# Treatment classification of posterior capsular opacification (PCO) using
automated ground truths

Raisha Shrestha, Waree Kongprawechnon, Toshiaki Kondo
School of Information, Computer and Communication Technology,
Sirindhorn International Institute of Technology,
Thammasat University, Pathum Thani 12000, Thailand
&Teesid Leelasawassuk
National Electronics and Computer Technology Center (NECTEC),
Pathum Thani 12120, Thailand
Nattapon Wongcumchang
Department of Ophthalmology, Faculty of Medicine
Thammasat University, Pathum Thani 12120, Thailand
&Oliver Findl, Nino Hirnschall
Department of Ophthalmology, Vienna Institute for Research in Ocular Surgery,
Hanusch Hospital,
Vienna, Austria
Work done while at SIIT, Thammasat University. Correspondence:
<EMAIL_ADDRESS>
###### Abstract

Determination of treatment need of posterior capsular opacification (PCO)– one
of the most common complication of cataract surgery – is a difficult process
due to its local unavailability and the fact that treatment is provided only
after PCO occurs in the central visual axis. In this paper we propose a deep
learning (DL)-based method to first segment PCO images then classify the
images into treatment required and not yet required cases in order to reduce
frequent hospital visits. To train the model, we prepare a training image set
with ground truths (GT) obtained from two strategies: (i) manual and (ii)
automated. So, we have two models: (i) Model 1 (trained with image set
containing manual GT) (ii) Model 2 (trained with image set containing
automated GT). Both models when evaluated on validation image set gave Dice
coefficient value greater than 0.8 and intersection-over-union (IoU) score
greater than 0.67 in our experiments. Comparison between gold standard GT and
segmented results from our models gave a Dice coefficient value greater than
0.7 and IoU score greater than 0.6 for both the models showing that automated
ground truths can also result in generation of an efficient model. Comparison
between our classification result and clinical classification shows 0.98
F2-score for outputs from both the models.

_K_ eywords deep learning, U-Net, posterior capsular opacification, treatment
classification, medical imaging, semantic segmentation

##  1 Introduction

Posterior Capsular Opacification (PCO) is an after-effect of cataract surgery
which causes blurriness in vision. During cataract surgery, the cloudy natural
lens of a patient’s eye is removed from the lens capsule and replaced with an
artificial intraocular lens (IOL). Residual lens epithelial cells (LECs) in
the capsular bag proliferates and move towards the posterior capsule, reaching
closer to the central visual axis. This results in opacification of the
posterior capsule, which makes the vision cloudy [1]. The necessity of
treatment is usually prescribed by any doctor after several visits of the
patient only after PCO becomes prominently visible and causes visual
obstruction. This study proposes an automated method to classify treatment
required and not yet required cases of PCO which can ultimately reduce the
number of follow-ups a patient has to make.

Few studies related to features, severity and grading of PCO have been carried
out for analysis of PCO. EPCO [2] is one of the oldest commercial software
programs which was freely available online for evaluation of PCO in terms of
density and area. It was a standard technique to compare rates of PCO [3].
However, it was highly subjective and is not currently available. In [4], an
experiment was carried out to verify if retroillumination images are reliable
for automated analysis by calculating density and percentage area of PCO, for
which comparison with values from slit lamp images gave a good correlation.
POCO system [1] was texture-based and carried out objective methods to
quantify PCO in terms of area but being commercially unavailable and
expensive, its usability was low [3]. Likewise, AQUA [5] was an automated
texture-based system which carried out fusion of images to remove
illuminations. Both AQUA and POCO systems were unable to identify complex
textures. Later, a PCO quantifying software called POCOman [6] was built,
which gave severity score and area of PCO. However, this system was highly
subjective and is no longer available on the internet. A texture-based system
to identify early development of PCO and determine visually significant PCO
was developed by [7]. This system was updated later by OSCA [8] where light
reflections were removed by combining two PCO images. [9] demonstrated a new
approach of taking roughness as a texture measure for quantification of PCO.
Likewise, [10, 11] considered roughness using the Hölder Exponent image to
quantify PCO but it is unable to detect smooth, large pearl shaped PCO with
uniform area. [12] used Pentacam tomogram images for PCO analysis based on
pixel intensity and compared their results with POCOman which resulted in good
correlation. Despite, this system was not practical enough, due to
unavailability of scheimpflug system in maximum hospitals resulting in
difficulty to receive Pentacam images. A recent system AQUA II [13] has
successfully classified PCO into six different types using texture analysis
and has automated the process of analyzing PCO. While, this system is unable
to detect large single pearl shaped PCO, as it is defined by boundaries rather
than textures.

The systems developed so far have quantified PCO in terms of score, severity,
area, types and intensity. Their aim is to understand PCO and its varying
effects with different types of IOL so that preventive measures of PCO can be
identified, lowering its occurrence. The methods used so far for PCO analysis
are mostly based on conventional image processing techniques which are
difficult and time consuming. Some machine learning algorithms have also been
implemented,however, deep learning (DL) [14] based models have not yet been
used for PCO analysis. This research for the first time proposes a DL based
approach to automate the segmentation of PCO images and classify PCO in terms
of treatment requirement. Despite the ongoing research on PCO, there is a lack
of study which focuses on patients of PCO and their prior hospital visit
before its treatment. Nd: YAG laser capsulotomy [15] which is the treatment
provided for PCO, is not available in all local hospitals. Also PCO is not
treated until it involves the visual axis and causes visual symptoms. As a
result, patients need to visit hospital for follow ups until the treatment is
provided. Therefore, to reduce these frequent follow-ups, we propose
classification of PCO into treatment required (positive PCO) and not yet
required (negative PCO) cases. In our previous research [16], we classified
PCO into treatment required and not yet required cases by implementing a
conventional method: “intensity based method” and an unsupervised machine
learning algorithm based method: “k-means clustering based method”. There was
a need of identification of pre-processing techniques in these methods which
was difficult and critical, as it plays an important role in feature
extraction of PCO. Therefore, to automate the process, we propose to implement
deep learning for segmentation of PCO in this research.

DL generally requires large dataset but for our purpose, it is difficult to
gather a big dataset. Furthermore, these images require labeled ground truths
which is challenging to obtain as it needs to deal with pixels rather than
objects [17]. In order to segment PCO images, we use U-Net [18]. It is an
architecture for semantic segmentation which implements data augmentation so
that limited labelled samples will be sufficient for training models
eradicating the need for large training dataset. U-Net has symmetric
architecture and skip connections, which allows it to give information
consisting of both localization and context. This enables it to predict a
reliable segmentation map.

Several medical studies have used U-Net with some variations made such as for
lung segmentation [19], gland segmentation [20], segmentation of proximal
femur [21], etc. Because of the increasing popularity of U-Net architecture
for semantic segmentation in the medical field and its advantages as mentioned
above, we use U-Net for segmentation of PCO images in this research. The rest
of the paper is structured as follows: Methods section describes the methods
used to obtain goals of this study; Results section presents the experimental
results and their validation; and Discussion section analyses the results and
discusses its performance.

## Methods

### Image Dataset

The images used in this study were received on request from a database of PCO
from Vienna, Austria. [13]. The same images have also been used in our
previous study in [16]. In total, 135 slit lamp based retro illumination
images of eyes with PCO were received, out of which 118 images are selected to
be used in this study. This research has considered the central 3mm area of
eye as the region of interest (ROI) as it is visually significant region which
determines the treatment requirement. Some of the images did not capture the
central 3mm area of eyes, so this kind of images were discarded. Images with
higher effect of glare were also discarded. This made a total number of 118
images for our research.

The ROI is cropped from the original images to obtain ROI-cropped images,
which are split into training and test image set using k fold cross validation
technique, as we have limited number of images. Generally, the value of k is
considered as 10 or 5 because these values have shown to produce a test error
rate which avoids problems of high bias and variance [22]. Hence, we chose the
value of k to be 5 allowing each fold to have approximately 23 images. In each
iteration, training set has approximately 95 images and test set has
approximately 23 images (4 negative case and remaining positive case images).
Furthermore, this training set is split into train and validation set in a
ratio of 9:1, allowing train, valid and test set to have approximately 85, 10
and 23 images respectively in each iteration.

### U-Net Architecture

U-Net is a Fully Convolutional Network (FCN) generally used for semantic
segmentation of biomedical images. Network architecture is U-shaped with a
down-sampling path and up-sampling path as shown in Fig 1. Architecture of the
down-sampling path follows Convolutional Neural Network (CNN). Each block
consists of two 3×3 convolution layers with rectified linear unit (ReLU) as
activation function and a 2×2 max pooling with stride 2. Number of feature
maps is doubled at each pooling. This path collects contextual information of
input image for segmentation which will be transferred to the up-sampling path
by skip connections. Each block in the up-sampling path consists of a 2×2 up-
convolution and two 3×3 convolutions which recovers the size of the
segmentation map. Number of feature channels is halved at each up-sampling
block. After each up-convolution, feature maps are concatenated which gives
localization information from down-sampling path to up-sampling path. In this
way, lower and higher information obtained from feature maps of each layer and
blocks of the network are concatenated to produce segmentation map.

Figure 1: Network architecture of U-Net.

### Ground Truth Preparation

Generation of a trained model using U-Net architecture require training images
along with their ground truths (GTs). Different models are obtained by using
two different sets of GTs which are generated manually and automatically. GT
images of eyes with PCO are not available, so we generate the GT images
ourselves for our entire image set. GTs are generated for our model in two
ways: first is manually, under supervision of an expert ophthalmologist and
second is automatically by using some image processing techniques. Manually
generated GTs, which we refer as GT1 are generated using the Image Labeler
tool in MATLAB where PCO regions are selected using pixel-wise labeling tools.
This is a difficult and time-consuming task as PCO has complex features like
shape, type and size which is why automated GTs are also generated and
performance of our models trained on these two sets of GTs are evaluated.

Automated GTs which we term as GT2 are generated by using output received from
a method from our previous work [16] using k-means clustering-based approach.
Outputs obtained from this method have some missing PCO regions, so
morphological operations like dilation and close of filter size 3×3 are
applied to cover the missing regions. GT2 generation process is shown below in
Fig 2 and the images are shown in Fig 3.

Figure 2: Automated ground truth (GT2) generation process. Figure 3: Automated
ground truth (GT2) images. First and second rows show positive PCO cases and
third row shows negative PCO case. (a) Original images. (b) Output from
k-means clustering based method [16]. (c) GT2 received after application of
morphological operations to Fig 3(b).

### Image Augmentation

Train and valid images along with their GTs are augmented as the available
image set is not enough to train a deep learning model with enough variance
required by a CNN model. Operations such as rotation, width/height shift,
shear, horizontal flip are applied to get augmented images. The values
assigned for these operations are as follows:

  * •

rotation angle: 0.2 degrees

  * •

width shift: 0.05 fraction of total width

  * •

height shift: 0.05 fraction of total height

  * •

shear range: 0.05 degrees

### Model Training and Validation

A general network architecture of the U-Net model shown above in Fig 1 is used
to train on our image set. Model is trained on train images along with their
respective GTs. We have two sets of GTs: GT1 and GT2 as mentioned above. Model
trained on train images along with GT1 is referred as Model 1 whereas model
trained on train images along with GT2 is referred as Model 2. These models
are validated on valid image set along with their respective GTs. Models with
best performance in reference to validation Dice coefficient are saved.
Overfitting is avoided by implementation of early stopping, data augmentation
and k-fold cross validation. Number of augmented images equal to the product
of number of epochs, batch size and steps per epoch. These images are
generated in batches during the training process. Further, saved models are
used on test set to get the segmented output images.

### Classification of PCO Treatment Requirement

Segmented images obtained from Model 1 and Model 2 are now evaluated for area
calculation of PCO, which is measured in pixels. Among the areas of negative
PCO cases belonging to GT1, a cutoff value is determined which is used to
distinguish positive PCO case from negative PCO case. In order to select an
optimal cutoff value from the list of negative PCO cases, we take a number of
area values from negative PCO case and create confusion matrix for each of the
selected values. Then, a precision-recall (PR) curve and a receiver operating
characteristic (ROC) curve are drawn using these confusion matrices. With the
help of these plots, we select an optimal value which gives us a higher recall
as our cutoff. The output images from DL model are classified based on this
cutoff value.

After talking about the method in this section, in next section we evaluate
the performance of our model, as well as classification results and
demonstrate the PR and ROC curves for optimal cutoff selection.

## Results

### Model Training and Performance

Model was trained using k-fold cross validation technique. As our chosen k is
5, we obtained 5 models which were trained and validated on different image
sets for each GT1 and GT2. We used Google Colab with Tesla T4 GPU, Intel(R)
Xeon(R) CPU @ 2.20GHz and RAM 12.72 GB to train our model. Frameworks used are
Keras and TensorFlow. Number of epochs, steps per epoch and batch size are
chosen in such a way that training and validation loss decreases and accuracy
increases. Binary cross-entropy is used as loss function and the weights are
updated by Adam [23] optimizer, with a 1e-4 learning rate. During training
model, weights are saved in HDF5 format. Average time required for training
each model was 1.6 hours and prediction time was 370 ms per image.

Segmented images from Model 1 is shown below in Fig 4. The metrics used for
performance evaluation are Dice coefficient (a.k.a., F1 score), intersection-
over-union (IoU) score (a.k.a., Jaccard index) and accuracy. As we used k-fold
cross validation, performance metrics for each fold were evaluated for the
validation image set. The average Dice coefficient acquired is 0.81, the
average IoU score is 0.68 and average accuracy is 0.87.

Similarly, segmented images from Model 2 is shown below in Fig 5. Average Dice
coefficient acquired is 0.8, average IoU score is 0.67 and average accuracy is
0.88. These performance metrics were obtained from evaluation on validation
image set.

Figure 4: Segmented results from Model 1. First row shows positive PCO and
second row shows negative PCO case. (a) Input image (b) GT1, and (c) Segmented
resulting images. Figure 5: Segmented results from Model 2. First row shows
positive PCO and second row shows negative PCO case. (a) Input image (b) GT2
(c) Segmented resulting images.

### Classification Results

Area percentage of PCO in segmented images are calculated to classify PCO
images into treatment required (positive) or not yet required (negative) PCO
cases. A cutoff value is obtained from negative PCO case of GT1, which is used
to classify images. Now, area of PCO for each segmented image is compared to
this cutoff value. If the area of PCO for the segmented image is greater than
the cutoff, then the image is classified as positive PCO case and if the area
of PCO for the segmented image is smaller than the cutoff then it is
classified as negative PCO case.

Classification results are in binary form (0 and 1). Confusion matrices formed
to draw PR and ROC curves contain metric values: True positive (TP), true
negative (TN), false positive (FP) and false negative (FN). These metrics are
obtained by comparing our classification with the clinical classification. The
obtained confusion metrics values are used for calculation of performance
metrics: recall/true positive rate (TPR), precision and false positive rate
(FPR). After calculation of the performance metrics for each selected values,
PR and ROC curves are plotted as shown in Fig 6 in order to select the optimal
cutoff value.

Figure 6: Curves plotting performance of models trained with GT1 and GT2 for
selection of best cut-off value. (a) Precision-recall (PR) curve (b) Receiver
operator characteristics (ROC) curve.

For the outputs from Model 1 in Fig 6(a), cutoff value of 4 at recall 0.989
and precision 0.94 was selected. Similarly, for outputs from Model 2, cutoff
value of 12 at recall 0.95 and precision 0.94 was selected. In the ROC curve
of Fig 6(b), for outputs from Model 1 and Model 2, selected cutoff value has
recall/TPR 0.989 whereas for FPR, outputs from Model 1 has value 0.35 and
outputs from Model 2 has value 0.3. As false negatives (FN) play a crucial
role in the medical field, we chose the cutoff value which gave us higher
recall/TPR such that FN is minimized.

### Comparison between Predicted Segmentation and GT

GT1 is the set of GT images prepared manually under expert supervision which
is why it is the gold standard GT. IoU and Dice coefficient are the metrics
used to evaluate the segmented images as these metrics are common and reliable
in semantic segmentation to evaluate the performance of the models [24].
Segmented results from Model 1 when compared to the gold standard GT, gave an
average Dice coefficient score of 0.74 and an average IoU score of 0.63.
Likewise, comparison between segmented results from Model 2 and gold standard
GT gave an average Dice coefficient value 0.73 and average IoU score 0.60. Fig
7 shows original images of eyes with PCO, their respective gold standard GT
along with segmented results from Model 1 and Model 2.

Figure 7: Comparison of gold standard ground truth, segmented result from
Model 1 and segmented result from Model 2.

Similarly, scatterplot of area covered by PCO for predicted images obtained
from Model 1 and Model 2 is shown below in Fig 8. It exhibits the area values
of positive and negative PCO cases, also few misclassifications (FPs and FN)
are clearly spotted in the plot.

Figure 8: Scatterplot of area covered by PCO for predicted images from model
trained with GT1 (Model 1) versus the predicted images from model trained with
GT2 (Model 2).

### Validation

Classification results obtained from our research is validated with
classifications made by an expert ophthalmologist (clinical classification).
TP, TN, FP, FN are identified and counted, then PR and ROC curves are obtained
by plotting classification metrics calculated from these values. These curves
help to select a cut off value which gives the best classification result. The
confusion matrices for the selected cutoff values for Model 1 and Model 2 are
shown in Table 1. It shows some misclassifications through FP and FN. However,
the value of FN is minimized, and the result is acceptable as FN plays a
critical role in biomedical imaging.

Table 1: Confusion matrices for outputs from Model 1 and Model 2. | Model 1 | Model 2
---|---|---
| Actual
---
Positive
| Actual
---
Negative
| Actual
---
Positive
| Actual
---
Negative
Predicted Positive | 97 (TP) | 6 (FP) | 97 (TP) | 7 (FP)
Predicted Negative | 1 (FN) | 14 (TN) | 1 (FN) | 13 (TN)

## Discussion and Conclusion

In this study, PCO is segmented semantically and quantified followed by a
cutoff selection which helps to classify images into treatment required or not
yet required cases. There are a number of PCO quantifying software and studies
which have been carried out as mentioned in introduction section. These
research works have served for tasks like PCO analysis, quantification,
severity identification and so on. However, the quantification has not been
utilized to determine if the PCO needs to be treated or not. This study uses
118 PCO images collected from VEC, Austria. [16] also used same images and
implemented treatment classification. [16] used two methods based on
conventional image processing and k-means clustering for the classification
which needed significant pre-processing. Identification of a right image
processing technique played an important role as it can cause loss of
important features. Also, cutoff value was not selected using a standard
method. The current study considers these problems and has used deep learning-
based model to segment images such that sensitive image processing task will
be carried out more effectively and automatically. It also implements standard
cutoff selection method by using ROC and PR curves. This research work can be
a base for new research activities related to PCO which will include DL since
DL has not been previously implemented for the study of PCO.

DL based method requires a set of ground truth images which are fed to the
model during the training process. We produced manual ground truths (GT1) for
our entire image set. It was a time consuming as well as difficult task
because of complex textures and features of PCO. So, we also aimed on
generating automated GTs (GT2) and compared the performance of our models
trained using both GT1 and GT2 with the gold standard GT (i.e, GT1).
Comparison between the gold standard GT and the predicted results from model
using GT1 (Model 1), resulted in Dice coefficient score 0.74 and IoU score
0.631. Likewise, comparing the gold standard GT with the predicted results
from model using GT2 (Model 2), resulted Dice coefficient and IoU score values
to be 0.73 and 0.6, respectively. Both Model 1 and Model 2 show good values of
Dice coefficient and IoU scores when compared with the gold standard GT which
implies that both of our models are eligible for segmentation of PCO.
Performance of Model 2 also shows that automated GTs are capable of replacing
the manual GTs.

The work of segmentation is followed by classification of PCO images into
treatment required and not yet required cases. Cutoff values for
classification is chosen with the help of PR and ROC curves as shown above in
Fig 6. For biomedical images, false negatives play an important role. Thus, we
refer to recall values and F2 scores [25] to evaluate performance of our
models. Table 2 shows the evaluation of classification through performance
metrics for intensity based method [16], k-means clustering based method [16]
and U-Net based models. k-means clustering based method shows the best metric
values. However, U-Net models also show promising recall and F2-score values.
Intensity and k-means clustering based methods classified PCO with performance
metrics as shown in Table 2, but these methods were semi-automated and it was
challenging to extract PCO features as mentioned above. Whereas, our current
approach is fully automated showing a reliable performance metric values as
shown in Table 2. Comparison between segmented results from DL models and gold
standard ground truth, as well as performance metrics of classification shows
that our current approach is reliable for segmentation of PCO and
classification of PCO into treatment required and not yet required cases.

Table 2: Evaluation of classification of PCO into treatment required and not
yet required cases by previous methods and current study.

|  | Intensity
---
based method [16]
| k-means clustering
---
based method [16]
| U-Net model
---
trained with GT1
(Model 1)
| U-Net model
---
trained with GT2
(Model 2)
Recall | 0.936 | 1.00 | 0.989 | 0.989
Precision | 1.00 | 0.963 | 0.942 | 0.933
FPR | 0.00 | 0.214 | 0.300 | 0.350
F1-score | 0.967 | 0.981 | 0.965 | 0.960
F2-score | 0.948 | 0.99 | 0.979 | 0.978

Now, comparing outputs from our current models: Model 1 and Model 2, we can
see performance metrics values of these two models do not vary much. Both the
models have recall value 0.989 and F2-score greater than 0.97 showing a good
classification performance. Results from segmentation and classification show
that model trained with automated labels (Model 2) is equally potential of
carrying out PCO segmentation and classification as the model trained with
manual labels (Model 1). Hence the performance shown by both of our models
show that DL can be implemented for automated segmentation of PCO and
automated ground truths also allows the DL model to perform well for the
segmentation process.

The finding of this research contributes in the study of PCO by implementing
DL for PCO segmentation whereas, it can contribute in the process of providing
treatment through the classification of treatment need. Implementation of this
classification can ease the patients with PCO such that their number of
hospital visit will be minimized. In further research, generative adversarial
networks (GAN) can be used for generative modelling. GAN is capable of
learning the regularities and patterns of input data and generating new data
using generator and discriminator model [26]. Implementing GAN to increase
training dataset and also augmenting alongside could help the model to learn
features from input data more efficiently and perform better.

## Acknowledgements

This research was supported by the Center of Excellence in Biomedical
Engineering of Thammasat University, Thailand.

## References

  * [1] Sarah A Barman, Emma J Hollick, James F Boyce, David J Spalton, Bunyarit Uyyanonvara, Giorgia Sanguinetti, and William Meacock.  Quantification of posterior capsular opacification in digital images after cataract surgery.  Investigative ophthalmology & visual science, 41(12):3882–3892, 2000.
  * [2] Manfred R Tetz, Gerd U Auffarth, Martina Sperker, Marcus Blum, and Hans E Völcker.  Photographic image analysis system of posterior capsule opacification.  Journal of Cataract & Refractive Surgery, 23(10):1515–1520, 1997\.
  * [3] T M Aslam, B Dhillon, N Werghi, A Taguri, and A Wadood.  Systems of analysis of posterior capsule opacification.  British journal of ophthalmology, 86(10):1181–1186, 2002.
  * [4] David S Friedman, Donald D Duncan, Beatriz Munoz, Sheila K West, and Oliver D Schein.  Digital image capture and automated analysis of posterior capsular opacification.  Investigative ophthalmology & visual science, 40(8):1715–1726, 1999.
  * [5] Hannes Siegl, Axel Pinz, Wolf Buhl, Michael Georgopoulos, Oliver Findl, and Rupert Menapace.  Assessment of posterior capsule opacification after cataract surgery.  In PROCEEDINGS OF THE SCANDINAVIAN CONFERENCE ON IMAGE ANALYSIS, pages 54–61, 2001.
  * [6] Lloyd Bender, David J Spalton, Bunyarit Uyanonvara, James Boyce, Catherine Heatley, Romina Jose, and Jaheed Khan.  POCOman: new system for quantifying posterior capsule opacification.  Journal of Cataract & Refractive Surgery, 30(10):2058–2063, 2004\.
  * [7] Tariq M Aslam, Niall Patton, and Jim Graham.  A freely accessible, evidence based, objective system of analysis of posterior capsular opacification; evidence for its validity and reliability.  BMC ophthalmology, 5(1):9, 2005.
  * [8] Tariq M Aslam, Niall Patton, and Christopher J Rose.  OSCA: a comprehensive open-access system of analysis of posterior capsular opacification.  BMC ophthalmology, 6(1):1–6, 2006.
  * [9] Naoufel Werghi, Rachid Sammouda, and Fatma AlKirbi.  An unsupervised learning approach based on a Hopfield-like network for assessing posterior capsule opacification.  Pattern Analysis and Applications, 13(4):383–396, 2010.
  * [10] Aruna Vivekanand, Naoufel Werghi, and Hussain Al-Ahmad.  Automated image assessment of posterior capsule opacification using Hölder exponents.  In 2013 IEEE 20th International Conference on Electronics, Circuits, and Systems (ICECS), pages 538–541, 2013.
  * [11] Aruna Vivekanand, Naoufel Werghi, and Hussain Al-Ahmad.  Multiscale roughness approach for assessing posterior capsule opacification.  IEEE journal of biomedical and health informatics, 18(6):1923–1931, 2014.
  * [12] Dilraj Grewal, Rajeev Jain, Gagandeep Singh Brar, and Satinder Pal Singh Grewal.  Pentacam tomograms: a novel method for quantification of posterior capsule opacification.  Investigative ophthalmology & visual science, 49(5):2004–2008, 2008.
  * [13] Martin Kronschläger, Hannes Siegl, Axel Pinz, Christoph Feichtenhofer, Wolf Buehl, Nino Hirnschall, and Oliver Findl.  Automated qualitative and quantitative assessment of posterior capsule opacification by Automated Quantification of After-Cataract II (AQUA II) system.  BMC ophthalmology, 19(1):114, 2019.
  * [14] Yann Lecun, Yoshua Bengio, and Geoffrey Hinton.  Deep learning, may 2015.
  * [15] Tariq M Aslam, Hilary Devlin, and Baljean Dhillon.  Use of Nd: YAG laser capsulotomy.  Survey of ophthalmology, 48(6):594–612, 2003.
  * [16] Raisha Shrestha, Waree Kongprawechnon, Teesid Leelasawassuk, Nattapon Wongcumchang, and Toshiaki Kondo.  Intensity and K-Means Clustering based Treatment Classification of Posterior Capsular Opacification.  In 2020 6th International Conference on Control, Automation and Robotics (ICCAR), pages 541–545. IEEE, 2020.
  * [17] Bipul Neupane, Teerayut Horanont, and Jagannath Aryal.  Deep learning-based semantic segmentation of urban features in satellite images: A review and meta-analysis.  Remote Sensing, 13(4):808, 2021.
  * [18] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.  U-net: Convolutional networks for biomedical image segmentation.  In International Conference on Medical image computing and computer-assisted intervention, pages 234–241. Springer, 2015.
  * [19] Yu Gordienko, Peng Gang, Jiang Hui, Wei Zeng, Yu Kochura, Oleg Alienin, Oleksandr Rokovyi, and Sergii Stirenko.  Deep learning with lung segmentation and bone shadow exclusion techniques for chest x-ray analysis of lung cancer.  In International Conference on Computer Science, Engineering and Education Applications, pages 638–647. Springer, 2018.
  * [20] Hao Chen, Xiaojuan Qi, Lequan Yu, and Pheng-Ann Heng.  DCAN: deep contour-aware networks for accurate gland segmentation.  In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 2487–2496, 2016.
  * [21] Guodong Zeng, Xin Yang, Jing Li, Lequan Yu, Pheng-Ann Heng, and Guoyan Zheng.  3D U-net with multi-level deep supervision: fully automatic segmentation of proximal femur in 3D MR images.  In International workshop on machine learning in medical imaging, pages 274–282. Springer, 2017.
  * [22] Jason Brownlee.  A Gentle Introduction to k-fold Cross-Validation, 2018.
  * [23] Diederik P Kingma and Jimmy Ba.  Adam: A method for stochastic optimization.  arXiv preprint arXiv:1412.6980, 2014.
  * [24] Ekin Tiu.  Metrics to Evaluate your Semantic Segmentation Model | Towards Data Science, 2019.
  * [25] Thomas Wood.  F-Score Definition | DeepAI, 2019.
  * [26] Jason Brownlee.  A Gentle Introduction to Generative Adversarial Networks (GANs), 2019\.


XXXX-XXXX

# Medium effects of charged-hadron production in $p+Pb$ and $Pb+Pb$ collisions
at LHC energies using modified Tsallis distribution

Kapil Saraswat<EMAIL_ADDRESS>Institute of Physics, Academia
Sinica, Taipei 11529, Taiwan  Prashanta Kumar Khandai<EMAIL_ADDRESS>Department of Physics, Ewing Christian College, Prayagraj-211003  Venktesh
Singh  Department of Physics, School of Physical & Chemical Science,Central
University of South Bihar, Gaya 428236, India

###### Abstract

The transverse momentum ($p_{T}$) spectra of charged hadrons in $p+p$, $p+Pb$
and $Pb+Pb$ collisions at $\sqrt{s_{NN}}=5.02$ TeV are presented here within
the rapidity range of $-2.5<y<2.0$. We study the medium effects, which is
produced by heavy ion collisions, on the behaviour of charged hadrons, by
using a phenomenological fit function. These effects are attributed to two
main factors: the transverse collective flow and the the energy loss of
charged hadrons due to multiple scatterings. We observe the transverse
collective flow at low and intermediate $p_{T}$ region and the energy loss at
high $p_{T}$ region. Here we take all the published data from the ATLAS
collaboration.

Charged hadron spectra, Quark Gluon Plasma, Collective Flow, Energy loss

##  1 Introduction

The RHIC (Relativistic Heavy Ion Collider) RHIC1 ; RHIC2  and the LHC (Large
Hadron Collider) LHC  are designed to study an elusive state of matter known
as Quark Gluon Plasma (QGP) Shuryak . The QGP is a thermalized state of matter
where quarks and gluons are deconfined and free to move independently.
Scientists believe that such a state would have existed in the early Universe
shortly after the Big Bang Olive ; Schwarz . There are many heavy-ion
collisions performed at RHIC such as $Au+Au$, $Cu+Cu$, $Cu+Au$ etc and many at
LHC as $Pb+Pb$, $Xe+Xe$ etc. The $p+p$ collisions are taken as a baseline for
collisions involving heavy ions PPPROD  . The particles produced in both $p+p$
and heavy ion ($A+A$) collisions are the results of multiple scatterings among
partons. The distribution of $p_{T}$ spectra in $p+p$ collisions provides
insights into the state of the matter when particles stop interacting at
freeze-out state. Recent high-multiplicity $p+p$ collisions data from the LHC
HighMul ; HighMul2  predicts the formation of a quark-gluon plasma-like
medium.

In high energy collisions between heavy ions (like lead or gold), the momentum
distribution of emitted hadrons exhibits extra effects that happen after the
initial collision. These effects include the collective flow Collective_flow
of hadrons, caused by the expansion of a hot Recom_Model1 ; Recom_Model2 ,
dense matter formed in the collision and jet quenching Jet_Quenching , where
high energy sprays of particles lose energy as they travel through the dense
environment.
In $p+p$ collisions, researchers analyze the $p_{T}$ spectra of hadrons using
a Tsallis distribution Tsallis:1987eu ; Biro:2008hz  characterized by two
parameters $T$ and $q$ PPG099 . While the $T$ parameter corresponds to the
kinetic freeze-out temperature in heavy ion collisions, where particles stop
interacting elastically, its interpretation in $p+p$ collisions is less clear
compared to different types of collisions. The parameter $q$, known as the
nonextensive parameter indicates how much the system deviates from complete
thermalization q_Tsallis . It captures variations in the system’s temperature
IJMPA_Khandai ; Wong:2012zr ; Wong:2013sca . The Tsallis distribution, which
models systems close to thermal equilibrium, resembles Hagedorn’s power law,
which is employed in hard scattering processes in QCD Hagedorn:1983wk ;
Blankenbecler:1974tm .

This work focuses on the $p_{T}$ distribution of charged hadrons in $p+p$,
$p+Pb$ and $Pb+Pb$ collisions at $\sqrt{s_{NN}}=5.02$ TeV from the data of
ATLAS collaboration ATLAS:2022kqu  in the rapidity range of $-2.5<y<2.0$. We
employ the new Modified Tsallis distribution to describe the transverse flow
and in medium energy loss JPC_Kapil ; Khandai  of hadrons in the medium. Both
systematic and statistical uncertainties are combined and incorporated into
the fitting process.

##  2 Tsallis/Hagedorn distribution function and the modification

The transverse mass ($m_{\rm{T}}=\sqrt{p_{\rm{T}}^{2}+m^{2}}$) distribution of
particles produced in hadronic collisions can be described by the Hagedorn
function which is a QCD-inspired summed power law Hagedorn:1983wk  given as

$\displaystyle
E~{}\frac{d^{3}N}{dp^{3}}=A~{}\Bigg{(}1+\frac{m_{\rm{T}}}{p_{0}}\Bigg{)}^{-n}~{}.$
(1)

This function describes both the bulk spectra in the low $m_{\rm{T}}$ region
and the particles produced in QCD hard scatterings reflected in the high
$p_{\rm{T}}$ region. Let us compare this function with the Tsallis
distribution Tsallis:1987eu ; Biro:2008hz  of thermodynamic origin given by

$\displaystyle
E\frac{d^{3}N}{dp^{3}}=C_{n}~{}m_{\rm{T}}~{}\Bigg{(}1+(q-1)\frac{m_{\rm{T}}}{T}\Bigg{)}^{-1/(q-1)}~{}.$
(2)

The Tsallis distribution describes near-thermal systems in terms of Tsallis
parameter $T$ and the parameter $q$ which measures degree of non-
thermalization q_Tsallis . The functions in Eq. 1 and in Eq. (2) have similar
mathematical forms with $n=1/(q-1)$ and $p_{0}=n\,T$. Larger values of $n$
correspond to smaller values of $q$. Both $n$ and $q$ have been
interchangeably used in Tsallis distribution Biro:2008hz ; Adare:2010fe ;
Cleymans:2012ya ; Adare:2011vy ; Abelev:2006cs . Phenomenological studies
suggest that, for quark-quark point scattering, $n\sim 4$ Blankenbecler:1975ct
; Brodsky:2005fza , which grows larger if multiple scattering centers are
involved. The study in Ref. Zheng:2015mhz  suggests that both the forms given
in Eq. 1 and in Eq. (2) give equally good fit to the hadron spectra in $p+p$
collisions. We use Eq. (2) in case of $p+p$ collisions.

Tsallis/Hagedorn function is able to describe $p_{\rm{T}}$ spectra in $p+p$
collisions practically at all generations of proton colliders. There have been
many attempts to use the Tsallis distribution in heavy ion collisions as well
by including the transverse collective flow Tang:2008ud ; Khandai:2013fwa ;
Sett:2015lja . In addition, in heavy ion collisions, particle spectra at high
$p_{\rm{T}}$ are known to be modified due to in-medium energy loss. The
Tsallis/Hagedorn distribution can be modified by including these final state
effects in different $p_{\rm{T}}$ regions as follows:

$\displaystyle E\frac{d^{3}N}{dp^{3}}$
$\displaystyle=A_{1}\Bigg{[}\exp\left(-\frac{\beta
p_{\rm{T}}}{p_{1}}\right)+\frac{m_{\rm{T}}}{p_{1}}\Bigg{]}^{-n_{1}}~{}:~{}p_{\rm{T}}<p_{\rm{T_{th}}}~{}.$
(3a) $\displaystyle E\frac{d^{3}N}{dp^{3}}$
$\displaystyle=A_{2}~{}\Bigg{[}\frac{B}{p_{2}}~{}\Bigg{(}\frac{p_{\rm{T}}}{q_{0}}\Bigg{)}^{\alpha}+\frac{m_{\rm{T}}}{p_{2}}\Bigg{]}^{-n_{2}}~{}:~{}p_{\rm{T}}>p_{\rm{T_{th}}}~{}.$
(3b)

The first function (Eq. 3a) is shown to govern the thermal and collective part
of the hadron spectrum with the temperature $T=p_{1}/n_{1}$ and the average
transverse flow velocity $\beta$ Khandai:2013fwa .

The second function (Eq. 3b) is obtained after shifting the distribution in
Eq.1 by energy loss $\Delta m_{\rm{T}}$ in the medium as

$E\frac{d^{3}N}{dp^{3}}=A_{2}~{}\Bigg{[}1+\frac{m_{\rm{T}}+\Delta
m_{\rm{T}}}{p_{2}}\Bigg{]}^{-n_{2}}~{}.$ (4)

The energy loss $\Delta m_{\rm{T}}$ is proportional to $p_{\rm{T}}$ at low
$p_{\rm{T}}$ and in general can be parameterized similar to the work in Ref.
Spousta:2016agr  as

$\Delta m_{\rm{T}}=B~{}\Big{(}\frac{p_{\rm{T}}}{q_{0}}\Big{)}^{\alpha}~{}.$
(5)

Here, the parameter $\alpha$ quantifies different energy loss regimes for
light quarks in the medium Baier:2000mf ; De:2011fe . The parameter $B$ is
proportional to the medium size and $q_{0}$ is an arbitrary scale set as 1
GeV. Using Eq. (5) in Eq. 4 and ignoring 1 we get Eq. (3b) applicable for high
$p_{\rm{T}}$. In our study, we find that this function describes the particle
spectra at $p_{\rm{T_{th}}}$ above 7 GeV/$c$. Fits to the data would constrain
the value of $B/p_{2}$ and thus $p_{2}$ is not an independent parameter. The
empirical parton energy loss in nuclear collisions at RHIC energies is found
to be proportional to $p_{\rm{T}}$ Wang:2008se .

##  3 Results and discussions

###  3.1 $p+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

Figure (1) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for $p+p$ collisions at $\sqrt{s}$ = 5.02 TeV measured by the
ATLAS experiment ATLAS:2022kqu . The solid curves are the Tsallis
distributions fitted to the spectra. The Tsallis distribution function gives
good description of the data for both the collision energies which can be
inferred from the values of $\chi^{2}/\rm{NDF}$ given in the Table (1).

Figure 1: The invariant yields of the charged particles as a function of
transverse momentum $p_{\rm{T}}$ for $p+p$ collision at $\sqrt{s}$ = 5.02 TeV
measured by the ATLAS experiment ATLAS:2022kqu . The solid curves are the
fitted Tsallis distribution functions.

Figure (2) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for different centrality classes in $p+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment ATLAS:2022kqu
. The solid curves are the fitted Tsallis distributions. Figure (3) shows the
ratio of the data and the fitted Tsallis distribution as a function of
$p_{\rm{T}}$ for $p+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. ATLAS
measured data of $p+Pb$ and $p+p$ collisions show deviations from the fit
which can be inferred from the values of $\chi^{2}/\rm{NDF}$ given in the
Table (2).

Figure 2: The invariant yields of the charged particles as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $p+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment
ATLAS:2022kqu . The solid curves are the fitted Tsallis distribution
functions. Figure 3: The ratio of the charged particles yields data and their
Tsallis fits as a function of the transverse momentum $p_{\rm{T}}$ for
different centrality classes in $p+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ =
5.02 TeV.

Figure (4) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for different centrality classes in $p+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment ATLAS:2022kqu
. The solid curves are the modified Tsallis distributions given by Eq. (3a and
3b). Figure (5) shows the ratio of the data and the fit function by the
modified Tsallis distribution as a function of $p_{\rm{T}}$ for different
centrality classes in $p+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.
The ratio of the data and the fit function shows that modified Tsallis
distribution function gives excellent description of the measured data in full
$p_{\rm{T}}$ range for all centrality classes. The parameters of the modified
Tsallis distribution are given in the Table (3). The values of the first set
of parameters ($n_{1}$, $p_{1}$, $\beta$) are constant for differnt ramge of
pseudo-rapidities. While fitting the second function, we fix the parameter
$n_{2}=7.67$ guided by $p+p$ value. The exponent $\alpha$ which decides the
variation of the energy loss of partons as a function of their energy remains
same. In conclusion, the function given in Eq. ( 3a and 3b) gives excellent
description of the hadron spectra over wide range of $p_{\rm{T}}$ with its
parameters indicating different physics effects in the $p+Pb$ collisions.

Figure 4: The invariant yields of the charged particles as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $p+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment
ATLAS:2022kqu . The solid curves are the modified Tsallis distributions (Eq.
3a). Figure 5: The ratio of the charged particle yield data and the fit
function (Modified Tsallis distribution Eq. 3a) as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $p+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.

###  3.2 $Pb+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

Figure (6) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for $p+p$ collisions at $\sqrt{s}$ = 5.02 TeV measured by the
ATLAS experiment ATLAS:2022kqu . The solid curves are the Tsallis
distributions fitted to the spectra. The Tsallis distribution function gives
good description of the data for both the collision energies which can be
inferred from the values of $\chi^{2}/\rm{NDF}$ given in the Table (4).

Figure 6: The invariant yields of the charged particles as a function of
transverse momentum $p_{\rm{T}}$ for $p+p$ collision at $\sqrt{s}$ = 5.02 TeV
measured by the ATLAS experiment ATLAS:2022kqu . The solid curves are the
fitted Tsallis distribution functions.

Figure (7) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for different centrality classes in $Pb+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment ATLAS:2022kqu
. The solid curves are the fitted Tsallis distributions. Figure (8) shows the
ratio of the data and the fitted Tsallis distribution as a function of
$p_{\rm{T}}$ for $Pb+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. ATLAS
measured data of $Pb+Pb$ and $p+p$ collisions show deviations from the fit
which can be inferred from the values of $\chi^{2}/\rm{NDF}$ given in the
Table (5).

Figure 7: The invariant yields of the charged particles as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $Pb+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment
ATLAS:2022kqu . The solid curves are the fitted Tsallis distribution
functions. Figure 8: The ratio of the charged particles yields data and their
Tsallis fits as a function of the transverse momentum $p_{\rm{T}}$ for
different centrality classes in $Pb+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ =
5.02 TeV.

Figure (9) shows the invariant yields of the charged particles as a function
of $p_{\rm{T}}$ for different centrality classes in $Pb+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment ATLAS:2022kqu
. The solid curves are the modified Tsallis distributions given by Eq. (3a and
3b). Figure (10) shows the ratio of the data and the fit function by the
modified Tsallis distribution as a function of $p_{\rm{T}}$ for different
centrality classes in $Pb+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.
The ratio of the data and the fit function shows that modified Tsallis
distribution function gives excellent description of the measured data in full
$p_{\rm{T}}$ range for all centrality classes. The parameters of the modified
Tsallis distribution are given in the Table (6). The values of the first set
of parameters ($n_{1}$, $p_{1}$, $\beta$) are constant for differnt ramge of
pseudo-rapidities. While fitting the second function, we fix the parameter
$n_{2}=7.70$ guided by $p+p$ value. The exponent $\alpha$ which decides the
variation of the energy loss of partons as a function of their energy remains
same. In conclusion, the function given in Eq. ( 3a and 3b) gives excellent
description of the hadron spectra over wide range of $p_{\rm{T}}$ with its
parameters indicating different physics effects in the $Pb+Pb$ collisions.

Figure 9: The invariant yields of the charged particles as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $Pb+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV measured by the ATLAS experiment
ATLAS:2022kqu . The solid curves are the modified Tsallis distributions (Eq.
3a). Figure 10: The ratio of the charged particle yield data and the fit
function (Modified Tsallis distribution Eq. 3a) as a function of the
transverse momentum $p_{\rm{T}}$ for different centrality classes in $Pb+Pb$
collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.

##  4 Conclusion

In the article, we carried out an analysis of transverse momentum spectra of
charged hadron in $p+Pb$, and $Pb+Pb$ collisions at $\sqrt{s_{\rm{NN}}}$ =
5.02 TeV. We first use the Tsallis distribution to describe the $p_{\rm{T}}$
spectra of charged hardons in $p+Pb$ and $Pb+Pb$ collisions. We found that
Tsallis distribution does not describe the $p_{\rm{T}}$ spectra properly. We
see a suppression of $p_{\rm{T}}$ spectra above 7GeV/c. To describe and
explain the $p_{\rm{T}}$ spectra of the charged hadrons, we use the modified
Tsallis distribution by incorporating the medium effects. Here we fit the
$p_{\rm{T}}$ spectra of charged hadrons in different centralities of $p+Pb$
and $Pb+Pb$ collisons at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV using modified
Tsallis distribution. We observe the effect of transverse flow in the low-to-
intermediate $p_{\rm{T}}$ region ($p_{\rm{T}}\leq$ 7.0 GeV/c) and in-medium
energy loss in the high $p_{\rm{T}}$ region ($p_{\rm{T}}>$ 7GeV/c). We found
that in the low (to intermediate) $p_{\rm{T}}$ region the parameters
$n_{1},p_{1}$ and $\beta$ are more in central collisions and are gradually
decreasing towards peripheral collisions. This is due to the larger number of
multi-scatterings phenomena occurring among partons in the central collisions
than the peripheral collisions. So there is a transverse collective flow
observed among particles in this region. In the high $p_{\rm{T}}$ region the
exponent $\alpha$ which decides the variation of the energy loss of partons as
a function of their energy remains within 0.59 to 0.73 in $p+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV, 0.32 to 0.59 in $Pb+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. So, finally, we can say that a simple
modification in the Tsallis distribution gives excellent description of
charged particle spectra with its parameters having potential to quantify
various in-medium effects in $p+Pb$ and $Pb+Pb$ collisions.

##  5 Appendix :: Table : $p+Pb$ collisions $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

Table 1: The parameters of the Tsallis function obtained by fitting the charged particles transverse momentum ($p_{\rm{T}}$) spectrum in $p+p$ collision at $\sqrt{s}$ = 5.02 TeV. Rapidity | $n$ | $q$ | $T$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---
($y$) |  |  | (MeV) |
\- 2.5 $<~{}y~{}<$ 2.0 | 7.67 $\pm$ 0.10 | 1.13 | 84.78 $\pm$ 5.46 | 0.05
Table 2: The parameters of the Tsallis function obtained by fitting the charged particles transverse momentum ($p_{\rm{T}}$) spectrum in $p+Pb$ collisions $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. Centrality | $N_{\rm{part}}$ | $n$ | $q$ | $T$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---|---
($\%$) |  |  |  | (MeV) |
0 - 5 | 16.50 $\pm$ 1.90 | 7.97 $\pm$ 0.11 | 1.13 | 116.26 $\pm$ 5.98 | 0.16
5 - 10 | 14.60 $\pm$ 1.20 | 7.88 $\pm$ 0.11 | 1.13 | 112.85 $\pm$ 6.14 | 0.15
10 - 20 | 13.10 $\pm$ 0.80 | 7.83 $\pm$ 0.10 | 1.13 | 110.56 $\pm$ 5.40 | 0.14
20 - 30 | 11.40 $\pm$ 0.60 | 7.77 $\pm$ 0.10 | 1.13 | 107.15 $\pm$ 5.54 | 0.13
30 - 40 | 9.80 $\pm$ 0.60 | 7.73 $\pm$ 0.10 | 1.13 | 104.12 $\pm$ 5.56 | 0.11
40 - 60 | 7.40 $\pm$ 0.40 | 7.65 $\pm$ 0.10 | 1.13 | 98.44 $\pm$ 5.70 | 0.09
60 - 90 | 4.00 $\pm$ 0.20 | 7.53 $\pm$ 0.10 | 1.13 | 86.65 $\pm$ 5.77 | 0.05
Table 3: The parameters of the modified Tsallis function Eq.( 3a and 3b)
obtained by fitting the charged particle spectra in $p+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.

System | Centrality | $N_{\rm{part}}$ | $n_{1}$ | $p_{1}$ | $\beta$ | $\alpha$ | $B$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---|---|---|---|---
| ($\%$) |  |  | (GeV/$c$) |  |  | (GeV/$c$) |
$p+Pb$ | 0 - 5 | 16.50 $\pm$ 1.90 | 7.69 $\pm$ 2.21 | 1.38 $\pm$ 1.55 | 0.13 $\pm$ 0.55 | 0.59 $\pm$ 0.27 | 2.40 $\pm$ 0.75 | 0.05
$p+Pb$ | 5 - 10 | 14.60 $\pm$ 1.20 | 7.36 $\pm$ 0.84 | 1.23 $\pm$ 0.28 | 0.15 $\pm$ 0.30 | 0.63 $\pm$ 0.25 | 2.76 $\pm$ 1.35 | 0.05
$p+Pb$ | 10 - 20 | 13.10 $\pm$ 0.80 | 7.32 $\pm$ 1.88 | 1.22 $\pm$ 0.79 | 0.13 $\pm$ 0.12 | 0.59 $\pm$ 0.13 | 2.72 $\pm$ 0.47 | 0.05
$p+Pb$ | 20 - 30 | 11.40 $\pm$ 0.60 | 7.20 $\pm$ 2.19 | 1.16 $\pm$ 0.81 | 0.13 $\pm$ 0.19 | 0.60 $\pm$ 0.05 | 2.90 $\pm$ 0.38 | 0.04
$p+Pb$ | 30 - 40 | 9.80 $\pm$ 0.60 | 7.12 $\pm$ 1.96 | 1.11 $\pm$ 0.68 | 0.13 $\pm$ 0.10 | 0.60 $\pm$ 0.05 | 2.95 $\pm$ 0.38 | 0.04
$p+Pb$ | 40 - 60 | 7.40 $\pm$ 0.40 | 6.99 $\pm$ 2.03 | 1.02 $\pm$ 0.64 | 0.13 $\pm$ 0.09 | 0.59 $\pm$ 0.05 | 3.08 $\pm$ 0.39 | 0.03
$p+Pb$ | 60 - 90 | 4.00 $\pm$ 0.20 | 6.78 $\pm$ 2.30 | 0.86 $\pm$ 0.45 | 0.12 $\pm$ 0.10 | 0.57 $\pm$ 0.04 | 3.26 $\pm$ 0.39 | 0.02
$p+p$ | - | - | 6.94 $\pm$ 2.36 | 0.86 $\pm$ 1.17 | 0.12 $\pm$ 0.10 | 0.61 $\pm$ 0.05 | 2.92 $\pm$ 0.37 | 0.01

##  6 Appendix :: Table : $Pb+Pb$ collisions $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

Table 4: The parameters of the Tsallis function obtained by fitting the charged particles transverse momentum ($p_{\rm{T}}$) spectrum in $p+p$ collision at $\sqrt{s}$ = 5.02 TeV. Rapidity | $n$ | $q$ | $T$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---
($y$) |  |  | (MeV) |
\- 2.5 $<~{}y~{}<$ 2.0 | 7.70 $\pm$ 0.09 | 1.13 | 86.19 $\pm$ 5.31 | 0.05
Table 5: The parameters of the Tsallis function obtained by fitting the charged particles transverse momentum ($p_{\rm{T}}$) spectrum in $Pb+Pb$ collisions $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. Centrality | $N_{\rm{part}}$ | $n$ | $q$ | $T$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---|---
($\%$) |  |  |  | (MeV) |
0 - 5 | 383.70 $\pm$ 0.40 | 7.71 $\pm$ 0.07 | 1.13 | 94.07 $\pm$ 5.22 | 1.05
5 - 10 | 331.40 $\pm$ 0.90 | 7.70 $\pm$ 0.10 | 1.13 | 95.28 $\pm$ 8.90 | 0.98
10 - 20 | 262.30 $\pm$ 1.50 | 7.70 $\pm$ 0.12 | 1.13 | 95.85 $\pm$ 11.09 | 0.87
20 - 30 | 188.20 $\pm$ 1.90 | 7.71 $\pm$ 0.10 | 1.13 | 96.33 $\pm$ 8.36 | 0.73
30 - 40 | 131.00 $\pm$ 2.00 | 7.67 $\pm$ 0.09 | 1.13 | 94.27 $\pm$ 7.99 | 0.58
40 - 50 | 87.20 $\pm$ 2.00 | 7.71 $\pm$ 0.10 | 1.13 | 95.12 $\pm$ 7.83 | 0.43
50 - 60 | 54.40 $\pm$ 1.80 | 7.68 $\pm$ 0.11 | 1.13 | 92.76 $\pm$ 7.63 | 0.29
60 - 80 | 23.40 $\pm$ 1.20 | 7.72 $\pm$ 0.11 | 1.13 | 92.51 $\pm$ 7.39 | 0.16
Table 6: The parameters of the modified Tsallis function Eq.( 3a and 3b)
obtained by fitting the charged particle spectra in $Pb+Pb$ collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV.

System | Centrality | $N_{\rm{part}}$ | $n_{1}$ | $p_{1}$ | $\beta$ | $\alpha$ | $B$ | $\frac{\chi^{2}}{\rm{NDF}}$
---|---|---|---|---|---|---|---|---
| ($\%$) |  |  | (GeV/$c$) |  |  | (GeV/$c$) |
$Pb+Pb$ | 0 - 5 | 383.70 $\pm$ 0.40 | 10.85 $\pm$ 0.43 | 2.00 $\pm$ 1.77 | 0.28 $\pm$ 0.10 | 0.49 $\pm$ 0.05 | 4.97 $\pm$ 0.57 | 0.06
$Pb+Pb$ | 5 - 10 | 331.40 $\pm$ 0.90 | 10.72 $\pm$ 0.44 | 2.00 $\pm$ 1.06 | 0.27 $\pm$ 0.10 | 0.50 $\pm$ 0.13 | 4.83 $\pm$ 0.71 | 1.29
$Pb+Pb$ | 10 - 20 | 262.30 $\pm$ 1.50 | 10.56 $\pm$ 0.47 | 2.00 $\pm$ 1.03 | 0.25 $\pm$ 0.11 | 0.32 $\pm$ 0.06 | 4.61 $\pm$ 0.67 | 1.28
$Pb+Pb$ | 20 - 30 | 188.20 $\pm$ 1.90 | 10.35 $\pm$ 0.72 | 2.00 $\pm$ 1.23 | 0.21 $\pm$ 0.12 | 0.50 $\pm$ 0.14 | 4.26 $\pm$ 0.60 | 0.07
$Pb+Pb$ | 30 - 40 | 131.00 $\pm$ 2.00 | 10.09 $\pm$ 0.96 | 2.00 $\pm$ 1.41 | 0.15 $\pm$ 0.09 | 0.52 $\pm$ 0.12 | 4.02 $\pm$ 0.55 | 0.02
$Pb+Pb$ | 40 - 50 | 87.20 $\pm$ 2.00 | 8.61 $\pm$ 4.54 | 1.43 $\pm$ 1.27 | 0.24 $\pm$ 0.13 | 0.48 $\pm$ 0.17 | 3.61 $\pm$ 0.49 | 1.16
$Pb+Pb$ | 50 - 60 | 54.40 $\pm$ 1.80 | 7.96 $\pm$ 4.10 | 1.22 $\pm$ 0.83 | 0.24 $\pm$ 0.17 | 0.55 $\pm$ 0.16 | 3.56 $\pm$ 0.77 | 0.02
$Pb+Pb$ | 60 - 80 | 23.40 $\pm$ 1.20 | 7.60 $\pm$ 2.35 | 1.11 $\pm$ 0.85 | 0.19 $\pm$ 0.13 | 0.52 $\pm$ 0.21 | 3.01 $\pm$ 0.52 | 0.02
$p+p$ | - | - | 7.45 $\pm$ 0.52 | 1.11 $\pm$ 0.74 | 0.02 $\pm$ 0.01 | 0.59 $\pm$ 0.12 | 2.86 $\pm$ 0.65 | 0.01

## References

  * (1) PHENIX Collab. (K. Adcox et. al.), Nucl. Phys. A 757, 184-283 (2005).
  * (2) STAR Collab. (J. Adams et al.), Nucl. Phys. A 757, 102-183 (2005).
  * (3) LHC Collab. (N. Armesto et al.), J. Phys. G 35, 054001 (2008).
  * (4) E. V. Shuryak,Phys. Rept. 61, 71$-$158 (1980).
  * (5) K. A. Olive, Science 251, 1194$-$1199 (1991).
  * (6) D. J. Schwarz, Annalen Phys 12, 220$-$270 (2003).
  * (7) F. Becattini and U. Heinz, Z. Phys. C 76, 269 (1997).
  * (8) V. Khachatryan et al., (CMS Collaboration), JHEP 09, 091 (2010) arXiv:1009.4122 [hep-ex].
  * (9) J. Adam et al. (ALICE Collaboration), Nature Phys. 13, 535 (2017).
  * (10) T. Hirano and Y. Nara, Phys. Rev. C 69, 034908 (2004).
  * (11) R. Fries et al. Ann. Rev. Nucl. Part. Sci. 58 177-205 (2008), Phys. Rev. Lett. 90, 202303 (2003), Phys. Rev. C 68, 044902 (2003), J. Phys. G 30 S223-S228 (2004).
  * (12) R. Fries J. Phys. G 30 S853-S860 (2004).
  * (13) X. N. Wang Phys. Lett. B 579, 299 (2004).
  * (14) C. Tsallis, J. Statist. Phys. 52 479 (1988).
  * (15) T. S. Biro, G. Purcsel and K. Urmossy, Eur. Phys. J. A 40 325 (2009. [arXiv:0812.2104 [hep-ph]].
  * (16) A. Adare et al, PHENIX Collaboration, Phys.Rev. D 83 052004 (2011).
  * (17) G. Wilk and Z. Wlodarczyk, Phys. Rev. Lett. 84, 2770 (2000).
  * (18) P. K. Khandai, P. Sett, P. Shukla and V. Singh, Int. J. Mod. Phys. A 28 (2013) 1350066.
  * (19) C. Y. Wong and G. Wilk, Acta Phys. Polon. B 43 2047 (2012), [arXiv:1210.3661 [hep-ph]].
  * (20) C. Y. Wong and G. Wilk, Phys. Rev. D 87 114007 (2013), [arXiv:1305.2627 [hep-ph]].
  * (21) R. Hagedorn, Riv. Nuovo Cim. 6N10 1 (1983).
  * (22) R. Blankenbecler and S. J. Brodsky, Phys. Rev. D 10 2973 (1974).
  * (23) [ATLAS], [arXiv:2211.15257 [hep-ex]].
  * (24) K. Saraswat, P. Shukla and V. Singh, J. Phys. Commun. 2 035003 (2018).
  * (25) P. Kumar, P. K. Khandai, Kapil Saraswat and V. Singh, Int. J. Mod. Phys. A 36 2150059 (2021),
  * (26) J. Cleymans and D. Worku, Eur. Phys. J. A 48 (2012) 160, [arXiv:1203.4343 [hep-ph]].
  * (27) A. Adare et al. [PHENIX Collaboration], Phys. Rev. C 83 (2011) 064903, [arXiv:1102.0753 [nucl-ex]].
  * (28) B. I. Abelev et al. [STAR Collaboration], Phys. Rev. C 75 (2007) 064901, [nucl-ex/0607033].
  * (29) A. Adare et al. [PHENIX Collaboration], Phys. Rev. D 83 (2011) 052004, [arXiv:1005.3674 [hep-ex]].
  * (30) R. Blankenbecler, S. J. Brodsky and J. F. Gunion, Phys. Rev. D 12 (1975) 3469.
  * (31) S. J. Brodsky, H. J. Pirner and J. Raufeisen, Phys. Lett. B 637 (2006) 58, [hep-ph/0510315].
  * (32) H. Zheng and L. Zhu, Adv. High Energy Phys. 2016 (2016) 9632126, [arXiv:1512.03555 [nucl-th]].
  * (33) Z. Tang, Y. Xu, L. Ruan, G. van Buren, F. Wang and Z. Xu, Phys. Rev. C 79 (2009) 051901, [arXiv:0812.1609 [nucl-ex]].
  * (34) P. K. Khandai, P. Sett, P. Shukla and V. Singh, J. Phys. G 41 (2014) 025105, [arXiv:1310.4022 [nucl-th]].
  * (35) P. Sett and P. Shukla, Int. J. Mod. Phys. E 24 (2015) 1550046, [arXiv:1505.05258 [hep-ph]].
  * (36) M. Spousta, Phys. Lett. B 767 (2017) 10, [arXiv:1606.00903 [hep-ph]].
  * (37) R. Baier, D. Schiff and B. G. Zakharov, Ann. Rev. Nucl. Part. Sci. 50 (2000) 37, [hep-ph/0002198].
  * (38) S. De and D. K. Srivastava, J. Phys. G 39 (2012) 015001, Erratum: [J. Phys. G 40 (2013) 049502], [arXiv:1107.5659 [nucl-th]].
  * (39) G. Wang and H. Z. Huang, Phys. Lett. B 672 (2009) 30, [arXiv:0810.2822 [nucl-ex]].