undersampling masks of the acceleration factor (AF) $\times
2$, $\times 4$ and $\times 8$. SSIM, PSNR and LPIPS results are quoted as
‘mean (standard deviation)’. ⋆ indicates the specific distribution is
significantly different ($p<0.05$) from the best results distribution by the
two-sample t-test.

Metrics | Test-S | Test-D
---|---|---
AF x2 | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
SSIM $\uparrow$ | 0.819 (0.042) ⋆ | 0.857 (0.026) ⋆ | 0.931 (0.030) | 0.919 (0.031) ⋆ | 0.819 (0.044) ⋆ | 0.851 (0.025) ⋆ | 0.932 (0.031) | 0.921 (0.034) ⋆
PSNR $\uparrow$ | 25.76 (2.27) ⋆ | 27.86 (1.59) ⋆ | 31.80 (2.45) | 30.83 (2.50) ⋆ | 26.41 (2.38) ⋆ | 28.25 (1.52) ⋆ | 32.43 (2.70) | 31.61 (2.80) ⋆
LPIPS $\downarrow$ | 0.149 (0.037) ⋆ | 0.060 (0.024) ⋆ | 0.050 (0.026) | 0.050 (0.023) | 0.148 (0.034) ⋆ | 0.066 (0.028) ⋆ | 0.053 (0.032) | 0.052 (0.029)
FID $\downarrow$ | 82.29 | 33.1 | 18.53 | 17.2 | 101.55 | 38.84 | 25.83 | 22.7
AF x4 | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
SSIM $\uparrow$ | 0.663 (0.068) ⋆ | 0.751 (0.039) ⋆ | 0.849 (0.044) | 0.842 (0.048) ⋆ | 0.668 (0.065) ⋆ | 0.783 (0.039) ⋆ | 0.860 (0.047) | 0.851 (0.051) ⋆
PSNR $\uparrow$ | 20.92 (2.34) ⋆ | 24.20 (1.62) ⋆ | 26.85 (2.14) | 26.56 (2.19) ⋆ | 21.86 (2.40) ⋆ | 25.66 (1.76) ⋆ | 28.27 (2.37) | 27.75 (2.51) ⋆
LPIPS $\downarrow$ | 0.321 (0.040) ⋆ | 0.117 (0.041) ⋆ | 0.092 (0.030) ⋆ | 0.090 (0.030) | 0.313 (0.041) ⋆ | 0.090 (0.032) | 0.091 (0.039) ⋆ | 0.089 (0.039)
FID $\downarrow$ | 218.18 | 61.9 | 38.42 | 29.48 | 212.61 | 51.41 | 44.64 | 36.17
AF x8 | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
SSIM $\uparrow$ | 0.529 (0.084) ⋆ | 0.579 (0.054) ⋆ | 0.680 (0.067) ⋆ | 0.719 (0.074) | 0.544 (0.080) ⋆ | 0.595 (0.054) ⋆ | 0.689 (0.064) ⋆ | 0.720 (0.072)
PSNR $\uparrow$ | 18.26 (2.37) ⋆ | 20.15 (1.73) ⋆ | 21.52 (2.09) ⋆ | 22.32 (2.19) | 19.33 (2.40) ⋆ | 20.99 (1.80) ⋆ | 22.63 (2.20) ⋆ | 23.17 (2.30)
LPIPS $\downarrow$ | 0.491 (0.040) ⋆ | 0.212 (0.059) ⋆ | 0.197 (0.049) ⋆ | 0.165 (0.046) | 0.473 (0.037) ⋆ | 0.199 (0.061) ⋆ | 0.196 (0.055) ⋆ | 0.166 (0.055)
FID $\downarrow$ | 375.91 | 100.14 | 127.37 | 62.28 | 368.58 | 85.79 | 137.83 | 72.96

Figure 5:  The visualised samples of the reconstruction on Test-S and Test-D
with the undersampling masks of AF $\times 2$, $\times 4$ and $\times 8$. Odd
Rows: the ground truth (GT), undersampled k-space zero-filled images (ZF) and
the reconstruction results of DAGAN, D5C5 and SwinMR. Even Rows: the
corresponding difference ($\times 10$) of ZF and the reconstruction results of
DAGAN, D5C5 and SwinMR. Row 1-2: AF $\times 2$; Row 3-4: AF $\times 4$; Row
5-6: AF $\times 8$; Col 1-5: the results on testing set Test-S; Col 6-10: the
results on testing set Test-D.

In this study, four metrics were considered to assess the reconstruction
quality. Peak Signal-to-Noise Ratio (PSNR) is a simple and commonly used
metric for measuring the reconstruction quality, which measures the ratio of
the maximum possible power of the signal to the power. Higher PSNR value
indicates a better reconstruction quality. Structural Similarity Index (SSIM)
is a perceptual-based metric that measures the similarity between two images
by comparing their structural information. Higher SSIM value indicates a
better reconstruction quality. Learned Perceptual Image Patch Similarity
(LPIPS) [41] is a learned metric that measures the perceptual similarity
between two images by computing the distance in the latent space using a pre-
trained deep neural network. LPIPS has shown a high correlation with human
perceptual judgements of the image similarity. Lower LPIPS value indicates a
better generated images quality. Fréchet Inception Distance (FID) [42] is a
learned metric that measures the similarity between two sets of images by
comparing their feature statistics, using a pre-trained deep neural network.
FID has also shown to have high correlation with human perceptual experience.
Lower FID value indicates a better generated images quality.

Quantitative reconstruction results on the Test-S and Test-D datasets are
presented in Table 2. The two-sample t-test was applied for the statistical
analysis, and ⋆ in Table 2 indicates the specific result distribution is
significantly different ($p<0.05$) from the best result distribution. Among
the evaluated models, D5C5 demonstrates superior fidelity in the
reconstruction, while SwinMR provides results with higher perceptual score.

Visualised samples of the reconstruction results on Test-S and Test-D datasets
are shown in Figure 5.

### Diffusion Tensor Parameter Quality Assessment

Table 3:  Differences of diffusion tensor parameter global mean values between
the reference and reconstruction results (undersampled k-space zero-filled
images ZF included), on systole testing sets Test-S and Test-MI-S. Mean
absolute error are applied for fractional anisotropy (FA), mean diffusivity
(MD), and mean absolute angular error are applied for helix angle gradient (HA
Slope) and second eigenvector (E2A). The results are quoted as ‘median
[interquartile range]’. ⋆ indicates the specific error distribution is
significantly different from the best results distribution by Mann-Whitney
Test ($p<0.05$). Data point with green background indicates that the specific
distribution of corresponding diffusion tensor parameter global mean values is
NOT significantly different from the reference distribution by Mann-Whitney
Test ($p>0.05$). Units: FA unitless; MD
$10^{-3}\cdot\text{mm}^{2}\cdot\text{sec}^{-1}$; HA Slope
$\text{degrees}\cdot\text{mm}^{-1}$ and E2A degrees.

DT Para. | Test-S | Test-MI-S
---|---|---
AF $\times 2$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.057 [0.021] ⋆ | 0.006 [0.008] | 0.005 [0.008] | 0.008 [0.007] | 0.075 [0.016] ⋆ | 0.011 [0.014] | 0.012 [0.009] | 0.012 [0.007]
MD | 0.011 [0.012] ⋆ | 0.008 [0.007] ⋆ | 0.003 [0.006] | 0.003 [0.009] | 0.027 [0.014] ⋆ | 0.005 [0.008] | 0.002 [0.006] | 0.004 [0.004]
HA Slope | 1.264 [0.849] ⋆ | 0.244 [0.279] ⋆ | 0.227 [0.228] | 0.165 [0.244] | 1.560 [0.630] ⋆ | 0.220 [0.503] | 0.208 [0.166] | 0.164 [0.161]
E2A | 2.658 [4.213] ⋆ | 0.850 [1.248] ⋆ | 0.570 [0.858] | 0.608 [1.197] | 3.093 [4.065] ⋆ | 0.904 [1.284] | 0.959 [0.993] | 0.699 [0.805]
AF $\times 4$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.140 [0.043] ⋆ | 0.014 [0.020] | 0.031 [0.028] ⋆ | 0.020 [0.019] | 0.167 [0.043] ⋆ | 0.029 [0.022] | 0.056 [0.020] ⋆ | 0.033 [0.029]
MD | 0.034 [0.028] ⋆ | 0.022 [0.041] ⋆ | 0.013 [0.020] | 0.011 [0.014] | 0.076 [0.041] ⋆ | 0.026 [0.028] ⋆ | 0.009 [0.017] | 0.013 [0.019] ⋆
HA Slope | 2.608 [1.080] ⋆ | 0.492 [0.569] | 0.747 [0.728] ⋆ | 0.274 [0.433] | 2.952 [1.084] ⋆ | 0.575 [0.546] | 0.905 [0.820] ⋆ | 0.229 [0.368]
E2A | 10.365 [9.306] ⋆ | 1.977 [2.747] | 1.661 [2.874] | 1.680 [1.557] | 7.372 [6.349] ⋆ | 2.393 [2.551] | 3.444 [3.890] ⋆ | 2.372 [1.878]
AF $\times 8$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.210 [0.049] ⋆ | 0.034 [0.036] | 0.048 [0.041] ⋆ | 0.045 [0.050] ⋆ | 0.240 [0.037] ⋆ | 0.051 [0.038] | 0.122 [0.043] ⋆ | 0.072 [0.053]
MD | 0.049 [0.045] ⋆ | 0.052 [0.054] ⋆ | 0.039 [0.051] ⋆ | 0.031 [0.028] | 0.139 [0.069] ⋆ | 0.066 [0.071] ⋆ | 0.028 [0.035] | 0.039 [0.057]
HA Slope | 3.839 [1.937] ⋆ | 2.073 [1.495] ⋆ | 2.537 [2.229] ⋆ | 1.273 [1.611] | 4.131 [1.124] ⋆ | 2.358 [1.666] | 2.767 [1.496] | 2.327 [1.973]
E2A | 14.346 [9.715] ⋆ | 6.359 [8.273] ⋆ | 4.847 [7.203] | 2.942 [5.299] | 11.849 [8.061] ⋆ | 4.916 [9.433] | 5.000 [6.944] | 4.334 [7.459]

We further evaluated the quality of DT parameters, including FA, MD, E2A and
HA, after post-processing.

Differences in DT parameter global mean values between the reference and
reconstruction, on systole testing sets (Test-S and Test-MI-S) and diastole
testing sets (Test-D and Test-MI-D) are presented in Table 3 and Table S2,
respectively. The mean absolute error for FA, MD and the mean absolute angular
error for the HA gradient (HA Slope) and E2A were employed to quantify the
difference. The Mann-Whitney test was utilised for the statistical analysis,
and ⋆ in Table 3 and Table S2 indicates that the specific error distribution
is significantly different ($p<0.05$) from the best results distribution. Data
point with green background indicates that the specific distribution of
corresponding DT parameter global mean values is NOT significantly different
($p>0.05$) from the reference distribution according to the Mann-Whitney Test.

Overall, SwinMR can achieve better or comparable (not significantly different)
MD, HA slope and E2A results on all testing sets. DAGAN can achieve better or
comparable (not significantly different) FA results on all testing sets. D5C5
has provided better results only on Test-S at AF $\times 2$, but it is not
significantly better than SwinMR (on MD, HA Slope and E2A) and DAGAN (on FA).

Some cases of visualised DT parameter maps are presented in this study,
including FA, MD, HA and absolute value of E2A (|E2A|). The DT parameter maps
of a systole healthy case from Test-S with different AFs are shown in Figure 6
(AF$\times 2$), Figure 7 (AF$\times 4$), and Figure 8 (AF$\times 8$). The DT
parameter maps of a diastole healthy case from Test-D with different AFs are
shown in Figure S2 (AF$\times 2$), Figure S3 (AF$\times 4$), and Figure S4
(AF$\times 8$). The DT parameter maps of a systole MI case from Test-MI-S with
different AFs are shown in Figure S5 (AF$\times 2$), Figure S6 (AF$\times 4$),
and Figure S7 (AF$\times 8$). The DT parameter maps of a diastole MI case from
Test-MI-D with different AFs are shown in Figure S8 (AF$\times 2$), Figure S9
(AF$\times 4$), and Figure S10 (AF$\times 8$).

Figure 6:  Diffusion parameter maps of the reconstruction results (AF $\times
2$) and the reference of a healthy systole case from testing set Test-S. Row
1: fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix
angle (HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure
7:  Diffusion parameter maps of the reconstruction results (AF $\times 4$) and
the reference of a healthy systole case from testing set Test-S. Row 1:
fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle
(HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure 8:
Diffusion parameter maps of the reconstruction results (AF $\times 8$) and the
reference of a healthy systole case from testing set Test-S. Row 1: fractional
anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle (HA); Row 4:
absolute value of the second eigenvector (|E2A|).

## Discussion

In this study, we have investigated the performance of deep learning-based
methods in the context of cDTI reconstruction. We have implemented three deep
learning-based MRI reconstruction methods, namely DAGAN, D5C5 and SwinMR, on
our cDTI dataset. Experimental results have been reported from the perspective
of reconstruction quality assessment and DT estimation quality assessments.

According to Table 2, for the reconstruction tasks with undersampling masks of
AF $\times 2$ and AF $\times 4$, D5C5 has achieved superior PSNR and SSIM,
while SwinMR has achieved better deep learning-based perceptual scores, i.e.,
LPIPS and FID. For the reconstruction tasks at AF $\times 8$, SwinMR has
outperformed other methods across all the metrics applied.

According to Figure 5, for the reconstruction task of AF $\times 2$, all three
methods have produced fairly good visual reconstruction results. For the
reconstruction task of AF $\times 4$, all three methods have successfully
recovered overall structure information, whereas they have behaved differently
in the recovery of the high-uncertainty area. For example in the experiment on
Test-S at AF $\times 4$ (Row 3-4, Col 1-5, Figure 5), the red arrows indicate
the high-uncertainty area on the LV myocardium due to the signal loss. DAGAN
has provided a noisy estimation while SwinMR has clearly preserved this part
of information. However, the results of D5C5 have missed the information in
this area. For reconstruction task of AF $\times 8$, neither of three methods
is able to produce visually satisfied reconstruction results. For example in
the experiment on Test-S with AF $\times 8$ (Row 5-6, Col 1-5, Figure 5), a
large amount of visible aliasing artefacts along the PE direction has remained
in the reconstruction results of both D5C5 and SwinMR, with D5C5 performing
relatively worse than SwinMR. DAGAN, to some extent, has eliminated the
aliasing artefacts at the expense of the increased noise, leading to a low-SNR
reconstruction. Regarding the recovery of high-uncertainty area, both DAGAN
and D5C5 have failed to preserve the information in this area. SwinMR can
retain most information of this area, but meanwhile it has produced ‘fake’
estimation (green arrow).

For the fidelity of the reconstruction, D5C5 has shown superiority on the
condition of a relative lower AF, whereas this superiority has been observed
disappearance on the condition of a relative higher AF. This phenomenon is
caused by the utilisation of DC module in D5C5 (Figure 3), which combines the
k-space measurements information with the CNN estimation to keep the
consistency. According to Figure S1, in the reconstruction task at a relative
lower AF, a large proportion of information in the final output of D5C5 is
provided by the DC module, whereas this proportion is significantly decreased
in a relative higher AF reconstruction task (AF $\times 8$). Therefore, this
kind of unrolling-based methods with DC module is more suitable for the
reconstruction at a relative lower AF.

For the perceptual score of the reconstructions, experiments have shown SwinMR
outperforms D5C5 and DAGAN on metrics LPIPS and FID. However, even though the
perceptual score has a high correlation with the observation of human, it is
not always equivalent to a better reconstruction quality [30]. According to
Figure 5 (green arrow), SwinMR has learnt to estimate a ‘fake’ reconstruction
detail for a higher perceptual score, which is totally unacceptable and
dangerous for clinical use. We believes this phenomenon is caused by the
nature of the Transformer applied in SwinMR, which is powerful enough to
estimate and generate details that does not exist originally. In addition, the
utilisation of the perceptual VGG-based loss restricts SwinMR to produce more
_perceptual-similar_ reconstruction instead of _pixel-wise-similar_
reconstruction.

In general, the differences for tensor parameter global mean values between
reference and reconstruction results tend to increase as the AF rises.
Concerning the global mean values of FA, DAGAN has demonstrated superiority on
the Test-S and Test-MI-S, with its superiority growing as the AF increases. On
the Test-D and Test-MI-D, the three methods have yielded similar results, with
no statistically significant difference observed. Regarding the global mean
values of MD, D5C5 and SwinMR have outperformed DAGAN across all the testing
sets. Specifically, D5C5 has delivered better results on Test-S, while SwinMR
has excelled on Test-MI-S. On the Test-D and Test-MI-D, SwinMR and D5C5 have
achieved similar results with no statistical difference at AF $\times 2$ and
AF $\times 4$, while SwinMR has surpassed D5C5 at a higher AF (AF $\times 8$).
For the global mean values of HA Slope, it is clear that SwinMR has
outperformed DAGAN and D5C5 on all testing sets, with its superiority being
statistically significant on Test-S and Test-D. In terms of the global mean
values of E2A, generally SwinMR has achieved better or comparable results
among the three methods, but the differences are typically not statistically
significant.

Generally, the quality of DT parameter maps have decreased as the AF
increases. We believes that at AF $\times 2$ and AF $\times 4$, the DT
parameter maps calculated by these three methods can achieve similar level
with the reference. For the MI cases from out-of-the-distribution testing set
Test-MI, these three methods can successfully preserve the information in
lesion area for clinical use. For example at AF $\times 2$, all three methods
have provided visually similar DT parameters maps with the reference (Figure 6
and Figure S5). At AF $\times 4$, all three methods can recover most
information of the DT parameters maps. DAGAN tends to produce noisier DT
parameter maps, while SwinMR and D5C5 tend to produce the smoother DT
parameter maps, which matches the results from the reconstruction quality
assessment. We can observed from the MD map and its corresponding error map,
that the vertical aliasing (along PE) direction has affected the DT parameter
maps (Figure 7, red arrows). The intensity of the MI area in the MD map of
DAGAN has had a trend to decrease, while D5C5 and SwinMR has clearly preserved
it (Figure S6, red arrows).

However, at AF $\times 8$, the quality of DT parameter maps have significantly
gone worse, which also matches the results from the reconstruction quality
assessment. For the FA map, a band of higher FA is expected to be observed in
the mesocardium for a healthy heart [43]. However, DAGAN and D5C5 have failed
to recover the band of higher FA, where DAGAN has produced very noisy FA map,
and D5C5 over-smoothed the FA map and wrongly estimate a highlight area
(Figure S7, blue arrows). For the MD map, the affect from the aliasing that
has been observed at AF $\times 4$, has gone more severe. In the healthy case,
the highlight area has wrongly appeared in MD maps from all three methods,
which is unacceptable for clinical use and may lead to misdiagnosis (Figure 8,
red arrows). In the MI case, the MI lesion area tends to decrease for all the
methods, especially for the results of DAGAN, where the lesion area has nearly
disappeared (Figure S7, red arrows). For the HA map, it has been observed that
SwinMR can produce relatively smooth HA map, while DAGAN can only reconstruct
very noisy one. However, the direction of HA has been wrong estimated in the
epicardium of the healthy case (Figure 8, green arrows). This is not
acceptable for clinical use and easier to lead to misdiagnosis such as MI. For
the |E2A| map, DAGAN tends to reconstruct a noisy map, while SwinMR tends to
produce a smooth map. All three maps can reconstruct similar results with
reference even at AF $\times 8$.

Through our experiments, we have demonstrated that the models discussed in
this paper can be effectively applied for clinical use at AF $\times 2$ and AF
$\times 4$. However, at AF $\times 8$, the performance of these three models,
including the best-performing SwinMR, has still remained limited.

We hope that this study will serve as a baseline for the future cDTI
reconstruction model development. Our findings have indicated that there are
still limitations when directly applying these general MRI reconstruction
methods for cDTI reconstruction.

_There is an absence of restrictions on diffusion._ The loss functions
utilised in the three models discussed in this study all rely on image domain
loss, with D5C5 and DAGAN additionally incorporating the frequency domain loss
and the perceptual loss. In other words, these is no diffusion information
restriction implemented during the model training stage. For further work, the
diffusion tensor or parameter maps can be jointly considered into the loss
function. Moreover, physical constraints on diffusion can be also incorporated
into the training stage.

_There is a trade-off between perceptual performance and quantitative
performance._ Cardiac diffusion tensor MRI is a quantitative technique, which
places greater emphasis on the accuracy contrast, pixel intensity range, and
pixel-wise fidelity, also referred to as pixel-wise distance. However, the
models discussed in this study were originally designed for structural MRI,
and they tend to pay more attention on the ‘perceptual-similarity’, which can
be regarded as the latent space distance. A trade-off exists between pixel-
wise fidelity and perceptual-similarity [44]. For example, blurred images
generally exhibit better pixel-wise fidelity, while the images with clear but
‘fake’ details tend to have better perceptual-similarity [30]. Such ‘fake’
details can sometimes be harmful for clinical use. Consequently, for further
work, more efforts should be made to consider about how to improve the pixel-
wise fidelity rather than the perceptual-similarity, or how to prevent the
appearance of the ‘fake’ information.

_There is a gap between current DT evaluation methods and the true quality of
cDTI reconstruction._ This study has revealed that the global mean value of
diffusion parameters is not always accurate or sensitive enough to evaluate
the diffusion tensor quality. For example, Table 3 indicates no statistically
significant difference in MD between reconstruction results (even including
ZF) and the reference on Test-S, whereas the Figure 8 shows that the MD maps
are entirely unacceptable. This discrepancy arises because the MD value
increases and decreases in different parts of the MD map, while the global
mean value maintains relative consistency, rendering the global mean MD
ineffective in reflecting the quality of the final DT estimation. For future
work, apart from the visualised assessment, we will applied the down-stream
task assessment, e.g., utilising a pre-trained pathology classification or
detection mdoel to evaluate the reconstruction quality. Theoretically, better
classification or detection accuracy corresponds to improved reconstruction
results.

There are still limitations for this study. 1) The size of testing sets is not
sufficiently large. The relative small testing sets enlarge the randomness of
experimental results and reduce the reliability of statistical tests. In
future studies, we will expand our dataset, and provide more accurate results.
2) Our simulation experiment is based on the retrospective k-space
undersampling on single-channel DWIs that have been reconstructed by the MR
scanner. The retrospective undersampling step itself has removed a large
amount of noise, leading to unrealistic post-processing results. In future
studies, we will conduct our experiment on prospectively-acquired k-space raw
data.

## Conclusion

In conclusion, we have investigated the application of deep learning-based
methods for accelerating cDTI reconstruction, which has significant potential
for improving the integration of cDTI into routine clinical practice. Our
study focuses on three different models, namely D5C5, DAGAN, and SwinMR, which
have been evaluated on cDTI datasets with the AF of $\times 2$, $\times 4$,
and $\times 8$. The results have demonstrated that the examined models can be
effectively utilised for clinical use at AF $\times 2$ and AF $\times 4$, with
SwinMR being the recommended optimal approach. However, at AF $\times 8$, the
performance of all models has remained limited, and further research is
required to improve their performance at a relative higher AF.

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## Acknowledgements (not compulsory)

This study was supported in part by the UKRI Future Leaders Fellowship
(MR/V023799/1), BHF (RG/19/1/34160), the ERC IMI (101005122), the H2020
(952172), the MRC (MC/PC/21013), the Royal Society (IEC/NSFC/211235), the
NVIDIA Academic Hardware Grant Program, EPSRC (EP/V029428/1, EP/S026045/1,
EP/T003553/1, EP/N014588/1, EP/T017961/1), and the Cambridge Mathematics of
Information in Healthcare Hub (CMIH) Partnership Fund.

## Author contributions statement

Must include all authors, identified by initials, for example: J.H. and G.Y.
conceptualised the study, P.F.F, A.A.R., C.B.S., A.D.S.,S.N.V., and G.Y.
guided the methodology development, J.H., P.F.F., S.N.V., and G.Y. conceived
the experiments, J.H., P.F.F., L.W., Y.W. and G.Y. conducted the experiments,
J.H., P.F.F., L.W., Y.W., S.N.V., and G.Y. analysed the results, Z.K., M.D.,
R.R., R.D.S., and D.J.P. provided the data including labelling and clinical
guidance. All authors reviewed the manuscript.

## Additional information

Competing interests D.J.P receives research support from Siemens and is a
stockholder and director of Cardiovascular Imaging Solutions. The RBH CMR
group receives research support from Siemens Healthineers. Others have no
Conflicts of Interest.

Supplemental Materials of
Deep Learning-based Diffusion Tensor Cardiac Magnetic Resonance
Reconstruction: A Comparison Studies

## Methodology

Figure S1:  Three Cartesian k-space undersampling masks applied in this work. AF: acceleration factor; CF: centre factor.  Table S1: The detailed information of the dataset. AMYLOID: amyloidosis; DCM: dilated cardiomyopathy; rDCM: in-recovery DCM; HCM: hypertrophic cardiomyopathy HCM G+P-: HCM genotype-positive–phenotype-negative MI: acute myocardial infarction.  Disease Type | Cardiac Phase | Total | Train & Validation | Test
---|---|---|---|---
All Folds | Fold-1 | Fold-2 | Fold-3 | Fold-4 | Fold-5
Health | Diastole | 112 | 90 | 18 | 18 | 18 | 18 | 18 | 22
Health | Systole | 129 | 103 | 21 | 21 | 21 | 20 | 20 | 26
AMYLOID | Diastole | 14 | 11 | 3 | 2 | 2 | 2 | 2 | 3
AMYLOID | Systole | 17 | 14 | 3 | 3 | 3 | 3 | 2 | 3
DCM | Diastole | 23 | 18 | 4 | 4 | 4 | 3 | 3 | 5
DCM | Systole | 24 | 19 | 4 | 4 | 4 | 4 | 3 | 5
HCM (G+P-) | Diastole | 24 | 19 | 4 | 4 | 4 | 4 | 3 | 5
HCM (G+P-) | Systole | 24 | 19 | 4 | 4 | 4 | 4 | 3 | 5
HCM | Diastole | 23 | 18 | 4 | 4 | 4 | 3 | 3 | 5
HCM | Systole | 16 | 13 | 3 | 3 | 3 | 2 | 2 | 3
rDCM | Diastole | 16 | 13 | 3 | 3 | 3 | 2 | 2 | 3
rDCM | Systole | 19 | 15 | 3 | 3 | 3 | 3 | 3 | 4
MI | Diastole | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 20
MI | Systole | 20 | 0 | 0 | 0 | 0 | 0 | 0 | 20

## Experiments and Results

Table S2:  Differences of diffusion tensor parameter global mean values
between the reference and reconstruction results (undersampled k-space zero-
filled images ZF included), on diastole testing sets Test-D and Test-MI-D.
Mean absolute error are applied for fractional anisotropy (FA), mean
diffusivity (MD), and mean absolute angular error are applied for helix angle
gradient (HA Slope) and second eigenvector (E2A). The results are quoted as
‘median [interquartile range]’. ⋆ indicates the specific error distribution is
significantly different from the best results distribution by Mann-Whitney
Test ($p<0.05$). Data point with green background indicates that the specific
distribution of corresponding diffusion tensor parameter global mean values is
NOT significantly different from the reference distribution by Mann-Whitney
Test ($p>0.05$). Units: FA unitless; MD
$10^{-3}\cdot\text{mm}^{2}\cdot\text{sec}^{-1}$; HA Slope
$\text{degrees}\cdot\text{mm}^{-1}$ and E2A degrees.

DT Para. | Test-D | Test-MI-D
---|---|---
AF $\times 2$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.044 [0.017] ⋆ | 0.007 [0.011] | 0.005 [0.006] | 0.004 [0.004] | 0.058 [0.019] ⋆ | 0.007 [0.011] | 0.010 [0.009] | 0.008 [0.007]
MD | 0.018 [0.023] ⋆ | 0.013 [0.013] | 0.007 [0.010] | 0.009 [0.009] | 0.042 [0.017] ⋆ | 0.007 [0.012] | 0.007 [0.006] | 0.004 [0.006]
HA Slope | 2.188 [1.105] ⋆ | 0.370 [0.522] ⋆ | 0.238 [0.295] | 0.218 [0.377] | 1.961 [1.065] ⋆ | 0.315 [0.290] | 0.173 [0.256] | 0.167 [0.274]
E2A | 3.124 [3.340] ⋆ | 0.887 [1.259] ⋆ | 0.524 [0.796] | 0.497 [0.661] | 3.609 [3.316] ⋆ | 0.693 [1.246] | 0.608 [0.781] | 0.895 [0.821]
AF $\times 4$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.120 [0.041] ⋆ | 0.013 [0.017] | 0.013 [0.022] | 0.009 [0.012] | 0.138 [0.024] ⋆ | 0.011 [0.009] | 0.016 [0.018] | 0.007 [0.014]
MD | 0.075 [0.057] ⋆ | 0.029 [0.030] ⋆ | 0.014 [0.022] | 0.015 [0.014] | 0.130 [0.032] ⋆ | 0.036 [0.031] ⋆ | 0.014 [0.018] | 0.015 [0.017]
HA Slope | 2.783 [1.035] ⋆ | 0.641 [0.977] ⋆ | 0.592 [0.896] ⋆ | 0.392 [0.643] | 3.076 [0.699] ⋆ | 0.508 [0.810] | 0.682 [0.294] ⋆ | 0.215 [0.483]
E2A | 5.093 [3.850] ⋆ | 1.464 [1.922] | 1.422 [1.881] | 1.036 [1.200] | 5.006 [5.062] ⋆ | 1.407 [1.566] | 1.296 [1.195] | 1.061 [1.690]
AF $\times 8$ | ZF | DAGAN | D5C5 | SwinMR | ZF | DAGAN | D5C5 | SwinMR
FA | 0.233 [0.080] ⋆ | 0.035 [0.044] | 0.030 [0.034] | 0.033 [0.042] | 0.248 [0.028] ⋆ | 0.026 [0.043] | 0.056 [0.061] | 0.034 [0.038]
MD | 0.103 [0.094] ⋆ | 0.085 [0.099] ⋆ | 0.068 [0.068] ⋆ | 0.041 [0.066] | 0.232 [0.065] ⋆ | 0.145 [0.067] ⋆ | 0.086 [0.067] ⋆ | 0.043 [0.069]
HA Slope | 3.282 [1.301] ⋆ | 3.014 [2.634] ⋆ | 2.145 [1.883] ⋆ | 1.116 [1.408] | 3.802 [1.226] ⋆ | 3.101 [1.453] ⋆ | 2.708 [1.591] ⋆ | 1.599 [1.909]
E2A | 3.538 [4.984] ⋆ | 4.100 [4.677] | 8.363 [10.243] ⋆ | 3.118 [6.069] | 4.025 [3.634] ⋆ | 2.476 [4.193] | 5.369 [6.679] ⋆ | 3.854 [5.843]

Figure S2:  Diffusion parameter maps of the reconstruction results (AF $\times
2$) and the reference of a healthy diastole case from testing set Test-D. Row
1: fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix
angle (HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure
S3:  Diffusion parameter maps of the reconstruction results (AF $\times 4$)
and the reference of a healthy diastole case from testing set Test-D. Row 1:
fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle
(HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure S4:
Diffusion parameter maps of the reconstruction results (AF $\times 8$) and the
reference of a healthy diastole case from testing set Test-D. Row 1:
fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle
(HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure S5:
Diffusion parameter maps of the reconstruction results (AF $\times 2$) and the
reference of a acute myocardial infarction (MI) systole case from testing set
Test-MI-S. Row 1: fractional anisotropy (FA); Row 2: mean diffusivity (MD);
Row 3: helix angle (HA); Row 4: absolute value of the second eigenvector
(|E2A|).  Figure S6:  Diffusion parameter maps of the reconstruction results
(AF $\times 4$) and the reference of a acute myocardial infarction (MI)
systole case from testing set Test-MI-S. Row 1: fractional anisotropy (FA);
Row 2: mean diffusivity (MD); Row 3: helix angle (HA); Row 4: absolute value
of the second eigenvector (|E2A|).  Figure S7:  Diffusion parameter maps of
the reconstruction results (AF $\times 8$) and the reference of a acute
myocardial infarction (MI) systole case from testing set Test-MI-S. Row 1:
fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle
(HA); Row 4: absolute value of the second eigenvector (|E2A|).  Figure S8:
Diffusion parameter maps of the reconstruction results (AF $\times 2$) and the
reference of a acute myocardial infarction (MI) diastole case from testing set
Test-MI-D. Row 1: fractional anisotropy (FA); Row 2: mean diffusivity (MD);
Row 3: helix angle (HA); Row 4: absolute value of the second eigenvector
(|E2A|).  Figure S9:  Diffusion parameter maps of the reconstruction results
(AF $\times 4$) and the reference of a acute myocardial infarction (MI)
diastole case from testing set Test-MI-D. Row 1: fractional anisotropy (FA);
Row 2: mean diffusivity (MD); Row 3: helix angle (HA); Row 4: absolute value
of the second eigenvector (|E2A|).  Figure S10:  Diffusion parameter maps of
the reconstruction results (AF $\times 8$) and the reference of a acute
myocardial infarction (MI) diastole case from testing set Test-MI-D. Row 1:
fractional anisotropy (FA); Row 2: mean diffusivity (MD); Row 3: helix angle
(HA); Row 4: absolute value of the second eigenvector (|E2A|).


# Searching for Long Lived Particles at LHC

A. Santocchia  INFN Sezione di Perugia and Perugia University, Via Pascoli,
06123-Perugia, Italy

###### Abstract

Search for BSM phenomena is one of the fundamental goals of the LHC
experiments. Many BSM models foresee long-lived particles which decay far from
the production vertex and a big effort has been done by the ATLAS and CMS
collaborations to detect these long-lived particles. A widely studied example
for such searches is made through the measurement of displaced vertices
produced by a narrow resonance decaying to a muon pair. This paper will first
analyze what has been done so far and will try to evaluate if there are
improvements which may allow enhancing the discovery potential at LHC for such
particles. The outcome of this analysis is that while long-lived particles
decaying very far from the primary vertex have been carefully studied by ATLAS
and CMS, the region within a few millimeters from the primary vertex may have
been partially neglected. A new approach for searching such resonances, fully
based on data, will then be proposed which allows minimizing systematic
uncertainties while keeping at a maximum the discovery potentiality for long-
lived neutral resonances which decay a few millimeters from the primary
vertex.

###### keywords:

Long-Lived Particles, Displaced Vertices, LHC.

††journal: Physics Letters B

##  1 Introduction

The standard model (SM) of particle physics is one of the most powerful theory
ever developed by mankind. His success has been experimentally confirmed in
the past 30 years in many sectors, the last one being the discovery of the
Higgs boson at LHC [1, 2]. There are however many aspects that suggest that
the SM is not the end of the story and a tremendous effort has been made to
find new phenomena beyond the SM which could give a hint on how to extend it.
One of the sectors where a big effort has been made recently by ATLAS [3] and
CMS [4], is the search of long-lived particles (LLPs). A LLP is an unstable
particle with a decay length big enough to decay far from the production
vertex and which travels a length measurable by a detector.

Neutral LLPs can produce unique signatures, such as displaced vertices (DV)
with respect to the primary vertex (PV), which could be detected by looking at
the decay products. A simple option is to consider an LLP producing a narrow
resonance decaying to $\mu^{+}\mu^{-}$. Generally speaking, this choice
doesn’t limit the power of the discussed method because it could be applied
also to other decay channels where a displaced vertex could be detected.

In this paper I will first discuss what has been done up to now for detecting
LLPs at ATLAS and CMS analyzing the different techniques used by both
experiments. Subsequently, a reference model implemented in pythia [5] will be
introduced and discussed, highlighting why it can be used to generate a LLP
decaying to a lepton pair and why all the results obtained with this process
could be used also for other models. Detector effects will be simulated using
the Delphes [6] parametric simulation with a dedicated configuration card used
to emulate the performance of the CMS detector. Finally, simulated events will
be used to show that there are LLP decaying to $\mu^{+}\mu^{-}$ which could go
undetected with the methods used to date at LHC experiments and a new
technique will be suggested to cover this unexplored area.

##  2 Review of LLP searches at LHC

There are two different complementary methods for searching LLP: inclusive or
exclusive searches. In the first case, the search is mainly based on looking
for a displaced vertex reconstructed from a muon pair or from one or more jets
using their charged components measured in the tracking detector. The total
invariant mass, build from the charged particles associated to the vertex, or
the number of events detected are then used to look for any excess with
respect to SM predictions. For exclusive searches, more complex topologies are
tested, and the DV is associated with other leptons, jets or missing
transverse energy. Topologies where the DV is not directly detectable because
the decay products are neutral are also studied. Most of the time, however,
inclusive and exclusive searches look for a DV: the farther the DV is from the
PV, the more difficult is to measure it because of the deterioration of the
track efficiency. On the other hand, there are very few, well known, SM
particles which produce a DV within LHC detectors acceptance so all these
searches are almost background free.

Figure 1: Schematic view of the $\phi_{3D}$ angle, $L_{xy}$ and $L_{xyx}$.

This paper discusses only the inclusive search of a neutral, narrow resonance
decaying to a muon pair. Both ATLAS and CMS use a set of standard requirements
to guarantee the quality of the muon sample. Special requirements are also
used to minimize the contribution from cosmic ray muons which may be
reconstructed as back-to-back muons. In the case of the two muons inclusive
LLP searches, the most common additional requirements are on the muon
transverse impact parameter $d_{0}$ or its significance $|d_{0}/\sigma_{d}|$
where $\sigma_{d}$ is the error on $d_{0}$.

After the selection of the muon pair, a secondary vertex finding algorithm may
be run. If the vertex is found, then typical additional requirements are on
the distance $L_{xy}$ in the transverse plane of the DV with respect to the PV
or its significance $L_{xy}/\sigma_{L_{xy}}$ where $\sigma_{L_{xy}}$ is the
error on $L_{xy}$. The distance $L_{xyz}$ of the DV with respect to the PV or
its significance $L_{xyz}/\sigma_{L_{xyz}}$ where $\sigma_{L_{xyz}}$ is the
error on $L_{xyz}$ is not used in any of the paper considered in this review.
The main reason is probably the poor resolution of the longitudinal coordinate
of the DV with respect to the resolution of the distance in the transverse
plane. Another aspect that could drive this choice is the presence of pile-up
events which could reduce the efficiency of selecting the correct PV: the PV-
DV distance in the transverse plane does not depend on the presence of
additional PVs along the beam direction while the 3D distance would change
dramatically if the wrong PV is selected. Another kinematic quantity used is
$\phi_{3D}$: the angle between the PV-DV vector $\vec{\Delta}_{3D}$ (which
represent the line of flight of the particle) and the momentum direction of
the muon pair $\vec{p}_{\mu\mu}$. In the hypothesis of a LLP 2-body decay,
this angle would be zero and any deviation would depend on the tracking
resolution of the detector only. Figure 1 shows a schematic view of these
quantities and table 1 reports a summary of the main pre-selection cuts
applied in all the papers published until 2023 by ATLAS and CMS for run 1 and
run 2.

Paper | Year | $\sqrt{s}$ |  $\mathcal{L}$ fb-1 | $d_{0}$ | $|d_{0}/\sigma_{d_{0}}|$ |  $\chi^{2}_{\mathrm{vtx}}$ fit | $d_{xy}$ | $d_{xy}/\sigma_{d_{xy}}$ | $\phi_{3D}$
---|---|---|---|---|---|---|---|---|---
CMS 1 [7] | 2013 | 7 | 5.1 | - | 2 | 5 | - | 5 | NO
CMS 2 [8] | 2015 | 8 | 20.5 | - | 12 | 5 | - | - | 2D
ATLAS 1 [9] | 2019 | 13 | 32.9 | - | - | NO | - | - | NO
ATLAS 2 [10] | 2020 | 13 | 32.8 | 2 | - | 5 | 2 | - | NO
ATLAS 3 [11] | 2020 | 13 | 136 | 2 | - | 5 | 4 | - | NO
CMS 3 [12] | 2022 | 13 | 113 | 0.1 | - | NO | - | - | NO
CMS 4 [13] | 2022 | 13 | 101 | - | 2/1 | 5 | - | - | 2D
CMS 5 [14] | 2023 | 13 | 97.6 | - | 6 | 10 | - | 6 | 2D
ATLAS 4 [15] | 2023 | 13 | 139 | 0.1 | - | NO | - | - | NO
Table 1:  CMS and ATLAS muon inclusive searches ($d_{0}$, $\sigma_{d_{0}}$,
$d_{xy}$ and $\sigma_{d_{xyz}}$ are measured in mm). 2D in the table indicates
that the $\phi_{3D}$ angle is used in the transverse plane.

##  3 Analysis Strategy

The capability to detect a BSM particle thanks to a DV depends on particle
properties and detector performances. While the detector performances are
somehow fixed, the particle properties could be of any type. Specifically, the
number of events with a DV that a detector could measure may be calculated
through

$N=\mathcal{L}\sigma\mathit{B}_{\mu\mu}\epsilon_{D},$ (1)

where $N$ is the number of events detected, $\mathcal{L}$ is the integrated
luminosity, $\sigma$ is the production cross-section of the resonance,
$\mathit{B}_{\mu\mu}$ is the branching fraction of the resonance decaying to
$\mu^{+}\mu^{-}$, and $\epsilon_{D}$ is the detector efficiency (including the
efficiency of the vertex finding algorithm) for the specific process. Going to
a more detailed analysis, it’s clear that the product
$\sigma\mathit{B}_{\mu\mu}$ depends on the particle properties (i.e., on the
chosen model), $\mathcal{L}$ is fixed and also $\epsilon_{D}$ is fixed
(depends on the detector). The efficiency to measure a DV depends on the
distance traveled by the resonance before its decay, which depends on its
momentum and lifetime $\tau$. If the resonance lifetime is very short (below
$\tau\sim 0.1$ ps, which correspond to a $c\tau$ lower than $\sim 30\mu m$),
then its decay products are named prompt, i.e. are consistent with the PV
because the detector can not distinguish such a small distance and its
discovery cannot happen thanks to a DV search. On the contrary, if the
lifetime is bigger, then the resonance may decay at a DV. The lifetime $\tau$
has an exponential distribution which depends on the mean lifetime
$\langle\tau\rangle$. Both $\sigma$ and $\langle\tau\rangle$ depend on the
coupling constants of the BSM but, for the specific process discussed in this
paper, they do not involve the same coupling constants (apart interference
effects and higher order contributions): $\sigma$ depends on the quarks-gluons
coupling constants at the production vertex while $\langle\tau\rangle$ depends
on the leptons coupling constants at the decay vertex.

There are several BSM models where a resonance with such characteristics is
present and, regardless of the model itself, is it often possible to set
independently the quarks-gluons coupling constants and the leptons coupling
constants to have a resonance with a lifetime in the interesting range. The
cross-section $\sigma$ could then be used to calculate the number of events
produced in run 2 at LHC using eq. 1. This is the common approach in all LHC
analyses: given a BSM model, a prediction for the number of expected events is
compared with the SM expected number and, in case of no deviation, a limit is
placed on the BSM parameters. The purpose of this paper is instead to evaluate
if what has been done so far could be improved and the key question becomes:
given a narrow resonance which decay to a $\mu^{+}\mu^{-}$ pair with a
specific lifetime (which depends only on the lepton coupling constants), what
is the minimal cross-section (times the branching ratio to $\mu^{+}\mu^{-}$)
$\sigma_{D}$ which may produce a discovery at the LHC run 2?

To answer this question, it is necessary to define what is the discovery
condition. The standard criterion for claiming a discovery in particle physics
is that the observed effect should have the equivalent of a five standard-
deviation discrepancy with the Standard Model. This paper will use this
discovery condition but inverting eq. 1: given the mass and mean lifetime of
the resonance and the SM background distribution, it is possible to calculate
the expected number of SM events at the resonance mass and, using the
5$\sigma$ criterion, extrapolate the minimum $\sigma\mathit{B}_{\mu\mu}$ which
will produce a 5$\sigma$ discrepancy with respect to the SM background.

##  4 LLP Model and simulation

The benchmark model used in this study is the one [16] implemented in PHYTIA
where a new $U(1)$ gauge group is considered for the production of a new
neutral gauge boson $Z^{\prime}$. By a suitable choice of the parameters, it
is possible to simulate just about any imaginable $Z^{\prime}$ scenario,
including one where the $Z^{\prime}$ lifetime is big enough to produce a DV.
To fully explore the LHC scenario, 5 different mass points have been generated
from 200 GeV to 1 TeV with 200 GeV steps. A proper choice of the model
parameters will allow to generate a $Z’$ with a mean lifetime in a range that
produce a DV up to few meters away from the PV. The generated events are then
passed to the Delphes simulation where a Vertex-Fitter code [17] is also run
to identify the PV and the dimuon vertex, which, depending on the model
parameters, may coincide with the PV or generate a DV.

To estimate the LHC sensitivity to the signal, the 3 main SM background
processes have been also generated and simulated using pythia and MadGraph
[18]: Drell-Yan production
$p\bar{p}\rightarrow\gamma/Z^{0}\rightarrow\mu^{+}\mu^{-}$, $t\bar{t}$
production and single-top production (_s_ -channel, _t_ -channel and $tW$).
All simulated samples include an average of 35 pile-up events. Because the
target is to analyze the effectiveness of the LHC run 2 analysis listed in
table 1, both the signal and the background processes have been normalized to
an integrated luminosity of $\mathcal{L}=$ 140 fb-1. A basic set of cuts has
been used to select $pp\rightarrow\mu\mu$ events:

  * 1.

at least 2 opposite-charge muons with $p_{T}>30(10)$ GeV/c for the most
(second) energetic muon;

  * 2.

$|\eta|<2.4$ for both muons;

  * 3.

relative charged isolation defined as $\sum{p_{T}}/p_{T}^{\mu}<0.3$ and the
sum is extended to all tracks with $\Delta
R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}<0.3$.

The width of the BSM resonance depends on its lifetime and above
$\langle\tau\rangle\sim 10^{-25}$ s, the $Z^{\prime}$ width is too small to be
measured with LHC detectors and the measured width is basically given by the
dimuon mass resolution of the detector (which therefore depends only on the
$Z^{\prime}$ mass). The mass resolution has been extracted from the simulation
of the $Z^{\prime}$ signal and varies from 3.8 GeV for $m_{Z^{\prime}}=200$
GeV up to 54.4 GeV for $m_{Z^{\prime}}=1$ TeV.

The last ingredient which allows to calculate $\sigma_{D}$ as a function of
the LLP mean lifetime $\langle\tau\rangle$ is then given by the equation

$\tau=\frac{L}{c\beta\gamma}=\frac{mL}{|\vec{p}|},$ (2)

where $m$ is the mass of the LLP, $L$ is the PV-DV distance and $\vec{p}$ is
the LLP momentum. At generator level, the mean $Z^{\prime}$ lifetime
$\langle\tau\rangle$ may be calculated from the exponential fit of the
$Z^{\prime}$ lifetime distribution obtained using eq. 2. From an experimental
point of view, equation 2 shows that, given a model which foresees a narrow
resonance with a given mass and mean lifetime, the number of LLPs which can be
detected depends on the momentum spectrum (the larger the LLP momentum, the
greater the distance traveled before the LLP decay). It’s important to
underline that the momentum spectrum of such narrow resonance produced in the
$s$-channel at LHC will be the same regardless of the chosen BSM model.

Putting together all the ingredients, the recipe to calculate $\sigma_{D}$ is
the following: for each set of the $Z^{\prime}$ pythia parameters, 50000
events have been generated and simulated with Delphes. The generated mean
lifetime is calculated using the distance between the generated PV and DV of
the $Z^{\prime}$. The number of signal events
$N_{S}=\sigma_{D}\epsilon\mathcal{L}$ which pass a specific selection for
$\mathcal{L}=$140 fb-1, is obtained from the integral of the invariant dimuon
mass distribution over the range $[m-2\Gamma,m+2\Gamma]$ where $\Gamma$ is the
experimental width of the $Z^{\prime}$. The number of background events
$N_{B}$ is obtained from the background only distribution integrated over the
same mass range used for the signal sample. Finally, the discovery cross-
section $\sigma_{D}$ is calculated from

$\sigma_{D}=\frac{5\sqrt{N_{B}}}{\epsilon_{D}\mathcal{L}},$ (3)

where the factor 5 is for the common 5$\sigma$ discovery condition.

##  5 Results

Table 1 indicates clearly that two quantities are mainly used to identify a
DV: the muons transverse impact parameter $d_{0}$ for both muons and the
distance $L_{xy}$. Different selection cuts are applied to these 2 quantities
or to their significance. A full scan of these parameters together with the 3D
distance $L_{xyz}$ and its significance could give an indication on what is
the best approach for a LLP search. For each variable, the discovery cross-
section is calculated using the base cuts described in section 4 together with
a single cut on one of the 6 variables considered. The chosen ranges for each
variable are the following: $d_{0}\in[0,0.1]$ mm, $L_{xy}$ and $L_{xyz}$ in
$[0,2]$ mm and all significance $|d_{0}/\sigma_{d_{0}}|$,
$L_{xy}/\sigma_{L_{xy}}$ and $L_{xyz}/\sigma_{L_{xyz}}$ in $[0,20]$.

Figure 2: Discovery cross-section $\sigma_{D}$ for different transverse impact
parameter cuts for a 400 GeV $Z^{\prime}$.

As an example, fig. 2 show the discovery cross-section $\sigma_{D}$ for a 400
GeV $Z^{\prime}$ as a function of its mean lifetime and transverse impact
parameter cut used in the selection. The area above each curve represents the
combination of $Z^{\prime}$ parameters that may lead to a discovery at LHC
while the area below indicates a combination of $Z^{\prime}$ parameters which
gives too few events to allow for a discovery using the 5$\sigma$ criterion.
It’s evident that below $\langle\tau\rangle\backsimeq 0.1$ ps (which
corresponds to a $c\langle\tau\rangle\backsimeq 30\mu$m) the distance between
the PV and the DV is too small for the actual generation of LHC detectors and
the 2 muons from the $Z^{\prime}$ decay are prompt: in this case the best
approach is to avoid any cut on the transverse impact parameter. Above
$\langle\tau\rangle\backsimeq 30$ ps (which corresponds to a
$c\langle\tau\rangle\simeq 10$ mm), the $\sigma_{D}$ shape is flat and there
is no improvement in going to a $d_{0}$ cut bigger than $\simeq 90\ \mu$m. A
full scan of the 6 variables listed above allows to identify the best
variable/cut value pair for different mean lifetime to maximize the discovery
potential of a $Z^{\prime}$ resonance with a mass up to 1 TeV.

Figure 3: Discovery cross-section $\sigma_{D}$ for the best combination and
CMS 1 paper [7].

A better understanding of the discovery potential could be seen in fig. 3
where (as an example) the best combination obtained from the full 6-variables
scan (extended to the 5 $Z^{\prime}$ masses discussed here) is compared to the
$\sigma_{D}$ obtained using the set of cuts applied in one of the CMS papers
normalized to $\mathcal{L}=$140 fb-1 (see table 1). This comparison shows that
the set of cuts used in ref. [7] is not optimized for a $Z^{\prime}$ with a
$c\langle\tau\rangle$ below few cm.

##  6 The $\cos\phi_{3D}$ approach

In the previous section it was shown that LHC experiments can improve LLP
searches by optimizing the main variables cuts used to select signal events.
The most interesting region to study is where the mean path of the LLP
resonance is at most few mm. In this region, however, the SM background due to
detectors effects is not negligible, and therefore it is necessary to develop
an appropriate strategy to evaluate it. All the previous plots have been
produced assuming that simulated events represents correctly data
distributions but for the 6 variables considered, it is known that also full
simulated events are not good enough to represent correctly background
distributions and all the papers listed in Tab. 1 prefer to evaluate it
directly from data. It is evident that the final result will depend on the
chosen procedure for the background evaluation and the capability to keep
under control all the systematic uncertainties.

Figure 4: Relative efficiency for the P- and M-distribution as a function of
$c_{\phi}$.

The new proposed approach will use the $\phi_{3D}$ angle introduced in section
1 for reducing at a minimum the systematic uncertainties: above $m(\mu\mu)=$
200 GeV there are no SM resonances and the
$\vec{\Delta}_{3D}/\vec{p}_{\mu\mu}$ vector directions will be fully
uncorrelated. The distribution of $\cos\phi_{3D}$ would be fully symmetric
around $\cos\phi_{3D}=0$ while for a LLP decaying far from the PV, the
$\cos\phi_{3D}$ distribution will be strongly peaked at 1. The dimuon
invariant mass distribution could be therefore divided in 2 different
distributions: the Plus-distribution (P-distribution: dimuon invariant mass
where $c_{\phi}\leq\cos\phi_{3D}\leq 1$) and the Minus-distribution
(M-distribution: dimuon invariant mass where $-1\leq\cos\phi_{3D}\leq-
c_{\phi}$), where $c_{\phi}$ may be chosen in the range $[0,1]$. For SM
background events, the P- and M-distribution will be identical within
statistical errors, while a LLP will produce a strong peak in the
P-distribution only. Fig. 4 shows this effect for SM background and a 400 GeV
$Z^{\prime}$ with $c\langle\tau\rangle=1.4$ mm: for SM background, the
relative efficiency is always 50% (within statistical error) and the
P/M-distribution are identical; for a LLP, the higher the cut on the
$c_{\phi}$, the greater will be the P- relative efficiency with respect to the
M-distribution (as an example, using Fig. 4, for $c_{\phi}$ = 0.90, the
relative efficiency will be 72% for the P- and 28% for the M-distribution).

An example is shown in fig. 5 where a 400 GeV $Z^{\prime}$ signal with
$c\tau=1.4$ mm and $\sigma B_{\mu\mu}=2.1$ fb has been added to the SM
background. The selection applied is $|d_{0}/\sigma_{d_{0}}|\geq 1.0$ and
$c_{\phi_{3D}}=0.90$. This approach offers an undisputed advantage with
respect to other methods: almost all systematic uncertainties could be
neglected because they will affect at the same level both the P- and
M-distributions. The only check needed (and eventually considered as a
systematic effect) is to verify that the P- and M-distributions are identical
within statistical errors.

Figure 5: Dimuon invariant mass P- and M-distributions for SM background
events only and with a 400 GeV $Z^{\prime}$ with $c\langle\tau\rangle=1.4$ mm
superimposed ($|d_{0}/\sigma_{d_{0}}|\geq 1.0$ and $c_{\phi_{3D}}=0.90$)

The discovery cross-section can be obtained using again eq. 3 calculating the
number of events $N_{P}$ and $N_{M}$ over the range $[m-2\Gamma,m+2\Gamma]$ in
the P- and M-distributions:
$N_{P/M}=B_{P/M}+\sigma_{D}\mathcal{L}\epsilon_{P/M}$ where $B_{P/M}$ are
respectively the SM background P- and M-distribution,
$\sigma_{D}\mathcal{L}\epsilon_{P/M}$ are the $Z^{\prime}$ P- and M-
contribution and the significance is $(N_{P}-N_{M})/\sqrt{N_{M}}$. Requesting
the $5\sigma$ discovery condition will allow calculating $\sigma_{D}$ as a
function of the $Z^{\prime}$ mass and lifetime. Six different $c_{\phi_{3D}}$
values in the range $[0,0.98]$ have been used together with the 6 variables
already discussed. A cross-check of the goodness of the P/M method can be done
using a profile likelihood ratio as a statistical test to calculate the
significance of the signal. Assuming a Gaussian signal with mean value equal
to 400 GeV and width equal to the mass resolution (11.4 GeV for a 400 GeV
$Z^{\prime}$), the background given by the M-distribution and the data by the
P-distribution, the calculated significance is 5.12 in agreement with the
$5\sigma$ discovery condition.

Fig. 6 compares the best combination and 3 different selections using the P/M
approach for a 400 GeV $Z^{\prime}$: again, below $\langle\tau\rangle\simeq
0.1$ ps, there is no improvement in using any specific selection: all muons
are prompt, and DVs are not distinguishable from the PV. Above $\simeq 10$ ps,
there are virtually no SM background events: all reconstructed DVs are due to
detector effects or mis-reconstructed cosmic rays. The full scan of the 6
considered variables shows that the significances $|d_{0}/\sigma_{d_{0}}|$ and
$L_{xyz}/\sigma_{L_{xyz}}$ are the most powerful quantities to selects LLPs.
In the region between 0.1 and 10 ps however, the P/M approach has many
advantages with respect to previous published methods: $\sigma_{D}$ is
comparable with the ideal best combination but virtually all systematic
uncertainties normally studied could be neglected because affect identically
both the P- and M-distribution.

Figure 6: Discovery cross-section $\sigma_{D}$ for a 400 GeV $Z^{\prime}$
resonance: ideal best combination and 2 possibile choice of the
$\cos\phi_{3D}$ approach.

##  7 Conclusions

Searches of LLPs are widely used at LHC experiments for BSM particles
discovery. This paper has shown that a careful study is needed to fully
exploit tracker detector potentialities for a DV within few mm from the PV and
a full-simulation study is needed to confirm the outcome of this paper: the
best approach to search for neutral resonances which decay within few mm from
the PV is to use $|d_{0}/\sigma_{d_{0}}|$ and $L_{xyz}/\sigma_{L_{xyz}}$ for
pre-selecting the muon pairs. However, also full simulation for these
quantities are far from being perfect and a robust search will have to be
based on data only. Finally, a new approach has been proposed which is based
on exploiting the properties of the $\phi_{3D}$ angle. This new approach,
fully based on data, allows reducing at a minimum the systematic uncertainties
while keeping at best the capability to detect a BSM resonance which decays
nearby the PV.

## Acknowledgments

I’m grateful to Gian Mario Bilei, Livio Fanó, Michele Gallinaro, Valentina
Mariani, Simone Pacetti, Matteo Presilla and Alessandro Rossi for many useful
discussions and comments on the manuscript.

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