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Optimization of cone-beam CT image quality for image guided radiotherapy MV Source kV Source Flat panel EPID Treatment couch Derbe Ejere October 2006

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Optimization of cone-beam CT image quality for image

guided radiotherapy MV Source

kV Source

Flat panel EPID Treatment couch

Derbe Ejere October 2006

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Thesis for Master of BIOMEDICAL PHYSICS from Academic Medical Center (AMC), university of Amsterdam (UvA).

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Optimization of cone-beam CT image quality for image

guided radiotherapy Derbe Ejere Amsterdam, October 2006 Research for the master thesis is done at the National Cancer Institute, Antoni van Leeuwenhoek Hospital (NKI-AVL) department of Radiotherapy Graduation Committee: Dr. Ir. Jan-Jakob Sonke Prof. Dr. Marcel van Herk Dr. Ir. Geert Streekstra

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Acknowledgments This thesis could not have been accomplished without the help of many. Hence, I would like to express my heartfelt gratitude to all those who have been involved in my work and assisted me. First and for most I would like to owe many thanks to my supervisors Dr. Ir. Jan-Jakob Sonke and Prof. Dr. Marcel van Herk for your incessant help and supervising. Jan-Jakob, thank you very much for all your time of more than a year and your patience. Thanks also for the inspiration you gave me to learn Matlab and Delphi for I was initially not versed in these fields. I got stuck so many times while using these tools and thanks for pulling me forward time and again. Thanks also for suggesting the idea of using beam stop array technique for scatter estimation and persistent help with it. I have learned from your positive critique about double-checking the intermediary results. Marcel, thanks for your help especially in exactly and quickly pinpointing bugs in XVI software program. Thanks also for your formula that we have used for the simulation of scatter. It has contributed a lot to the result and also to the insight of the problem. I have enjoyed your amicable nature and the respect you owe others. Particularly, I was beholden when you owe me your comfortable chair while you sat on tabouret during our discussion in your working room. I will not forget that and will practice it myself. You both have taken your precious time for proofreading my thesis and your invaluable feedback and corrections are indispensable in finalizing this thesis. Thanks also to Dr.Ir. Geert Streekstra for all information and of course for the whole coordination. I want to thank also ALL my colleagues at the NKI-AVL. Special thanks go to my roommates who gave me the comfort I needed, made my work easier and helped me with many things. Josien de Bois, Anja Betgen, Michel Frenay, Jurjen van Dijk, Joost Voogt, Danny Minkema and Jurgen Mourik I thank you a lot for everything. Josien thank you for your precious time you have taken for meticulously proof-reading my thesis. I would like to thank also the Radim workers Peter, Nico , Henk, Karin and others for your help with the networking system. Lastly, I would like to thank ALL those who helped me with arranging my new house. I am greatly indebted among others to Josien de Bois, Manon velderman, and Patricia Fewer. Josien, due to you, my house is not only livable but also comfortable. Thanks for initiating ideas and also transporting things to my house in that scorching sun. Besides your helping character, I have enjoyed your frank, sociable and affable personality. Patricia, thanks for your contribution. Besides your friendly character, I have also enjoyed your efficient and quick way of handling things. Manon, my special thanks to you because of not only giving me those luxurious and beautiful chairs but also for the inconvenience you have experienced every morning in transporting them to the NKI-AVL one by one. Thank you all whom I have not mentioned here.

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Summary Radiotherapy is a treatment of cancer using high-energy radiation at a high dose in order to damage cancerous cells and stop them from growing and dividing. The high-energy beam is delivered by medical linear accelerator with a megavolt radiation source. Tumor changes in position due to for example respiration, bowel movement or filling. In order to locate clearly the tumor position, imaging before and during treatment is thus indispensable. Therefore, it is important to combine irradiation protocols with online imaging techniques working towards the so-called "image guided radiotherapy" (IGRT) [1-3]. With IGRT, tracking and targeting tumors becomes a simultaneous process. Such a development will boost efficiency, reliability and safety in radiation therapy. To this end, Elekta, Inc. has manufactured and put into operation a clinically applicable solution for IGRT. The two main additional components of IGRT system, the low-energy (kV) conventional diagnostic X-ray tube and the flat panel detector, form a cone-beam online imaging unit that moves in a circular orbit around the patient. Using the acquired images, the radiotherapist can determine the location of the tumor, which allows him/her to precisely point the radiation beam to the right spot, sparing the surrounding healthy tissues. This implies that image quality plays a pivotal role in patient cure rate since one can increase dose without the risk of targeting the surrounding healthy cells. Although cone-beam computed tomography has several advantages over the conventional computed tomography, for image guidance its image quality is not as good as the conventional one especially when imaging low contrast, wide and thicker part of a body such as chest and pelvis. The X-ray scatter deteriorates image quality of cone-beam CT the most since cone-beam CT is scatter prone in comparison to Fan beam CT. The aim of this thesis is to deal with one such important process: optimizing image quality for a cone-beam guided linear accelerator for the purpose of radiotherapy image guidance. Essential elements that contribute to the deterioration of image quality are targeted. The fundamental factor that impairs image quality is the image artifact. An artifact is any distortion or error in the image that is unrelated to the subject being studied. CT image artifacts arise from the inherent polychromatic nature of medical x-ray source that results in what is known as beam hardening, scatter of x-ray photons from the patient that causes detection of photons from other than straight-line trajectory from source through the object, and non-ideal detector performance that results in incomplete and inaccurate detection of the attenuated x-ray photons. Beam hardening and scatter induce artifacts such as streaks and cupping while detector poor performance results in ring artifacts. Cone-beam CT suffers a lot from artifact that is caused by scatter. Estimating and correcting artifacts that arise from the flat-panel detector and scatter is the subject of this thesis. The performance of a flat panel detector is investigated for linearity, uniformity and bad-pixels. Especially at lower and higher exposures, the flat-panel detector is non-linear. The same exposures that cause non-linearity of the flat-panel detector have also caused non-uniformity. The detector is linearized piecewise and exposures that cause non-uniformity are sought and calibrated. Pixel-by-pixel standard deviation investigation is used in

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correcting non-uniformity. We have compromised between the number of calibration points and the costly calculations involved. Bad-pixels are also examined and replaced by the average of the neighboring pixels. The effect of X-ray scatter on image quality is examined. X-ray scatter has the most detrimental effect on cone-beam CT image quality. Streaks and cupping are the most prominent artifacts caused by x-ray scatter. On top of that, scatter reduces contrast and increases noise. Scatter is estimated and corrected using three methods: Estimating scatter as uniform given by scatter-to-primary ratio (SPR) of 0.33, beam stop array technique, and simulation. Scatter is estimated by choosing a constant SPR of 0.33. This way of estimating and correcting scatter has reduced the cupping artifact to some extent but does not eliminate the artifact completely. To estimate scatter more accurately, scatter is sampled using beam stop array technique and interpolated to generate scatter-only image. Scatter-only image is then subtracted from the total image to give the scatter corrected primary image. The total image and the scatter corrected primary image are then compared in order to inspect the effect of scatter on image quality and evaluate the scatter correction methods. Effect of scatter on CT number accuracy, contrast and noise is investigated using an image quality phantom. The method of simulation considered the effect of X-ray source fluctuation, area occupied by the beam stoppers, transmission of the lead discs and transmission of the phantom itself. Inclusion of these factors in our beam stop array correction method has produced considerable result. CT number inaccuracy or cupping artifact is corrected after the employment of the scatter correction methods. Contrast is improved satisfactorily. However, all the scatter correction methods incurred noise.

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Table of Contents Part I: General principles of computed tomography 1. Introduction ....…………………………1 1.1 History of CT scanning ....…………………………1 1.2 Generations of CT scanners ……………………………2 1.3 Cone-beam CT ……………………………4 2. CT Instrumentation ....…………………………5

2.1 X-ray tube ……………………………5 2.2 Linear accelerator ……………………………6 2.3 Flat-panel detector ……………………………7 3. X-ray interaction with matter ....…………………………9 3.1 Photoelectric effect ……………………………10 3.2 Compton scattering ……………………………12 3.3 Pair production ……………………………13 4. CT imaging ....…………………………15 4.1 Intensity profile ……………………………16 4.2 Attenuation (projection) profile ……………………………18 4.3 Image reconstruction ……………………………19 Part II: Image quality optimization for cone-beam CT 5. Factors that affect image quality ....…………………………25 5.1 Flat-Panel detector performance ……………………………26 5.2 Beam-hardening ……………………………28 5.3 Scatter ……………………………31 5.3 Objective: Optimization of cone-beam image quality …………..36

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6. Materials and methods ....………………….39

6.1 Experimental setup …………………….39 6.2 Optimization of detector performance …………………….41 6.3 Beam hardening correction method …………………….43 6.4 Scatter estimation and correction method …………………….44

7. Results ....………………….51 7.1 Optimization of detector performance results …………….51 7.2 Beam hardening and scatter estimation and correction results ….56 8. Discussion …………………………………………….67

9. Conclusion …………………………………………….71 10. Appendix …………………………………………….73 Bibliography …………………………………………….77

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Part I General principles of

Computed Tomography

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Chapter 1 Introduction

1.1 History of CT scanning Radiography (or conventional X-ray imaging) is the recording of a shadow image of an optically opaque object using penetrating radiation and a recording medium. In 1895 Konrad Roentgen records the first radiography soon after he discovered one of the penetrating radiations, the X-ray. In conventional X-ray imaging an object is irradiated by photons generated from an X-ray source and the transmitted photons are registered on a photographic plate. Since the X-ray attenuation of the irradiated object is proportional to its electron density, the X-ray intensity transmitted after traversing a region of lower density (for example muscle) is greater than the X-ray intensity transmitted after traversing a region of higher density (for example bone). In other words, the gray level of the image recorded is inversely proportional to the attenuation of the object in the ray path. This gives contrast and thus imaging of the object as it has been applied for diagnostic purpose. This clinical method of imaging has however, some drawbacks. The three dimensional structure of an object is collapsed onto a two dimensional film of detector which causes the loss of depth information. Furthermore, conventional X-ray images produce a two dimensional image of the anatomy only perpendicular to the X-ray beam. All these obstacles found its solution in the Computed Tomography (CT). CT refers to the cross-sectional imaging of an object from its projection data slice by slice. The origin of the word "tomography" is from the Greek word "tomos" meaning "slice" or "section" and "graphe" meaning "drawing." It is a diagnostic procedure in which a large series of cross-sectional (2 dimensional) X-ray images or “Tomographic slides” are used to generate three-dimensional images of the internals of an object non-invasively. Generally, the principle of CT consists of measuring the spatial distribution of a physical quantity of an object to be examined from different directions and to compute superposition free images from these data. The mathematical basis for CT was first discovered by Radon in 1917. However, it was not until 1972 that the first CT scanner was invented, for which G.N. Hounsfield and Alan McCormack received the Nobel Prize. Since then, many improvements have been made in scanner technology as well as in the algorithms used for CT reconstruction. Today CT scanners are used extensively in many applications such

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as medicine, astronomy and non-destructive testing to realize a three dimensional volume visualization of objects and treatment purpose. An important application of CT is in medicine, both to help with diagnosis (by taking images with x-rays as described above), and for radiotherapy. Radiotherapy uses carefully measured doses of high-energy radiation to treat cancer. Radiotherapy works because the high dose of radiation damages cancerous cells and stops them from growing and dividing. Normal cells that are affected usually recover or repair themselves quite quickly. Cancer cells, which are abnormal cells, do not recover. High dose radiation can be achieved by linear accelerator that delivers radiation energy in MV. The NKI/AVL hospital has co-developed a linear accelerator for therapeutic purpose with integrated cone beam CT for diagnostic and planning purpose. Generations of CT scanners including cone-beam scanner will be discussed below. 1.2 Generations of CT scanners A variety of CT geometries have been developed for visualization, diagnostic and therapeutic purposes. These geometries are commonly called generations. The imaging geometry of CT is of fundamental importance in designing a CT scanner system and a reconstruction algorithm. Popular types of CT geometries are depicted in figure 1.1 [5]. The first generation scanner is characterized by an assembly of an X-ray source and a single detector (Figure 1.1(a)). For a given projection angle, a parallel-beam projection profile is collected while the assembly is translated along a straight-line. To acquire the next projection, the frame rotated by 10, and then translated. The second-generation scanner is also in a translation-rotation mode but multiple detectors are employed that extend a small fan-beam angle (Figure 1.1(b)) in an attempt to reduce the scan time that lasts approximately four minutes in the first generation. In the third generation scanners, scan time is reduced further by eliminating linear acquisition completely. An arc-shaped array of detectors and the X-ray source form fan-beam geometry with a scanning field of view large enough to cover the slice of interest completely. The whole arrangement rotates around the object at a very high speed, often completing a full 360o rotation in less than one second (figure 1.1(c)). In the fourth generation design, stationary detectors are distributed along a full circle forming a detector ring, and only an X-ray source is orbited (Figure 1.1(d)). The major drawbacks of the fourth generation scanner arrangement are the very high cost of the detector array and the difficulty of scatter rejection. Cost and acquisition time consciousness have propelled the development of CT scanners from fan beam to cone beam geometry.

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(b)

(a) (c) (d)

FIGURE 1.1. Generations of CT scanners, (a)first, (b)second, (c)third, and (d)fourth

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1.3 Cone-beam CT Another generation of CT scanners is Cone-Beam CT. In all the aforementioned CT generations, volumetric image reconstruction is achieved through scanning a series of cross-section, and by stacking these slices. In cone-beam geometry, instead of scanning an object with a planar beam of X-rays, the entire object is illuminated with a point source that delivers a cone-shaped beam, and the X-ray flux is measured by a detector plane behind the object (Figure 1.2).

FIGURE 1.2. Cone-beam CT geometry The primary advantages of cone-beam geometry include compact construction, reduced data acquisition time since data for the entire region of interest can be acquired in one single rotation, improved image spatial resolution, and higher efficiency in x-ray usage (optimized photon utilization) [6]. CBCT also helps to better understand geometric uncertainties that could lead to improved dose distributions and integrating imaging and treatment modules in CBCT.

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Chapter 2

CT instrumentation 2.1 kV Source X-rays stem from the conversion of the kinetic energy attained by electrons accelerated under a potential difference into electromagnetic radiation, as a result of collisional and radiative interactions with the target material. An X-ray generator and an X-ray tube are the necessary components for control and X-ray production. The X-ray generator provides the source of electrical voltage and user controls to energize the X-ray tube whereas the X-ray tube provides the proper environments and components to produce X-rays. Basic components of an X-ray system is illustrated in figure 2.1 [7]

FIGURE 2.1. X-ray generator and x-ray tube components are illustrated. The x-ray generator provides operator control of the radiographic techniques, including tube voltage (kV), tube current (mA), and exposure duration (mS), and delivers power to the x-ray tube. The x-ray tube provides the environment (evacuated x-ray tube insert and high-voltage cable sockets), source of electrons (cathode), source of x-rays (anode), induction motor to rotate the anode (rotor/stator), transformer oil and expansion bellows to provide electrical and heat build-up protection, and the tube housing to support the insert and provide protection from leakage radiation.

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The basic components of the X-ray tube are the cathode and anode. Connected to the cathode and the anode are negative and positive high voltage cables, respectively from the X-ray generator. Electrons are produced by thermionic emission from electrically heated cathode filament in a vacuum tube, focused and accelerated toward the positively charged anode target by applied high voltage usually rated in “peak-kilo voltage” kVp supplied by the X-ray generator, the cathode and anode. These electrons collide at high speed with the target area of the anode. This collision produces the X-rays. Two types of X-rays are produced: Bremsstrahlung and Characteristic X-rays. The coulomb interaction between the incident electrons and the target nuclei results in continuous X-ray radiation called Bremsstrahlung X-ray with energy ranging from zero to the kinetic energy of the incident electron (maximum energy) depending on the distance of the electron to the nucleus. Characteristic X-rays produced as a result of interaction of incident electron with the electrons in the target material. The incident electron ejects an inner shell electron from the atom. The vacancy causes electrons from higher levels to fall down to occupy the lower levels, emitting characteristic radiation (line spectra). 2.2 Linear accelerator A medical linear accelerator (linac) is a highly specialized machine in which a high frequency alternating electric field is used to accelerate electrons, producing high energy X-ray photons and electrons. Linacs are cyclic accelerators that accelerate electrons to kinetic energies from 4 to 25 MeV using microwave RF fields in the frequency range from 103 MHz (L band) to104 MHz (X band), with the vast majority running at 2856 MHz (S band). Linacs are usually mounted isocentrically and the operational systems are distributed over five major and distinct sections of the machine. In operation, an electron gun with a tungsten filament produces electrons thermionically, which ride on microwaves (produced by a magnetron) down an accelerating waveguide. A powerful bending magnet is used to select the appropriate electron energy and send these electrons to impact on a thin heavy metal target producing deeply penetrating X-ray photons. On the way to the patient, the chosen type of radiation beam is made flat and symmetric by using a Flattening Filter with quadrant monitor chambers. The dose is monitored using ionization chambers, and the beam is constrained and shaped using the 120 leaf multi-leaf collimator, to cover only the target area. A schematic diagram of a typical modern S band medical linac is shown in Fig. 2.2 [8]. Also shown are the connections and relationships among the various linac components listed above. The diagram provides a general layout of a linac’s components; however, there are significant variations from one commercial machine to another, depending on the final electron beam kinetic energy as well as on the particular design used by the manufacturer.

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Fig 2.2 Diagram illustrating parts of medical linear accelerator 2.3 Flat-panel detector

X-ray imaging for medical diagnosis began soon after Roentgen discovered x-rays in 1895, but it has taken 100 years to replace x-ray film with a digital imaging technology. Around 1990, researchers in X-ray physics recognized that the development of a digital flat panel X-ray detector would be a major technological breakthrough in X-ray imaging, and soon began to develop such detector. The essential elements are a sensor that absorbs X-rays and creates a corresponding electric charge, a capacitor to store the charge, and the active matrix addressing that organizes the readout of the signal to external electronics, which amplify, digitize and display the image. Flat-panel X-ray imagers are based on solid-state large area integrated circuit (IC) technology that exist in the form of amorphous-silicon or cadmium selenide thin-film-transistors (TFT). They incorporate a two dimensional matrix of thin-film switches (one switch per pixel) made to control the readout. The pixel switch is a single TFT made from amorphous silicon. Flat-panel detectors are categorized as either direct or indirect based on the materials used for X-ray detection, X-ray photoconductors or scintillators, respectively [10].

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With indirect digital X-ray imaging, an X-ray tube sends a beam of X-ray photons through a target. X-ray photons not absorbed by the target strike phosphor layer such as layer of scintillating material like cesium iodide (CsI) or gadolinium oxysulfide (Gd2O2S) that excites a high-energy electron in the phosphor by the photoelectron effect. The electron loses energy by ionization creating many low-energy electron-hole pairs, which then recombine to emit visible light photons. The intensity of the light emitted from a particular location of the phosphor is a measure of the intensity of the X-ray beam incident on the surface of the detector at that point. These visible light photons then strike an array of photodiodes, which converts them into electrons that can activate the pixels in a layer of amorphous silicon. The activated pixels generate electronic data that a computer can convert into a high-quality image of the target. The detection process is indirect in that the image information is transferred from the X-rays to visible light photons and then finally to electrical charge. In the direct detection, X-ray photoconductors are used instead of scintillators to convert the X-rays into electron-hole pairs. The fundamental x-ray conversion chain is shown in figure 1 below [9]

Figure 2.3: Flat-panel detector signal chain The imaging system must be able to record the transmitted x-ray signal over the projected area of the anatomy under investigation. The flat panel detector used in our system has a square field of view of 41 x 41 cm2.

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Chapter 3 X-ray interaction with matter X-ray projection imaging and CT are made possible by the differential attenuation of X-rays through the body. Attenuation is the result of the interaction of X-ray photon with matter. Also the interaction of photons with the detector converts an x-ray image into one that can be viewed or recorded. Some of the X-ray interactions with matter that are pertinent to medical X-ray applications are [10]: • Photoelectric effect • Compton Scattering • Pair production Which process among the above dominates is dependent on the mass absorption characteristics of the target (directly related to the atomic number, Z) and the energy of the X-rays, as shown schematically in the graph below [10]

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FIGURE 3.1. Regions of relative predominance of the three main forms of photon interaction with matter. The left curve represents the region where the atomic coefficients for the photoelectric effect and Compton effect are equal, the right curve is for the region where the atomic Compton coefficient equals the atomic pair production coefficient.

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3.1 Photoelectric effect X-rays not only exhibit continuous wave properties but also exhibits discrete particle characteristics as described by wave-particle duality in quantum physics. Photoelectric effect involves the absorption of an incident discrete X-ray photon by the absorbing atom. The absorption results in the ejection of an inner shell (usually the K-shell) electron from the atom with a kinetic energy Epe equal to the difference of the incident photon energy E0 and the electron shell binding energy φ. E0 = φ + Epe 3.1 The vacated electron shell is subsequently filled by either radiative process, in which electrons with less binding energy from the outer shell fall down emitting characteristic X-ray equal in energy to the difference in electron binding energies of the source electron shell and the final electron shell, or by radiation less (Auger) processes. The production of photoelectric and auger electron is shown diagrammatically in figure 3.2 (from Jenkins and Snyder, 1996).

FIGURE 3.2. Diagram (a) shows the incident X-ray photon interacting with target atom, (b) shows the production of a high-energy primary photoelectron. In (c) a lower energy electron moves into the vacated K-shell resulting in the production of an X-ray photon that lthe atom, and in (d) the X-ray photonabsorbed by an outer shell electron resulting in the emission of a Augeelectron.

eaves

is

r

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As it is evident from equation 3.1, the photoelectric interaction cannot occur if the incident photon energy is less than the binding energy of the electron, but if the X-ray is equal to the electronic binding energy (E0 = φ) the photoelectric effect becomes energetically feasible and a large increase in attenuation occurs. Probability P of the photoelectric absorption is given by:

P ∝ Z3 ∝ 3

1E

3.2

Where Z is the atomic number of the absorbing atom and E is the incident photon energy.

FIGURE 3.3. Probability of photoelectricabsorption.

The K absorption edge refers to the sudden jump in the probability of photoelectric absorption when the K-shell interaction is energetically possible. Similarly, the L absorption edge refers to the sudden jump in photoelectric absorption occurring at the L-shell electron binding energy (at much lower energy). Tissues in the human body contain mostly low atomic number elements (e.g., hydrogen, Z = 1; carbon, Z = 6; nitrogen, Z = 7; and oxygen Z = 8), which have low K-shell binding energies, and the “yield” of characteristic x-ray production for Z = 10 is close to zero. Auger electron events predominate in these elements. Thus, for photoelectric absorption in tissues, the energy of the incident photon is locally deposited (resulting in effectually “total” absorption). However, for higher atomic number elements the characteristic X-ray is higher. Such phenomenon enhances signal contrast. It also means that contrast agents, X-ray detectors, and protection devices are preferably made of high Z elements, such as iodine, gadolinium, and lead, respectively.

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3.2 Compton scattering Compton scattering is an inelastic interaction between an x-ray photon of energy E0 that is much greater than the binding energy of an atomic electron (in this situation, the electron is essentially regarded as “free” and unbound). Partial energy transfer to the electron causes a recoil and removal from the atom at an angle, φ. The remainder of the energy, ES, is transferred to a scattered x-ray photon with a trajectory of angle θ relative to the trajectory of the incident photon as shown in figure 3.4 below.

FIGURE 3.4. Compton (incoherent scattering) From the conservation of momentum and energy follows:

0EEs =

)cos1(511

1

10 θ−+keV

E 3.3

This is known as the Klein–Nishina equation, where 511 keV is the energy equal to the rest mass of the electron.

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3.3 Pair production If a photon enters matter with energy in excess of 1.022 MeV, it may interact with a process called pair production. 1.022 MeV represents the rest mass energy equivalent of 2 electrons (i.e., E = 2 m0c2, where m0 is the rest mass of the electron [9.11 × 10-31 kg] and c is the speed of light [3.0 ×108 m/s]).

Figure 3.5. Pair production The interaction of the incident photon with the electric field of the nucleus results in the production of an electron (e-)–positron (e+) pair, with any photon energy in excess of 1.02 MeV being transferred to the kinetic energy of the e-/e+ pair equally. Pair production is a rare process and only occurs at high X-ray photon energies with high atomic weight targets.

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Chapter 4 Computed tomography image This chapter deals briefly with the basic steps of acquiring CT image: irradiating the patient, recording the transmitted intensity profile, calculating attenuation profile, and reconstructing an image from its projections. These steps are summarized in figure 4.1. In CT imaging, a patient is irradiated by photons from an X-ray source and the transmitted photons are registered on a detector giving transmitted intensity profile. Taking the logarithm of the intensity profile gives a 2-dimensional projection image of the tissues within the patient's body. The differential absorption of x-rays as they pass through the different parts of a patient's body delivers the contrast needed for imaging.

FIGURE 4.1. Principle of CT scanning. A cross-section of an object is probed with x-rays from various directions; attenuated signals (intensity profiles) are recorded and converted to projections of the linear attenuation coefficient distribution (attenuation profile) of the cross section. These X-ray shadows are directly related to the Fourier transform of the object and, can be processed to reconstruct the object.

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4.1 Intensity profile A CT image is formed by the collision of X-ray photons with a patient and a detector, as discussed in chapter 2 and 3. The detected photons can either be primary photons, which have passed through the patient without interacting, or secondary photons, which result from an interaction in the patient. The secondary photons are deflected from their original path and carry little useful information about the patient. The primary photons do carry useful information. They give a measure of the probability that a photon will pass through the patient without interacting and this probability will itself depend upon the sum of the X-ray attenuating properties of all the tissues the photon traverses giving the image of the patient. The probability can be expressed per thickness of the attenuator, by Lambert-Beer equation as: I = I0exp(- ) 4.1 ∫

L

dlzyxf ),,(

This gives the transmitted X-ray intensity I along a specific line L from the source to a detector element. I0 is the initial intensity from the source. The function ƒ(x,y,z) is the X-ray linear attenuation coefficient of the object at various points in the objects cross section. By fixing the third dimension, z, to a particular value, we measure a two dimensional slice ƒ(x,y) of the three dimensional object. For different values of z, we obtain different slices. For homogenous object and monochromatic radiation, equation 4.1 simplified to: I = I0exp(-µl) 4.2 Where l is the length of the object traversed by the photon.

FIGURE 4.2. Intensity profile for homogenous object and monochromatic radiation. HVT, the half value thickness, is the thickness at which the intensity has fallen to half its initial value.

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For inhomogeneous object and polychromatic radiation, equation 4.1 becomes: I = ∫I0(E)exp(-∫ µ(E)dl)dE 4.3

FIGURE 4.3. Photon energy dependence of the mass attenuation coefficient At a given photon energy, the linear attenuation coefficient can vary significantly for the same material if it exhibits differences in physical density. The mass attenuation coefficient, µ/ρ, compensates for these variations by normalizing the linear attenuation by the density of the material ρ. For the mass attenuation coefficient, “thickness” becomes the product of the density and linear thickness of the material, or ρx. This is known as the mass thickness with units of g/cm2. The reciprocal of the mass

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FIGU4.4. Mass attenuation coefficie

thickness represents the units of mass attenuation coefficient, cm /g.

RE

nts f several aterials

d ic

omencounterein diagnostx-ray imaging are illustrated as functionof energy.

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From these plots it can be determined that mass attenuation decreases at a rate of pproximately 1/E3 for low energy (∼10 to ∼100 keV) and increases as a function of tomic number (Z) of attenuating material as approximately Z3. With higher Z, presence

ent.

he negative natural logarithm of the relative attenuation of equation 4.1 gives us the ra ing each intensity profile to the

orresponding attenuation profile p (r) is given by:

aaof “absorption edges” results from increased attenuation of x-rays by photoelectricabsorption event at energies equal to binding energies of electrons in the specific elem 4.2 Attenuation (Projection) profile Tinteg l value of the object function transformc θ

pθ(r) = -lnIoI = ∫

L

dlyxf ),( = ∫L

dlyx ),(µ 4.4

) is called projectionIt is the line integral of the image intensity, ƒ(x,y), along a line L that is distance r from

e origin and at angle θ off the x-axis. For simplicity, we consider parallel projection as

IGURE 4.5. Parallel beam rojection. θ and r represent the rojection angle and the detector

a on formula [7]:

The function pθ(r the of the function ƒ(x,y) along the fixed angle θ.

thdepicted in Figure 4.5 and define two different coordinate systems: one that is fixed in theobject being X-rayed (x,y) and another that rotates with the source and detector (r,s).

Fppposition, respectively.

The coordinate change obeys the following transform ti

4.5 and

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4.6 An object can be visualized as a stack of two-dimensional slices or it can be examined in its natural three-dimensional representation. Stacking all the one-dimensional projections p (r) together results in a two-dimensional data set p(r, θ). Edholm and Jackson (1975)

n the Cartesian (r,θ) spaceontributes to projection values along a sinusoidal curve in this space. The transformation

= −+ dxdyryxyxf )sincos(),( θθδ 4.7

he Radon transform (or the sinogram) of the two-dimensional object implies us that if ber of one-dimensional projections of an object taken at an infinite

umber of angles, we could perfectly reconstruct the opractice there is a limited number of projections or views and a limited number of

etector samples.

ill examine the methods available to us for reconstructing a slice, x,y), from its projection data, p(r,θ). As p(r,θ) is the Radon transformation of f(x,y), this eans that what we actually need is an expression of the inverse Radon transform:

e line ρ inclined at an angle θ. This is the famous central section theorem and is derived

θ

named this projection i a sinogram since each object point cof a function ƒ(x,y) into the sinogram p(r,θ) is called the radon transformation. From equation 4.4 and 4.5 the Radon transform is given by: p(r, θ) = ∫ +−

L

dlsrsrf )cossin,sincos( θθθθ

∞ ∞

∫ ∫ ∞− ∞−

Twe had an infinite numn riginal object, f(x,y). However, in

d 4.3 Image reconstruction In this section we wf(m f(x,y) = ℜ-1{p(r,θ)} 4.9 One of the important properties of the Radon transform, p(r,θ), of an object, f(x,y), is its

lationship to the Fourier transform, F(u,v) of f(x,y) usually termed as the central section retheorem. Taking the 1D Fourier transform of the projections of an object at an angle θ is equivalent to obtaining the two dimensional Fourier transform of the density f(x,y), along thin appendix 1 and given schematically in figure 4.6.

19

Figure 4.6. Schematic illustration of the central section theorem Therefore if we take these projections at many angles, then we can get two-dimensional Fourier transform of the projections at many such lines inclined at various angles. If the number of angles is large enough, we will get many lines of two-dimensional Fourier transforms of the object.

The most dominant CT reconstruction technique is filtered backprojection (FBP). Theoretically, FBP has its roots in the Radon transform and the Fourier slice theorem, which link a function and its projections to its Fourier transform. FBP is therefore analytic, and its practical implementations take advantage of the fast Fourier transform (FFT). FBP is fast and deterministic, and its properties are very well understood. In backprojection the measurements obtained at each projection are projected back along the same line (or equivalently same θ) and summed. The image one calculates with the backprojection method is the blurred version of the actual image. In order to get the final image, we need to undo this blurring function and the artifact. To undo the 1/r blurring function that occurs as a result of backprojection, the data can be two-dimensionally filtered by a truncated Ram-Lak, or ramp filter, assuming parallel-beam geometry. Derivation of the backprojection algorithm for parallel beam reconstruction is given in appendix 2.

20

21

22

Part II Cone-beam CT Image quality optimization

23

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Chapter 5 Factors that affect image quality Ideal x-ray image acquisition assumes a point and monochromatic x-ray source, a straight-line trajectory from the source through the object and complete detection of the x-ray beam that is attenuated due to photoelectric effect. The image of an object obtained by such acquisition is an exact representation of the object. However, ideal world exists only on paper and practical image acquisition suffers from image quality deterioration. The ability of a radiologist or other medical specialist to detect sign of pathologically significant processes depends among other things on image quality of a CT system. Image quality can be roughly described as the ability of the imaging device to record faithfully each point in the object as a point in the final image characterized mainly by CT numbers in the object and image. This transfer of information is never perfect. Image artifacts affect image quality. An artifact is any distortion or error in the image that is unrelated to the subject being studied. For x-ray CT, artifacts are any systematic discrepancy between the CT numbers in the reconstructed image and the expected CT number based on the true linear attenuation coefficient of the object. Image artifacts degrade image quality as well as hide areas of pathology. Thus it is necessary to understand which types of artifacts occur; why these artifacts occur; and how they can be prevented or suppressed. Deviation from the idealities described at the outset above is the main cause of image artifacts. The inherent polychromatic nature of medical x-ray source results in what is known as beam hardening; scatter of x-ray photons from the patient causes detection of photons from other than straight-line trajectory from source through the object; and non-ideal detector performance results in incomplete and inaccurate detection of the attenuated x-ray photons. Consequently, the following image artifacts can occur: (a) streaking, which is generally due to an inconsistency in a single or a few measurement; (b) cupping, which is due to preferential attenuation of low energy photons; and (c) rings, which are due to errors in an individual detector calibration etc. CT image artifacts originate from a range of sources [12]. (a) physics-based artifacts, which result from the physical processes involved in the acquisition of CT data. Artifacts caused by beam hardening and X-ray scatter are examples of physics-based artifacts; (b) detector-based artifacts, which result from imperfections in detector function. Ring artifact is the most prominent artifact in this category. (c) patient-based artifacts, which are caused by such factors as patient movement or the presence of metallic materials in or on the patient; (d) artifacts which are produced by the image reconstruction process. This thesis deals with the first two categories. The concepts of detector performance, beam-hardening and scatter will be introduced in chapter 5 and the methods to estimate and correct poor detector performance, beam hardening and scatter are the subject of chapter 6. In chapter 7 the results of applying these methods will be presented and discussed in chapter 8. Conclusions in chapter 9 will finish the thesis.

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5.1 Flat panel detector performance Detector-based artifacts are among common source of artifacts that affect image quality. Flat panel detector non-linearity, non-uniformity and bad pixels are the main causes for such artifacts. The value of any pixel within the object acquired by a detector should be linear function of the amount of x-ray photon transmitted through that part of the object. This is not always the case since there are many effects that distort linearity. These effects include saturation (nonlinear pixel effects above certain light levels), dark current (charge that accumulate over time with or without light exposure), non-uniformity (pixel-to-pixel gain variation), bad pixels (pixels that do not react to light exposure), and many other effects that are not uniform across the scene. Fortunately many of these effects can be modeled and corrected. Measurements of the mean pixel as a function of exposure or measurement of dose as a function of detector response indicates the degree of linearity of the detector and show the exposure that result in saturation of the flat-panel detector. For a linear regime of flat-panel detector where exposure is not too high or not too small, there is a linear relationship between the number of photon hitting a pixel and the output signal recorded. As the exposure increases, the electric field across the photodiode decreases which increases the effect of charge trapping and eventually saturation, thus leading to progressive loss of linearity. Beyond lower linear value limit where the detector is insensitive and between the upper linear value limit and before the pixel becomes fully saturated, the detector response can be nonlinear, but still well behaved and thus can be modeled and linearized by the technique of piecewise linearization. Any nonlinear detector response function F can be modeled with a constrained piecewise linear function Fpl. A piecewise linear function Fpl is defined with N linear segments over the interval [X0, XN] as: F1 X0 ≤ x < X1

F2 X1 ≤ x < X2 . Fpl = { . 5.1 . FN XN-1 ≤ x < XN Where Fn(x)= anx +bn, 5.2 is the equation of the nth linear segment. Fpl is linear in each of those separate intervals. For that reason it is said to be piecewise linear. The “y-intercepts” bn can be written as a function of the slopes an and exposure Fn defining each previous linear segment. The points where the slope of the function changes are known as breakpoints. n=1,…, N is a linear segment and xn represents N+1 break points in [X0, XN]. In a continuous piecewise linear function, the endpoint of one segment has the same coordinates as the initial point

26

of the next segment. It is possible to get slope and intercepts of a segment between any two exposures by taking the ratio of the difference of response to the difference of exposure at the required points. After slope is calculated in this manner, it is a matter of substituting it, the response and the respective exposure into equation 5.2 to calculate the corresponding intercept. Error of approximating the flat-panel detector response curve by linear segments can be determined from: RMS = ∑ ns /)(Re 2 5.3 Where RMS is the root mean square error, Res is the difference between measurement and fit data (Residue) and n is the number of exposures used. The sum is over the number of exposures. Figure 5.1 shows an example of piecewise linearization. A non-linear curve is represented by a number of linear segments.

Figure 5.1. Piecewise linearization Non-uniformity measures the gain variation among different pixels.

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5.2 Beam hardening When an x-ray hits matter, it can go through or it can interact with the matter. If it interacts, all of the energy can be absorbed due to photoelectric interaction or some of it can scatter off due to Compton interaction as discussed in section 3.1 and 3.2. The probability of photoelectric interaction increases with increasing atomic number of the absorber as discussed in section 3.1 (equation 3.2). The photons transmitted through a voxel of an object give rise to a pixel signal that the detector registers. The pixel value corresponds to the linear attenuation coefficient, i.e., the amount of x-rays taken away from the beam (attenuated) as x-rays pass through the voxel. This implies that the pixel values registered by a detector depend on the average composition of the voxel and density of matter within the voxel. Low atomic number (Z) and low-density materials generally attenuates less than materials of higher Z and higher density. Thus, in addition to enabling nondestructive visualization of the internal structure of the objects, CT allows quantitative analysis of the thickness or composition of the material. The reconstruction algorithms that make this analysis possible assume that the attenuation of the incident x-ray beam is exponentially related to the thickness of the object, cfr. Beer’s law. This relationship will no longer hold because attenuation coefficient depends not only on Z and density of the voxel but also on the energy of the x-ray source (see section 3.1) and the energy of the x-ray source used in medical CT are inherently polychromatic. Consequently, the different x-ray energy levels traversing the object are not attenuated in the same way. In fact, the lower x-ray energies will be easily absorbed due to photoelectric effect, while the higher or harder energies are less attenuated. In other words, lower energy photons are removed preferentially from the polychromatic x-ray beam as it passes through the object, resulting in the change of energy spectrum towards a profile with a higher average energy which penetrates more efficiently than does the initial spectrum, hence the term “beam hardening” since the beam gradually becomes harder, i.e. its mean energy increases (figure 5.2, [12])

a) b) c) Figure 5.2 When a polychromatic X-ray beam passes through matter, low energy photons are preferentially absorbed, and the beam gradually becomes harder, i.e. its mean energy increases. Energy (keV) and initial intensity are depicted on horizontal and vertical axis respectively. a)shows no attenuation( 0 cm) in water; b)30 cm in water and; c)60 cm in water. The energy is not corrected for the detector sensitivity.

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When the energy of the beam is changed, the linear attenuation coefficients of the object that the “harder” beam travels across are also changed. The harder a beam, the less it is further attenuated. Therefore, the total attenuation is no longer a linear function of absorber thickness. This is schematically depicted in figure 5.3 for a homogeneous circular phantom.

Figure 5.3 Schematic representations of monochromatic situation shown in gray line (no beam hardening effect) and polychromatic measurements drawn in black line (a) intensity profile, (b) attenuation profile (projection), r is the location on the detector array and (c) projection value as a function of path length s through the absorber material. For the monochromatic case (gray line), the projection value increases linearly with path length. For the polychromatic case (black line), the projection value is a non-linear function of the path length. At least two types of artifacts can result from beam hardening effect: cupping artifacts (this is the situation where the plot of CT number against thickness or cross section of a reconstructed image shows depression of the CT number values from the nominal value, with the depression greatest in the center and least at the edges, producing a ‘cupping’ effect) and the appearance of dark bands or streaks between dense objects in the image. The numbers of photons taken away from the beam in the same voxel in different projection angles will be different. When the voxel is close to the edge of the object on the side turned towards the source, the average energy of the x-ray spectrum will be lower and the number of photons attenuated from the beam as it passes through the voxel is larger than in the later projections when the x-ray source is in the opposite direction since there is more object material between the voxel and the source, which causes the average energy to rise since the low energy photons are taken away from the beam before reaching the voxel. This indicates a less dense material for the voxel in this projection than in the former. It also indicates that pixel values acquired with polychromatic source are dependent on voxels itself and where it is located. Upon reconstruction, the attenuation coefficients and the CT numbers appear to have been decreased (figure 5.4 [12]). This results in CT images showing less dense materials in the center of the object, indicating a false linear attenuation coefficient gradient. The consequence is image artifacts.

29

(c) . Figure 5.4 CT number profiles obtained across the center of a uniform water phantom with polychromatic radiation (a) and with monochromatic radiation (b). CT number value against distance from the center (orange is for monochromatic) (c).

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5.3 Scatter Not only photoelectric effect plays a role in medical CT imaging. At diagnostic imaging energies in soft tissues and bone, a large fraction of the attenuation occurs by Compton scattering rather than by photoelectric absorption, chiefly because of the low atomic number of the tissues as discussed in section 3.2. The amount of scatter increases with object thickness and field size. An important measure of scatter is the scatter-to-primary ratio, S/P, indicating the ratio of the scattered x-ray fluence to the primary x-ray fluence incident on the detector. The effect of scatter on the detected intensity and attenuation profile is schematically depicted in figure 5.5. Due to the non-linear behavior, the qualitative nature of the scatter artifacts is similar to beam hardening. Streaks in the direction of highest attenuation and cupping are the most prominent artifacts caused by x-ray scatter.

Figure 5.5 schematic representations of measurements with and without scatter: (a) intensity profile, (b) attenuation profile (projection), and (c) projection value as a function of path length. As the directional information of scattered photons is largely lost in the scattering process, scatter profiles will be relatively smooth. Sometimes the scatter contribution is even approximated by a constant. In the intensity profile, the scatter contribution results in a simple addition of a constant Isc. Quantities that often used as a measure of physical image quality are the contrast, the CT number accuracy of the reconstructed image and the signal to noise ratio (SNR). X-ray scatter deteriorate image quality by reducing image contrast; by increasing image noise; and by introducing incorrect CT number to the reconstructed image. The inaccurate CT numbers are called streaks and cupping artifacts. The influence of scatter on each of these factors will be discussed briefly below. Contrast is the spatial variation of the x-ray photon intensities that are transmitted through the patient and thus gives a measure of difference between regions in an image. The variation in transmitted intensities is a result of differential attenuation of x-rays by tissues that differ in density, atomic number and thickness. The probability of x-ray scatter is also proportional to these factors (see chapter 3) and therefore x-ray scatter has a direct effect on subject contrast. X-ray scatter reduces subject contrast by adding background signals that are not representative of the anatomy thereby reducing image quality.

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Contrast reduction by scatter is illustrated in Figure 5.6 [13], where a uniform incident fluence I0, and transmitted fluences through tissue, air, and bone I1,I2, and I3 respectively are indicated.

Figure 5.6: Ideal projection radiograph is representation of transmitted primary x-ray fluence from point source through object and incident on detector, as depicted on left for a uniform incident fluence, I0, and transmitted I1, I2, and I3 fluences through tissue, air, and bone, respectively. The effect of scatter is shown on the right. Subject contrast is the relative difference in signals of an object to its background.

Contrast = signalbackground

signalbackgroundalobjectsign − 5.4

For example taking air as a background, the contrast of soft tissue in the absence of scatter in the above figure is given by:

Ctissue = tyAirIntensi

IntensitySofttissuetyAirIntensi − 5.5

32

Ctissue = 2

12

III − 5.6

Where Ctissue is the contrast of soft tissue without scatter (SPR=0). As displayed in figure 5.7, contrast of an image without scatter is a linear function of thickness. With the presence of scatter, contrast is not any longer linear with thickness and besides it is decreases as scatter (given by scatter to primary ratio (SPR)) increases. Equation 5.7 (Kieran Maher, 2001) and figure 5.7 show this fact.

Ctissue,scatter = ln[SPRdD

SPR+−−

+)(exp(

1

21 µµ] 5.7

Where µ1 is the linear attenuation coefficient of the soft tissue, µ2 is the linear attenuation coefficient of air, D is the thickness of the soft tissue and d is the thickness of air.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Con

trast

CONTRAST WITH AND WITHOUT SCATTER

SPR=0

SPR=0.5

SPR=1.5

SPR=3

µ1D - µ2d Figure 5.7 Effect of scatter on contrast showing plot of equation 5.7

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Another problem caused by x-ray scatter is noise. Noise is defined as the uncertainty or imprecision with which a signal is recorded. Noise will diminish our ability to observe objects of low contrast that are close to the visibility threshold as illustrated in figure 5.8.

Figure 5.8 The impact of noise on the ability to image objects of varying contrast. As the object contrast decreases the ability to observe any detail is masked by the noise While there are several ways to quantify noise, the simplest one is to determine the standard deviation of the image pixel values about their mean in a uniform region of an image. The noise in a reconstructed image has a direct relationship with the noise in the projections images. The noise-to-signal-ratio in a reconstructed image, σ/µ, is related to the noise-to-signal-ratio of the projection images, σI/I, [14]:

2

⎟⎟⎠

⎞⎜⎜⎝

⎛µσ = κ2

2

⎟⎠⎞

⎜⎝⎛

IIσ 5.8

with κ2 = na 222

5.9

Where µ is the true value of the attenuation coefficient at a point, and σ is the variance in a set of measurements of the attenuation at that point, I is the transmitted x-ray intensity, σI is the standard deviation of the transmitted x-ray intensity, a is the linear sampling distance and n is the number of views. The noise-to-signal-ratio of the projection images, σI/I, in turn is given by [14]:

2

⎟⎠⎞

⎜⎝⎛

IIσ = 2

222

)()(

pp

paddsspp

ENENENEN

><><+><+><

5.10

Where Np and Ns are the numbers of primary and scattered x-ray photons detected per view, respectively. Nadd is the additive electronic noise in a detector and E is the photon energy. The operator <…> denotes the statistical expectation (mean) value of the energy. Equations 5.10 shows that the square of noise in the projection image increases directly with the number of scattered x-ray photons and equation 5.8 shows that the square of noise in the reconstructed image is directly proportional to the square of noise in the projection image. Therefore, x-ray scatter increases noise in both projection and reconstructed images.

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The third effect of scatter on CT image quality is that it introduces CT number inaccuracies in a reconstructed image with the consequence of streaks and cupping artifacts exactly the same way as it happens in beam hardening. However, in effect, the beam becomes “more penetrating,” not because of the increased x-ray energy (as it is the case in beam hardening) but because scattered radiation is detected together with the attenuated primary radiation and the effect is the increased detected signal that gives false linear attenuation value and thus false object density. Acceptance of scattered radiation by a detector reduces the estimate of the linear attenuation coefficient. This results in cupping artifacts. To see how scatter causes cupping, let us write the detected intensity I with the presence of scatter as I = P+S where P and S are primary and scattered radiations respectively. Linear attenuation coefficient with the presence of scatter is thus given by:

∫µdl = ln(I0/I) = ln[SP

I+0 ] = ln[

)/1(0

PSPI

+] = ln

PI 0 + ln[

)/1(1

PS+] 5.11

The first term in equation 5.11 is the linear attenuation coefficient with the absence of scatter. The second term is the contribution of scatter to the linear attenuation coefficient. This term is always negative; causing the value of linear attenuation coefficient with the presence of scatter near the centre of the object, where less primary radiations present due to a high attenuation, to appear smaller than it actually is. The magnitude of this cupping can be estimated as:

cupping = 100×edge

centeredge

µµµ −

5.12

Where µ is the linear attenuation coefficient.

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5.4 Objective: Optimization of cone-beam image quality

The effect of flat-panel detector nonlinearity, beam hardening and scatter artifacts on image quality is clear from the discussions so far. The aim of this thesis is to investigate these effects on the image quality of Cone-beam CT.

Cone-beam CT uses a flat-panel detector and as such its image quality depends to some extent on the performance this detector. Besides, the inherently polychromatic nature of medical x-ray sources introduces artifacts due to beam hardening. Furthermore, as a volumetric imaging system, Cone-beam CT has greater field of view than Fan-beam CT which makes it to be much more vulnerable to scatter. To capitalize the potential advantages of Cone-beam CT, the poor image quality of Cone-beam CT that can be caused by these factors should be tackled.

To that end, the thesis addresses the following problems: can we linearize the amount of x-ray photons transmitted through a body to the pixel values they generate upon striking the flat-panel detector? Given the attenuation of polyenergetic x-ray beams through a body, can we give estimates of the attenuation of the corresponding monoenergetic x-ray beams through the same part of the body that can result to the precise CT numbers upon reconstruction? Given the total registered x-ray photons by the detector, can we extract the intensity of the photons that has followed straight-line trajectory from the source through a body and to the detector, i.e., can we estimate the intensity of the scattered photons and reject it from the total intensity?

Researchers have been investigating the performance of a flat-panel detector [26-27]. It has been shown that the detector performs fairy well. We have used extra calibration points in correcting the non-uniformity and non-linearity of the flat-panel detector. In addition, we have also investigated the optimum trade-off between the calibration points and the costly calculations involved in the optimization of the detector.

Artifacts caused by beam hardening can be estimated and corrected using either hardware correction (filters) or software correction (linearizing) [16]. Hardware filtering is the most popular method to reduce the beam hardening effect. The main disadvantage of hardware filtering technique is the decreased amount of X-rays, which results in a decrease of the image signal-to-noise ratio (SNR). We have used the method of software correction in tackling the artifact caused by beam hardening. We fitted the data from a poly-energetic source with power function, and then every value on power function is corrected towards the linear trendline that is expected in the monochromatic case.

Several techniques and devices have been implemented to estimate and correct scatter in cone-beam CT. Examples of such techniques include modeling the scatter mathematically and correcting it using digital filtration technique [17-19]. Air gaps are also standard methods for suppression of scatter by increasing the ration of the primary to scattered photons [20]. However, it does not completely remove scatter. Collimation of the incident x-ray beam before and after the patient can also be used as a scatter suppression method. This method is not applicable for acquisition where larger field of view (e.g., chest) is required. Scatter can also be controlled using anti-scatter grids [21]. Anti-scatter grids although effective in reduction of scatter, do not remove all scattering.

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Another demerit of anti-scatter grid is that the dose level in the patient must be increased in order to cope with the attenuation caused by the grids.

Here, we have used three different scatter correction methods and evaluated these methods based on the results achieved. The three scatter correction methods we have used here include: simple uniform assumption of scatter, Scatter sampling using the beam stop array technique and simulating the scatter sample of beam stop array technique. We have used these scatter correction methods together with beam hardening correction method to get the best image quality.

The materials and the techniques we have used in optimizing the flat-panel detector, in correcting the beam hardening and the scatter artifacts are given in the next chapter. The results achieved by using these materials and methods are given in chapter 7.

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Chapter 6 Materials and methods 6.1 Experimental setup Volumetric imaging system for the purpose of guided radiation therapy uses the medical linear accelerator with kilovoltage (kV) cone-beam computed tomography system orthogonal to the Megavoltage (MV) source. A medical linear accelerator with an on-board imaging system consisting of an X-ray tube and an amorphous silicon flat-panel image detector on a pair of arms is illustrated in figure 6.1

MV Source

kV Source Flat panel EPID

Treatment couch Figure 6.1. Illustration of cone-beam computed tomography incorporated in an Elekta medical linear accelerator for the purpose of image guided radiotherapy.

39

For the online imaging, a conventional diagnostic x-ray tube generates the kV x-rays and is mounted on a retractable arm that extends from the drum of an Elekta linear accelerator. The focal spot of the x-ray tube is located at 90 degrees to the MV source and 100 cm from the treatment system’s axis of rotation, sharing a common rotational axis with the MV treatment source. A 41 cm by 41 cm flat panel x-ray detector is mounted on an extendable robotic arm opposite the kV tube with a nominal detector-to-focal spot distance of 153.6 cm. This setup combines irradiation protocols with online imaging technique by rotating the beam around the patient and making an image every desired angle so that the area of interest is observed and imaged from many different angles. Using the setup displayed in figure 6.2 (a), we scanned a quality investigation phantom Catphan®600 illustrated in figure 6.2 (b) (The Phantom Laboratory, Salem, NY) in order to investigate the image quality of the cone-beam CT. The acquisition setting of X-ray energy of 120 kV, a tube current of 20 mA and exposure duration of 20 msA was used.

(a) (b) Figure 6.2. (a)Experimental setup for the investigation of image quality of cone-beam CT: (b) Catphan®600 phantom, (c) reconstructed slice of the phantom corresponding to CTP528 high-resolution module of the Catphan600

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Module CTP528 of Catphan®600 phantom has a diameter of 15 cm and thickness of 4 cm. The design of the CTP528 module of the phantom minimizes visual artifacts by reducing the amount of high contrast material. The 2mm thick aluminum contrast figures are cast into position on the radial gauge, which has resolution sections ranging from 1 to 21 line pairs per cm. This radial design pattern eliminates the possibility of streaking artifacts from other test objects. 6.2 Optimization of detector performance To characterize the performance of the flat-panel detector, we measured and corrected its linearity and uniformity using the following methods. 6.2.1 Linearization To correct the nonlinear behavior of a flat-panel detector, piecewise linearization as discussed in section 5.1 was employed. The linear segments were found by using equation 5.2. Difference between the measured flat-panel detector response curve and the linear fit to it at all the measurement points is determined and the maximum deviation is selected as a breakpoint for piecewise linearization. The error made in approximating the detector response curve was found by using equation 5.3. In linearizing the flat-panel detector piecewise, we followed the following procedure.

1. Start with one linear segment between zero and maximum exposure. 2. Use the exposure that resulted in maximum difference between the measured non-

linear curve and the linear fit represented by the segment obtained in step number 1 as a breakpoint to approximate the non-linear response with two new segments.

3. Use the exposure that resulted in maximum difference between the measured non-linear curve and the linear fit represented by the two segments obtained in step number 2 as breakpoint to approximate the non-linear response with new three segments.

4. Etc. 6.2.2 Uniformity The non-uniformity was measured using image pixel-by-pixel standard deviation as function of calibration level, which gives the spatial variation of the digital values. Each image was subdivided into a 16 by 16 matrix of contiguous subregions, each 64 by 64 pixels. The mean and standard deviation of pixel values in each subregion were computed, providing a distribution of a subregional intensity values. In addition for each,

41

the mean and standard deviation of intensity values for all subregion is computed. This mean of the mean values provides the global image mean intensity. The standard deviation of the distribution of mean intensity values for the subregions is a measure of the large-scale spatial non-uniformity of the corrected image when sampled and averaged every 64 pixels. We have corrected the non-uniformity of flat panel detector response by pixel-by-pixel piecewise linearization as follows:

1. Take the maximum exposure as the first correction point and correct it for non-uniformity.

2. Determine the standard deviation of the in step number 1 corrected image over

all exposures and get the index of the exposure that resulted in the maximum standard deviation.

3. Correct the detector response at the exposure represented by the index found

in step number 2 for non-uniformity by pixel-by-pixel piecewise linearization

4. Determine the standard deviation of the in step number 3 corrected image over all exposures and get the index of the exposure that resulted in the maximum standard deviation.

5. Correct the detector response at the exposure represented by the index found

in step number 4 for non-uniformity by pixel-by-pixel piecewise linearization 6. Etc.

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6.3 Beam hardening correction method Artifacts caused by beam hardening can be estimated and corrected using software correction (linearization) method. We have used the method of software linearization in correcting the cupping artifact caused by beam hardening. In the method of linearization, one transforms data values from the curved function to a value on the linear function for the same object thickness. Ideally, this method requires a homogenous object material in order to be corrected because a particular value of the linear attenuation coefficient obtained by poly-energetic source must correspond to a specific thickness. However, the data from a poly-energetic source can be best fitted with parametric equations. To investigate the attenuation error caused by the beam-hardening artifact, we acquired multiple sets of scan data for Perspex at multiple thicknesses. Perspex is a transparent plastic also known as Plexiglass or acrylic. Perspex of multiple slabs was positioned on the scanner table in a horizontal orientation. Horizontal slabs are removed from or added to the stack of slabs to vary the thickness of the Perspex. We have measured the x-ray attenuation of Perspex of different thicknesses. A relationship can be established between a detector response at a given exposure and Perspex attenuation length. A correction function, typically power function is then fitted to provide the corrected attenuation length from the raw detector response. The plot of the logarithm of the measured attenuation versus thickness is fitted to the power function as shown in figure 6.3. The exponent of the power function obtained from the fit is used to adjust attenuation measurements. Exponent equal to unity shows no correction. The power fit given in figure 6.3 gave us the power exponent of 0.87. Alternatively different values of the power exponents that can best remove cupping artifacts can be found experimentally.

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

350

400

450

log(Attenuation)

Thic

knes

s (m

m)

dataPower fit

Figure 6.3 Attenuation vs thickness of a Perspex data is fitted to power function

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6.4 Scatter estimation and correction methods Scatter-to-primary ratio (SPR) is an important measure of scatter, which indicates the ratio of scattered x-ray fluence to the primary x-ray fluence incident on the detector. A simpler method to estimate scatter is assuming that scatter is uniform. This method of estimating and correcting scatter is applied to the Cathphan®600 phantom scan. Cone beam CT scan of this phantom is taken with a collimator of 2cm and 10cm width. The scene of the scan of the phantom has got two distinct parts: the portion covered by the phantom and the uncovered portion that belongs to air scan. Portion that belongs to air scan Figure 6.4 The entire image consists of portion occupied the phantom, collimator and air. In order to estimate the magnitude of scatter by the phantom, a uniform scatter with an SPR value equal to 0.33 is attributed to the region occupied by the phantom. Scatter in this region is then assumed to be the product of the SPR value and the intensity registered there. The portion that belongs to air scan is assumed to produce no scatter since there is no object that scatters x-ray photons in this region. The scatter in the whole scene is then the area weighted average of scatter in these two regions.

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A potentially more accurate and direct method to estimate and correct scatter is by sampling scatter directly through out the projection images using the beam stop array techniques [22]. The probability of photoelectric interaction increases with increasing atomic number of the absorber (see section 3.1) that implies that lead with certain thickness absorbs virtually all x-rays incident upon it and thus can be used as beam stopper.

Arrays of beam stops such as lead disks are placed between the source and the object so as to determine the scatter component on a pixel-by-pixel basis. The standard beam stop array technique is given schematically in figure 6.5.

(a)

(b) FIGURE 6.5. The geometric configuration of the standard beam stop array technique approach to estimate x-ray scatter distribution. (a) Representation of the beam stop sampler array with a center-to-center spacing of 8.5 mm, length of 4.2 mm and diameter of 3.0 mm, (b) The beam stop array technique in a typical imaging configuration with distance of source to beam stop Dsb=196 mm, distance of source to object Dso= 1000 mm, and distance of source to detector Dsd=1536 mm. The lead discs block (absorb) primary photons and thus x-rays going straight cannot be detected behind the beam stops if the inconsequential transmission of the beam stops is neglected. Only the scattered radiations that are bouncing in all directions are detected in the shadows of these beam stops as illustrated in figure 6.6.

FIGURE 6.6. The radiation striking the detector in the shadow of the beam stop is virtually scattered radiation if the insignificant transmission of the beam stop is neglected. Radiation recorded on the detector surrounding the shadow includes both scattered and primary radiation.

45

Intensities at the shadow of beam stop array were assumed to be that of scatter. To estimate the scatter distribution over the whole projection image, scatter can be directly measured at the locations of the shadow of regularly spaced beam stops and interpolated into a smooth surface to generate a scatter-only image. Scatter image obtained by the technique of beam stop array above is a result of the assumption that all intensities registered at the shadow of the beam stop arrays are attributed to scatter. This assumption is based on the expectation that the lead discs from which the beam stop arrays are made absorb all x-ray photons incident upon them completely. In reality the lead discs up to certain thickness transmit some amount of x-ray photon fractions that are incident upon them. This transmission can be approximated theoretically from the length and diameter of the lead discs provided that the half value length for lead is known. Consequently the intensities registered at the shadows of the beam stop array belong not only to scatter but also to transmission of the lead discs and phantom itself. Therefore, scatter image obtained by the beam stop array technique above can be overestimated. The assumption was necessary as it is difficult due to scatter to measure the transmission of lead discs and phantom. Hence we had to resort to theoretical simulation that incorporates these transmissions to estimate the intensities of lead discs and phantom. Transmission of lead discs can be determined from intensities registered at the shadows of beam stop arrays when air is scanned with the beam stop arrays in place. Tbs = IBSA(air)/I(air) 6.1 Where Tbs is the fraction of transmission of beam stops and IBSA is intensity registered at the shadows of beam stop array in the scan of air with beam stops and I(air) is the intensity of air scan. We have found a typical mean value of Tbs = 0.12. The transmission of beam stops and phantom are not the only factors that affect the accurate estimation of scatter. Directly adapting the standard beam stop array technique for the correction of scatter in cone-beam CT requires two sets of X-ray cone-beam projections for each projection angle to accurately estimate scatter: one with a beam stop array for estimating scatter and the second without the beam stop array for reconstruction to obtain the scatter plus primary image. The x-ray photon intensity on the second acquisition is different from the first one due to among other things the warming up of the x-ray photon source and the absorption of plastics upon which the beam stop arrays are placed. As a result scatter intensities present in the second acquisition is not the same as the first since the intensity of the primary photon is changed. This fluctuation factor (f-factor) of the primary photon intensity should be taken into consideration if scatter is to be estimated accurately. The f-factor ƒ can be determined by comparing the intensities registered in the two acquisitions. It is equal to the ratio of scan of phantom with beam stop arrays in air (PWB(air)) to scan of phantom without beam stop arrays (PWoB(air)). PWoB(air) is the portion of the phantom scan that belongs to air.

ƒ = )()(

airPWoBairPWB 6.2

46

We have found a typical mean value of ƒ = 0.8934. Another factor that affects the precise estimation of scatter is the change of the scatter intensity on the first acquisition due to absorption by the lead discs. This change is assumed to be directly proportional to the area occupied by the beam stop arrays. The larger the area occupied by the beam stop arrays, the more the primary photons absorbed, and thus the less the scattered photons compared to the second acquisition. This area factor (A-factor) should also be compensated for. The A-factor can be determined by comparing the intensities registered from the only air scan and intensities registered from air scan with beam stop arrays. It is equal to the ratio of area occupied by all beam stop arrays to the total area of the scene.

A = t

bs

AA

6.3

Where A is the A-factor, Abs is the total area occupied by the beam stops and At is the total area of the scene. We have found a typical mean value of A = 0.8922. With all these factors included, the simulated scatter samples S at the shadows of the beam stop arrays is then equal to (see appendex 3 for the derivation):

S = ATT

bs

bs

ƒ− )1(W-PwB n 6.4

Where PwB is the scan of phantom with beam stop arrays in place and Wn is the intensity near the beam stop arrays. The following procedures were followed in measuring the effect of scatter on image quality. We have scanned the quality phantom using the imaging cone-beam CT module incorporated in medical linear accelerator. The acquisition is done with 5.5 frames per second in a time of 2 min. The flat-panel detector is partially displaced for the purpose of greater volume acquisition. Two types of collimator width, 10cm and 2cm, were used. The scan of the phantom with 10 cm width collimator is taken with and without beam stop array. Around 690 projection images of the phantom were acquired within one complete rotation of the gantry. The projection images acquired with the beam stop array are processed in order to estimate the amount of scatter present in the image. To that end, the locations (pixel coordinates) of the beam stop arrays are identified using the distinct intensity profile detected at their shadows and the method of thresholding.

47

Figure 6.7 image of the Catphan phantom acquired by the presence of beam stop array. Nine beam sarray shadows fall upon the phan

top

tom.

tensities at the shadow of the nine beam stop array shown in figure 6.7 were assumed to

ages.

s

t were

.8 on

ll the three methods mentioned above were used for scatter correction. The scatter the

atter

t

Inbe that of scatter. Scatter intensities at the shadows of beam stop arrays are measured and averaged over a diameter of 14 pixels (the lead discs used as beam stoppers have a diameter of about 25 pixels) to give the scatter samples over the whole projection imThese scatter samples are interpolated using two-dimensional linear interpolation to generate scatter only projection images. We have also interpolated the scatter sampleobtained by the method of simulation explained above (equation 6.4). Scatter-only projection images are then subtracted from the corresponding projection images thataken without the beam stop arrays to give the scatter corrected primary images. Two sets of scatter corrected projection images (by beam stop array technique and by the simulation methods) were obtained. These procedures are summarized in figure 6next page. Scatter is estimated only at the shadows of the nine beam stoppers that fall onthe phantom. The other beam stoppers are outside or on the edge of the phantom and thusdo not contribute much to the estimation of scatter that is caused by the phantom. For reconstruction purpose, the side values of the scatter samples are extrapolated to the respective four corners to cover the scene of 512 by 512. Acorrected primary images acquired by the 10cm collimator were reconstructed using standard one-dimensional weighted filtered back projection algorithm [15]. We have alsoacquired and reconstructed the same data by 2cm collimator. The scan of the same phantom with the fan beam CT was acquired and reconstructed for the purpose of reference. Data acquired by fan beam CT are almost scatter-free. The reconstructedimage was stored at a resolution of 400×400×104 (1mm isotropic voxel size). The scuncorrected and corrected reconstructed images of the phantom were investigated for cupping artifacts and compared. Quantification of the scatter uncorrected and correctedreconstructed images for CT numbers, contrast and noise were also compared. The resulof these procedures is given in chapter 7.

48

P+S image BSA image Local scatter estimate

-

=

- = age catter corrected image

Reconstructi n

s in scatter correction by the method of beam stop array technique. P+S

Interpolated scatter im S o Display Analysis, etc Figure 6.8 Steprefers to primary plus scatter image.

49

50

Chapter 7

esults

this chapter only the results are given. Detailed discussion of these results is the

.1 Optimization of detector performance results

.1.1 Linearization results

e measured a series of dose values with an ionization chamber at different tube current

R Insubject of the next chapter. 7 7 Wvalues. Figure 7.1 shows the measured flat-panel detector response as a function of dose. Our detector response data is fitted to a linear line by means of linear least square fit (red).

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dose (Centigray)

Res

pons

e

datalinear fit

Figure 7.1 Dose-response curve and linear fit to it for the flat-panel detector

he measured dose corresponds to the tube current values given in table 7.1.

able 7.1: Tube current (mA) used to detect the linearity and uniformity of the flat-panel

16 20 25 32 40 50 64 80 100 125 160

T Tdetector 10 12

51

At first glance it appears that the flat-panel detector response is overall perfectly linear.

igure 7.2 ce

data

e

is

ata

n in

able 7.2 Index that belong to new breakpoint is given in red 7.1

However, a closer look at the differences between the flat-panel detector response data and the linear fit to it at each point (residue) in figure 7.2 reveals that this appearance is only illusory especially at lower and higher exposures. The residue (difference between linear fit and response data of flat-panel detector) indicated in figure 7.2 shows that the flat-panel detector response deviates more from linearity at lower and higher exposures.

FThe differenbetween measuredand linear fit or residue revealing thnon-linearity of flat-panel detector. Error bar equal to one standard deviation below and above the d

We have linearized the flat-panel detector by the method of piecewise linearization according to the procedures given in section 6.2.1. The indices (or number of columtable 7.1) used as breakpoints obtained by the procedures of section 6.2.1 is given in table7.2. TNumber of breakpoints Index corresponding to values in table

1 0, 13 2 0, 2, 13 3 0, 2, 7, 13 4 0, 2, 7, 12, 13 5 0, 2, 7, 11, 12, 13 6 0, 2, 4, 7, 11, 12, 13 7 0, 2, 4, 5, 7, 11, 12, 13 8 0, 1, 2, 4, 5, 7, 11, 12, 13 9 0, 1, 2, 4, 5, 7, 9, 11, 12, 13 10 3 0, 1, 2, 4, 5, 6, 7, 9, 11, 12, 111 0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13 12 0, 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 13 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

0 20 40 60 80 100 120 140 160 180-8

-6

-4

-2

0

2

4

6

8

10x 10-3

Tube curent (mA)

Res

i=Y

data

-Yfit

52

For example index number “13” represents a tube current of 160 mA (maximum tube

the 0

ure 7.3 shows the accuracy of linearization in the form of root mean square (RMS)

the

owever, taking all the thirteen exposure measurement points as listed in table 7.1 f

de

.

that give us the best compromise between costly calculations and the errors made.

current) given in table 7.1. Determining the difference between detector response curveand a line through 0 and 13 gives index number 2 (12 mA) as the index with the maximum difference between measurement and fit results. Similarly determiningdifference between detector response curve and a curve represented by a line through and 2, and 2 and 13 gives index number 7 (40 mA) as the index with the maximum difference. The indices that give maximum difference between the detector responsecurve and a curve represented by linear segments in each step are given in red in table7.2. Indices that give maximum difference between measurement and fit results in eachstep are taken as breakpoints. Figerror as function of number of breakpoints. After taking all the exposure measurement points as a breakpoint, the flat-panel detector response is fully linearized for the measurement points listed here and the error is of course zero.

0 2 4 6 8 10 12 140

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Break points

RM

S E

rror

Figure 7.3. Error one can make in approximating the non-linear response curve by piecewise linear segments as function of the number of piecewise linear segments or number of breakpoints. Hbreakpoints, results in expensive calculation. It is possible to take only the number obreak points that result in the lower RMS error and affordable calculation. This is a traoff we considered seriously. After the first five breakpoints, the RMS errors approaches zero and the difference between the fifth, sixth, etc is insignificant as shown in figure 7.3Therefore, we can safely take just the first five breakpoints in the process of linearization

53

7.1.2 Uniformity results We ha e investigated the flv at-panel also for non-uniformity. The procedures listed in

ction 6.2.2 produce the results given in table 7.3. The indices give column number of

calibration point is given in red

ure 7.4 illustrate the s given in sec n 6.2.2.

sethe tube current listed in table 7.1. Table 7.3 Index that belong to new

Fig results obtained by the procedure tio

0 2 4 6 8 10 120

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

stan

dard

dev

iatio

n

Index of tube currents

2

3

4

5 6

7 8

9 10 11 12

Figure 7.4 Standard deviation of the flat-panel detector response as a measure of non-uniformity. Uniformity improved with the number of calibration points (given on

umber of calibration Index corresponding to values in table 7.1 Npoints 1 0, 13 2 0, 7, 13 3 0, 7, 10, 13 4 0, 7, 10, 12, 13 5 0, 2, 7, 10, 12, 13 6 0, 1, 2, 7, 11, 12, 13 7 0, 1, 2, 3, 7, 10, 12, 13 8 0, 1, 2, 3, 4, 7, 10, 12, 13 9 0, 1, 2, 3, 4, 5, 7, 10, 12, 13 10 , 13 0, 1, 2, 3, 4, 5, 6, 7, 10, 1211 2, 13 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 112 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13 13 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

respective lines)

54

The standard deviation curve value decreases with increasing correction (calibration) oints. Decrease in standard deviation implies that the detector response becomes more

rity are also volved in non-uniformity and vice versa. For this purpose, exposures that cause non-

t he

ar

of the flat-panel etector do also cause non-uniformity. To this end, exposures that cause non-linearity are

puniform. As in the case of linearity, taking only the first five correction points that do result in the most deviation is enough to correct for non-uniformity of the flat-panel detector because of the consideration of trade off between the error one can make in correcting non-uniformity and the costly calculations involved. After the first five calibration points, the errors approaches zero and the difference between the errors infifth, sixth, etc calibration points is insignificant as shown in figure 7.4. It is interesting to investigate whether the exposures that cause non-lineainuniformity are taken as breakpoints to linearize the flat-panel detector piecewise. We have exchanged the first five indices of Table 7.2 with the first five of Table 7.3 thaare not common to both tables. As the comparison of Table 7.2 and 7.3 indicates, tsame exposures that caused non-uniformity have also caused non-linearity. Taking the first five exposures that caused non-uniformity as breakpoints to linearize the non-lineflat-panel detector response curve results also in the same RMS error that obtained by taking the first five measurement points that caused non-linearity. These points correspond to exposures of 160, 12, 40, 125, and 100 mA respectively. We have also investigated whether the exposures that cause non-linearitydtaken as correction points for non-uniformity and the procedures of section 6.2.2 are applied to them. Those exposures that caused non-linearity have also caused non-uniformity of the flat-panel detector.

55

7.2 Beam hardening and scatter estimation and correction results

age quality is iscussed in section 5.3. Scatter causes streaks and cupping artifacts on the reconstructed

e.

gated

.2.1 Result of linear attenuation coefficient measurement

aken only for the uniform art of the phantom that comprises a diameter of 15 cm. Measurement values are

and the near attenuation coefficient of the reconstructed phantom image was investigated.

The detrimental effect of artifacts caused by scatter on cone-beam CT imdimage. Moreover, scatter reduces contrast and increases noise of the reconstructed imagTo investigate the effect of scatter on cone-beam CT image quality, we took scans of Catphan600 phantom described in section 6.1 and applied the methods discussed in section 6.4 in order to estimate and correct X-ray scatter. We measured the linear attenuation coefficient, contrast, and noise of the reconstructed Catphan600 phantomimage. The effect of scatter estimation and correction on these quantities is investiand the results are given below. 7 All analysis of the measured linear attenuation coefficient is tpnormalized at the edge of the uniform phantom cylinder of diameter 15 cm. The projection data of the phantom acquired by Fan-beam CT is reconstructed liFigure 7.5 gives the result.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1000

1200

Distance in the image plane (in pixels)

CT

num

ber (

in H

U)

With Fan beam CT

(a) (b)

Catphan600 phantom ata acquired by Fan-beam CT. (a) Cross sectional image of the phantom, (b) linear

Figure 7.5 The linear attenuation coefficients of reconstructed dattenuation coefficients (CT number) of the image across the diameter.

56

Figure 7.5 shows that the linear attenuation coefficient of the phantom image as acquired y Fan-beam CT is constant from edge to edge. This result confirms the expectation since

ntom was acquired by Cone-beam CT and a 2cm wide collimator to vestigate the image quality of the Cone-beam CT. The projection data of the phantom

ated

bmaterial and density or thickness of the selected parts of the phantom is uniform throughout. The same phainacquired by the Cone-beam CT and 2cm collimator was reconstructed and investigfor linear attenuation coefficient values across the cross its section. The result is given in figure 7.6.

0 50 100 150 200 250 300 350 4000

200

400

600

800

1000

1200

Distance in the image plane (in pixel)

CT

num

ber (

in H

U)

Cone beam with 2cm collimator

Figure 7.6 The linear attenuation coefficient of the original data of the reconstructed Catphan600 phantom acquired by cone-beam CT and a 2cm wide collimator.

e-beam T does not agree with the expectation. In spite of the fact that the reconstructed part of

r,

Unlike the Fan-beam CT data displayed in figure 7.5, the data acquired by ConCthe phantom is uniform in thickness and density, the linear attenuation coefficient valuefor the cone-beam CT data decreases as one advances from the edge of the phantom towards the center. This result gives a false impression that the phantom is less dense in the center than the edge. However, the cupping artifact due to beam hardening, scatteghosting, etc and not a true linear attenuation coefficient value should cause this. Since the data was acquired by a 2cm wide collimator and therefore almost no scatter should present in it, the most likely cause of the cupping artifact is beam hardening.

57

We have corrected the Cone-beam CT data for beam hardening artifact using the method

e linear attenuation coefficient of the original reconstructed Catphan600 hantom data (not corrected for beam hardening) acquired by cone-beam CT and a 2cm

he beam hardening correction has removed the cupping artifact and the values of the near attenuation coefficient of the uniform phantom is now uniform from edge to edge.

e

given in section 6.3 with the power exponent of 1.25. The result is given in figure 7.7 together with linear attenuation coefficient of Fan-beam CT for the purpose of comparison.

Figure 7.7 Th

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 00

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

D is t a n c e in t h e im a g e p la n e (p ix e l)

CT

num

ber (

HU

)

F a n b e a m2 c m , C B C TB H = 1 . 2 5

pwide collimator (red) and by the fan beam CT (blue). The result of beam hardening correction on the reconstructed Cone-beam data is given in green. TliThe correction of beam hardening has also made the linear attenuation coefficient of thCone-beam data similar to that of Fan-beam as indicated in figure 7.7. However there is some over correction near the edge of the phantom

58

To investigate the effect of scatter on the image quality of Cone-beam CT, we acquired

e Catphan600 phantom projection data of Cone-beam CT by 10cm wide collimator.

thThe linear attenuation coefficient or the CT number of data acquired by Cone-beam CT and by 2cm and 10cm collimator is compared in figure 7.8. The data acquired by 10cmhas more cupping than the data acquired with a 2cm wide collimator. Unlike the 2cm wide collimator, the 10cm wide collimator introduces a significant amount of scatter to the projection data since scatter increases with field of view.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1000

1200

Distance in the image plane (pixel)

CT

num

ber (

HU

)

Fan beam CT2cm CBCT10cm, CBCT

Figure 7.8 The linear attenuation coefficient of the original reconstructed Catphan600 hantom data acquired by cone-beam CT with a 2cm wide collimator (green) and by the

catter thus causes the extra cupping artifact displayed in figure 7.8 on the data acquired y the 10cm wide collimator. From figure 7.8 it is clear that for this test object the

p10cm wide collimator (red). Fan beam CT data is in blue. Sbcontribution of scatter to the cupping is moderate.

59

We have applied the method of assuming scatter as uniform and represent it with

PR=0.33, discussed in section 6.4, to the original data acquired by 10 cm wide Scollimator. The result is given in figure 7.9.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1000

1200

Distance in the image plane (pixel)

CT

num

ber (

HU

)

Fan beam CT10cm CBCTscatter (SPR=0.33)

Figure 7.9 The effect of scatter correction by assuming uniform scatter and

PR=0.33(black curve) on the original data acquired by Cone-beam CT and 10cm wide phantom.

igure 7.9 show that the simple scatter correction method has indeed improved the upping artifact. However, the cupping artifact is not removed completely. Possible

ng sent

Scollimator (red curve) of the linear attenuation coefficient of the Catphan600 Fcreasons can be that beam hardening might be responsible for the remaining of cuppiartifact or the simple scatter correction method has not removed the entire scatter prein the projection data.

60

To get rid of the remaining cupping artifact, we have included the beam hardening orrection method in conjunction with the simple scatter correction method. Figure 7.10

en for

igure 7.10 The effect of scatter correction by assuming uniform scatter and PR=0.33(black curve) on the original data acquired by Cone-beam CT and 10cm wide

phantom.

is possible to obtain similar results with a higher beam hardening correction and aller scatter correction.

cgives the result of such corrections. Fan beam data and the original data are also givcomparison.

0 50 100 150 200 250 300 350 400 450 5000

200

400

600

800

1000

1200

Dis tance in the im age plane (pixel)

CT

num

ber (

HU

) Fan beam CT10cm CBCTscatter (S PR=0.33)BH=1.1+SPR=0.33

FScollimator (red curve) of the linear attenuation coefficient of the Catphan600 Additional beam hardening correction (green curve) has almost removed all cupping artifact since it is almost same as the Fan-beam data (blue curve). Itsm

61

We have also applied the beam hardening correction method directly to the data acquired

y 10cm wide collimator. As it was the case with the data acquired 2cm wide collimator,

eam

cupping artifact is not completely removed after the maximum beam ardening correction with power exponent of 1.25 is applied. To remove the remaining

in r

bthe beam hardening correction method has removed the cupping artifact almost completely as shown in figure 7.11. This result shows that the cupping artifact caused by beam hardening is independent of the collimator width.

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 00

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

D is t a n c e in t h e im a g e p la n e (p ix e l )

CT

num

ber (

HU

)

F a n b e a m C T d a t a 1 0 c m , C B C T d a t aB H = 1 . 2 5 + S P R = 0B H = 1 . 2 5 + S P R = 0 . 1 0B H = 1 . 2 5 + S P R = 0 . 2 0B H = 1 . 2 5 + S P R = 0 . 3 3

Figure 7.11 Maximum possible beam hardening correction (black curve) is applied to CBCT data acquired by 10cm wide collimator (red curve). The effect of scatter correction by assuming a uniform scatter represented by different SPR values is appliedto remove the cupping artifact remained from the beam hardening correction. Bhardening corrections with power exponent of 1.25 and extra scatter correction by SPR=0.1 (green curve) produce the same CT number profile as that of Fan beam CT (Blue curve). However, the hcupping artifact, we varied the scatter correction from SPR=0 to SPR=0.33 as shownfigure 7.11. Beam hardening correction with power exponent of 1.25 and uniform scatterepresented with SPR=0.1 is the best combined correction that removed all cupping artifact (given by green curve in figure 7.11).

62

A more accurate and direct method of scatter correction is sampling it using the beam stop array technique given in section 6.4. Figure 7.12 shows the result of scatter correction by the technique of beam stop array.

0 50 100 150 200 250 300 350 400 450 500200

400

600

800

1000

1200

1400

Dis tance in the image plane (in pixel)

CT

num

ber (

in H

U)

Fan beam CTCBCT with 10cm col.Scatter (BSA)

Figure 7.12 The effect of scatter correction using the method of beam stop array technique (black curve) on the original data (red curve) of the linear attenuation coefficient of the Catphan600 phantom. Fan beam data (blue curve) is given along with for the comparison purpose. As illustrated in figure 7.12, improvement of the cupping artifact is clear. However, the improvement is not constant. There is still cupping artifact in the centre of the phantom and overcorrection on both sides of the middle point. This is a significant overcorrection in particular if one takes into account that beam hardening is not corrected here.

63

To get rid of the remaining cupping artifact and over correction, we have included factors

igure 7.13 The effect of scatter correction using simulation (green curve) on the

that are associated with the beam stop array technique in our correction method. Those factors are included using simulation method given in section 6.4. Scatter samples usedfor interpolation of scatter image are obtained purely by the mathematical formulas givenin appendix 3. Figure 7.13 shows the result of subtracting scatter image obtained by simulation. This time the scatter correction is more uniform and the central depressiondisplayed in figure 7.12 is improved.

0 50 100 150 200 250 300 350 400 450 500200

400

600

800

1000

1200

1400

D is tanc e in the im age p lane (in p ix e l)

CT

num

ber (

HU

) F an beam C TC B C T w ith 10c m c o l.S c a t te r(S im u la t ion )

Foriginal data (red curve) on the linear attenuation coefficient of the Catphan600 phantom. The scatter free Fan-beam data is also given (blue curve).

64

A comparison of the result of scatter correction with only the beam stop array technique

ray m.

ince scatter correction by beam stop method gives already an overcorrection, extra beam

he cupping artifact is quantified according to equation 5.9. The degree of cupping is t

able 7.4 on each side of the middle slice)

d al Original data Fan

and the simulation result is given in figure 7.14

Figure 7.14 The effect of scatter correction using simulation (black curve) on the original data (blue curve) and on method of scatter correction by the beam stop artechnique (green curve) on the linear attenuation coefficient of the Catphan600 phanto Shardening correction by on this scatter correction method is not useful. Tcalculated for the average of 40 pixels at the edge and center of the phantom. The effecof all scatter correction methods on cupping is given in table 7.4. The negative value shows overcorrection. TFor average of 11 slices (five slices Original Original data, Scatter Simulation Origin

data by 10cm

by 10cm and SPR=0.33

correctevia BSA

data data by 2cm

by 2cm and SPR=0.33

beamdata

Cupping -2.52 6.01 2.94 -1.00 4.24 4.04 0.51

0 50 100 150 200 250 300 350 400200

400

600

800

1000

1200

1400

Dis tanc e in the im age plane (in pix el)

Line

ar a

ttenu

atio

n co

effic

ient

(in

HU

)

O riginal c orrec ted by B S As im ulat ion data

65

7.2.2 Contrast, nois contrast-to ratio rement results

ontrast, noise and contrast-to-noise ratio of the scatter uncorrected original data and the

ontrast is measured according to equation 5.4 applied to the slice of the Catphan600

igure 7.15: reconstructed slice of

high-” is

re

s

able 7.5 on each side of the middle slice)

d al Original data Fan

e and -noise measu Cscatter corrected data by different methods for the 2cm and 10cm collimator acquisition was measured. The results of these measurements are given in table 7.5. C(CTP528), figure 7.15 where resolution inserts are visible as indicated in the figure.

Fthe Catphan600 phantom corresponding to CTP528 resolution module. “Backgroundthe region not occupied by the inserts whereas “object” refers to the largest insert as indicated. The average intensity of the central region whethe inserts are absent is taken as the background intensity for the inserts and this intensity is subtracted from the intensity of one of the inserts, designated in the above diagram a“object”.

TFor average of 11 slices (five slices Original Original data, Scatter Simulation Origin

data by 10cm

by 10cm and SPR=0.33

correctevia BSA

data data by 2cm

by 2cm and SPR=0.33

beamdata

Contrast 0.62 0.54 0.56 0.62 0.58 0.59 0.68 Noise 3.64 4.15 6.21 6.03 4.65 4.82 4.08 CNR 0.15 0.14 0.10 0.10 0.12 0.12 0.17

66

Chapter 8

iscussion

he difference between the flat-panel detector’s response curve and the linear fit to it es

ment

own

igure 7.5 (b) and figure 7.6 compare the linear attenuation coefficient of the selected

ed g

he acquisition of Cone-beam CT by a 10cm wide collimator has increased the cupping

D Tdisplays that flat–panel detector is non-linear and more so at lower and higher exposuras disclosed in figure 7.2. At higher exposures non-linearity is imputed to saturation. The flat-panel detector loses its linearity gradually as it nears the full saturation point. At lower exposures the flat-panel detector could be non-linear before it becomes fully sensitive. However, potentially there is also an impact of the accuracy of the currentcontrol of the x-ray unit. Accuracy in approximating the detector’s response by linearsegments is directly proportional to the number of the linear segments involved as displayed in figure 7.3. However, there is a trade off between taking all the measurepoints as breakpoints and the calculations involved. The more linear segments chosen in linearizing the detector, the more costly it is calculation wise. The non-uniformity of the flat-panel detector is evident from its response’s standard deviation. Correcting the measurement points that cause the most deviations can improve the uniformity as shin figure 7.4. As stated above, however, there is trade off between number of calibration points and the calculations involved. It is investigated whether the flat-panel detector is also non-uniform at the same exposures that resulted in non-linearity and vice versa. To check this, measurement points that cause non-uniformity are used as breakpoints for piecewise linearization. This is indeed the case for five and above calibration points asshown in table 7.2 and table 7.3. Fregion of the reconstructed Catphan600 acquired by Fan-beam and cone-beam CT. Thedisparity in the values of linear attenuation coefficient of the same material by these two acquisitions method is clear. The gradual decrease of the linear attenuation coefficient, asone progress from the edge to the centre, of the phantom acquired by cone-beam CT should be imputed to the cupping artifact since the density and thickness of the selectpart of the phantom is entirely uniform. The cupping artifact is caused by beam hardeninsince scatter is rejected from the data by 2cm wide collimator. Beam hardening correction has indeed removed the cupping artifact as indicated in figure 7.7. Tartifact (figure 7.8). The extra cupping artifact that emerged by using a 10cm collimator is imputable to scatter since scatter increase with a field of view. The simple scatter

67

correction by approximating it with a uniform value and SPR=0.33 has indeed improthe cupping artifact significantly as indicated in figure 7.9. However, the cupping artifact is not removed completely and thus there are other factors that might be responsible for the remaining cupping artifact or this method of scatter correctiondoes not remove scatter amply.

ved

method

xtra beam hardening correction with the power exponent of 1.1 on the data corrected for

he data acquired by 10cm wide collimator was corrected only for beam hardening. The

e

he technique of using beam stop arrays for scatter correction has yielded a better result

nd ct

on of

ore m

ly

ince it is difficult to measure some of these factors, their theoretical simulation is gure

,

ter even

Escatter by assuming a uniform scatter has almost removed the remaining cupping artifact as shown in figure 7.10. However, the cupping artifact was not removed entirely. Tmaximum beam hardening correction with power exponent of 1.25 has almost removed the cupping as shown in figure 7.11. Combination of maximum beam hardening correction and uniform scatter correction represented by SPR=0.1 has removed thcupping entirely. Tas shown in figure 7.12. Cupping artifact is removed completely except in the middle of the phantom where almost an abrupt depression in the values of linear attenuation coefficient is observed. This implies that scatter contribution is underestimated arouthe middle or there are other factors that cause cupping artifact there. The cupping artifathat caused by beam hardening is the primary candidate here. There are also other interesting factors that may have led to the underestimation of scatter. The acquisitiphantom with beam stop arrays for scatter estimation purpose was earlier than the acquisition of phantom alone. It is possible that the X-ray source has warmed up mand thus delivered more primary photons during the acquisition of phantom without beastop arrays. More primary photons implies more scatter than it has observed at the shadows of the beam stop arrays and thus under estimation. Another factor that may have led to scatter underestimation is the reduction of primary photons due to absorption by the lead discs that are used as beam stoppers. The reduction in primary photons is directproportional to the area occupied by the beam stop arrays. Less primary photons implies less scatter estimation. On both sides of the middle point the linear attenuation coefficienthas increased beyond the normal value. This entails that scatter is overcorrected on both sides of the middle of the phantom. Scatter overestimation suggests that there are signals at the shadow of the beam stop array that are not only of scatter. These signals could arisefrom transmission of the beam stop arrays and the phantom itself. It is also possible that there is more scatter during acquisition with the beam stop array than without it. Sinspired in order to incorporate them into this correction method. Figure 7.13 and fi7.14 shows the result of incorporating X-ray fluctuation factor, beam stoppers area factortransmission of the lead discs, and transmission of phantom in the simulation. The severe artifact in the middle of the phantom is removed significantly showing that the fluctuation and area factors have indeed played a role in underestimation of scatthough there are also other factors that might have caused the underestimation. However,

68

the overcorrection on both side of the middle of the phantom persists and thus the incorporation of transmissions in the simulation is has got no effect. May be the edthe beam stoppers has increased scatter that would not present in acquisition without the beam stoppers. Extra beam hardening will only make the overcorrection more serious.

ge of

or the purpose of comparison, the result of scatter correction by all the three forementioned scatter correction methods and beam hardening correction method,

depicted in

er exponent according to the method of section 6.3 is equal to 0.89. owever this resulted in under correction. A power exponent of 1.25 is good enough as

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 00

2 0 0

4 0 0

6 0 0

8 0 0

1 0 0 0

1 2 0 0

D is t a n c e in t h e im a g e p la n e (p ix e l )

CT

num

ber (

HU

) F a n b e a m C T d a t aC B C T , 1 0 c mS c a t t e r (S P R = 0 . 3 3 )S c a t t e r (B S A )S c a t t e r (S im u la t io n )S P R = 0 . 1 + B H = 1 . 2 5S P R = 0 . 3 3 + B H = 1 . 1

Fatogether with the original data acquired by a 10 cm collimator and fan beam is the above figure. The measured powHshown in figure 7.7 to remove cupping from the data acquired by 2cm wide collimator where scatter is almost absent. In case of the data acquired by 10cm, an extra scatter correction by the simple uniform assumption represented by 0.1 and beam hardening correction with the power exponent of 1.25 as indicated in figure 7.11 is also good enough to remove all the cupping.

69

Both the beam hardening and scatter correction methods adjust the attenuation measurements. Both beam hardening and scatter cause cupping artifact almost in a

milar manner. It is difficult to draw a line between cupping artifact caused by beam e as to

y

. have improved contrast. The degree of

provement, however, depends on the method used for scatter correction. Contrast is

ise. ver,

e increase of noise is not surprising since a lot of calculations and interpolations are on

mal ever, directly adapting the

eam stop array technique for the correction of scatter in cone-beam CT requires two sets ter:

y

sed by

chnique needs to be used with another technique or modified so that the extra exposure

sihardening and the one caused x-ray scatter. Therefore, it is not easy to determinwhich correction method should be applied for the correction of particular cupping artifact. Probably it is better to use both beam hardening and scatter corrections flexibltogether until the cupping artifact is removed. Scatter produces not only cupping artifact but also decreases contrast and increases noiseAll the scatter correction methods applied hereimincreased by approximately 5%, 16%, 15% upon scatter correction by the method of simple uniform scatter estimation, beam stop array technique, and simulation respectively. Contrast of the fan beam data is 26% greater than the cone beam data. According to equation 5.8, scatter correction would have led to reduction in noContrary to the expectation, all scatter correction methods have incurred noise. Howethinvolved in the scatter estimation methods. Especially the method of scatter correctiusing beam stop array technique has incurred noise the most. Lots of calculations including interpolation are involved in this method. All these calculations and interpolations might be responsible for the increased noise. The technique of beam stop array is very useful in estimating scatter before norpatient scan for cone-beam CT reconstruction is taken. Howbof X-ray cone-beam projections for each projection angle to accurately estimate scatone with a beam stop array for estimating scatter and another without the beam stop arrato obtain the scatter plus primary image. This will not be practical because it results in double volume scanning time and double total patient x-ray exposure and thus significantly increases patient dose in cone-beam CT as well as data acquisition time. To solve the problem of double exposure and double acquisition time that is caudirectly using the beam stop array technique for scatter sampling, the beam stop array tecan be reduced or eliminated. Some of the modifications researchers employed include:using beam stop array technique together with image sequence processing [23]; using moving beam stop array [24]; and using the collimator leaves as beam stoppers [25].

70

Chapter 9

or is less linear at lower and higher exposures. This non-linearity is e method of piecewise linearization. Taking all the measurement points

costly calculation wise. The first five measurement points that resulted in the most non-een

between the rror one can make in correcting non-uniformity and costly calculations are investigated.

tent. pping artifact is reduced and contrast is improved. However, all the

atter correction methods have also incurred noise. Noise has increased the most in the

s

form.

Conclusion Flat-panel detectcorrected using thislinearity of the detector and thus taken as breakpoints give the optimum trade off betwcostly calculation and the error one can make in the linearization process. Flat-panel detector is also not perfectly uniform. The non-uniformity is corrected by calibrating measurements that give the largest standard deviation. Trade offeThe same five points that give optimum trade off for linearization give also optimum trade off for non-uniformity correction and vice versa. Beam hardening correction by the method of power function fit has improved the cupping artifact. All the scatter correction methods employed here have reduced scatter to certain exConsequently, cusccase of scatter correction using beam stop array technique. Scatter assumption by constant SPR has less effect in scatter correction compared with the other two methodbut it corrected cupping artifact uniformly. Beam stop array has removed the cupping artifact. However, scatter correction by the technique of beam stop array is not uni

71

There is overcorrection on both sides of the middle of the phantom and underestimation of scatter at the middle of the phantom. The latter is corrected by the method of simulation that included the following. The X-ray source fluctuation and the area occupied by the beam stop arrays playimportant role in estimating and correcti

ed an ng scatter. Incorporation of these factors through

mulation into the correction method has significantly improved the estimation and

to

ologically significant decisions. From the results f this thesis, combining the beam hardening and scatter correction by searching the right

e

sicorrection of scatter by the technique of beam stop array. Underestimation of scatter bythe technique of beam stop array is corrected when these factors are included to it through simulation. However, the inclusion of phantom and lead disc transmissions inthe correction method has got no effect. The cupping artifact caused by beam hardening and scatter deteriorate image quality severely hampering the judgment of pathoSPR value for scatter and power fit exponent for beam hardening experimentally is thbest way to get rid of the cupping artifact. The method of beam stop array is potentially accurate way of estimating scatter. Further investigation can improve the overcorrection.

72

Chapter 10

ppendix

central slice theorem

rojection as defined in section 4.2:

)cosθ

ryxyxf )sincos(),( θθδ

Taking the one-dimensional Fourier transform of the projection, we get: P(θ,ρ) = F1D(r){p(r,θ)} = ∫∫∫ ƒ(x,y)δ(xcosθ + ysinθ - r)exp(-i2πρr)dxdydr

A1.1

A Appendix 1: the P p(r, θ) = ∫ +− srsrf sin,sincos( θθθ dl

L

= y ∫ ∫∞

∞−

∞−

−+ dxd

= ∫∫ ƒ(x,y)exp(-i2πρ( xcosθ + ysinθ ))dxdy = ∫∫ ƒ(x,y)exp(-i2π(ρcosθx + ρsinθy ))dxdy The two-dimensional Fourier transform of f(x,y) is: F(u,v) = ∫∫ f(x,y)exp(-i2π(ux +vy))dxdy A1.2

73

Observing that (u,v) in polar coordinates is (ρcosθ, ρsinθ), we can see that:

ppendix 2: The backprojection alogorithm for parallel beam reconstruction

athematically the backprojection of a single measured projection along an unkown density is iven by:

θ(x,y) = ∫ pθ(r) δ(xcosθ + ysinθ - r)dr A2.1

,y) is the backprojected density due to the projection pθ(x,y). The total backprojected age, also called laminogram, is the integral (sum) of this over all angles:

(x,y) = bθ(x,y)dθ

δ(xcosθ + ysinθ - r)drdθ A2.2

Applying the CST to substitute the inverse transform of P(θ,ρ), p (r) is given by:

pθ(r) = F-11D,ρ{ P(θ,ρ)} = P(θ,ρ)exp(i2πρr)dρ A2.3

Then the backprojected image can be written as:

fb(x,y) = ∞ π

P(θ,ρ)δ(xcosθ + ysinθ - r) exp(i2πρr)drdθdρ

P A2.4

Let us compare this equation with the 2D inverse Fourier transform formulation in polar

A2.5

Equation A2.4 can be modified to conform to equation A2.5 with the following changes:

P(θ,ρ) = F(u,v) = F(θ,ρ) A1.3 A Mg b Where bθ(xim

∫π

fb

0

= ∫ p∫∞

∞−

π

0

θ(r)

θ

∫∞

∞−

∫∫∫∞

∞−∞− 0

= (θ,ρ)exp(i2πρ(xcosθ + ysinθ)dθdρ ∫∫∞

∞−

π

0

coordinates:

ƒ(x,y) = ∫∫ F(ρ,θ)exp(i2πρ(xcosθ + ysinθ)ρdρdθ ∞

0

2

0

π

74

a. Limits of integration should be (0, 2π) and (0, ∞) and eed ρ

We can address the first issue by recognizing that F(-ρ,θ)=F(ρ,θ+π) and we can address the

b. We n dρdθ for an integration in polar coordinates

second issue by multiplying and dividing by ρ.

fb(x,y) = ρ00

θρ ),(F∫∫ exp(i2πρ(xcosθ +ysinθ)ρdρdθ

π2 ∞

= F 12−

D {ρ

θρ ),(F }

ƒ(x,y)** F 12−

D {ρ1

= }

yr1

= ƒ(x, )** A2.6

s ni convolution. Practically this means that the actual image ƒ(x,y) is blurred by the 1/r term.

PTph + S = Pw/oB A3.1

ƒ(PT + AS) = W A3.3

ubtract equation A4.2 from A4.3:

T ) = W – PwB A3.4

Where the double asterisk (**) ig fies

Appendix 3: simulation of scatter sample at the shadows of beam stop arrays (description of the symbols is given below)

ƒ(PTphTbs + AS) = PwB A3.2 ph n S ƒPTph (1- bs n

ƒPTph = bsT-1

PwB -Wn A3.5

ion A4.5 gives: Equation A4.2 into equat

Tbs-1 Pw -Wn

S = PwB A3.6 B

Tbs + ƒA

he simulated scatter is then equal to: T

S = ATTbsn A3.7

bs ƒ− )1(W-PwB

75

ƒ can be determined from air scan as:

ƒ =

oBPwPwB

/ A3.8

rimary pTph=Transmission of phantom

er Intensity of phantom without beam stop

here W

P = Intensity of the p hotons

S=ScattPw/oB=ƒ=Source fluctuation factor Tbs=Transmission of beam stops A=Area factor PwB=Intensity of phantom with beam stop W =Intensity values near the beam stops n

76

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