# Introduction of Medical Imaging Modalities

S. K. M Shadekul Islam
Research and Development Department, Pioneer Alpha,
Dhaka, Bangladesh
<EMAIL_ADDRESS>
&MD Abdullah Al Nasim
Research and Development Department, Pioneer Alpha,
Dhaka, Bangladesh
<EMAIL_ADDRESS>
&Ismail Hossain
University of Alabama at Birmingham
Alabama, USA
<EMAIL_ADDRESS>
&Dr. Md Azim Ullah
University of Memphis
<EMAIL_ADDRESS>
&Dr. Kishor Datta Gupta
Clark Atlanta University
Georgia, USA
<EMAIL_ADDRESS>&Md Monjur Hossain Bhuiyan
School of Aerospace and Mechanical Engineering
University of Oklahoma
Norman, OK-73019, USA
<EMAIL_ADDRESS>

###### Abstract

The diagnosis and treatment of various diseases had been expedited with the
help of medical imaging. Different medical imaging modalities, including
X-ray, Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Nuclear
Imaging, Ultrasound, Electrical Impedance Tomography (EIT), and Emerging
Technologies for in vivo imaging are widely used in today’s healthcare
throughout the world. A review of medical imaging modalities is presented in
this chapter, in addition to these modalities, some advanced techniques such
as contrast-enhanced MRI, MR approaches for osteoarthritis, Cardiovascular
Imaging, and Medical Imaging data mining and search. Despite its important
role and potential effectiveness as a diagnostic tool, reading and
interpreting medical images by radiologists is often tedious and difficult due
to the large heterogeneity of diseases and the limitation of image quality or
resolution. Besides the introduction and discussion of the basic principles,
typical clinical applications, advantages, and limitations of each modality
used in current clinical practice, this chapter also highlights the importance
of emerging technologies in medical imaging and the role of data mining and
search aiming to support translational clinical research, improve patient
care, and increase the efficiency of the healthcare system.

_K_ eywords X-ray, Computed Tomography, Electrical Impedance Tomography,
Magnetic Resonance Imaging

##  1 Introduction

Medical imaging is a very essential tool in the diagnosis and treatment of
various diseases. It refers to the non-invasive techniques and approaches used
to create visual representations of the internal organs and tissues of the
human body. These visual representations, also known as images, can be used to
detect and diagnose a variety of diseases, guide disease treatment strategies
and monitor treatment efficacy. Specifically, medical imaging is widely used
to examine and visualize different parts of the body, including bones,
muscles, organs, blood vessels, and other internal structures. Medical imaging
can also be divided into two main categories: diagnostic imaging and
therapeutic imaging [1]. Diagnostic imaging, which is used to detect and
diagnose diseases, and determine disease severity, include modalities such as
X-ray radiography, computed tomography (CT), magnetic resonance imaging (MRI),
ultrasound (US), and nuclear medicine. Therapeutic imaging, which is used to
guide procedures such as surgery or radiation therapy, includes modalities
such as fluoroscopy, angiography, and interventional radiology. With the
advancements in technology, a wide range of medical imaging modalities have
been developed to provide detailed visual information about the internal
structures and functions of the body.

In current clinical practice, the routinely used imaging modalities include
X-ray, CT, MRI, ultrasound, and nuclear medicine. Each imaging modality has
its unique advantages for different applications and diagnostic purposes, and
limitations to reveal the internal structure of the different organs of the
body and their functions [2]. This chapter aims to provide an overview of the
most commonly used medical imaging modalities, including X-ray, CT, MRI,
ultrasound, and nuclear medicine. We will discuss the basic principles,
typical clinical applications, advantages, and limitations of each modality.
Additionally, we will explore the latest developments and future directions in
the field of medical imaging. The goal of this chapter is to provide a
comprehensive understanding of the various medical imaging modalities
available today and their impact on the field of medicine.

##  2 Background Study

Medical imaging modalities refer to the various techniques and technologies
used to create visual representations of the human body for disease diagnosis
and treatment purposes. In recent years we have seen a good deal of research
to further improve the efficacy and image quality of many existing medical
imaging modalities including X-ray, CT, MRI, ultrasound, nuclear medicine,
cardiovascular imaging, electrical impedance tomography (EIT), and positron
emission tomography (PET).

Additionally, Chapman’s (2009) X-ray imaging, also known as radiography, is
one of the oldest and most widely used imaging modalities. It is based on the
principle that different types of tissue absorb different amounts of X-ray
radiation, allowing for the creation of detailed images of the body’s internal
structures [3]. However, X-ray imaging has some limitations, such as exposure
to ionizing radiation and the lack of tissue contrast.

Ole Petter Wennberg et. al 2018 [4] CT scans, also known as computed
tomography, use X-ray radiation to create detailed cross-sectional images of
the body. CT scans are highly effective in detecting bone and soft tissue
abnormalities, and they can also be used to guide biopsies and other
procedures. However, CT scans also involve exposure to ionizing radiation and
may not be suitable for certain patient populations.

Paul Glover 2013 [5] MRI uses a magnetic field and radio waves to create
detailed images of the body’s internal structures. MRI is particularly
effective at imaging soft tissue and is often used to detect tumors, injuries,
and other abnormalities in the brain, spine, and other parts of the body. MRI
is non-invasive and does not involve ionizing radiation, making it a safer
option for certain patients.

Yobo Xie 2021 [6] Ultrasound imaging uses high-frequency sound waves to create
images of the body’s internal structures. It is particularly effective at
imaging soft tissue and is often used to image the fetus during pregnancy, as
well as to evaluate organs such as the liver, gallbladder, and kidneys.
Ultrasound is non-invasive, does not involve ionizing radiation, and has no
known harmful effects.

Sandip Basu et. al 2011 [7] PET imaging uses small amounts of radioactive
material to create detailed images of the body’s metabolic processes. It is
often used in conjunction with other imaging modalities, such as CT and MRI,
to create more detailed images of certain diseases and conditions. However,
PET imaging involves exposure to ionizing radiation and may not be suitable
for certain patient populations.

In conclusion, various medical imaging modalities are available, each with its
own advantages and disadvantages. X-ray, CT, MRI, ultrasound, and PET are some
of the most commonly used modalities. X-ray and CT use ionizing radiation to
create images, which can be harmful to the patient, while MRI and ultrasound
use non-ionizing radiation, which is safer. PET uses ionizing radiation, but
it is used in conjunction with other imaging modalities to create more
detailed images of certain diseases and conditions [8].

##  3 Methodology

First of all in the research methodology for this chapter on "Introduction of
Medical Imaging Modalities" I identified the different medical imaging
modalities that will be covered in the paper such as X-ray, CT Scan, MRI,
Ultrasound imaging, Nuclear medicine imaging, Electrical Impedance Tomography,
Cardiovascular Imaging. The paper is focused on introducing these modalities
in terms of their basic working principle, applications, limitations, and
advantages.

In the second step, we conducted a comprehensive literature search using
relevant databases such as Google, Google Scholar, PubMed, Scopus, and Web of
Science using keywords such as "Medical Imaging Modalities", "X-ray", "CT
Scan", "MRI", "Ultrasound imaging", "Nuclear medicine imaging", "Electrical
Impedance Tomography", "Cardiovascular Imaging" used to find relevant studies.
Inclusion/exclusion criteria such as the date range of the studies and the
language of the studies are applied to narrow down the search.

Then thirdly, we carefully review and assess the quality of the studies found
in the literature search, and select the most relevant and high-quality
studies for the review. The selected studies covered the basic working
principle, applications, limitations, and advantages of the different medical
imaging modalities. Relevant information is extracted from these studies. The
results are presented in a clear and concise manner, including a discussion of
the strengths and limitations of the studies.

In the fourth step, we provided a conclusion that summarizes the main findings
of the review. This section provides a clear summary of the key takeaways from
the review.

And finally, we included a reference list of all the studies that were
included in the review, following the Springer citation style. This is an
important step that allows other researchers to easily find and access the
studies that were used in the review.

Overall, the methodology for this review paper on "Introduction of Medical
Imaging Modalities" is a systematic and thorough process that will help to
ensure that the review is comprehensive, accurate, and up-to-date.

Figure 1: The categorization of medical imaging modalities [9].

Figure 1 depicts the five groups into which the diagnostic pictures are
divided. As a result of the various medical technology each of these groups
employs, the output images they create vary. Although all of the images fall
under the same general category structure, each one has unique qualities that
aid in the production of various kinds of information. The information
generated by various picture modalities to deal with registration and/or
fusion procedures is described in this section. In accordance with the kind of
output photos, this part is divided into five subsections. These divisions are
X-ray imaging, visible light photography, printed signals (waves), microscopy,
and three-dimensional representation.

##  4 Definition of medical imaging

Medical imaging refers to the techniques and technologies used to create
visual representations of the inside of the body. [10] These visual
representations, also known as images, can be used to diagnose, monitor, or
treat various medical conditions. Medical imaging can be used to examine and
visualize different parts of the body, including bones, muscles, organs, blood
vessels, and other internal structures.

Medical imaging can be divided into two main categories: diagnostic imaging
and therapeutic imaging [1]. Diagnostic imaging, which is used to diagnose and
determine the severity of a condition, includes modalities such as X-ray, CT,
MRI, ultrasound, and nuclear medicine. Therapeutic imaging, which is used to
guide procedures such as surgery or radiation therapy, includes modalities
such as fluoroscopy, angiography, and interventional radiology.

Some common modalities include X-ray, CT, MRI, ultrasound, and nuclear
medicine, each has its own indications and limitations, these modalities are
powerful diagnostic tools that can reveal the internal structure of the body
and its functions [11].

##  5 Overview of different modalities

###  5.1 X-ray imaging:

X-rays are a form of electromagnetic radiation that can pass through many
solid objects, including the human body. X-ray imaging uses X-rays to create
detailed images of internal structures, such as bones [12].

###  5.2 Computed Tomography (CT) imaging:

CT scans use X-rays to create detailed cross-sectional images of the body.
These images can be used to diagnose and monitor a wide range of conditions,
including cancer, heart disease, and injuries to the bones and internal
organs.[13]

###  5.3 Magnetic Resonance Imaging (MRI) imaging:

With the use of a strong magnetic field, radio waves, and a computer, MRI
scans can produce very detailed images of internal structures, such as the
brain and spinal cord, muscles, tendons, ligaments, and blood vessels.[5]

###  5.4 Ultrasound imaging:

Ultrasound employs acoustic waves of very high frequency to create images of
internal structures, such as the uterus and ovaries during pregnancy. It is
also used to examine the organs and blood vessels in the abdomen and to guide
needle biopsies.[10]

###  5.5 Nuclear medicine imaging:

Nuclear medicine uses small amounts of radioactive material to create images
of internal structures and to identify abnormalities, such as tumors or
infections. This modality is also used to diagnose and monitor certain
diseases such as cancer, heart diseases, and thyroid diseases.[14]

###  5.6 Electrical Impedance Tomography (EIT):

EIT imaging uses electrical signals to image internal structures and monitors
changes in their electrical properties, such as the lungs and heart.[15]

###  5.7 Cardiovascular Imaging:

It is a subspecialty of Medical imaging that uses various modalities to
visualize the structure and function of the heart and blood vessels. It
includes techniques such as Echocardiography, Cardiac Computed Tomography
(CCT), and Magnetic Resonance Imaging (CMR). These techniques help in the
diagnosis of heart diseases, such as Coronary artery disease, valvular heart
disease, and congenital heart disease. They also play an important role in
guiding interventional procedures such as angioplasty and stent placement.

##  6 X-ray Imaging

###  6.1 Basic principles

X-ray imaging, often known as radiography, is an imaging technique used in
medicine for diagnosis that creates high-resolution images of internal
anatomical structures like bones. The basic principles of X-ray imaging are as
follows:

X-rays are a form of electromagnetic radiation that can pass through many
solid objects, including the human body. This type of imaging works by
directing a beam of X-rays at the area of the body being examined.[3] The
X-rays pass through the body and are then detected by a special detector, such
as an X-ray film or digital detector. Because bones are denser than other
tissues, they absorb more X-rays and appear white on the final image, while
softer tissues such as muscles and organs absorb fewer X-rays and appear
darker. In order to produce a clear image, the machine must use the
appropriate amount of radiation and the body part must be positioned correctly
in relation to the X-ray beam. The image produced is a two-dimensional
representation of the three-dimensional structure of the body, so to obtain a
more complete picture, different views are taken to examine the same area from
different angles. The images are then interpreted by a radiologist, who is a
medical doctor who specializes in interpreting medical images, to identify any
abnormalities or issues, such as broken bones, tumors, or other conditions,
and provide a diagnosis.

Figure 2: X-ray Imaging Principle[16]

Electromagnetic radiation, or X-rays, are created in an evacuated glass tube
(the bottle in Figure 2) [16]. High-speed electrons that have been propelled
across the tube’s vacuum by a significant voltage difference between the
cathode and the anode are aimed at a rotating tungsten anode. X-rays and heat
are released when the electrons hit the anode. The X-ray tube is inactive
until the Medical Radiation Technologist (MRT) turns it on because X-rays can
only be created when a voltage differential is introduced across the cathode
and anode. All imaging modalities that utilise X-rays (X-ray, CT, Fluoroscopy,
Angiography) use this physical, on/off configuration.

###  6.2 Typical Clinical Applications

X-ray imaging, also known as radiography, has a wide range of clinical
applications in medicine.[17] Some typical clinical applications of X-ray
imaging include:

X-ray imaging is a widely used medical tool that has a variety of
applications. It is commonly used to examine bones for signs of fractures,
dislocations, or other injuries. It can also detect bone diseases such as
osteoporosis, osteoarthritis, and bone tumors. In addition to bone imaging,
X-ray imaging is also used to evaluate the chest and detect lung conditions
such as pneumonia, lung cancer, and emphysema. It can also be used to evaluate
the size, shape, and position of the heart and blood vessels. X-ray imaging
can also be used to examine the abdominal organs such as the liver, spleen,
and kidneys, as well as detect conditions such as gallstones, kidney stones,
and intestinal obstruction. In dentistry, X-ray imaging is commonly used to
detect tooth decay, abscesses, and impacted teeth, as well as evaluate the jaw
and temporomandibular joint. In emergency departments and trauma centers,
X-ray imaging is used to quickly and accurately diagnose injuries such as
fractures, dislocations, and foreign bodies. Additionally, X-ray imaging can
be used to monitor the progress of treatment for certain conditions, such as
monitoring the healing of a fracture or the response of a tumor to radiation
therapy.

X-ray imaging, also known as radiography, is a widely used and valuable
diagnostic tool in medicine. Some advantages of X-ray imaging include:

X-ray imaging is a popular diagnostic tool because of its wide availability,
speed, and low cost. X-ray imaging equipment is widely available in hospitals,
clinics, and doctors’ offices, making it a convenient and accessible
diagnostic tool. Additionally, X-ray imaging can be performed quickly, and
results are usually available within a short period of time. Moreover, X-ray
imaging is relatively inexpensive compared to other imaging modalities such as
MRI or CT. Another advantage is that X-ray imaging produces high-resolution
images that can reveal fine details of bone and other internal structures.
Additionally, X-ray imaging is a non-invasive technique, meaning that it does
not involve the use of incisions or injections, making it relatively safe and
comfortable for the patient.

However, there are also some limitations of X-ray imaging to consider:

X-ray imaging has some limitations, one of them being limited tissue contrast.
It is not as good as other modalities at distinguishing between different
types of tissue, such as tumors and normal tissue. Additionally, X-ray imaging
has low soft tissue contrast, meaning it is not as good as other modalities at
visualizing soft tissue, such as muscles and organs. Another limitation is
that X-ray imaging produces a 2D representation of the 3D structure of the
body, which can make it difficult to visualize certain internal structures,
particularly in areas that are not easily accessible. This 2D representation
can make it difficult to understand some areas of internal structures and make
it hard to diagnose certain issues.

##  7 Computed Tomography (CT) Imaging

###  7.1 Basic principles

CT imaging, commonly called CAT scanning due to its use of X-rays to make
cross-sectional pictures of the body, is a common medical imaging procedure.
The basic principles of CT imaging are as follows:

CT scans use X-rays to create detailed images of internal structures by
directing the X-rays at the body from different angles and measuring the
intensity of the X-rays that pass through the body with detectors. A special
type of X-ray detector called a multi-slice detector is used in CT scans,
which can acquire multiple images at the same time from different angles,
allowing for the creation of detailed cross-sectional images of the body. To
improve image quality, CT scans also use techniques such as spatial filtering,
which removes noise and improves contrast, and multi-energy imaging, which
uses different energy levels of X-ray beams to capture different information
and increase image contrast. Additionally, CT scans can use a technique called
dose modulation, which adjusts the radiation dose according to the size,
shape, and composition of the body part being scanned, reducing the risk of
side effects. The images obtained from CT scans are then processed by a
computer and can be viewed in different ways, such as cross-sectional slices,
3D images, and even virtual reality images [18, 19, 20]. CT scans are
performed by radiologic technologists and the images are interpreted by
radiologists, who are medical doctors who specialize in interpreting medical
images.

Figure 3: Computed Tomography Imaging Principle[4]

A CT (Computed Tomography) scan is a medical imaging technique that uses
X-rays and advanced computer algorithms to produce detailed, cross-sectional
images of the body. The patient lies down on a table that slides into a donut-
shaped machine, called a CT scanner. The X-ray tube, located inside the
scanner, rotates around the patient, emitting a series of X-ray beams at
different angles [4]. As the X-ray beams pass through the body, they are
absorbed by different tissues in varying degrees, creating a pattern of
attenuated X-rays. The pattern of attenuated X-rays is detected by an array of
detectors located opposite the X-ray tube, which converts them into electrical
signals. These electrical signals are then sent to a computer that uses
sophisticated algorithms to reconstruct the data into a 3D image.

###  7.2 Typical Clinical Applications

Computed axial tomography (CAT) scanning, is a medical imaging technique that
has a wide range of clinical applications in medicine. Some typical clinical
applications of CT imaging:

CT scans can be used for a variety of purposes, including imaging the brain
and head for abnormalities such as tumors, bleeding, or trauma. It can also be
used to evaluate the sinuses, ears, and eyes. In the chest, CT scans can be
used to evaluate the lungs for conditions such as pneumonia, lung cancer, and
emphysema. Additionally, it can be used to evaluate the heart and blood
vessels. Another specialized CT technique, called CT angiography (CTA), uses
contrast material and imaging techniques to visualize blood vessels. This
technique can be used to evaluate blood vessels in the head, neck, chest,
abdomen, and pelvis and can also be used to evaluate for blood clots,
aneurysms, and blockages in the blood vessels.

Some advantages of CT imaging:

CT imaging is a widely used diagnostic tool that offers several advantages.
One of the major advantages of CT imaging is its high resolution, which allows
it to reveal fine details of internal structures such as bones, organs, and
blood vessels. This makes it useful for identifying a wide range of medical
conditions. CT imaging equipment is also widely available in hospitals,
clinics, and doctors’ offices, making it a convenient and accessible
diagnostic tool. Another advantage of CT imaging is advantageous since it does
not need any kind of incisions or injections, making it relatively safe and
comfortable for the patient. Furthermore, CT imaging is fast and efficient,
with results usually available within a short period of time. This allows for
quick identification of any issues and prompt treatment.

However, there are also some limitations of CT imaging to consider:

While CT imaging has many advantages, it also has some limitations and risks.
One major risk associated with CT imaging is exposure to ionizing radiation,
which can cause cancer and other health problems, especially in the case of
multiple scans or in pregnant women. Another limitation of CT imaging is that
it is limited in dynamic imaging, it can’t capture dynamic processes such as
blood flow or organ function, and other modalities such as fluoroscopy or
angiography can be better suited for those situations. Additionally, CT
scanners are relatively expensive and require a significant investment, which
can be a limitation for some facilities. This may limit access to this imaging
modality for some patients. It’s important to balance CT scans’ benefits and
risks and consider alternative imaging modalities when appropriate.

##  8 MRI and Magnetic Resonance Microscopy (MRM)

###  8.1 Basic principles

Magnetic Resonance Imaging (MRI) and Magnetic Resonance Microscopy (MRM) are
medical imaging techniques that use a magnetic field and radio waves to
produce detailed images of internal structures [17]. The basic principles of
MRI and MRM are:

Magnetic Resonance Imaging (MRI) and Magnetic Resonance Angiography (MRA) are
advanced imaging modalities that use a powerful magnetic field to align the
nuclei of hydrogen atoms in the body [21, 22]. This creates a small magnetic
moment, which can be used to create detailed images of the body’s internal
structures. The process begins by using radiofrequency (RF) pulses to change
the alignment of the hydrogen nuclei, which causes them to emit a weak radio
signal. These signals are picked up by a detector and processed by a computer
to create detailed images of the internal structures of the body.

Figure 4: MRI Imaging Principle[5]

MRI is a non-invasive and safe imaging technique, and it can provide detailed
images of soft tissues, such as organs, muscles, and nerves[5]. It is commonly
used to diagnose and monitor a wide range of medical conditions, including
brain and spinal cord disorders, tumors, joint and bone problems, and
cardiovascular disease. When a patient enters the MRI machine, they are
exposed to a strong magnetic field that aligns the hydrogen atoms in their
body. Radio waves are then sent through the body, causing the aligned hydrogen
atoms to produce a weak radio signal [5]. The radio signal is detected by an
antenna, and the data is sent to a computer for analysis. The computer uses
complex algorithms to process the data and construct a detailed image of the
body’s internal structures. Different tissues in the body have varying levels
of hydrogen atoms, which produce different signals, allowing for the
differentiation between different types of tissue. MRI can produce images in
different planes, such as axial, sagittal, and coronal, providing a three-
dimensional view of the body.

###  8.2 Typical Clinical Applications

Magnetic Resonance Imaging (MRI) and Magnetic Resonance Microscopy (MRM) are
medical imaging techniques that have a wide range of clinical applications in
medicine. Some typical applications:

MRI and MRM are used in a variety of imaging applications to evaluate
different parts of the body for abnormalities or injuries. These imaging
techniques are particularly useful for evaluating the brain and spinal cord,
as they can detect tumors, stroke, injury, and degenerative diseases [5].
Additionally, MRI and MRM are commonly used to evaluate bones, joints,
ligaments, and muscles for injuries, arthritis, or tumors. They are also used
to evaluate the heart, blood vessels, and blood flow, as well as the liver,
spleen, pancreas, kidneys, and other abdominal and pelvic organs. In addition
to these applications, MRI is also used as a supplementary tool for
mammography for breast cancer detection, especially for dense breast tissue.

Magnetic Resonance Imaging (MRI) and Magnetic Resonance Microscopy (MRM) are
medical imaging techniques that have several advantages and limitations.

MRI and MRM have several advantages as imaging modalities. One of the main
advantages is that they do not use ionizing radiation, making them safer than
other imaging modalities such as X-ray or CT. They also have the ability to
provide high-resolution images of soft tissues such as muscles, tendons, and
ligaments. Additionally, they can create images in multiple planes, which
allows for a more complete picture of internal structures. Furthermore, MRM
can provide microscopic resolution with the advantage of being non-destructive
and non-invasive.[23] However, MRI and MRM also have several limitations. One
of the main limitations is the cost, as scanners are relatively expensive and
require a significant investment, which can be a limitation for some
facilities. Additionally, some patients can experience discomfort or
claustrophobia during the examination, which can limit the use of MRI in
certain patients. Furthermore, the examination can take longer than other
imaging modalities such as X-rays or CTs.

##  9 Nuclear Imaging

###  9.1 Basic principles

Nuclear imaging is a kind of medical imaging that creates high-resolution
pictures of the body’s inner workings by using minute quantities of
radioactive material, called radiotracers.[24] The basic principles of nuclear
imaging are:

Figure 5: Ultrasound Imaging Principle[24]

Nuclear imaging is a medical imaging technique that uses radiotracers that are
introduced into the body, either by injection, inhalation, or ingestion,
depending on the type of exam. The radiotracers emit gamma rays, which are
detected by a special camera, called a gamma camera. This camera creates an
image of the distribution of the radiotracer in the body. The gamma camera
detects the gamma rays and converts them into an image, which is then
processed by a computer and can be viewed in different ways, such as 2D
images, 3D images, or functional images [14, 25]. The images obtained from
nuclear imaging provide functional information about the body, such as blood
flow, metabolism, or chemical activity. This information can be used to
evaluate certain disorders such as cancer, inflammation, or heart function.

###  9.2 Typical Clinical Applications

Nuclear imaging is a medical imaging technique that has a wide range of
clinical applications in medicine. Some typical applications:

Nuclear imaging is used in a variety of applications to evaluate different
parts of the body. One of the main uses of nuclear imaging is cardiac imaging,
where it can be used to evaluate blood flow to the heart and detect any
blockages or abnormalities, such as coronary artery disease or heart failure.
Additionally, it is used in cancer imaging, especially in cases where
traditional imaging modalities like X-ray or CT is not as effective to detect
and staging cancer. Nuclear imaging can also be used to detect bone disorders
such as osteoporosis or bone cancer and to evaluate bone healing after injury
or surgery. Furthermore, it can be used to evaluate the function of the
thyroid gland and detect any abnormalities such as thyroid cancer or
hyperthyroidism.

Nuclear imaging is a medical imaging technique that has several advantages and
limitations.

Nuclear imaging has several advantages as a medical imaging modality. One of
the main advantages is its high sensitivity, which allows it to detect small
tumors or abnormalities that can not be visible on other imaging modalities.
Additionally, it can create images in multiple planes, which allows for a more
complete picture of internal structures. Nuclear imaging is also non-invasive,
requiring no incisions or injections, making it relatively safe and
comfortable for the patient. Furthermore, it provides functional information
about the body, such as blood flow, metabolism, or chemical activity, which
can be used to evaluate certain disorders such as cancer, inflammation, or
heart function. Positron Emission Tomography (PET) is a type of nuclear
imaging that uses short-lived radionuclides to create detailed functional
images of the body [26], it’s used to evaluate a wide range of conditions such
as cancer, heart disease, brain disorders, and others.

However, nuclear imaging also has several limitations. One of the main
limitations is the cost, as nuclear imaging equipment is relatively expensive
and requires a significant investment, which can be a limitation for some
facilities [20]. Additionally, nuclear imaging requires specialized equipment
and trained personnel, which can limit its availability in some areas.
Furthermore, nuclear imaging involves exposure to ionizing radiation, which
can cause cancer and other health problems [27], especially in the case of
multiple scans or in pregnant women.

##  10 Ultrasound Imaging

###  10.1 Basic principles

Ultrasound imaging is a medical imaging technique that uses high-frequency
sound waves to produce detailed images of the internal structures of the body.
The basic principles of ultrasound imaging are:

Ultrasound imaging is a medical imaging technique that uses a transducer, a
device that emits high-frequency sound waves and detects the echoes of these
waves as they bounce back from internal structures. The sound waves are sent
into the body and as they hit a boundary between different types of tissue,
some of the sound waves are reflected back to the transducer. These echoes are
picked up by the transducer and converted into electrical signals, which are
then processed by a computer to create detailed images of the internal
structures of the body. These images can be used to evaluate organs, blood
vessels, and fetuses during pregnancy, among other things.

Figure 6: Ultrasound Imaging Principle[28]

A diagram of the ultrasound imaging method’s basic principles can be found in
Figure 5 [28]. It gives depth information that is then used to create a two-
or three-dimensional image of the sample based on the measurement of the time
delay between the red-coloured ultrasound wave that is being emitted and the
blue-coloured waves that are being reflected back from each layer of the
sample.

###  10.2 Typical Clinical Applications

Ultrasound imaging is a widely used medical imaging technique that has a range
of clinical applications. Some typical applications include:

Ultrasound imaging is used for a variety of applications to evaluate different
parts of the body. One of the main uses of ultrasound imaging is abdominal
imaging, where it is used to evaluate the liver, gallbladder, pancreas,
spleen, and kidneys. Additionally, it is commonly used in musculoskeletal
imaging, to evaluate muscles, tendons, ligaments, and joints to detect injury,
inflammation, or tumors [6]. Ultrasound imaging is also used to evaluate the
thyroid gland and detect any abnormalities such as nodules or cysts.
Furthermore, it’s commonly used in Obstetrics and Gynecology, to evaluate
pregnancy, including the detection of the fetus, placenta, and amniotic fluid,
as well as to evaluate the female reproductive organs, such as the ovaries,
uterus, and fallopian tubes.

Ultrasound imaging is a medical imaging technique that has several advantages
and limitations.

Ultrasound imaging has several advantages as a medical imaging modality. One
of the main advantages is that the equipment is portable, which allows for
imaging to be performed at the patient’s bedside or in remote locations.
Additionally, ultrasound equipment is relatively inexpensive, which makes it
widely available in medical facilities. It is also preferable to other imaging
modalities like X-ray and CT since it does not use harmful ionizing radiation.
Furthermore, it can create images in real-time, which allows for the dynamic
evaluation of internal structures and functions. Image segmentation has been
addressed in [29] which has numerous applications in medical imaging

However, ultrasound imaging also has several limitations. One of the main
limitations is the limited penetration depth, which can make it difficult to
image deep structures or structures behind bone or gas. Furthermore,
aberrations such as bone or air echoes can distort pictures, making it hard to
discern interior structures. Additionally, the ultrasound operator’s degree of
knowledge might alter the quality of the ultrasound pictures and the accuracy
of the diagnosis.

##  11 Emerging Technologies for in Vivo Imaging

###  11.1 Electrical Impedance Tomography (EIT)

####  11.1.1 Basic principles

Electrical Impedance Tomography (EIT) is a medical imaging technique that uses
electrical currents to produce images of the interior of the body. The basic
principles of EIT are:

EIT (Electrical Impedance Tomography) is a medical imaging technique that uses
a small number of electrodes placed on the surface of the body. These
electrodes are used to apply small electrical currents to the body and to
measure the resulting voltage changes. When the electrical currents pass
through the body, they encounter different types of tissue with different
electrical properties, such as conductivity and permittivity. These properties
affect the distribution of the electrical current, which can be used to
produce images of the internal structure of the body. Electrodes measure
electrical activity in various parts of the body, and the computer uses this
information to build representations of organs including the lungs, heart, and
brain [30]. EIT is a non-invasive, radiation-free imaging technique that has
the potential to be used in various medical applications, such as lung
imaging, brain imaging, and breast imaging.

Figure 7: EIT Imaging Principle[31]

The functional EIT image (fEIT) and the electrical impedance tomography (EIT)
principle (Figure 6). Between pairs of nearby surface electrodes (1 to 16),
electrical excitation currents are applied; the resulting voltages are
monitored between the other electrodes (U) [31]. The impedance change brought
on by the tidal volume is separated into a 32 by 32 matrix in the fEIT image.
An image of the ventilation distribution is produced because each pixel
carries a unique tidal impedance variation. In the right panel, the ROIs that
are ventral to dorsal are highlighted in gray.

####  11.1.2 Limited Typical Clinical Usage

Electrical Impedance Tomography (EIT) is a relatively new medical imaging
technique that is still being researched and developed, and its clinical
applications are still being explored. However, some potential clinical
applications include:

EIT is a promising imaging modality that has been explored in various medical
applications. One of the main applications of EIT is in lung imaging, where it
has been shown to have the potential to evaluate lung function and detect
changes in lung structure, for instance, among those who suffer from
respiratory disorders like COPD or ARDS (Acute Respiratory Distress Syndrome).
Additionally, EIT has been proposed as a method to evaluate heart function,
such as cardiac output and blood flow, which could be useful for patients with
heart failure or cardiac arrhythmias. Furthermore, it has been proposed as a
method to evaluate brain function, such as cerebral blood flow, which could be
useful for patients with stroke, brain injury, or brain tumors. Lastly, EIT
has been proposed as a method to evaluate the function of the gastrointestinal
tract, such as motility, which could be useful for patients with
gastrointestinal disorders.

However, it’s important to note that EIT is still a new technology and there
is still ongoing research to explore its potential further.

Electrical Impedance Tomography (EIT) is a medical imaging technique that has
several advantages and limitations.

EIT is a medical imaging modality that has several advantages. One of the main
advantages is that the equipment is portable, which allows for imaging to be
performed at the patient’s bedside or in remote locations. Additionally, EIT
equipment is relatively inexpensive, which makes it widely available in
medical facilities. It is also preferable to other imaging modalities like
X-ray and CT since it does not use harmful ionizing radiation. In addition, it
does not need any incisions or injections, thus it is a generally safe and
pleasant option for the patient. On the other hand, EIT has several serious
drawbacks. The inability to see through bone or through gas might be
problematic when trying to scan deeper structures. Additionally, images can be
affected by interference from other sources of electrical activity, such as
muscle activity or electronic devices. Furthermore, the quality of EIT images
can vary depending on the skill of the operator, and the level of expertise
can affect the accuracy of the images and the diagnosis.

###  11.2 Advancements and new modalities

Several in vivo imaging methods are in development. These technologies might
enhance medical imaging and provide new body assessment methods. New
technologies include:

There are several new imaging modalities that are being developed to produce
high-resolution images of the internal structures of the body[32].
Photoacoustic imaging creates high-resolution pictures of bodily structures
using laser light and ultrasound. It has potential applications in cancer
detection, vascular imaging, and optoacoustic imaging. Another technology is
Optical Coherence Tomography (OCT), which uses infrared light to produce high-
resolution images. It has potential applications in ophthalmology, cardiology,
and oncology. Magnetomotive Optical Coherence Tomography (MM-OCT) combines
magnetic fields with optical coherence tomography to provide pictures of
anatomical components within the body. Possible uses include vascular imaging
and the detection of malignancy. Another method that combines lasers and
ultrasound to provide quantitative pictures of the body’s interior
architecture is called quantitative photoacoustic tomography (QPAT). It might
be used for things like functional imaging and cancer diagnosis. Another
method that employs infrared light to create pictures of the body’s interior
architecture is diffuse optical tomography (DOT). Both cancer and functional
brain imaging might benefit from this discovery.

Each of these technologies has its own advantages and disadvantages and the
selection of the appropriate technique depends on the clinical situation and
the patient’s condition.

##  12 Comparative Analysis

Table 1: Comparative Analysis

Modality |  Working Principle |  Applications |  Advantages |  Limitations
---|---|---|---|---
X-ray |  Using ionizing radiation to produce images of the internal structure of a body |  Detecting broken bones, monitoring treatment of conditions such as pneumonia, monitoring the healing of fractures |  Inexpensive, widely available, quick results |  Low-resolution images, ionizing radiation exposure
CT Scan |  X-ray technology combined with computer processing to produce detailed images |  Detecting cancers, identifying blood clots, assessing organ damage, diagnosing spinal problems |  High-resolution images, non-invasive |  Ionizing radiation exposure, high cost
MRI |  Combination of powerful magnetic fields and radio waves allows for the creation of high-resolution photographs of hidden structures |  Detecting tumors, brain and spinal cord injuries, joint problems, and monitoring the progression of conditions such as multiple sclerosis |  Non-ionizing radiation, detailed images |  Long examination time, high cost, not suitable for patients with metal implants
Ultrasound Imaging |  Using high-frequency sound waves to produce images |  Monitoring the growth and development of a fetus, evaluating organs and tissues, detecting tumors and cysts |  Non-invasive, no ionizing radiation exposure |  Operator dependent, limited view of deep structures
Nuclear Imaging |  Using radioactive tracers to produce images |  Detecting diseases and conditions such as cancer, heart disease, and neurological conditions |  High specificity for certain conditions, non-invasive |  Limited view of the structure, exposure to ionizing radiation
Electrical Impedance Tomography |  Using electrical currents to produce images |  Monitoring changes in tissue, measuring organ function |  Non-invasive, portable |  Limited spatial resolution, operator dependent
Cardiovascular Imaging |  Various techniques to produce images of the heart and blood vessels |  Diagnosing heart disease, monitoring treatment |  High-resolution images, non-invasive |  High cost, operator dependent

The functioning principles, uses, benefits, and limitations of each modality
are compared in this table (Table 1) along with their advantages and
disadvantages.

##  13 Specialized Techniques

###  13.1 Contrast-Enhanced MRI

Contrast-enhanced magnetic resonance imaging (MRI) employs a contrast agent to
highlight bodily structures. Before the MRI, a gadolinium-based contrast agent
is injected into the patient’s bloodstream. MRI scanners can detect magnetic
field changes caused by the contrast agent once it reaches the region of
interest [33]. This helps see blood arteries, tumors, and inflammation.

The contrast agent reduces target tissue water proton T1 relaxation time. This
enhances T1-weighted imaging and distinguishes healthy from diseased tissue.

Contrast-enhanced MRI is used to evaluate a wide range of medical conditions
[34], including:

  * •

Cancer diagnosis and staging

  * •

Evaluating blood vessels and blood flow

  * •

Evaluating the liver, kidney, and other organs

  * •

Evaluating the brain and spinal cord

There are dangers connected with using the contrast agent, such as an allergic
response or renal damage, thus not all individuals are good candidates for
contrast-enhanced MRI. Before deciding to schedule a checkup, your physician
will assess the benefits and risks.

###  13.2 MR Approaches for Osteoarthritis

Magnetic resonance imaging (MRI) is a powerful tool for evaluating
osteoarthritis, a condition characterized by the degeneration of the cartilage
in joints. There are several MR approaches that can be used to evaluate
osteoarthritis:

Magnetic Resonance Imaging (MRI) is a powerful imaging modality that can be
used to evaluate the cartilage and detect early signs of osteoarthritis [35].
T1-weighted imaging is one technique that is sensitive to changes in the water
content of the cartilage and can be used to detect early signs of
osteoarthritis, such as cartilage loss, changes in the bone marrow, or the
presence of bone spurs. Another technique is T2-weighted imaging, which is
sensitive to changes in the biochemical composition of the cartilage and can
be used to detect later stages of osteoarthritis, such as increased water
content, increased proteoglycan loss, or increased collagen content. Sodium
MRI is a new technique that uses sodium ions to detect changes in the
cartilage, it can detect changes in the matrix of the cartilage and bone
marrow edema, which is indicative of early osteoarthritis [36]. Another
technique is Dynamic contrast-enhanced MRI, which can be used to evaluate the
blood flow in the joint, which can be useful in detecting inflammation or
synovitis. Each technique has its own advantages and disadvantages and the
selection of the appropriate technique depends on the clinical situation and
the patient’s condition.

It’s important to keep in mind that MRI is a diagnostic tool and the results
should be interpreted by a radiologist in conjunction with clinical
examination and other tests.

###  13.3 Cardiovascular Imaging

Cardiovascular imaging is a subspecialty of medical imaging that focuses on
the evaluation of the heart and blood vessels [37]. Several imaging modalities
are used in cardiovascular imaging, including:

There are several imaging modalities that can be used to evaluate the blood
vessels and the heart. Computed Tomography Angiography (CTA) uses X-rays and
computer processing to produce detailed images of the blood vessels and is
used to detect blockages or aneurysms in the coronary, carotid, and peripheral
vessels. Another modality is Magnetic Resonance Angiography (MRA) which uses
magnetic fields and radio waves to produce detailed images of the blood
vessels and is also used to detect blockages or aneurysms in the coronary,
carotid, and peripheral vessels [38, 39]. Both CTA (Computed Tomography
Angiography) and MRA (Magnetic Resonance Angiography) are imaging techniques
used to visualize blood vessels in the body. However, CTA uses X-rays to
create detailed images of blood vessels, while MRA uses magnetic fields and
radio waves to produce images. Additionally, CTA may expose patients to
ionizing radiation, while MRA does not use ionizing radiation. Another
modality is Cardiac Magnetic Resonance Imaging (CMR) which uses magnetic
fields and radio waves to produce detailed images of the heart and blood
vessels and is used to evaluate the heart’s structure and function and detect
heart diseases such as cardiomyopathies or valvular diseases. Echocardiography
is another modality that uses ultrasound waves to produce detailed images of
the heart and blood vessels, it is also used to evaluate the heart’s structure
and function and detect heart diseases such as cardiomyopathies or valvular
diseases. Each modality has its own advantages and disadvantages and the
selection of the appropriate modality depends on the clinical situation and
the patient’s condition.

The use of ML and AI has taken these modalities to new heights. These imaging
modalities can be used alone or in combination to provide a comprehensive
evaluation of the cardiovascular system, and to detect and diagnose
cardiovascular diseases.

###  13.4 Medical Imaging Data Mining and Search

Medical imaging data mining and search are techniques that are used to extract
useful information from large amounts of medical imaging data [40]. This
information can be used to improve patient care, to support research, and to
improve the efficiency of the healthcare system. Also has its own limitations
as well cite884767. Following are some techniques [41]:

  1. 1.

Content-based Image Retrieval (CBIR): CBIR is a technique that uses image
features such as color, texture, or shapes to search for similar images in a
large database of medical images [42]. This technique can be used to find
images of similar anatomy, pathology, or disease progression.

  2. 2.

Text-based Search: Text-based search allows to search images based on
associated metadata like patient’s demographics, diagnosis, anatomic location,
etc [43].

  3. 3.

Machine learning: Machine learning is a technique that uses algorithms to
automatically extract features from images and classify them into different
categories. This technique can be used to identify specific features of a
disease, such as a tumor, or to predict patient outcomes [44].

  4. 4.

Natural Language Processing (NLP): NLP is a technique that uses algorithms to
extract information from unstructured text data, such as radiology reports, to
identify relevant patient information and to support decision-making [45].

  5. 5.

Radiomics: Radiomics is the process of extracting quantitative information
from medical images using mathematical algorithms. It enables the extraction
of information from medical images that would otherwise be invisible to the
human eye and can be used to predict patient outcomes or to evaluate treatment
response [46].

These techniques can be used to improve patient care by providing physicians
with more accurate and detailed information about a patient’s condition, to
support research by providing a large amount of data for analysis, and to
improve the efficiency of the healthcare system by reducing the need for
repeat imaging and helping to identify patients at risk of certain diseases
[40].

##  14 Discussion and Conclusion

X-ray imaging is one of the oldest and most widely used medical imaging
techniques. It is commonly used for the diagnosis of bone fractures and other
conditions affecting the skeleton. However, its use is limited in the
visualization of soft tissue structures.

CT scans use X-rays and computer algorithms to generate cross-sectional images
of the body. This modality is widely used for the diagnosis of conditions
affecting the brain, chest, abdomen, and pelvis. However, CT scans use
ionizing radiation, which may increase the risk of cancer.

MRI uses a strong magnetic field and radio waves to generate detailed images
of the body’s internal structures. It is widely used for the diagnosis of
conditions affecting the brain and spine, as well as joints and muscles. MRI
is a safe and non-invasive alternative to CT scans, but it is often more
expensive and time-consuming.

Ultrasound imaging uses high-frequency sound waves to produce images of the
body’s internal structures. It is widely used for the diagnosis of conditions
affecting the abdomen, pelvis, and reproductive system. Ultrasound imaging is
safe and non-invasive, but its resolution may be limited in certain
applications.

Nuclear medicine imaging involves the use of radioactive isotopes to produce
images of the body’s internal structures. It is widely used for the diagnosis
of cancer and other conditions affecting the bones and organs. The amount of
radiation exposure during nuclear medicine imaging is generally low and
comparable to the amount of radiation exposure received during a standard
X-ray. The radiopharmaceuticals used in nuclear medicine imaging are designed
to be quickly eliminated from the body, minimizing the amount of radiation
exposure.

Cardiovascular imaging is a specialized field that uses various imaging
techniques to visualize the heart and blood vessels. And, techniques such as
echocardiography, angiography, and magnetic resonance angiography.
Cardiovascular imaging is essential for the diagnosis and treatment of
cardiovascular diseases.

The results of this review demonstrate the importance of medical imaging
modalities in the diagnosis and treatment of various diseases. Each modality
has its strengths and limitations, and the choice of modality should be based
on the specific needs of each patient. Because of the radiation risk
associated with CT and X-rays, MRI is not used. They use a different imaging
process to produce image contrast. Therefore, while MRI is helpful for
detecting some diseases, CT is good for detecting others.

The results of this review also highlight the need for continued research and
advancements in the field of medical imaging. The development of new and
improved modalities will play a critical role in the early detection and
treatment of diseases. The increasing availability of specialized imaging
techniques, such as Electrical Impedance Tomography and Cardiovascular
Imaging, has improved the accuracy of diagnoses and treatment plans.

In conclusion, Medical imaging modalities are a crucial tool in modern
healthcare and play a vital role in the diagnosis, management, and treatment
of various diseases. There are several different modalities available, each
with its own advantages and limitations. X-ray, Computed Tomography (CT),
Magnetic Resonance Imaging (MRI), Nuclear Imaging, Ultrasound, Electrical
Impedance Tomography (EIT), and Emerging Technologies for in vivo Imaging are
the common medical imaging modalities.

In addition to these modalities, there are also several advanced techniques,
such as contrast-enhanced MRI, MR approaches for osteoarthritis,
Cardiovascular Imaging, and Medical Imaging Data Mining and Search, that can
provide additional information and improve the accuracy of diagnosis.

However, it’s important to keep in mind that medical imaging is not without
risks and limitations and the results should be interpreted by a radiologist
or physician in conjunction with clinical examination and other tests.

## References

  * [1] Imaging and radiology: Medlineplus medical encyclopedia.  https://medlineplus.gov/ency/article/007451.htm.  (Accessed on 01/15/2023).
  * [2] Tonmoy Hossain, Fairuz Shadmani Shishir, Mohsena Ashraf, MD Abdullah Al Nasim4&, and Faisal Muhammad Shah.  Brain tumor detection using convolutional neural network.
  * [3] Henry N. Chapman.  X-ray imaging beyond the limits.  Nature Materials, 8(4):299–301, Apr 2009.
  * [4] Ole Petter Wennberg and Lars Rennan.  A brief introduction to the use of x-ray computed tomography (ct) for analysis of natural deformation structures in reservoir rocks.  Geological Society, London, Special Publications, 459(1):101–120, 2018.
  * [5] Paul Glover.  Magnetic Resonance Imaging (MRI) Methodology, pages 1368–1377.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2013.
  * [6] Yubo Xie.  Research on application of ultrasound medical imaging technology in big data mining of regional medical imaging.  Journal of Medical Imaging and Health Informatics, 11(3):930–937, 2021.
  * [7] Sandip Basu, Thomas C. Kwee, Suleman Surti, Esma A. Akin, Don Yoo, and Abass Alavi.  Fundamentals of pet and pet/ct imaging.  Annals of the New York Academy of Sciences, 1228(1):1–18, 2011\.
  * [8] Hossain Tonmoy, Shishir Fairuz Shadmani, Ashraf Mohsena, MD Al Nasim Abdullah, and Muhammad Shah Faisal.  Brain tumor detection using convolutional neural network.  In 2019 1st International Conference on Advances in Science, Engineering and Robotics Technology (ICASERT), IEEE, pages 1–6, 2019.
  * [9] Fatma El-Zahraa Ahmed El-Gamal, Mohammed Elmogy, and Ahmed Atwan.  Current trends in medical image registration and fusion.  Egyptian Informatics Journal, 17(1):99–124, 2016.
  * [10] Harris A Ahmad, Hui Jing Yu, and Colin G Miller.  Medical imaging modalities.  In Medical imaging in clinical trials, pages 3–26. Springer, 2014\.
  * [11] Md Shahin Ali, Md Sipon Miah, Jahurul Haque, Md Mahbubur Rahman, and Md Khairul Islam.  An enhanced technique of skin cancer classification using deep convolutional neural network with transfer learning models.  Machine Learning with Applications, 5:100036, 2021.
  * [12] X-ray imaging | fundamentals, industrial techniques and applications |.  https://www.taylorfrancis.com/books/mono/10.1201/9781315375199/ray-imaging-harry-martz-peter-shull-daniel-schneberk-clint-logan.  (Accessed on 02/02/2023).
  * [13] Thorsten M Buzug.  Computed tomography.  Springer, 2011.
  * [14] Nuclear medicine imaging: What it is & how it’s done.  https://my.clevelandclinic.org/health/diagnostics/4902-nuclear-medicine-imaging.  (Accessed on 02/02/2023).
  * [15] BH Brown.  Electrical impedance tomography (eit): a review.  Journal of Medical Engineering & Technology, 27(3):97–108, 2003\.  PMID: 12775455.
  * [16] X-rays – undergraduate diagnostic imaging fundamentals.  https://pressbooks.pub/undergradimaging/chapter/x-rays/.  (Accessed on 01/18/2023).
  * [17] Charles L Epstein.  Introduction to the mathematics of medical imaging.  SIAM, 2007.
  * [18] Sharon L. Brooks.  Computed tomography.  Dental Clinics of North America, 37(4):575–590, 1993.
  * [19] Md Khairul Islam, Md Shahin Ali, Md Sipon Miah, Md Mahbubur Rahman, Md Shahariar Alam, and Mohammad Amzad Hossain.  Brain tumor detection in mr image using superpixels, principal component analysis and template based k-means clustering algorithm.  Machine Learning with Applications, 5:100044, 2021.
  * [20] MD Nasim, Aditi Dhali, Faria Afrin, Noshin Tasnim Zaman, and Nazmul Karim.  The prominence of artificial intelligence in covid-19.  arXiv preprint arXiv:2111.09537, 2021.
  * [21] A Biswas and MS Islam.  Ann-based brain tumor classification: Performance analysis using k-means and fcm clustering with various training functions.  In Explainable Artificial Intelligence for Smart Cities, pages 83–102. CRC Press, 2021.
  * [22] Md Abdullah Al Nasim, Abdullah Al Munem, Maksuda Islam, Md Aminul Haque Palash, Md Mahim Anjum Haque, and Faisal Muhammad Shah.  Brain tumor segmentation using enhanced u-net model with empirical analysis.  In 2022 25th International Conference on Computer and Information Technology (ICCIT), pages 1027–1032. IEEE, 2022.
  * [23] J Michael Tyszka, Scott E Fraser, and Russell E Jacobs.  Magnetic resonance microscopy: recent advances and applications.  Current Opinion in Biotechnology, 16(1):93–99, 2005.  Analytical biotechnology.
  * [24] Nadine Barrie Smith and Andrew Webb.  Introduction to medical imaging: physics, engineering and clinical applications.  Cambridge university press, 2010.
  * [25] Faisal Muhammad Shah, Tonmoy Hossain, Mohsena Ashraf, Fairuz Shadmani Shishir, MD Abdullah Al Nasim, and Md Hasanul Kabir.  Brain tumor segmentation techniques on medical images-a review.
  * [26] Positron emission tomography scan - mayo clinic.  https://www.mayoclinic.org/tests-procedures/pet-scan/about/pac-20385078#:~:text=A%20positron%20emission%20tomography%20(PET)%20scan%20is%20an%20imaging%20test,normal%20and%20abnormal%20metabolic%20activity. (Accessed on 02/02/2023).
  * [27] Nuclear medicine.  https://www.nibib.nih.gov/science-education/science-topics/nuclear-medicine.  (Accessed on 02/02/2023).
  * [28] Stephane Perrin.  Development and characterization of an optical coherence tomography micro-system: Application to dermatology.  PhD thesis, Université de Franche Comté, 2016.
  * [29] Nazmul Karim, Niluthpol Chowdhury Mithun, Abhinav Rajvanshi, Han-pang Chiu, Supun Samarasekera, and Nazanin Rahnavard.  C-sfda: A curriculum learning aided self-training framework for efficient source free domain adaptation.  arXiv preprint arXiv:2303.17132, 2023.
  * [30] E. L. V. Costa, R. Gonzalez Lima, and M. B. P. Amato.  Electrical Impedance Tomography, pages 394–404.  Springer Berlin Heidelberg, Berlin, Heidelberg, 2009.
  * [31] Ido Bikker, Steffen Leonhardt, Dinis Miranda, Jan Bakker, and Diederik Gommers.  Bedside measurement of changes in lung impedance to monitor alveolar ventilation in dependent and non-dependent parts by electrical impedance tomography during a positive end-expiratory pressure trial in mechanically ventilated intensive care unit patients.  Critical care (London, England), 14:R100, 05 2010.
  * [32] Ralph Weissleder.  A clearer vision for in vivo imaging.  Nature Biotechnology, 19(4):316–317, Apr 2001.
  * [33] Jessica Lohrke, Thomas Frenzel, Jan Endrikat, Filipe Caseiro Alves, Thomas M. Grist, Meng Law, Jeong Min Lee, Tim Leiner, Kun-Cheng Li, Konstantin Nikolaou, Martin R. Prince, Hans H. Schild, Jeffrey C. Weinreb, Kohki Yoshikawa, and Hubertus Pietsch.  25 years of contrast-enhanced mri: Developments, current challenges and future perspectives.  Advances in Therapy, 33(1):1–28, Jan 2016.
  * [34] Steven P. Sourbron and David L. Buckley.  Classic models for dynamic contrast-enhanced mri.  NMR in Biomedicine, 26(8):1004–1027, 2013.
  * [35] Thomas M. Link, Lynne S. Steinbach, Srinka Ghosh, Michael Ries, Ying Lu, Nancy Lane, and Sharmila Majumdar.  Osteoarthritis: Mr imaging findings in different stages of disease and correlation with clinical findings.  Radiology, 226(2):373–381, 2003.  PMID: 12563128.
  * [36] D. Hayashi, F.W. Roemer, and A. Guermazi.  Imaging for osteoarthritis.  Annals of Physical and Rehabilitation Medicine, 59(3):161–169, 2016\.  Special Issue: Osteoarthritis / Coordinated by Emmanuel Coudeyre and François Rannou.
  * [37] Leslee J. Shaw, James K. Min, Rory Hachamovitch, Eric D. Peterson, Robert C. Hendel, Pamela K. Woodard, Daniel S. Berman, and Pamela S. Douglas.  Cardiovascular imaging research at the crossroads.  JACC: Cardiovascular Imaging, 3(3):316–324, 2010.
  * [38] Machine learning approaches in cardiovascular imaging | circulation: Cardiovascular imaging.  https://www.ahajournals.org/doi/10.1161/CIRCIMAGING.117.005614.  (Accessed on 02/02/2023).
  * [39] Damini Dey, Piotr J. Slomka, Paul Leeson, Dorin Comaniciu, Sirish Shrestha, Partho P. Sengupta, and Thomas H. Marwick.  Artificial intelligence in cardiovascular imaging.  Journal of the American College of Cardiology, 73(11):1317–1335, 2019.
  * [40] Rui Wang, Jinguo Wang, and Na Wang.  Application of data mining technology in medical image processing.  In Proceedings of the 2016 International Conference on Engineering and Advanced Technology, pages 18–21. Atlantis Press, 2016/05.
  * [41] Xianzhe Cao and Shimin Wang.  Research about image mining technique.  In Maotai Zhao and Junpin Sha, editors, Communications and Information Processing, pages 127–134, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
  * [42] Mohd Afizi Mohd Shukran, Muhamad Naim Abdullah, and Mohd Sidek Fadhil Mohd Yunus.  New approach on the techniques of content-based image retrieval (cbir) using color, texture and shape features.  Journal of Materials Science and Chemical Engineering, 9(1):51–57, 2021.
  * [43] Surbhi Aggarwal, Venkatesh Babu RADHAKRISHNAN, and Anirban Chakraborty.  Text-based person search via attribute-aided matching.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV), March 2020.
  * [44] Harsurinder Kaur, Husanbir Singh Pannu, and Avleen Kaur Malhi.  A systematic review on imbalanced data challenges in machine learning: Applications and solutions.  ACM Computing Surveys (CSUR), 52(4):1–36, 2019.
  * [45] William Leeson, Adam Resnick, Daniel Alexander, and John Rovers.  Natural language processing (nlp) in qualitative public health research: a proof of concept study.  International Journal of Qualitative Methods, 18:1609406919887021, 2019.
  * [46] Mathieu Hatt, Chintan Parmar, Jinyi Qi, and Issam El Naqa.  Machine (deep) learning methods for image processing and radiomics.  IEEE Transactions on Radiation and Plasma Medical Sciences, 3(2):104–108, 2019.


# Study of Einstein Equivalence Principle with Sagnac Effect in Lorentz and
Galilean Frame

Shouvik Sadhukhan  Email<EMAIL_ADDRESS>
(Dated : )

###### Abstract

Sagnac effect has been studied in terms of Gyroscopic system in both Lorentz
frame as well as flat Einstein frame. The Einstein equivalence principle has
been used to determine the phase shift due to pseudo force in the
transformation from rotating earth frame to stationary frame. A polychromatic
broadband source has been considered for the discussion. The Langevin-Landau-
Lifschitz metric has been used during the incorporation of General theory of
relativity. The square type sagnac interferometer has been used in theory
establishment.

Keywords: Einstein Equivalence Principle (EEP), Langevin-Landau-Lifschitz
metric, Lorentz Frame, Sagnac Effect, Rotating Frame of Reference

##  1 Introduction

General relativity and its explanations are known and used widely in many
papers [1]. Einstein’s Equivalence principle is one of the general
relativistic approach available in literature [2, 3]. It states the
equivalence between the gravitational force and inertial mass [4]. In other
words, it can be tell that the equivalence principle basically provides a
correspondence relation between gravitational force experienced by a massive
system and pseudo force experienced by a body under non inertial frame of
references [5]. This equivalence has been reestablished with Sagnac effect and
Coriolis force with gyroscopic interferometer [6].

The flat space time geometry has been used here which is also similar as
Minkowskian geometry [7]. In our work we have considered rotating frame of
reference to provide pseudo force and hence the metric or line element gets a
transformation due to rotation. This modified form of rotating geometry is
named as Langevin-Landau-Lifschitz metric geometry [8]. This metric space
system can provide rotational pseudo force on a body under motion on this
frame. Our present work in mainly focused on this metric space system [9].

During the development of Einstein gravity it was necessary to provide some
link between Einstein curvature based gravity model and Newtonian
gravitational force theory. Thus, the idea of linearized gravity has been
brought in literature. The gravity is nothing but the effect of scale time
curvature produced by mass and hence this force is force is equivalent to the
pseudo effect of inertial mass [10]. now large distance weak gravity can be
compared with Newtonian gravity theory and thus the weak gravity approximation
has been designed in theory. The space time curvature is nothing but a weak
gravity perturbation happened due to mass of the system which also provides
the gravitational wave as a solution of propagating wave. This linearized
gravity can be equivalent to Sagnac effect under non inertial frame of
reference [11].

Sagnac interferometer experiments are used widely in many measurements [12,
13, 14, 15]. Sagnac Interferometer is a square structured interferometric set
up where three reflecting mirror is used (figure 2) [16, 17, 18, 19, 20]. The
position of source and detector remains perpendicular to each other. The whole
system base can be provide gyroscopic motion [21, 22, 23, 24, 25]. The
rotation may be relativistic or non relativistic. In our present work we have
also tried to incorporate polychromatic source spectral interferometry based
of gyroscopic Sagnac Interferometer [26, 27]. The polychromatic source and its
spectral content interference output can provide highly sensitive displacement
measurements and hence this has been used to analyze the pseudo effect of
rotation in our work [28]. Some details analysis of Einstein gravity can also
be found in the references [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].

The paper has been proceeded accordingly, Section 2 we have given the
mathematical overviews and theoretical analysis of Sagnac effect using Lorentz
and Geometric frame. In section 3 we have mage our conclusion of our work.

##  2 Mathematical analysis of Gravitational Equivalence of Sagnac Effect

In this section we discuss the mathematical analysis of special theory of
relativity with white light interferometry. We start from the standard theory
for relativistic formalism with temporal delay in dynamic interferometer. The
Minkowski space time or in Lorentz frame the line element using cylindrical
coordinate system can be written as follows. [1, 2, 3, 4, 5, 6, 7, 8, 9]

$ds^{2}=c^{2}dt^{2}-dr^{2}-r^{2}d\phi^{2}-dz^{2}$ (1)

Our experimental study contains a planner gyroscope and hence with respect to
a rotating disc we should have symmetry around z axis. Thus, the line element
should be reduced to the following equation.

$ds^{2}=c^{2}dt_{0}^{2}-R^{2}d\phi^{2}$ (2)

Here, $t_{0}$ represent the time with respect to inertial frame and $R$ is the
measure of radius of the gyroscope. Now after applying rotation in the frame,
we must have $\phi=\phi_{0}-\omega t$ where $\omega$ is the angular speed of
the gyroscope. Hence, substituting this into the reduced line element, we can
find the following form.

$ds^{2}=(c^{2}-\omega^{2}R^{2})dt^{2}-R^{2}d\phi^{2}-2R^{2}\omega d\phi dt$
(3)

Now comparing this relation with the generalized line element
$ds^{2}=g_{00}dt^{2}-g_{\phi\phi}d\phi^{2}-2g_{0\phi}\omega d\phi dt$ we can
write the differential time for light as follows. For light we must have to
consider null geometry and hence we have $ds=0$.

$dt=\frac{-g_{0\phi}d\phi\pm\sqrt{(g_{0\phi}^{2}d\phi^{2}-g_{\phi\phi}g_{00}d\phi^{2})}}{g_{00}}$
(4)

For light like solutions with future directed system we must have $dt>0$. For
clock wise or anti-clock wise rotation we must have $d\phi>0$ and $d\phi>0$
respectively. Hence the temporal difference due to rotation can be established
as follows.

$\Delta t=-2\oint_{l}{\frac{g_{0\phi}}{g_{00}}Rd\phi}$ (5)

Here $l$ is the circumference of the gyroscope. For proper time frame the
relation should be $d\tau=\sqrt{g_{00}}dt$. Hence we can write as follows.

$\Delta\tau=-2\sqrt{g_{00}}\oint_{l}{\frac{g_{0\phi}}{g_{00}}Rd\phi}$ (6)

Now converting these generalized form into real line elements, we can re-write
them as follows.

$\Delta t=4\frac{\pi\omega R^{2}}{c^{2}(1-\frac{R^{2}\omega^{2}}{c^{2}})}$ (7)

and,

$\Delta\tau=4\frac{\pi\omega
R^{2}}{c^{2}\sqrt{(1-\frac{R^{2}\omega^{2}}{c^{2}})}}$ (8)

The pictorial diagram for the relativistic frame can be given as follows.

Figure 1: Rotating Relativistic Frame. Here, C and D are before rotation and
C’ and D’ are after rotation. O is the present frame, O’ is past and O” is
future frame reference point.

We are now providing our experimental diagram for the polychromatic gyroscopic
Sagnac Interferometer. Here, we have used the Lorentz frame and hence the
input signal should stay in broad band radio frequency region. The figure is
as follows. [13]

Figure 2: Sagnac Interferometer setup with the Gyroscopic table

In the experiments of relativistic sagnac effect we have used the broadband
polychromatic source of radio frequency ranges. The central wavelength of that
wave has been considered 6.12 m and bandwidth 2.38 m. The angular motions have
been taken of the range $10^{7}$ rps and the radius is in meter range. Now we
are providing the mathematical basis of our work that can produce the results
discussed in this paper. Primarily in this work we have discussed the system
with Gaussian optical beam source. In general we have considered the Gaussian
beam with central peak intensity $H_{0}$ and the spectral domain intensity has
been used as equation 1. (Spectral domain intensity should be represented )
Here we shall provide different angular motion in the table to calibrate the
spectral switch with motion. [1, 2, 3] Suppose the input spectral domain field
is $\varepsilon_{0}(\lambda)$. Therefore, the input intensity given in
equation 10 can be rewritten as follows. [27, 28]

$I_{0}(\lambda)=\varepsilon^{*}_{0}(\lambda)\varepsilon_{0}(\lambda)$ (9)

Now we have considered that our input spectral is Gaussian in nature and
that’s why the intensity is a Gaussian spectral content input. We have assumed
that the spectral content input may have asymmetry as well as excess in
nature. Thus, the input signal can be written as follows.

$I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})}g(\lambda)$
(10)

The term $g(\lambda)$ brings the asymmetry in the wave form. we have taken
that function as in following equation.

$\begin{split}g(\lambda)=1+\frac{A}{6}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}-3(\frac{\lambda-\lambda_{0}}{\Delta\lambda}))\\\
-\frac{C}{24}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{4}-6(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}+3)\end{split}$
(11)

The constant $A$ is the asymmetry factor and $C$ is the excess. The wave form
should follow the following condition.

$g(\lambda)\rightarrow\begin{cases}=1&\text{for symmetric wave form
spectrum}(A=B=0)\\\ \neq 1&\text{for asymmetric wave form spectrum}(A\neq
B\neq 0)\end{cases}$

For our present work we have used both the symmetric and non-excess singlet
Gaussian Beam in spectral domain and hence for the first part of our work we
must have $A=B=0$ and second part we have varied $A$ and $B$ with $A\neq B\neq
0$. We have used the frequency domain of the spectrum in this present work.
The amplitude on the two arms of the interferometer can be taken as $a(\nu)$
and $b(\nu)$ and both of them are dependent on the transmittance and
reflectance of the Beam splitter. We have considered the complex degree of
coherence for this system is 1. therefore, the wave field for the two arms of
the Interferometer can be taken as equation 3 and 4.
($a(\nu)=rt\exp{(ikL_{1})}$ and $b(\nu)=tr\exp{(ikL_{2})}$)

$\varepsilon_{1}(\lambda)=a(\lambda)\varepsilon_{0}(\lambda)$ (12)

And,

$\varepsilon_{2}(\lambda)=b(\lambda)\varepsilon_{0}(\lambda)$ (13)

The Field after interference will be as in next equation.

$\varepsilon(\lambda)=\varepsilon_{1}(\lambda)+\varepsilon_{2}(\lambda)$ (14)

The intensity of the output can be found with
$I(\lambda)=\varepsilon^{*}(\lambda)\varepsilon(\lambda)$. Finally the
spectral domain intensity output with arbitrary phase $\theta$, can be written
as follows.[1, 2, 3, 23]

$I(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})}(1+\cos{(\theta)})$
(15)

Therefore, all those equation will be used to determine the interfered
intensity spectrum. This kind of Interferometer has same arm lengths for both
directions of the interfering lights. The phase difference comes only with the
rotation of the set up bench. Therefore in every cases we have considered
$L_{1}=L_{2}$. Now we present calculation we shall only provide the results
for Gaussian beam. We can not introduce the geometrical path difference in
Sagnac Interferometer due to its structure. The rotation is the only process
that can provide optical path difference and thus we can get spectral shift
due to Gyroscopic motion of the interferometric set up. The mother equation
for the whole work is the equation 15. Consider the relativistic angular
velocity is $\omega$ and the tangential velocity due to this rotation is
$\omega R$. Here $R$ is the radius of the interferometric set up.

From the reference [23] we can write the phase difference due to optical path
difference with the motion as follows using the relativistic temporal delay
derived above. The local time delay provides the following phase difference.

$\theta=\frac{8\pi^{2}}{c}\frac{R^{2}\omega}{\lambda(1-\frac{\omega^{2}R^{2}}{c^{2}})}$
(16)

The proper time delay provides the path difference as follows.

$\theta=\frac{8\pi^{2}}{c}\frac{R^{2}\omega}{\lambda\sqrt{(1-\frac{\omega^{2}R^{2}}{c^{2}})}}$
(17)

The path and phase difference we found are due to the rotation of gyroscope.
Hence, there must have some kind of Doppler shift in the polychromatic wave
lenghts. For the tangential velocity $\omega R$ we have the Doppler shift as
follows.

$z=\gamma(1+\frac{\omega R}{c})-1=\frac{\Delta\lambda}{\lambda}$ (18)

For any wavelength $\lambda_{i}$, the Doppler shifted wavelength $\lambda_{f}$
can be written as follows.

$\lambda_{f}=\lambda_{i}\sqrt{(\frac{1+\frac{\omega R}{c}}{1-\frac{\omega
R}{c}})}=\lambda_{i}\gamma(1+\frac{\omega R}{c})$ (19)

Here all $\lambda^{\prime}$s can be substituted with $\lambda_{i}$ and we can
re-write the input signal in relativistic frame as follows.

$I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega R}{c}})}G(\lambda)$ (20)

Here $G(\lambda)$ is the relativistic form of $g(\lambda)$ which can be
written as follows.

$\begin{split}G(\lambda)=1\\\
+\frac{A}{6}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega
R}{c}})-3(\frac{\lambda-\lambda_{0}}{\Delta\lambda})\sqrt{(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega R}{c}})})\\\
-\frac{C}{24}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{4}(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega
R}{c}})^{2}-6(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega R}{c}})+3)\end{split}$ (21)

Where the conditions for $G(\lambda)$ can also be written like this.

$G(\lambda)\rightarrow\begin{cases}=1&\text{for symmetric wave form
spectrum}(A=B=0)\\\ \neq 1&\text{for asymmetric wave form spectrum}(A\neq
B\neq 0)\end{cases}$

Now the interference intensity follows the above equations. This can be
written as follows. We have shown the intensity in terms of both local and
proper time frame. For local time frame the intensity is as follows.

$\begin{split}I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega R}{c}})}G(\lambda)\\\
(1+\cos{(\frac{8\pi^{2}}{c}\frac{R^{2}\omega}{\lambda(1-\frac{\omega^{2}R^{2}}{c^{2}})})})\end{split}$
(22)

For proper time frame we can write the intensity as follows.

$\begin{split}I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})(\frac{1+\frac{\omega
R}{c}}{1-\frac{\omega R}{c}})}G(\lambda)\\\
(1+\cos{\frac{8\pi^{2}}{c}\frac{R^{2}\omega}{\lambda\sqrt{(1-\frac{\omega^{2}R^{2}}{c^{2}})}}})\end{split}$
(23)

Now using this spectral content intensity we can calibrate the Doppler shift
also with spectral switch shift. The details of the analysis have been given
in physical analysis section. In the discussed set up we don’t need to
incorporate Coriolis force or any kind of pseudo forces. But if the experiment
is done in space in any orbits, we must have to consider some pseudo force and
its effect on the phase shift measurements. This is because the laboratory
frame w.r.t. us is basically another rotating frame of reference due to the
rotation of earth and hence for synchronizing this effect we must have to
include pseudo forces due to earth rotation. the diagram can be drawn as
follows. [1]

Figure 3: Sagnac Interferometer setup with new geometry in space

In this orientation of experimental set up we can find total three phase
shift. Firstly, the interferometer should have some phase difference between
counter propagating waves due to its rotation w.r.t. its center of mass.
Second shift should come due to the pseudo effect of the rotating earth frame
and its Coriolis effect on rotating gyroscopic interferometer. In this case
the phase should oscillate between some positive and negative maximas. The
third shift should also come due to pseudo effect of earth rotation on orbital
motion of whole interferometric system. This effect should change with the
change of radial distance of elliptical orbit. The shift can be discussed
mathematically as follows. The total pseudo force can be written as follows in
terms of relativistic frame.

$F=-\frac{m(\overrightarrow{\omega}\times\overrightarrow{R})+2m(\overrightarrow{\omega}\times\overrightarrow{v})}{1-\frac{v^{\prime
2}}{c^{2}}}$ (24)

Here $\overrightarrow{\omega}$ is the angular speed of earth frame,
$\overrightarrow{v}$ is the velocity of the interferometer cum satellite and
$\overrightarrow{R}$ is the radial vector of satellite in rotating earth
frame. from this function we can also write the followings,

$F_{Coriolis}=-\frac{2m(\overrightarrow{\omega}\times\overrightarrow{v})}{1-\frac{v^{\prime
2}}{c^{2}}}$ (25)

And,

$F_{Centrifugal}=-\frac{m(\overrightarrow{\omega}\times\overrightarrow{R})}{1-\frac{v^{\prime
2}}{c^{2}}}$ (26)

The term $\overrightarrow{v^{\prime}}$ is the velocity of satellite on
stationary frame and this can be written as follows.

$\overrightarrow{v^{\prime}}=\overrightarrow{v}+(\overrightarrow{\omega}\times\overrightarrow{v})$
(27)

In the present configuration of the system, the centrifugal can not provide
work done for circular orbit and hence the Coriolis force should provide the
phase shift. The potential can be written as follows.

$\phi=2m\overrightarrow{R}.(\overrightarrow{v}\times\overrightarrow{\omega})=2mv\omega
R\cos{\theta}\sin{\psi}$ (28)

Where, $\sin{\psi}=\left|\widehat{\omega}\times\widehat{v}\right|$ and
$\cos{\theta}=\left|\widehat{R}.\widehat{n}\right|$. Here,
$\widehat{n}=\widehat{\omega}\times\widehat{v}$. For the derivation of phase
shift due to this Coriolis effect can be done using Einstein Equivalent
Principle. According to that principle the perturbed gravitational field
should be equivalent to this pseudo force into linearized approximation.
Hence, the time delay can be measured with the following relation.

$d\tau=\sqrt{(1+\frac{2\phi}{c^{2}})}dt\approx(1+\frac{\phi}{c^{2}})dt$ (29)

Hence, considering the both counter propagating lights towards beam splitter
we get the following formalism. For the light towards motion the time delay
should be as follows. (Non-Relativistic Frame)

$\tau_{1}=T+\frac{2mv\omega R\cos{\theta}\sin{\psi}T}{c^{2}}$ (30)

For opposite wave this should be as follows. Here $T$ is the time period for
the orbit w.r.t. the stationary frame.

$\tau_{2}=T-\frac{2mv\omega R\cos{\theta}\sin{\psi}T}{c^{2}}$ (31)

Hence the time delay can be measured as follows.

$\Delta\tau=\tau_{1}-\tau_{2}=\frac{4mv\omega
R\cos{\theta}\sin{\psi}T}{c^{2}}$ (32)

In Relativistic frame it should be as follows.

$\tau_{1}=T+\frac{2mv\omega R\cos{\theta}\sin{\psi}T}{c^{2}(1-\frac{v^{\prime
2}}{c^{2}})}$ (33)

and,

$\tau_{2}=T-\frac{2mv\omega R\cos{\theta}\sin{\psi}T}{c^{2}(1-\frac{v^{\prime
2}}{c^{2}})}$ (34)

Therefore, the resultant time delay can be written as follows.

$\Delta\tau=\tau_{1}-\tau_{2}=\frac{4mv\omega
R\cos{\theta}\sin{\psi}T}{c^{2}(1-\frac{v^{\prime 2}}{c^{2}})}$ (35)

here the velocity of satellite w.r.t. the rotating earth frame can be written
as a sum of its orbital motion and rotation motion about its axis. Hence this
can be formalized as follows.

$\overrightarrow{v}=\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R})$
(36)

Here, $\overrightarrow{v}_{1}$ is the orbital velocity w.r.t. rotating earth
frame and $\overrightarrow{\Omega}$ is the angular speed of the satellite
about its own axis w.r.t. to rotating earth frame. The term
$\overrightarrow{\Omega}\times\overrightarrow{R}$ can provide oscillating
velocity of the system in rotating frame of reference. Hence the final version
of time delay can be written as follows in both Non-relativistic and
Relativistic frame a follows.

$\Delta\tau=\tau_{1}-\tau_{2}=\frac{4mT\overrightarrow{R}.((\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R}))\times\overrightarrow{\omega})}{c^{2}}$
(37)

For Relativistic frame the velocity in stationary frame can be written as
follows.

$\overrightarrow{v^{\prime}}=(\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R})+(\overrightarrow{\omega}\times(\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R}))$
(38)

Hence the time delay in Relativistic frame can be written below.

$\Delta\tau=\tau_{1}-\tau_{2}=\frac{4mT\overrightarrow{R}.((\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R}))\times\overrightarrow{\omega})}{c^{2}(1-\frac{v^{\prime
2}}{c^{2}})}$ (39)

Now this phase shift should provide additional spectral shift in polychromatic
interferometer in its spectral content interference spectrum. The
polychromatic light should feel different Doppler effect due to this Coriolis
force. Hence, for the present case the Doppler shift should be as follows.

$z=(g_{00})^{-1/2}-1=\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}}-1$ (40)

Now final form of the intensity spectrum can be formalized as follows for both
in relativistic and non relativistic frame using the phase relation
$\Delta\phi_{phase}=\frac{2\pi}{\lambda}\Delta\tau$ and satellite radius $r$.

$\begin{split}I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}})^{2}}G^{\prime}(\lambda)\\\
(1+\cos{(\frac{4mT\overrightarrow{R}.((\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R}))\times\overrightarrow{\omega})}{c^{2}(1-\frac{v^{\prime
2}}{c^{2}})})})\end{split}$ (41)

And for non relativistic case we must have the following relation.

$\begin{split}I_{0}(\lambda)=H_{0}\exp{(-\frac{1}{2}(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2})(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}})^{2}}G^{\prime}(\lambda)\\\
(1+\cos{(\frac{4mT\overrightarrow{R}.((\overrightarrow{v}_{1}+(\overrightarrow{\Omega}\times\overrightarrow{R}))\times\overrightarrow{\omega})}{c^{2}})})\end{split}$
(42)

The asymmetric and excess term $G^{\prime}(\lambda)$ can be written as
follows.

$\begin{split}G^{\prime}(\lambda)=1\\\
+\frac{A}{6}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}})^{2}\\\
-3(\frac{\lambda-\lambda_{0}}{\Delta\lambda})(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}}))\\\
-\frac{C}{24}((\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{4}(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}})^{4}\\\
-6(\frac{\lambda-\lambda_{0}}{\Delta\lambda})^{2}(1+\frac{1}{\sqrt{(1+\frac{2\phi}{c^{2}})}})^{2}+3)\end{split}$
(43)

Where the conditions for $G(\lambda)$ can also be written like this.

$G^{\prime}(\lambda)\rightarrow\begin{cases}=1&\text{for symmetric wave form
spectrum}(A=B=0)\\\ \neq 1&\text{for asymmetric wave form spectrum}(A\neq
B\neq 0)\end{cases}$

##  3 Concluding Remarks

The Sagnac effect on relativistic frame with the effect of pseudo force have
been discussed here in this paper. The primary structure of the interferometer
is a square type Sagnac interferometer. The relativistic phase shift have been
introduced in the interference spectrum. Relativistic Doppler shift has been
incorporated in the spectrum analysis. Equation 20 represents the input
Doppler shifted broadband intensity profile. From equation 21 we observed that
the asymmetry and excess of the non isotropic gaussian spectrum has been
shifted due to relativistic motion of the gyroscopic frame. Finally equation
22 represents the interference spectral content output. Equation 2 represents
same intensity output with proper time frame. Here, this effect or phase shift
has also been discussed as a result of pseudo effect and Einstein equivalence
principle.

The general theory of relativity has provided the idea of time delay due to
linearized gravity and weak gravity approximations. This weak approximation
can be used to measure the pseudo effect on any orbital satellite due to
rotating earth frame. Hence, this idea has been incorporated into our new
synchronization methods. The figure 3 explains a satellite based Sagnac
interferometer. There are three motion in that figure. Firstly, the phase
shift is happening due to the axial rotation of the satellite in space.
Secondly, that phase shift is modifying due to Coriolis effect of the rotating
earth frame. These Coriolis effects have come on the orbital motion and
angular motion of the satellite about rotating earth frame. the Coriolis
effect has been incorporated to measure the phase shift through time delay
using linearized gravity theories. Equation 39 is the time delay for the
pseudo force discussed here. For our set up centripetal force do not providing
any additional effect. Equation 42 represents the interference spectrum output
with this effect. Finally, the Sagnac effect has been established for pseudo
effect in terms of Einstein Equivalence Principle.

## Limitations and Future scope of this work

We have established the theory on Sagnac Effect on Relativistic Frame (Both
General Relativity and Special Relativity) and its equivalence with Einstein
gravity. The simulated and experimental results analysis will be done in
future.

## Acknowledgement

Shouvik Sadhukhan thanks to Prof. C S Narayanamurthy for important discussion
on this work and Acknowledges the SERB/DST(Govt. Of India) for providing
financial support via the project grant CRG/2020/003338 to carry out this
work.

## References

  * [1] Benedetto, E., Feleppa, F., Licata, I., Moradpour, H. and Corda, C., 2019. On the general relativistic framework of the Sagnac effect. The European Physical Journal C, 79(3), pp.1-5.
  * [2] Corda, C., 2015. Interpretation of Mössbauer experiment in a rotating system: A new proof for general relativity. Annals of Physics, 355, pp.360-366.
  * [3] Kündig, W., 1963. Measurement of the transverse doppler effect in an accelerated system. Physical Review, 129(6), p.2371.
  * [4] Corda, C., 2018. New proof of general relativity through the correct physical interpretation of the Mössbauer rotor experiment. International Journal of Modern Physics D, 27(14), p.1847016.
  * [5] Jonsson, R.M., 2006. An intuitive approach to inertial forces and the centrifugal force paradox in general relativity. American journal of physics, 74(10), pp.905-916.
  * [6] Logunov, A.A. and Chugreev, Y.V., 1988. Special theory of relativity and the Sagnac effect. Soviet Physics Uspekhi, 31(9), p.861.
  * [7] Malykin, G.B., 2000. The Sagnac effect: correct and incorrect explanations. Physics-Uspekhi, 43(12), p.1229.
  * [8] Ridgely, C.T., 2010. Forces in general relativity. European journal of physics, 31(4), p.949.
  * [9] Rincón, Á. and Villanueva, J.R., 2020. The Sagnac effect on a scale-dependent rotating BTZ black hole background. Classical and Quantum Gravity, 37(17), p.175003.
  * [10] Giblin Jr, J.T., Mertens, J.B., Starkman, G.D. and Tian, C., 2019. Limited accuracy of linearized gravity. Physical Review D, 99(2), p.023527.
  * [11] Chamseddine, A.H. and Mukhanov, V., 2018. Mimetic massive gravity: beyond linear approximation. Journal of High Energy Physics, 2018(6), pp.1-20.
  * [12] Sagnac, G., 1905. Sur la propagation de la lumière dans un système en translation et sur l’aberration des étoiles. CR Acad. Sci. Parisn, 141, p.1220.
  * [13] Tartaglia, A. and Ruggiero, M.L., 2015. The Sagnac effect and pure geometry. American Journal of Physics, 83(5), pp.427-432.
  * [14] Ashtekar, A. and Magnon, A., 1975. The Sagnac effect in general relativity. Journal of Mathematical Physics, 16(2), pp.341-344.
  * [15] Pascoli, G., 2017. The Sagnac effect and its interpretation by Paul Langevin. Comptes Rendus Physique, 18(9-10), pp.563-569.
  * [16] Minguzzi, E., 2003. Simultaneity and generalized connections in general relativity. Classical and Quantum Gravity, 20(11), p.2443.
  * [17] Gogberashvili, M., 2002. Coriolis force and Sagnac effect. Foundations of Physics Letters, 15(5), pp.487-493.
  * [18] Post, E.J., 1967. Sagnac effect. Reviews of Modern Physics, 39(2), p.475.
  * [19] Licata, I. and Benedetto, E., 2018. The charge in a lift. A covariance problem. Gravitation and Cosmology, 24(2), pp.173-177.
  * [20] Malykin, G.B. and Pozdnyakova, V.I., 2015. Quadratic Sagnac effect—the influence of the gravitational potential of the Coriolis force on the phase difference between the arms of a rotating Michelson interferometer (an explanation of DC Miller’s experimental results, 1921–1926). Physics-Uspekhi, 58(4), p.398.
  * [21] Rizzi, G. and Ruggiero, M.L., 2003. A direct kinematical derivation of the relativistic Sagnac effect for light or matter beams. General Relativity and Gravitation, 35(12), pp.2129-2136.
  * [22] Corda, C., 2018. New proof of general relativity through the correct physical interpretation of the Mössbauer rotor experiment. International Journal of Modern Physics D, 27(14), p.1847016.
  * [23] Tamburini, F., De Laurentis, M. and Licata, I., 2018. Radiation from charged particles due to explicit symmetry breaking in a gravitational field. International Journal of Geometric Methods in Modern Physics, 15(07), p.1850122.
  * [24] Schwartz, E., 2016. Sagnac effect in an off-center rotating ring frame of reference. European Journal of Physics, 38(1), p.015301.
  * [25] Malykin, G.B., 2002. Sagnac effect in a rotating frame of reference. Relativistic Zeno paradox. Physics-Uspekhi, 45(8), pp.907-909.
  * [26] Spectrally resolved phase shifting interferometry for accurate group velocity dispersion measurements . S.K.Debnath , N.K.Viswanathan , M.P.Kothiyal ; Optics/ol-31-21-3098
  * [27] Brundavanam, M.M., Viswanathan, N.K. and Desai, N.R., 2007. Spectral anomalies due to temporal correlation in a white-light interferometer. Optics letters, 32(16), pp.2279-2281.
  * [28] Sadhukhan, S., Sadhukhan, S., Brundavanam, M.M. and Narayanamurthy, C.S., 2022. Spectral Switch Anomalies in Sagnac Interferometer with respect to Galilean Frame. arXiv preprint arXiv:2204.06190.
  * [29] Kar, A., Sadhukhan, S. and Chattopadhyay, S., 2021. Energy conditions for inhomogeneous EOS and its thermodynamics analysis with the resolution on finite time future singularity problems. International Journal of Geometric Methods in Modern Physics, Vol. 18, No. 08, 2150131 (2021)
  * [30] Sadhukhan. S, Quintessence Model Calculations for Bulk Viscous Fluid and Low Value Predictions of the Coefficient of Bulk Viscosity, International Journal of Science and Research (IJSR) 9(3):1419-1420, DOI: 10.21275/SR20327132301
  * [31] A. Kar, S. Sadhukhan, Hamiltonian Formalism for Bianchi Type I Model for Perfect Fluid as Well as for the Fluid with Bulk and Shearing Viscosity. Basic and Applied Sciences into Next Frontiers (New Delhi Publishers, 2021) (ISBN: 978-81-948993-0-3)
  * [32] A. Kar, S. Sadhukhan, Quintessence model with bulk viscosity and some predictions on the coefficient of bulk viscosity and gravitational constant, recent advancement of mathematics in science and technology (2021) (ISBN: 978-81-950475-0-5)
  * [33] Kar, A., Sadhukhan, S. and Chattopadhyay, S., 2021. Thermodynamics and energy condition analysis for Van-Der-Waals EOS without viscous cosmology. Physica Scripta. https://doi.org/10.1088/1402-4896/ac2f00
  * [34] Kar, A., Sadhukhan, S. and Debnath, U., 2021. Condensed body mass accretion with DBI-essence dark energy and its reconstruction with f (Q) gravity. arXiv preprint arXiv:2109.10906.
  * [35] Sadhukhan, S., Kar, A. and Chattopadhay, S., 2021. Thermodynamic analysis for Non-linear system (Van-der-Waals EOS) with viscous cosmology. The European Physical Journal C, 81(10), pp.1-21. arXiv:2110.13831
  * [36] Kar, A., Sadhukhan, S. and Debnath, U., 2021. Reconstruction of DBI-essence dark energy with f ( R ) gravity and its effect on black hole and wormhole mass accretion. Modern Physics Letters A, Vol. 36, No. 38 (2021) 2150262 (17 pages) ; DOI: 10.1142/S021773232150262X
  * [37] Malaver, M., Kasmaei, H.D., Iyer, R., Sadhukhan, S. and Kar, A., 2021. A theoretical model of Dark Energy Stars in Einstein-Gauss-Bonnet Gravity. arXiv preprint arXiv:2106.09520.
  * [38] Kar, A. and Gupta, N., 2021. Ultra-high Energy Gamma-rays from Past Explosions in our Galaxy. arXiv preprint arXiv:2112.08757.
  * [39] Sadhukhan, S., Kar, A. and Chattopadhyay, S., 2022. Multiple fluid theory of cosmic evolution and its thermodynamic analysis. Indian Journal of Physics, pp.1-27.