università degli studi di napoli federico ii patient lies prone on a table with a cutout for the...

Università degli Studi di Napoli Federico II

Scuola Politecnica e delle Scienze di Base

Area didattica di Scienze Matematiche, Fisiche e Naturali

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica

TESI DI LAUREA SPERIMENTALE IN FISICA MEDICA

Monte Carlo simulations for breast computed tomography

with synchrotron radiation

Relatori Candidato:

Prof. Paolo Russo Giulio Richichi

Prof. Giovanni Mettivier matr. N94/191

Anno Accademico 2013/2014

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Contents

Introduction ...................................................................................................... 3

1. Breast Computed Tomography ........................................................... 6

1.1 Breast Computed Tomography ............................................................... 6

1.2 Cone Beam Breast CT ................................................................................ 9

1.3 Phase Contrast Breast imaging with Synchrotron Radiation ............ 13

1.3.1. SYRMA-CT Project ........................................................................... 14

1.3.2 Phase contrast X-ray imaging .......................................................... 17

2. Monte Carlo simulations and Geant4 ..................................................... 20

2.1 Monte Carlo methods .................................................................................. 20

2.2 Geant4 simulation toolkit ............................................................................ 23

2.3 Geant4 simulations for SR BCT .................................................................. 26

3. Results of the simulations ......................................................................... 31

3.1 Validation of the code .................................................................................. 31

3.1.1 CTDI measurements.......................................................................... 31

3.1.2 CTDI simulations ............................................................................... 32

3.2 Results of the simulations............................................................................ 37

3.2.1 Dose Spread Functions ..................................................................... 37

3.2.2 Cumulative and equilibrium dose .................................................. 41

3.2.3 Dose radial profiles ........................................................................... 44

3.2.4 DgN ..................................................................................................... 47

3.2.5 Compton multiplicity........................................................................ 53

3.2.6 Fluence at detector surface ............................................................... 55

3.2.7 Comparison with the literature ....................................................... 61

Conclusions ..................................................................................................... 65

Appendix A. Code user’s guide .................................................................. 67

References ....................................................................................................... 77

3

Introduction

The aim of this thesis is to perform Monte Carlo simulations to characterize

the main dosimetric aspects of synchrotron radiation breast CT. This work

is in the framework of the SYRMA-CT project, funded by INFN, based at

Elettra Sincrotrone (Trieste, Italy), which is the first project aimed at

investigating the use of synchrotron radiation for computed tomography of

the breast with phase-contrast imaging techniques. To date, mammography

is the gold standard for breast imaging. In the last 20 years, it has

enormously lowered mortality for breast cancer, but it has some limitations.

First, it is a 2-D projection of 3-D structures. Thus, most important

anatomical details may not be well resolved because of the noise caused by

normal anatomical structures. Moreover, it is limited by the tumour size it

can detect; it can be a painful exam for women. So, in the years,

experimental 3-D techniques have been developed, like cone beam breast

CT, which is currently being studied in a few research institutes, including

the Department of Physics at Federico II University. In general, breast CT

as an experimental technique has potential to represent the future for

mammographic exams because of its full 3-D capabilities, but a lot of

aspects, like the dose issue and the resolution of the images acquired, have

to be examined.

Mammography with synchrotron radiation has already been

performed in 2000s with the SYRMA project; the first encouraging studies

on tomography in 2004 were at the basis of the SYRMA-CT project, which

started in 2014 and will end in 2016 with the examination of the first patient.

Before this, it is necessary to develop the dosimetry protocol and to study

the best imaging and reconstruction techniques to ensure a good diagnosis.

The dosimetric protocol is based on two dose indexes: CTDI and

MGD. CTDI is the computed tomography dose index, used in CT exams

since ‘70s for characterizing the output of an X-ray scanner. MGD is the

mean glandular dose, the mean dose that the glandular fraction of the breast

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absorbs during a mammographic exam. Since breast CT is a novel exam,

still at the experimental stage, which has characteristic of both CT and

mammography, these indexes could be linked in some way. The main issue

here is the fact that CTDI can be experimentally measured, while MGD can

be obtained only by means of computer simulations: it is not possible to

perform an in vivo dose measurement, without using invasive methods. To

evaluate MGD in a CT scan with synchrotron radiation, a Monte Carlo code

with Geant4 vers. 10.0 has been developed in this thesis work. The

experimental setup of SYRMA-CT experiment has been simulated in order

to link CTDI and MGD by means of DgN, the dose coefficients used in

mammography. As a second approximation, the code also permits to

evaluate dose distributions in breast phantoms. In fact, at present,

radiological risk models are based on MGD, which is an integral value;

having a detailed dose distribution within the breast could help to improve

these models, by taking into account also higher orders of the frequency

distribution of absorbed dose values in the glandular mass of the breast.

This work is organized as follows: in the first chapter,

mammography with its pros and cons is presented and then the

experimental CT techniques for imaging of the breast is illustrated. In

particular, cone beam breast CT and synchrotron radiation breast CT are

presented. The chapter ends with the description of the SYRMA-CT project

and phase-contrast imaging techniques. The second chapter deals with

Monte Carlo techniques and simulations. After a general introduction on

Monte Carlo, Geant4 is presented with particular attention to the most

important concepts useful to implement a simulation. Then, it is described

how SYRMA-CT experimental setup is described in Geant4 simulations.

The third chapter presents the results of the simulation, in particular code

validation and dose distributions. These results are compared to those

found in literature and the contribute of these results for the continuation

of the research activity are discussed. An appendix describes the code and

how to use it to carry out a simulation.

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1. Breast Computed Tomography

1.1 Breast Computed Tomography

Breast cancer is the most common form of cancer in the woman,

representing 30% of all cancers diagnosed among them. In USA, it causes

about 40000 deaths per year [1], and statistics say that almost one woman

on eight will develop breast cancer in the whole life [2]. There is no known

cause of it and so in the years many efforts have been accomplished in order

to reduce mortality, both with early detection and treatments. In the last 20

years, mortality has reduced by about 30% also thanks to screening with X-

ray mammography. First mammographic exams, in 1950s and 1960s,

produced images with poor diagnostic information; in 1970s,

mammography with xeroradiographic processes became very popular but

it was dismissed because of the high radiation dose and the poor image

quality. Later, screen-film imaging replaced Xerox processes and now,

thanks to continuous improvements in technology, the use of digital

mammography is very diffuse because of its greater accuracy and the

possibility to process data post acquisition [2]. Now, mammography is the

standard for diagnosing breast cancer.

Fig. 1.1: Relationship between 15-year-survival rate (%) and tumor size [3]. It underscores the importance of developing screening devices that can detect smaller tumors: a new imaging technology that could reliably detect breast tumors 5 mm in diameter would increase survival rates by 8%–10%.

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Despite the progresses, mammography lacks sensitivity, in particular if

breast tissue is too dense. Moreover, not all malignant lesions are seen;

almost all biopsies performed because a suspicious lesion detected with this

exam turn out to be negative. It would be desirable to detect the tumour at

an early stage; mammography is limited respect the tumour size that can be

detected [4]. Moreover, it is a 2D image, which results from the

superposition of 3D structures on a plane, and so it combines useful

diagnostic information with anatomical structures of no importance,

reducing lesion visibility; it is the so-called “anatomical noise”. Thus, a

number of technologies are currently being investigated in order to reduce

mortality improving breast cancer detectability. Between these, there are

PET, SPECT and MRI. This last one is very interesting because of its full 3-

D capability, but it is not so good at detecting microcalcifications.

Tomosynthesis, or limited-angle tomography, is another candidate, which

could be easily implemented by a simple upgrade of a digital

mammography system, but it is not really a 3-D imaging modality.

Moreover, it causes a very large blurring.

Fig. 1.2: Standard imaging techniques for the breast: from the left to the right, conventional mammography, tomosynthesis and MRI.

Computed tomography is probably one of the best candidate to substitute

conventional mammography : 3-D imaging is ensured by principle and can

eliminate the above-mentioned superposition of tissue; it can provide

adequate soft-tissue differentiation, which is often important, and it has

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been also demonstrated that contrast-enhanced imaging is feasible. The

greatest challenges in developing a CT dedicated to the breast are the costs,

patient comfort, dose radiation and image quality. In fact, mammography

is performed using a compression pad: to improve image quality, the

patient’s breast is compressed in order to have a size of about 4-5 cm, and

this results to be very painful for the women. Raiation dose is another

important question, because the AGD (Average Glandular Dose) has to be

less than the limit of 2x2.5 mGy indicated in Europe for a two-view digital

mammographic screening exam. It is also fundamental to detect

microcalcifications at a resolution similar to the one in mammography, i.e.

100 µm, and soft tissue lesions in the order of the mm.

First attempts to use a CT completely dedicated to the detection of

breast cancer were done in 1975 in Rochester, at Mayo Clinic, with the so

called CT/M scanner. It used a fan beam geometry to acquire 1 cm CT slices

in 10 s, typically performing the scanning with 120 kVp and 20 mA. The

mid-breast dose for an acquisition of six slices was about 1.75 mGy, and the

reconstruction was done with voxels of 1.56 mm size: poor spatial

resolution, high costs and the need to use an iodine contrast infusion made

the authors conclude that the use of CT for the screening of the breast was

undesirable.

There were also attempts to use whole-body scanners for breast

imaging, but they proved to have slightly worse diagnostic accuracy than

the dedicated scanner and greater dose to the breast: Miyake measured a

breast dose of 23.5 mGy and other researchers reported doses in the range

between 19 and 25 mGy: ten times the one of a conventional

mammography. Actually, the whole thorax has to be penetrated by X-rays

and so there is a general dose inefficiency. It has also to be said that Seo

found that reducing dose resulted in a minimal penalty in diagnostic

accuracy. The general conclusion is that CT has excellent sensitivity and so

more efforts have to be made to improve this technique.

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1.2 Cone Beam Breast CT

Whole-body CT could be useful to show possible metastases, as well as the

primary cancer. Moreover, such types of scanners are already available and

so no further cost for installation is needed. The main problems that concern

the whole-body CT are the high radiation dose, due to the greater

penetrability that X-rays must have to penetrate the entire thorax, and the

reduce image quality due to attenuation of non-breast tissues.

Fig. 1.3: Illustration of the geometry of a breast CT scanner.

Therefore, in the last years researchers have been developing CT systems

entirely dedicated to the imaging of the breast. In the basic design, the

patient lies prone on a table with a cutout for the breast (fig. 1.3). Only a

breast is imaged at a time and no compression is needed, thus improving

the comfort during the exam. Under the table, the X-ray source and the

detector rotate around the breast in order to acquire cone beam projection

images. One of the leaders in this field is John Boone with his group at

University of California, Davis; another group, which is dedicating its

efforts into developing breast CT, is the one of Prof. R. Ning, at Rochester

University. They are still investigating the feasibility of such an exam, and

there are many issues that need more investigation, like radiation dose and

image quality.

Among the advantages of a dedicated breast CT scanner with

respect to whole-body CT, there is the possibility to minimize the exposure

to thorax, and a better spatial resolution. This can help the diagnosis by

permitting the visualization of microcalcifications and of tumour margins,

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so improving the early detection of malignant lesions: it has been estimated

that 29-48% of nonpalpable carcinomas can be detected only by the basis of

microcalcifications. Using CT, the disadvantages of tissue superposition in

mammography could be overcome; this may allow a better visualization of

tumour margin which, if irregular, are often a malignancy signal .

Another advantage of a breast CT scanner is that less tissue has to

be penetrated, so lower energies can be used; since tumour contrast is

higher at lower energies, a dedicated breast CT scanner could provide

higher contrast and SNR than with the conventional CT used for the breast.

Since ductal and glandular tissue can extend out of the breast, in

the axilla, chest wall or abdomen, a breast CT scanner is needed to image

very close to the chest wall. In traditional mammography, compression

helps to pull the breast away from the chest wall, and in BCT gravity makes

something similar, but it is still a challenging problem. To do this, the

distance between the X-ray focal spot and the bottom of the table has to be

minimized; another way could be the use of complex non-circular orbits

during the scan. Moreover, the scans have to be very fast in order to

minimize the effects of patient motion: on one hand, breathe holding is

needed, on the other, the use of slip-ring gantries , already used in most

conventional CT scanners, could be a great advantage.

X-ray tubes used in breast CT must be small and powerful enough

to allow the collection of many projections in a few seconds. Small end

windowed X-ray tubes are suitable for this scope. They also have the focal

spot very near to the end of the tube, and this can facilitate the proximity of

the tube to the bottom of the table. As McKinley et al. showed , using a

tungsten anode X-ray tube with kVp settings between 50-70 kV could

produce an optimal SNR per dose.

As regards the detector, currently all the systems use indirect flat

panels using CsI:Tl scintillator coupled to TFT switches and photodiodes

based on a-Si [4]. One of the most important issues is to determine the

optimal voxel size to reach a tradeoff between resolution and noise, in

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particular for microcalcification detectability and tumour margin visibility.

A great challenge in cone beam breast CT is the reduction of the scattered

radiation on the detector. In fact, scatter reduces contrast in the

reconstructed image and cause cupping artifacts in it. Kwan et al. and Liu

et al. have studied the effects of scatter in flat-panel breast CT systems, and

they found that SPR varies with several factors, like kVp settings, breast

dimensions, the presence of an air gap or an antiscatter grid, and can be

greater than 1; also detector height can affect it. Antiscatter grids, however,

reduce scattered radiation but also primary photons, with greater noise in

the reconstructed image.

The reconstruction algorithms are similar to the ones used in whole-

body CT, with the difference that the half-cone angle is larger. Among them,

we mention the filtered back projection and the iterative solutions; these are

computationally demanding but may perform better in terms of image

quality at a given dose value.

A very important question to consider is the radiation dose to the

breast in a breast CT scan. The mean glandular dose (MGD) is an estimate

of the mean dose imparted to the glandular fraction of the breast tissue,

useful to evaluate the risks connected to the exam. Knowledge of the MGD

is important for optimizing scanner design and imaging acquisition

parameters. First prototypes imparted a very high dose, and using total-

body CT dose cannot be diminished because X-rays have to pass through

the whole thorax. This prevents a good imaging of the breast, due to the

attenuation of non-breast tissues, which receive a high and useless dose

radiation. Monte Carlo simulations, performed by Boone [5] and Thacker

[6], showed that the dose distribution through the breast is more

homogeneous with cone beam breast CT than the one reported for

conventional mammography (fig. 1.4). Most research groups are trying to

limiting the MGD to the one given in a typical two-view mammography

exam.

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Fig. 1.4: Distributions of glandular dose as functions of position for both mammography and breast CT.

In the development of breast CT, it is very important to verify the diagnostic

performance. Several approaches can be adopted: phantom studies,

computer simulations and clinical trials, imaging both symptomatic and

asymptomatic patients. One would evaluate radiologist performance for a

specified task using images acquired with different techniques, but it is not

always a practical way. Chen et al. and Vedula et al. simulated the

geometry of cone beam breast CT and in particular the flat-panel detector.

Simulations are the only way to explore the effects of radiation dose and its

distribution in the patient. However, simulations rely on a realistic

modeling of the breast, the lesions, the X-ray source and the detector, which

is not always possible: breast is usually modeled as a homogeneous cylinder

or semiellipsoid, and only a few software can describe accurately breast

anatomy. The other approach requires the use of physical phantoms with

the prototypes of scanners. They can be made of breast-tissue equivalent

materials, i.e. materials that mimic as well as possible breast tissue X-ray

attenuation and density. They can be used to evaluate MTF, NPS and other

important parameters for image quality, as well as for the dose radiation if

combined with an ionization chamber. Some of these materials can be

water, PMMA or epoxy resin.

The imperative for further developments of this technology is the

detection of breast cancer at earlier stages, before it has the possibility to

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metastasize, in order to reduce the lethality of this disease. Computer

simulations suggest that lesions of the size between 3 and 5 mm could be

seen with high resolution with a flat-panel breast CT system, but ultimate

answer will come from clinical testing of these systems. Another important

issue is the detection of microcalcifications, but in this case computer

simulations suggest that conventional mammography is better, to date.

Anyway, it is likely that the great contrast of calcium will make

microcalcifications of 200 µm diameter visible, if the pixels of the detector

are small enough. It has great importance the reduction of discomfort that

most women observe with conventional mammography and breast

compression. This could encourage women to undergo regular screening of

the breast. Another great challenge is the comparison between dedicated

breast CT and other imaging techniques like MRI and tomosynthesis.

Tomosynthesis has good qualities like fast scan time and low cost, and it is

available commercially, but further studies are necessary in order to

compare image quality and diagnostic accuracy of these techniques.

1.3 Phase Contrast Breast imaging with Synchrotron Radiation

As previously mentioned, even if mammography is the standard for

diagnosing breast cancer at an early stage, the reported positive predictive

value of screen-film mammography or DM is usually in the order of 80%

and so further studies necessary to help reducing the false-positive rate.

Mammography with synchrotron radiation (SR) is an innovative technique.

It uses the monochromatic, tunable and laminar radiation generated by a

synchrotron, and because of its spatial coherence it permits to employ phase

contrast imaging techniques. Synchrotrons can be considered, in this sense,

the evolution of Coolidge tubes, providing very high flux X-ray (laminar)

beams. The high brilliance allows the use of monochromators, thus

avoiding the usual beam hardening of polychromatic beams while passing

through the matter. Moreover, if source-to-object distance is large enough,

14

the focal spot can be considered pointlike. Therefore, such a kind of source

has the spatial coherence necessary for phase contrast imaging.

1.3.1. SYRMA-CT Project

SYRMA (Synchrotron Radiation Mammography) is the first project which has

studied the feasibility of breast imaging with a synchrotron radiation

source. It is based in Trieste, Italy, at Elettra facility . It hosts a storage ring

for electrons of 2-2.4 GeV of 259.2 m diameter; the beam lines, like SYRMA,

are located in correspondence of the curve sections of the machine. The X-

ray beam has tunable energy in the range 8.5-35 keV, laminar shape and

flux up to 108 photons/(s.mm2). The beam transfer along the line (fig. 1.5) is

performed in several steps:

-The beam passes along the optical hutch, where monochromatization and

optimization of the beam happen;

-then, there is the experimental hutch, used in experimental mode, where also

the dosimetric system is placed;

-the radiological room, where mammography and tomography are

performed;

-the medical examination room, where the console for data acquisition is

located.

Fig. 1.5: Elettra schematic beamline representation.

The original project has been presented in 1991; early in vitro studies (2000-

2003) showed the great potential of SR phase contrast imaging; the spatial

coherence is ensured by the distance (25 m) between the source and the

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object to irradiate, and the object-to-image distance (2 m) is big enough to

observe phase effects. The great advantage is the possibility to obtain a high

quality image with a great reduction of radiation dose to the patient [7].

After the in vitro studies, soon the in vivo studies on patients started and in

2006-2007 49 patients were scanned [8]. SR mammography was performed

moving at the same time the breast and the detector in a vertical scan; this

permitted to reject scatter photons on the detector, a screen-film system,

with a soft breast compression: thus, image quality and patient comfort are

improved. Preliminary studies of SR breast CT were conducted in 2004 [9].

They scanned a 4 mm slice of post-mortem excision breast, and it came out

that the MGD was of about 0.8 mGy (fig. 1.6). These encouraging results led

to the SYRMA-CT project, funded by INFN, which started in 2014 with the

collaboration of Trieste, Bologna, Cagliari, Ferrara, Pisa, Napoli and Sassari

Universities, and will end in 2016, when first clinical studies will take place.

The main intermediate objectives are the dosimetric characterization and

the study of the best reconstruction algorithm to achieve good image

quality. The detector used is Pixirad, developed by INFN Pisa and a related

spinoff company, which is a pixel detector with single photon counting

capability and whose pixel pitch is 60 µm.

Fig. 1.6: Tomogram of a frozen breast, obtained from post mortem excision. Beam energy is 28 keV; MGD is 0.8 mGy.

The radiation dose workpackage, under the responsibility of the INFN

Naples' group, refers to the dose evaluation procedure on the basis of the

beamline dosimetry system, which is composed of a system of two

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ionization chambers placed at 27 m from the light source and 3 m away

from the patient. Their function is to monitor the beam flux, the radiation

dose to the patient and to signal possible anomalies in order to stop the

examination. Since SR breast CT is an experimental technique, it is

necessary to establish a dosimetric protocol. Two parameters will be

evaluated: the Mean Glandular Dose (MGD) and CTDI (Computed

Tomography Dose Index). MGD will be calculated with the help of a Monte

Carlo code, using the formula [10]:

𝑀𝐺𝐷 = 𝐷𝑔𝑁 ⋅ 𝐾

𝐷𝑔𝑁 = 𝑓(𝐸) ⋅ 𝐸 ⋅ 1.6021 ⋅ 10−8 ⋅ Φ(𝐸) ⋅ 𝐺 ⋅𝐴

𝑀

where f(E) is the ratio between energy absorbed per incident photon by

breast tissue and their energy, E is the monochromatic energy employed in

the exam, Φ(𝐸) is the photon fluence, G is a factor which takes in account

the glandular fraction of the breast tissue, A is the irradiated area and M the

irradiated breast mass.

CTDI is the current standard index for CT examinations, and

provides information on the radiation dose to the scan volume. It is

important because it takes in account the scatter dose, which makes

radiation dose from a multiple acquisition greater than that of a single one.

CTDI is measured in a polymethil methacrylate cylinder of 15 cm in length

and diameters of 16 cm (head phantom) and 32 cm (body phantom); they

have one central and four peripheral holes in which an ionization chamber

is inserted. In particular, if a 100 mm ionization chamber is used, we can

talk about CTDI100 which is determined as:

𝐶𝑇𝐷𝐼100 =1

𝑛𝑇∫ 𝐷(𝑧)𝑑𝑧

50𝑚𝑚

−50𝑚𝑚

where T is the section thickness, D(z) the dose distribution along the z-axis

and n the number of section or slice measured simultaneously.

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1.3.2 Phase contrast X-ray imaging

Phase contrast X-ray imaging techniques [11] include different methods

used to transform phase shifts caused by the passage of an X-ray wavefront

in a sample to intensity variation, in order to acquire an image of the object.

Actually, not only the amplitude but also the phase of an electromagnetic

wave changes when it passes through a medium (fig. 1.7), because of its

complex refractive index given by the formula

𝑛 = 1 − 𝛿 + 𝑖𝛽

where δ is the decrement of the real part of the refractive index and β is

related to the attenuation coefficient µ. The phase shift and attenuation of a

plane wave passing through a medium of refractive index n can be clearly

understood with the help of the formula:

𝐸(𝑧) = 𝐸0𝑒𝑖𝑛𝑘𝑧 = 𝐸0𝑒𝑖(1−𝛿)𝑘𝑧𝑒−𝛽𝑧

The total phase shift, after a distance z, can be computed as

ΔΦ(𝑧) =2𝜋

𝜆 ∫ 𝛿(𝑧′) 𝑑𝑧′

This means that the phase shift is related to the decrement in imaging

direction of the real part of n, so one could use this information in

tomography to map δ, as traditional tomography does with β, which is

related to X-ray attenuation. For soft tissues and in the diagnostic energy

range, δ is about three order of magnitude greater than β, and this has the

important consequence that contrast can be enhanced, the effect growing

with increasing energy; moreover, the absorbed dose can be reduced using

high X-ray energies. Because of the small deviation of the refractive index

from unity for X-rays, which is in the order of 10−8, the refraction angles

caused at the boundary between two media are also very small. The

consequence is that refraction angles cannot be detected directly and are

18

usually determined indirectly using interference techniques, which

requires high coherence in the X-ray source.

There are various methods to obtain information about the phase

shift, like crystal interferometry, analyzer-based or diffraction-enhanced

imaging (ABI or DEI), and propagation-based imaging (PBI). Crystal

interferometry uses three beam splitters: the first divides the X-ray into two

parts, a reference beam, which passes undisturbed, and another one, which

passes through the sample; the second crystal makes the rays converge at

the surface of the analyzer, thus creating interference.

Fig. 1.7: Drawing of attenuation and phase shift of electromagnetic wave propagating in medium with complex index of refraction n.

Crystal interferometry allows to register the phase shift itself; tomographic

acquisition can be obtained simply rotating the sample and acquiring

projections. In ABI technique, an analyzer crystal put after the sample is

used to produce Bragg diffraction. The intensity of the reflected radiation

varies with the angle following the so-called “Rocking curve”, and the

typical acceptance angle is a few microradians. Thus, the contrast in the

image of the detector is determined by the diffraction angle, related to the

phase shift after the X-rays pass through the sample:

Δ𝜃 =1

𝑘

𝜕𝜙(𝑧)

𝜕𝑧.

19

Because of the presence of the first derivative of φ, this technique is less

sensitive than the crystal interferometry at low spatial frequencies, but it

can be used also with a polychromatic X-ray source because it does not need

the temporal coherence of the beam. As for the PBI, the experimental setup

of this technique is basically the same of the conventional radiography, with

the difference that the detector is not placed immediately behind the

sample, but at some distance, so that the radiation refracted by the sample

can interfere with the unrefracted beam. This simple setup is the greatest

advantage of this method over the other already discussed. If the beam has

spatial coherence and the distance between object and detector is large

enough, an interference pattern in Fresnel regime is created, and the fringes’

intensity are related to the second derivative of the wave front. This leads

to an increased contrast between the internal structures of the sample (edge

enhancement) that can be used to improve the contrast of an absorption

image but can also used to produce a different image. The phase retrieval is

usually obtained by acquiring images at two distances of the detector and

using linearization algorithms of the Fresnel diffraction integral.

Fig. 1.8: Scheme of phase contrast imaging via free-space propagation.

20

2. Monte Carlo simulations and Geant4

2.1 Monte Carlo methods

Monte Carlo (MC) methods are a broad class of computational algorithms

which rely on repeated generation of pseudo-random numbers to obtain

numerical results. They are often used when having an analytical solution

of a problem is difficult or impossible, for example in the evaluation of

multidimensional integrals in mathematics, modeling of chaotic systems

like fluids in physics, study of proteins and membranes in biology and

biophysics, or in finance to evaluate investments in projects. Actually, for

most of these problems, there are analytical expressions via differential

equations, which cannot be always easily solved, in particular when there

is a strong coupling between several degrees of freedom or complicated

domains of integration.

Probably, the first use of sequences of random numbers to solve a

problem was done by G. L. Leclerc in the 18th century, the so-called Buffon’s

needle; in 1930s, E. Fermi used random numbers to study neutron diffusion

. The modern version of Monte Carlo dates to 1940s by S. Ulam while

working on nuclear weapons projects at Los Alamos Laboratory Nicholas

Metropolis named these methods after the well-known casino due to the

prominence of randomness in both .

There is not only one Monte Carlo method, but all of them follow a

general scheme:

-Definition of a domain of possible inputs

-Generation of random inputs over the domain following a pdf

-Performation of deterministic computations on the inputs

-Aggregation of the results.

It is to distinguish between simulation, MC Methods and MC simulations.

A simulation is a fictitious representation of reality, described via a

mathematical medium; a MC method is only a technique to solve statistical

problems, while a MC simulation uses repeatedly a MC method to solve

21

problems. For example, one could model the toss of a coin with the

extraction of a random number between the interval (0, 1], assigning head

if the number is greater than 0.5 and tail if not. This is a simulation, but to

become a Monte Carlo simulation the sampling must be done many and

many times aggregating the results in order to obtain a probability

distribution for the phenomenon.

A high-quality MC simulation must have some characteristics.

First, its pseudo-random number generator must have a long period of

repetition. Second, there must be samples enough to ensure accurate results;

these things, together with the right sampling technique, appropriate for

what is being modeled and well-simulating the phenomenon in question,

can ensure a good result.

Pseudo-random numbers are so called because they are generated

via deterministic algorithms: given in input a number called seed, the

generator gives in output numbers in a recursive way. They are way easier

to generate than real random numbers and so require only a little

computational time. An example of algorithm is the multiplicative

congruential generator:

𝑅𝑛 = 75𝑅𝑛−1𝑚𝑜𝑑(231 − 1) → 𝑥𝑛 = 𝑅𝑛(231 − 1)

𝑅𝑛 is the n-th seed and 𝑥𝑛 the n-th pseudo-casual number. The intrinsic

defect of this kind of algorithms is that, after a period, the numbers

sequence repeat in the same way, but nowadays the most of them have

periods greater than 1018, virtually inexhaustible. The numbers generated

in this way are uniformly distributed in the interval (0, 1]. It is often

desirable to have a sequence of numbers distributed following a given pdf.

This can be done in several ways, for example with the rejection method: if

𝑝(𝑥) is a pdf with a maximum 𝑝𝑚𝑎𝑥 in the interval [𝑎, 𝑏], given two uniform

pseudo-random sequences of number between 0 and 1, {𝑠𝑖} and {𝑠𝑖′}, let

𝑥𝑖 = (𝑏 − 𝑎)𝑠𝑖 + 𝑎

𝑦𝑖 = 𝑝𝑚𝑎𝑥𝑠𝑖

22

Then, 𝑥𝑖 is accepted only if 𝑦𝑖 < 𝑝(𝑥𝑖), otherwise it is rejected; the new

sequence of numbers follows the pdf 𝑝(𝑥).

Another important method makes use of the cumulative probability

distribution function 𝑃(𝑥) of a pdf 𝑝(𝑥). 𝑃(𝑥) is a non-decreasing function

and so it has an inverse, 𝑃−1(𝑦). If 𝑃(𝑥) is randomly sampled, the

corresponding values 𝑥 = 𝑃−1(𝑦) will be randomly distributed but

following the pdf 𝑝(𝑥) (fig. 2.1). This is called inverse-transform method.

Fig. 2.1: Random sampling from a distribution 𝑝(𝑥) using the inverse-transform method.

Actually, even if Monte Carlo methods are easy to implement, their

drawback is their random nature, which affects the results with statistical

uncertainties, and reducing them could require a great amount of

computational time. It is then important to know how much accurate is an

estimate obtained via MC and how many repetitions are needed to reach a

desired precision. If 𝑥 is an unknown quantity and 𝑥𝑛 are the results of

repeated simulations, due to the law of large numbers,

lim𝑛→∞

1

𝑛∑ 𝑥𝑛

𝑛

= �� → 𝜇

where 𝜇 is the expected value. Because of the central limit theorem, the

convergence order is 1/√𝑁 so, to halve the error, one should quadruple the

trials (and hence the computational time). Anyway, it is easy to estimate the

accuracy of a series of results using the estimator

23

𝑆𝑁2 =

1

𝑁∑ 𝑥𝑖

2

𝑁

𝑖=1

− [1

𝑁∑ 𝑥𝑖

𝑁

𝑖=1

]

2

𝑆𝑁 √𝑁⁄ is called statistical error. Many techniques can be used to reduce

variance and then statistical error, for example the splitting technique: while

processing an event which has a given weight 𝑤 for the simulation, it can

be splitted in n events whose weight is 𝑤/𝑛, thus helping to reduce variance

of the average.

2.2 Geant4 simulation toolkit

Geant4 (fig. 2.2) is a MC simulation toolkit developed by CERN to simulate

the interaction of particles with matter. It was proposed to and approved

by the Detector Research and Development Committee (DRDC) of CERN

at the end of 1994, and the first production version was delivered at the end

of 1998 [12]. It is the successor of GEANT3, developed since 1974 and

written in FORTRAN; Geant4, instead, exploits object-oriented

programming (OOP) in C++. The word Geant stands for GEometry ANd

Tracking, due to the possibility to define a sensitive volume and to track the

particles passing through it, scoring some interesting quantities. Its

applications range from high-energy physics to astrophysics, but also

biophysics and medical physics, thanks to the possibility to simulate

particles with energy greater than 250 eV [13].

Fig. 3.1: Geant4 logo.

The most important concepts of OOP needed for understanding and

writing a Geant4 code are classes and inheritance. A class is the type of all

the objects of OOP. It represents a common set of attributes (for example,

the number of sides or angles of a polygon) and functionalities (for example,

the possibility to calculate the area of each polygon with a function, i.e. a

C++ method). Inheritance means that every object can derive methods and

24

attributes from another object, like in a hierarchic structure. For example, a

rectangle can inherit from the polygons the fact that it has an area and a

perimeter. Not always all the classes represent real objects in C++; some

objects could not present enough information to be instantiated and used;

thus, they are called virtual. In this case, inheritance must be used to

implement such missing functionalities: for example, it is not possible to

calculate the area of a generic polygon without knowing more information

about it.

Geant4 [14] provides a series of virtual classes in order to define the

geometry, i.e. the experimental setup, and all the parameters needed to

implement a simulation, like the type of particles which one wants to track

or the quantities to score. Geometry must be defined in three steps using a

class derived by the virtual class G4VUserDetectorConstruction. First, one

has to create a solid, which defines the shape and the dimensions of the

volume, for example a cylinder with a given height and radius, with the

class G4SolidVolume. The second step is the specification of the material

and the hierarchical position in the whole geometry using the class

G4LogicalVolume; for example, the cylinder can be made of water and can

be placed into another volume. Finally, the volume must be placed in the

space, and this is realized with the class G4PhysicalVolume. The only

mandatory volume is the world, which encloses all the other volumes, and

so must be big enough to contain all the elements that one needs to use. This

is how one can define the absorbers and detectors of the experiment.

Then, it is necessary to define the physics of the experiment, i.e. the particles

to be used in the simulation and their interactions with the materials

previously defined. For example, a photon is represented by the class

G4Gamma and must be instantiated as an object of the virtual class

G4Particle. Its interactions, like photoelectric effect, Compton and Rayleigh

scattering and pair production, must be instantiated too; one could also

choose to deactivate one or more interactions. The physics lists, or packages,

manage all the physics of Geant4. They are the classes that collect all the

25

particles and the physics processes. There is some need to define more than

one physics list: even if physics is always the same, there are different

models and approximation that can be used in different situations; in

particular, not all the simulations require all the physics that Geant4 can

offer. The main packages are those for electromagnetic, hadronic and weak

interactions; these last ones are useful for radioactivity and decays. In

particular, five main sub-packages treat electromagnetic interactions:

Standard, Penelope, Livermore, ICRU73 and Geant4-DNA. Standard sub-

package is the first package developed in the ambit of high-energy physics

and it is valid in the range 1 𝑘𝑒𝑉 − 1 𝑃𝑒𝑉. For low-energy applications, like

in biophysics and medical physics, it is recommended the use of a specific

package like Penelope or Livermore. Penelope (PENetration and Energy Loss

of Positrons and Electrons) has been developed by the Barcelona group

(Salvat et al.) and it is useful in particular for simulations of electromagnetic

showers at low energies. Penelope sub-package is based on NIST database

and for total cross section calculations it uses a mixed approach, analytical,

parameterized and database-driven. Livermore sub-package is based on

publicly available evaluated data tables (Evaluated Photon Data Libraries,

EPDL) from Lawrence Livermore National Laboratory (LLNL) and is

reliable in the energy interval 250 𝑒𝑉 ÷ 100 𝐺𝑒𝑉. At high energies, the

results obtained with those two packages match with the ones of Standard

library, but increasing computational time. ICRU73 sub-package is the one

dedicated to ions and their stopping power, from ICRU-73 report; Geant4-

DNA is the most used for microdosimetric and biophysics applications.

Finally, the class G4VUserPrimaryGeneratorAction allows the user

to choose a previously defined particle and to place it at a certain position

and with a certain momentum before the simulation starts. This permits to

define a beam with a user defined form, for example a uniform or a

Gaussian beam. It is to observe that scoring is neither automatic nor

mandatory. Other user’s classes, like G4UserSteppingAction, allow the

registration of data which are interesting for the further analysis: position,

26

energy, momentum, absorbed dose, etc. The whole simulation can be

schematized as follows: the greatest unit is the run, which can be thought as

the whole experiment or a part of it that repeats many times. For example,

a tomography performed with a 360° rotation and a step of 1° is composed

by 360 runs. Every run comprehends one or more events, i.e. the single

particle that one wants to track. Tracking is performed step by step: MC

methods are used to choose where a particle of the beam starts from,

following the distribution previously defined by the user, its path and what

interactions it will undergo; this depends on its energy and, so, on the cross

sections of the processes involved. Particles are tracked until they exit the

world volume or reach a minimum energy threshold; the same is done for

the secondary particles that could be created at an interaction point. Finally,

the user has the possibility to manage the simulation by macro, defining

commands with the messenger classes or using the ones already set. This

makes easier, for example, to change the position or material of a volume

between two runs without changing and compiling the code every time.

2.3 Geant4 simulations for SR BCT

As previously described, in BCT the patient lies on a bed with the pendant

breast and without compression. In the case of cone beam BCT, the X-ray

tube with the detector rotates synchronously under the patient bed to

acquire tomographic projections. In the case of SR BCT, as it happens in

SYRMA-CT project, there is a particular geometry: the beam from the

synchrotron has a fixed position in space and so, in order to realize a

tomography, the patient bed has to rotate. Moreover, because of the laminar

beam, for a complete vertical scan, patient translation has to be

accomplished. In order to describe this situation as accurately as possible in

MC simulations, it has been schematized as follows: the breast, which is the

rotating element, is assumed to have a cylindrical shape, as usually done in

simulation involving the breast due to its symmetry. The photon beam is

laminar with a size of 150x1 mm2 and monochromatic photon energy of 38

27

keV. Beam penumbra is neglected because the Elettra synchrotron radiation

beam, in the experimental hutch, has a divergence of only 7 mrad, and can

be approximated as a parallel beam. Finally, the detector is placed 2 m away

from the phantom, along y axis; this is done if one wants to use the PBI

technique (cap. 1), which needs great distances to highlight phase effects.

Geometry of the simulations is well described by fig. 2.3, which illustrates

a photon beam irradiating a breast phantom; 100 photons of 38 keV have

been simulated to obtain this image. It is noted that the point of view of the

simulation is rotated with respect to the real situation, where the vertical

axe coincides with the direction of the pendant breast and the horizontal

one indicates the direction of the beam.

Fig. 2.3: Drawing of a simulation which illustrates a photon beam irradiating a breast phantom; 100 photons of 38 keV have been simulated to print this image. Vertical axe of the image corresponds to the horizontal one of the experimental hutch.

In particular, I used phantoms of 50 cm length (z axis), diameters of 8, 10,

12 and 14 cm (x, y plane). They were composed of breast tissue of 0%, 50%

and 100% glandularity and surrounded by 1.45 mm of skin (fig. 2.4); the

composition of these materials were obtained from [15]. The version of the

software employed in the simulations is the 10.0.

Fig. 2.4: Reference system of Geant4 world (a). On the right, breast phantom of 50 cm length, oriented along z axis (b).

28

The unusual length and beam dimensions are justified by the intent to study

the dose spread functions (DSF) [16]. If we denote with 𝑃(𝑧) the dose due to

the interactions of primary photons with the phantom and with 𝐷(𝑧) the

dose due to photons which have already undergone scatter interactions,

𝐷𝑆𝐹(𝑧) is defined as follows:

𝐷(𝑧) = 𝑃(𝑧) + 𝑆(𝑧)

The name “spread function” is an analogy with the line spread function,

used in imaging to characterize the response of a system to a pulse entry:

the same can be done with dose, studying how dose due to an infinitesimal

thickness beam spreads in a phantom. This is why beam dimensions must

be much smaller than phantom dimensions, so the choice of a laminar beam

of 1 mm height. Phantom dimensions allow the assumption that, at great

distances from the center of the phantom, dose tails due to the scattered

radiation tend to zero: since it is not possible to model a phantom of infinite

length, 50 cm can be a good approximation of infinite in this case. DSFs,

thanks to their definition, can be used to evaluate the absorbed dose due to

a beam of arbitrary shape [17]:

𝐷𝐿(𝑧) =1

𝑏⋅ 𝑓(𝑧) ∗ Π (

𝑧

𝐿) =

1

𝑏∫ 𝑓(𝑧 − 𝑧′)𝑑𝑧′

𝐿/2

−𝐿/2

𝐷𝐿 is called cumulative dose and represents the absorbed dose along z

direction, where 𝑓(𝑧) corresponds to 𝐷𝑆𝐹(𝑧), the dose due to a single scan

centered at 𝑧 = 0, 𝐿 = 𝑁𝑏 is the total scan length, with 𝑁 scans of length 𝑏,

and ∗ stands for the convolution of 𝑓(𝑧) with the rectangular function of

unit height and length 𝐿, denoted by Π(𝑧/𝐿). As the scanning length

increases, the contributions of scatter dose to the total distribution at 𝑧 = 0

can be neglected and so 𝐷𝐿 reaches an equilibrium value called equilibrium

dose [17]:

𝐷𝑒𝑞 =1

𝑏∫ 𝑓(𝑧′)𝑑𝑧

+∞

−∞

′

29

Another parameter that can be obtained from DSF is the dose scatter to

primary ratio (dSPR) [16]:

𝑑𝑆𝑃𝑅 =∫ 𝑆(𝑧)𝑑𝑧

+∞

−∞

∫ 𝑃(𝑧)𝑑𝑧+∞

−∞

This is useful to evaluate the relative amount of scatter and primary dose

deposited in the whole phantom. In this work, DSF’s primary role is the

computation of 𝐷𝑔𝑁 , dose index used in mammography, mentioned in

cap.1. 𝐷𝑔𝑁 has been evaluated as follows:

𝐷𝑔𝑁 =(𝐷𝑆𝐹)𝑏𝑟𝑒𝑎𝑠𝑡 𝑡𝑖𝑠𝑠𝑢𝑒 ∗ Π (

𝑧𝐿)

(𝐷𝑆𝐹)𝑎𝑖𝑟 ∗ Π (𝑧𝐿)

After the acquisition of the DSFs of a phantom of breast tissue and one of

corresponding dimensions and made of air, the convolution has been

calculated for 𝐿 = 1.5 ⋅ 𝑅, which is the most common size for a human

breast. Then, 𝐷𝑔𝑁has been evaluated from the ratio of the average values of

the two dose profiles obtained from the convolutions illustrated above in

the scanning region. This calculation must take in account the different

percent composition of breast tissue in terms of adipose and glandular

tissue, which is the radiosensitive fraction and so must be safeguarded.

Thus, only the glandular weight fraction is taken in account for the

evaluation of the dose. So, the energy deposition is corrected using a factor,

𝐺, which gives the fraction of energy deposited in glandular tissue; its

expression is [10]

𝐺 =

𝑓𝑔 ⋅ (𝜇𝑒𝑛

𝜌 )𝑔

[𝑓𝑔 ⋅ (𝜇𝑒𝑛

𝜌 )𝑔

+ (1 − 𝑓𝑔) ⋅ (𝜇𝑒𝑛

𝜌 )𝑎

where 𝑓𝑔 is the glandular fraction, 𝑎 and 𝑔 subscripts stand for adipose and

glandular tissue, and 𝜇𝑒𝑛/𝜌 is the mass energy-absorption coefficient; this

30

coefficient is computed for every interaction and multiplied for the

corresponding energy deposition value.

The scoring of absorbed dose in the phantom was performed with

an implementation of G4UserSteppingAction class. Dose has been stored in

vectors in order to compute DSFs and in maps to obtain the dose

distributions. G4UserSteppingAction class was also used to score Compton

multiplicity, i.e. the mean number of Compton interactions a photon

undergoes while passing through the phantom: actually, there is some

interest in limiting photon scattering during an examination, in order to

safeguard the tissues not involved in the scan.

As previously mentioned, 2 m away from the phantom, a detector

was posed in order to evaluate beam transmission and integral energy

spectrum; its dimensions were 2.5x25 cm2, like Pixirad [18], and its material

was air, since it was interesting for us studying the properties of photons

entering the detector and not the interaction inside of it.

31

3. Results of the simulations

3.1 Validation of the code

Before proceeding with the simulations of the dose distribution in

the irradiated phantom, the first, necessary step is the validation of the code.

The better way to provide this is to compare the values obtained from the

measurements of CTDI (cap. 1) with the ones simulated. For this scope,

PMMA phantoms of 8, 10, 12 and 14 cm diameters and fixed height of 15

cm were simulated. The ionization chamber simulated was the Radcal

AccuPro 10x6-3CT model [19]. Since the measurements were performed

using a build-up cap, the same was implemented in order to obtain a better

agreement between measurements and simulations. Two kinds of beam

were employed: one laminar and with uniform fluence, and another

obtained from the scan of a radiochromic film exposed to Elettra beam [20];

energies used were 18, 20, 24, 28, 32, 35, 38 and 40 keV. The quantity scored

was air dose in the chamber, and the output of the simulation were

elaborated to compute the quantities:

(1)

13 𝐶𝑇𝐷𝐼100,𝑐

𝑃𝑀𝑀𝐴 +23 𝐶𝑇𝐷𝐼100,𝑝

𝑃𝑀𝑀𝐴

𝐶𝑇𝐷𝐼100,𝑐𝑎𝑖𝑟

(2) 𝐶𝑇𝐷𝐼100,𝑐

𝑃𝑀𝑀𝐴

𝐶𝑇𝐷𝐼100,𝑐𝑎𝑖𝑟

which are the 𝐶𝑇𝐷𝐼𝑤𝑃𝑀𝑀𝐴 and 𝐶𝑇𝐷𝐼𝑐

𝑃𝑀𝑀𝐴 normalized to 𝐶𝑇𝐷𝐼100,𝑐𝑎𝑖𝑟 . These

values are necessary for the study of the dosimetric protocol, since CTDI is

one of the two dose indexes used to evaluate MGD: this is only the first step,

the second will be the study of dose distributions.

3.1.1 CTDI measurements

CDTI measurements were performed at the experimental room of

the SYRMEP beamline by other members of the SYRMEP collaboration [20,

21]. Four PMMA cylinders of 15 cm length and diameters of 8, 10, 12, 14 cm

32

(fig. 3.1) and a polyethylene cylinder of 15 cm length and diameter of 10 cm

were used. Phantoms had five holes along the vertical direction (one central

hole and four peripheral holes) for the insertion of the ionization chamber.

The synchrotron laminar beam had a size of 170 × 3.94 𝑚𝑚2 and its energy

varied in the range 18 − 40 𝑘𝑒𝑉. It was used to irradiate the phantoms with

the ionization chamber inside them, posed at a distance of about 23 m from

the source. During the measurements, the phantoms were kept in rotation

at a speed of 3°/s to complete a whole rotation of 360°. The 𝐶𝑇𝐷𝐼100 values

were estimated by irradiating a single slice 3.94 mm thick; then, these values

were normalized to beam current time product and quoted in units of

𝑚𝐺𝑦/100 𝑛𝐴𝑠. Both peripheral and central 𝐶𝑇𝐷𝐼 (cap. 1) were estimated. In

particular, 𝐶𝑇𝐷𝐼𝑝 was evaluated as average of two measures performed in

two different peripheral holes. For every measure, estimated uncertainties

were smaller than 10%.

Fig. 3.1: PMMA phantom for the measurement of CTDI [20].

3.1.2 CTDI simulations

Fig. 3.2 shows the comparison between experimental results

(labeled with exp) and simulations (labeled with sim) for the 𝐶𝑇𝐷𝐼𝑤𝑃𝑀𝑀𝐴

normalized to 𝐶𝑇𝐷𝐼100,𝑐𝑎𝑖𝑟 versus phantom diameter. The other parameter of

the curves is photon energy; two energy per graph are shown. Statistic

errors are smaller than 1% and so they are omitted; experimental errors take

in account accuracy error of the ionization chamber of 4%.

33

8 10 12 14

0.2

0.3

0.4

CT

DI w

, P

MM

A/C

TD

I c, air (

nG

y/n

Gy)

Phantom diameter (cm)

18 exp

20 exp

18 sim

20 sim

(a)

8 10 12 140.3

0.4

0.5

0.6

0.7

CT

DI w

, P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


24 exp

28 exp

24 sim

28 sim

(b)

8 10 12 140.5

0.6

0.7

0.8

0.9

CT

DI w

, P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


32 exp

35 exp

32 sim

35 sim

(c)

8 10 12 140.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

CT

DI w

, P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


38 exp

40 exp

38 sim

40 sim

(d)

Fig. 3.2: Weighted CTDI in PMMA normalized to CTDI center in air versus phantom diameter at the energies of 18 and 20 (a), 24 and 28 (b), 32 and 35 (c), 38 and 40 (d) keV. Empty points and dotted lines represent simulations, while full points and continuous lines represent experimental results.

34

Simulations show the same trend of the experimental results. CTDI values

diminish at greater phantom dimensions because radiation has to pass

through a greater thickness before reaching the ionization chamber. This

decrement appears to be exponential at energies lower than 28 keV and

linear at higher energies. This is due to higher penetrability of photons at

higher energies, which compensates phantom attenuation, and to the

contribute of scattering: scattering make the photons spread into the

phantom, so that they can reach more easily the ionization chamber, even if

it is at the center of the phantom. This is also due to the dependence of dose

deposition upon the mass energy coefficient, which has a strong non linear

component at low energies. This is due to the preponderance of

photoelectric effect, whose cross section is proportional to E-3, with respect

to the Compton, whose cross section depends on E-1 and is prominent at

energies higher than 30 keV. Finally, CTDI values become higher when

energy increases. This is due to the energy deposition of secondary

electrons, which increases with photon energy.

Fig. 3.3 shows the comparison of experimental results (labeled with

28 exp), and the simulations with the laminar, ideal beam (labeled with 28

sim, i) and the one of Elettra (labeled with 28 sim, r), versus phantom

diameter at the energy of 28 keV. At this energy, the two simulated beams

behave substantially in the same way. So, they approximate very well the

trend of the measurements, showing the same decrement at higher

phantom diameters. Fig. 3.4 shows the trend of CTDI100 values both at the

center and in periphery versus phantom diameter at the energy of 28 keV.

It is noticeable how CTDI100,p is always higher than CTDI100,c. This reflects

the trend of dose distributions in tomography: energy deposit is higher at

the periphery of the phantom, due to exponential attenuation law, while at

the center it is less probable that a photon interact with the ionization

chamber. The decrement of CTDI in periphery is less prominent probably

because the distance of the ionization chamber from phantom border is

fixed. It only diminishes because, increasing the diameter, the ionization

35

chamber can reach great distances from photon beam during phantom

rotation. This increases the mean path that photon have to travel towards

the ionization chamber.

8 9 10 11 12 13 140.4

0.5

0.6

0.7

0.8

CT

DI

w P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


28 exp

28 sim, r

28 sim, i

Fig. 3.3: Weighted CTDI in PMMA normalized to CTDI center in air versus phantom diameter for the energy of 28 keV. Shown is the comparison of the simulations with the ideal laminar beam (28 sim, i), the beam obtained from the scan of a Gafchromic film directly exposed to Elettra beam at Trieste (28 sim, r) [20, 21] and the experimental points (28 mis).

8 10 12 1410

15

20

25

30

CTDI 100 values

E = 28 keV

CT

DI 100

(G

y)


c, PMMA

p, PMMA

Fig. 3.4: CTDI100 simulated in the central (lower curve) and peripheral (upper curve) position of the phantoms versus phantom diameter at the energy of 28 keV.

Fig. 3.5 shows the ratio of CTDI100,c in PMMA and air versus phantom

diameter at the several energies. The trend is substantially the same already

seen for CTDI100,w: the ratios decrease with increasing phantom diameter

and increase with increasing energy. Also in this case, the decrement with

phantom diameter is exponential at lower energies and tends to be linear at

high energies due to the different preponderance of photoelectric and

Compton effects.

36

8 10 12 140.0

0.1

0.2

0.3

CT

DI

c P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


18 exp

20 exp

18 sim

20 sim

(a)

8 10 12 140.1

0.2

0.3

0.4

0.5

0.6

24 exp

28 exp

24 sim

28 sim

CT

DI

c P

MM

A/C

TD

I c, air (

nG

y/n

Gy)

Phantom diameter (cm) (b)

8 10 12 140.3

0.4

0.5

0.6

0.7

0.8

0.9

CT

DI

c P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


32 exp

35 exp

32 sim

35 sim

(c)

8 10 12 140.5

0.6

0.7

0.8

0.9

1.0

1.1

CT

DI

c P

MM

A/C

TD

I c, air (

nG

y/n

Gy)


38 exp

40 exp

38 sim

40 sim

(d)

Fig. 3.5: CTDI100,c in PMMA normalized to CTDI100,c in air versus phantom diameter at the energies of 18 and 20 keV (a), 24 and 28 keV (b), 32 and 35 keV (c), 38 and 40 keV (d). Simulated data have dotted lines, while experimental data have full lines.

37

3.2 Results of the simulations

3.2.1 Dose Spread Functions

Figure 3.6 shows the primary, scatter and total dose spread functions of a

50% glandularity, 12 cm diameter and 50 cm height phantom at the energy

of 38 keV. x-axis is the direction along phantom axis, while y axis, which is

in semi-log scale, represents the three DSF normalized to their maximum

value, DSF(z=0).

-20 -10 0 10 2010

-6

10-5

10-4

10-3

10-2

10-1

100

total

primary

scatter

E=38 keV

fg=50%

d=12 cm

Re

lative D

SF

(z)

(G

y/

Gy)

z position (cm)

Fig. 3.6: Primary, scatter and total dose spread function versus z position (along the axis of the phantom) for a 50% glandularity, 12 cm diameter and 50 cm height phantom at the energy of 38 keV. The plot is in semi-log scale; all the values are normalized to their value at z=0.

Primary DSF is a function peaked in the origin, i.e. the position where

photon beam enters the phantom. It is considered primary the first energy

deposition of a photon, due to a photoelectric or Compton interaction, along

beam direction. All these energy depositions are localized around the center

of the phantom because electrons range at 38 keV is in the order of some

tens of microns, so they can not spread into the whole phantom. Instead,

this happens for scattered radiation, which comprehends all the energy

depositions of the photons which underwent more than one interaction.

Rayleigh and Compton scattering make radiation reach great distances

from the center of the phantom; multiple Compton events combined with

Rayleigh interactions and photoelectric effect make dose spread in the

whole phantom. The trend of this distribution is an exponential decrease,

38

underlined by the linear trend in semi-log scale. It rapidly falls off but is

never zero, even at great distances such as 25 cm from the center of the

phantom, where it is 10-4 times the maximum value. Total dose spread

function results from the sum of primary and scatter distributions; thus, it

has the same behavior of the scatter function. The only difference is the

contribute of primary radiation at the center, which is about an order of

magnitude greater than the scatter radiation. Thus, DSF(z=±25cm) is about

10-5 times DSF(z=0). To better investigate the behavior of scattering

radiation into the phantoms, the dose maps obtained from the simulations

were elaborated dividing all the circular slices of the phantom into three

annuli having the same area, denoted by the radii R1, R2 and R3, as shown

in fig. 3.7.

Fig. 3.7: The three concentric regions in which scatter dose was evaluated. If R3 is the radius

of the circle, 𝑅1 = √1/3 ⋅ 𝑅3 and 𝑅2 = √2/3 ⋅ 𝑅3 [16], so that the three annuli have the same

area.

These radii divide the phantoms into three regions, its center, an

intermediate region and its periphery, which have the same volume but

slightly different scatter dose distributions. The image in fig. 3.8 shows the

dose axial distribution of a 50% glandularity, 14 cm diameter, 50 cm height

at the energy of 38 keV.

Fig. 3.8: Axial scatter dose profile of a 14 cm diameter, 50 cm height, 50% glandularity phantom at the energy of 38 keV from which sDSF was evaluated. The image results from the sum of all the axial projections and it was applied decimal logarithm to emphasize its trend.

39

The image is the sum of all axial projections and is in log scale in order to

emphasize the trend of dose distributions, whose profile is shown in fig. 3.9

a. It reports the scatter dose distribution in semi-log scale versus z position

along phantom axis. All the values are normalized to their maximum,

sDSF(z=0).

-25 -20 -15 -10 -5 0 5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

100

R=R3

R=R1

E=38 keV

fg=100 %

d=14 cm

Re

lative

sD

SF

(z)

(G

y/

Gy)

z position (cm) (a)

-25 -20 -15 -10 -5 0 5 10 15 20 2510

-5

10-4

10-3

10-2

10-1

100

sDSF

r=R1

fg=100 %

fg=0 %

Re

lative s

DS

F (

z)

(G

y/

Gy)

z position (cm)

E=38 keV

d=8 cm

(b)

Fig. 3.9: (a) Scatter dose spread functions for 14 cm diameter, 50 cm height, 50% glandularity phantom in semi-log scale versus z position, along phantom axis. The three radii R1, R2 and R3 are chosen in order to divide the circular slices of the cylinders in three zones of equal area and then the whole cylinder in three volumes of identical value. All the functions are normalized to their maximum value. (b) sDSF for a 8 cm diameter, 50 cm height phantom and different glandularities at the energy of 38 keV. From the top to the bottom, the curves represent the scatter dose distributions for 0%, 50% and 100% glandularities, in semi-log scale versus z position along phantom axis. All the values are normalized to their maximum value.

Similarly to the profile shown in fig. 3.6, the trend is exponentially

decreasing from the center towards the periphery for all the three zones. It

can be seen, however, that in the peripheral zone (R=R3) the dose

40

distribution falls off less rapidly than in the other zones; in particular, the

difference between R1 and R3 functions is about half an order of magnitude

for all the distances from the center. This is due to a solid angle effect.

Scattered photons can easily reach the inner zone from the extern and

scatter many times, because Compton and Rayleigh cross sections are

peaked forward: they travel along beam direction losing many times a part

of their energy. After many scatter interactions, the probability of a

photoelectric interaction increases and so it is more probable that after a few

steps the photon loses all its energy in the inner zone.

Fig. 3.9 b shows the scatter DSF for a 8 cm diameter, 50 cm height

phantom and different glandularities at the energy of 38 keV. From the top

to the bottom, the curves represent the scatter dose distributions for 0%,

50% and 100% glandularities, in semi-log scale versus z position along

phantom axis.

All the values are normalized to their maximum value. As it can be

seen, in the inner zone of the phantom the trend of the three functions is

substantially the same, but the different characteristics of the three

glandularities emerge at great distances from the center: sDSF of the 0%

glandularity (=adipose tissue) phantom decreases less rapidly than the one

of 100% glandularity. Actually, adipose tissue is less dense than glandular

tissue (0.93 g/cm3 vs 1.04 g/cm3), which favors scattering interactions. The

combination of Rayleigh and Compton scattering, whose cross section is

higher for low glandularity tissues, makes photons reach phantom zones

far from the center and deposit their energy far from the point of their first

interaction. Moreover, glandular tissue is more attenuating, so that

scattered radiation can not travel great distances through the phantom

before to be absorbed.

Another important parameter that can be obtained from dose

spread functions is the dose scatter to primary ratio, dSPR. It can be used to

quantify the relative importance of scatter with respect to primary radiation

41

in the total dose distribution. It has been obtained summing over the whole

phantom the contributions of primary and scatter radiation. Figure 3.10

shows the SPR values versus phantom diameter for the three phantom

composition of 0%, 50% and 100% glandularity. Scatter contributions

increase with increasing diameter: a greater phantom allows radiation

spread more than in a phantom of minor radius because there is more

possibility for photon to interact, even more than one time. Thus, the dSPR

of a 14 cm diameter phantom can be higher than the dSPR for a 8 cm

phantom of about 20-30%. Phantom composition influences scatter

distribution in a preponderant way. Actually, for lower glandularities the

dSPR increases due to the Compton cross section, higher for adipose tissue.

For example, for a 8 cm diameter phantom, the dSPR is a little smaller than

1, but for a 14 cm diameter it varies from 1.3 for a 100% glandular tissue to

1.5 of an adipose tissue.

8 10 12 14

1.0

1.2

1.4

E=38 keV

dS

PR

(G

y/

Gy)


0 %

50 %

100 %

Fig. 3.10: dSPR values obtained from the ratio of the integrals of primary and scatter DSF evaluated between -25 and +25 cm versus phantom diameter.

3.2.2 Cumulative and equilibrium dose

It is not realistic to scan a phantom with a beam of infinitesimal height and

only at its center. A more realistic situation can be simulated convolving the

total DSF with rectangular functions of different length and unit height. In

this way, a multi slice scan is simulated, with the phantom translated of 1

mm, the height of the simulated photon beam, after a complete rotation.

This allows the study of the cumulative dose for different scan lengths. Scan

42

length was varied in order to study the increase in dose at z=0 due to the

contributions of dose tails. If scatter radiation did not exist, the dose at z=0

would be the same despite the scan length. Actually, as seen before, the dose

distribution in the phantom has an exponential decrease from the centre to

the borders, which never reaches the zero value. Thus, while scanning a

portion of the phantom, the DSF tails contribute to dose distribution in the

central slice even for scans of considerable length. This happens until a

certain scan length is reached, called 𝐿𝑒𝑞, the equilibrium length, after which

the increase in dose at z=0 is negligible. This length corresponds to the

equilibrium dose, 𝐷𝑒𝑞, the maximum dose a phantom can absorb for every

scan length.

-20 -10 0 10 200

1

2

0.1

4

8

1220

16

Cu

mu

lative

do

se

(G

y/

Gy)

z position (cm)

D0 = 3.7GyE=38 keV, d=8 cm, fg=0 %

24

(a)

4 8 12 16 20 241.6

1.8

2.0

E=38 keV

d=8 cm

fg=0 %

Deq

=1.96 Gy/Gy

Cu

mu

lative

do

se

(G

y/

Gy)

Scan length (cm) (b)

Fig. 3.11: (a) The graphs show the cumulative dose for different scan lengths in a 8 cm diameter, 0% glandularity at the energy of 38 keV versus z position along the phantom. The values are normalized to DSF(z=0). The scan lengths are indicated by labels on the graphs and vary in the range 0.1-24 cm. These values were obtained from the convolution of the dose profile of the pulse beam and rectangular functions of different lengths. All the values are normalized to the absorbed dose due to a single scan with the pulse beam at z=0 (D0). (b) The rise to equilibrium: the peak values of the several distributions versus scan length are reported in order to show the exponential growth of the peak value to the equilibrium. The values are normalized to DSF(z=0).

43

Fig. 3.11 (a) shows the cumulative dose distribution for a 0% glandularity,

8 cm diameter and 50 cm height breast phantom at the energy of 38 keV.

The tags on the curve stands for the scan lengths, which varie from 0.1 cm

(single scan with the pulse beam at the centre of the stationary phantom,

which gives the DSF) to 24 cm, with a step of 4 cm. All the values are

normalized to the dose at z=0 of the DSF, which is the inner curve in the

graph and whose value is labelled with D0, which is 3.7 µGy in this case.

Proceeding from the centre to the periphery of a curve, the trend is

decreasing due to the contribute of the tails, which becomes smaller and

smaller, with a rapid fall of at the border of the scan zone. It is interesting

to notice that out of the scan zone dose in not zero, but can be even 40% of

the maximum, since DSF(z) far from the centre is small but never zero.

Increasing scan length the curves tend to flatten and at the border of the

scan zone they reach a value which is about an half of the maximum. A

really flat curve would be obtained with a scan of infinite length, i.e. a length

greater than the one of equilibrium.

Proceeding from the inner to the outer curve, it can be noticed how

the peak value increases in a non-linear way until an asymptotic value of

equilibrium is reached. This increase is greater for smaller scan lengths. This

is due to the fact that the greatest contributions are the one from the values

of the DSF around z=0. Actually, even if these values are several order of

magnitude smaller than the peak, summing them over a finite number of

scans make their contribute rise substantially. This increase becomes

smaller increasing the scan length, because the tails of the distribution give

a smaller contribute. The graph in fig. 3.11 (b) reports the peak values of the

several distributions versus scan length, in order to show the exponential

growth of the peak value to the equilibrium. Equilibrium dose for this

phantom is 1.96 µGy/µGy, which means that dose tails contributions,

summed over a length of 24 cm, make dose at z=0 almost duplicate with

respect the one due to a single infinitesimal scan.

44

As shown in fig. 3.12, equilibrium dose increases with increasing

phantom diameter, because dose can spread even far from the center of the

phantom. If a single scan would have been performed, scatter dose tails

would be negligible, but summing it over a number of scans, their

contributes make dose rise substantially. Thus, equilibrium values are in

the range 2-2.5 µGy/µGy. In combination with the diameter, phantom

composition influences the equilibrium value, and in particular it is higher

for 0% glandularity than for 100% glandularity phantoms: glandular tissue

absorbs radiation and does not allow it to spread into the whole phantom

as the adipose tissue does.

8 10 12 14

2.0

2.2

2.4

2.6

E=38 keV

Equ

ilib

riu

m d

ose

(G

y/

Gy)


0 %

50 %

100 %

Fig. 3.12: The graph shows the equilibrium dose for the 50 cm phantoms versus their diameter. The three curves, from the top to the bottom, represent the equilibrium values for the 0%, 50% and 100% glandularity breast phantoms.

3.2.3 Dose radial profiles

After investigating the dose distributions along z direction, i.e. along the

phantom axis, it is also useful to study the distribution in the x,y plane, i.e.

in the coronal plane. Dose maps were obtained from the simulations by

dividing the phantoms in 1 mm3 cubic voxels and scoring the energy

depositions. An example of dose planar map for a 8 cm diameter, 0%

glandularity breast phantom is shown in fig. 3.13 (a). It represents, in order,

the primary, scatter and total dose distribution in the central slice of the

phantom, skin excluded. It has been obtained from the 3-D dose map

elaborated with ImageJ. The slice of the phantom is the circle inscribed into

45

the square, which appears black because dose has not been scored out of

the phantom. As previously said, high energy depositions corresponds to

white, so it can be seen –qualitatively- that the primary and total

distributions have their maximum at the border of the phantoms and reach

their minimum at the center, while the secondary one is quite uniform and

assumes lower values. These images make also evident the cylindrical

symmetry due to the scan geometry.

(a)

0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

E= 38 keV

d=8 cm

fg=0 %

primary

total

Re

lative

do

se

(G

y/

Gy)

x position (cm)

scatter

(b)

Fig. 3.13: (a) Image showing the dose distributions of –respectively- primary and scatter radiation and their sum for a 8 cm diameter, 0% glandularity breast phantom at the energy of 38 keV. Black zones out of the circle indicate that there is no scoring of dose. White color stands for greater dose deposition with respect to black. (b) The graph shows, from the top to the bottom, total, primary and scatter dose depositions versus horizontal position for a 8 cm diameter, 0% glandularity breast phantom at the energy of 38 keV. The curves are normalized to the maximum value of the total dose profile.

The profiles in fig. 3.13 (b) and 3.14 have been traced along a diameter of

the central slice of the primary, scatter and dose planar maps of a 8 cm

diameter, 0% glandularity (3.13 b) and a 14 cm diameter, 100% glandularity

(3.14) phantoms. The profiles report dose values normalized to the

maximum value of the total map versus horizontal position along phantom

diameter. The graphs show that total dose distribution has a maximum at

the periphery and a minimum in the center of the phantom; incidentally,

46

this is why in CTDIw evaluation CTDIp has a weight which is double respect

to CTDIc. This fact is due to the exponential attenuation of the photons,

which makes them deposit their energy more in periphery than in the

center. This causes a cupping profile for dose in the horizontal plane; for the

8 cm phantom, dose at the center can be about 70% of the maximum, but for

a 14 cm diameter in can also be the 30%. Great part –about 80%- of the dose

deposition is due to primary photons, whose distribution has the same

trend of the total one. Scatter dose distribution appears to be very

homogeneous, in particular in the 8 cm phantom: there is no cupping

artifact; the contribution of scatter dose to the total appears to be about 20%.

In the greatest phantom, the one of 14 cm (fig. 3.14), even scatter distribution

has a valley at the center, where photons come principally after many

scatter interactions and with lower energy.

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0

E= 38 keV

d=14 cm

fg=100 %

totalprimary

scatter

Re

lative d

ose

(G

y/

Gy)

x position (cm)

Fig. 3.14: The graph shows, from the top to the bottom, total, primary and scatter dose depositions versus horizontal position for a 14 cm diameter, 100% glandularity breast phantom at the energy of 38 keV. The curves are normalized to the maximum value of the total dose profile.

Phantom composition is another important parameter for dose planar

distributions. Fig. 3.15 shows the scatter (a) and total (b) dose distributions

at 38 keV for the 14 cm diameter phantoms with three different

glandularities. Since scatter interactions are favored in adipose tissue, their

contribute in dose deposition is about 7% higher than in glandular tissue,

which absorbs more scattered radiation, while 50% glandular breast tissue

47

behaves in an intermediate way. Total dose profile shows that the three

tissues have the same behavior in periphery, but at the center of the

phantoms, the 0% glandular curve is higher than the one of 100% of about

10%, which means that glandular tissue attenuates radiation more than the

adipose.

0 2 4 6 8 10 12 140.00

0.05

0.10

0.15

0.20

E= 38 keV, d=14 cm

fg=100 %

Re

lative

do

se

(G

y/

Gy)

x position (cm)

fg=0 %

(a)0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

fg=100%

fg=0%

E= 38 keV, d=14 cm

Re

lative

do

se

(G

y/

Gy)

x position (cm) (b)

Fig. 3.15: Dose horizontal profiles in the central slice of scatter (a) and total (b) distribution for a 14 cm diameter phantom, at the energy of 38 keV for three different glandularities: from the top to the bottom, 0%, 50% and 100% glandular breast tissue phantoms.

3.2.4 DgN

DgN is an indispensable parameter for a mammographic exam. Its value,

multiplied by the air kerma gives the amount of dose to the glandular tissue,

which is to be safeguarded since it is the radiosensitive fraction of the breast.

Determination of DgN values is not possible via a direct measurement, so

Monte Carlo simulations are indispensable for this purpose. To compute

DgN, the idea is to take advantage of the DSFs already calculated. Breast

mean thickness is about 1.5 times its radius. Thus, by means of the DSF, it

is easy to compute the cumulative dose for a breast of such thickness,

convolving the DSF with a rectangular function whose length is 1.5 times

breast radius and then evaluating the mean value over the whole scan

region. For the calculation of air kerma, an air phantom of the same

dimension of the breast has been simulated for every case: that dimensions

should be big enough to ensure secondary electron equilibrium and then to

compute with precision the air kerma. The evaluation of air kerma has been

48

done computing the mean value of the cumulative dose obtained in the

same way as done with the breast phantom.

Fig. 3.16 shows a plot of the DgN values obtained for the breast

phantoms versus their diameter; these values are reported in tab. 3.1. From

the top to the bottom, the curves represent the values found for the 0%, 50%

and 100% glandularity phantoms. All the values are in the range 0.6-0.9

µGy/µGy. These values decrease increasing phantom diameter and

increasing glandularity: increasing the diameter, also the mass of the

phantom increases. Thus, the same amount of energy deposited in the same

tissue results in a smaller dose for bigger phantom dimensions. Moreover,

it is very important to consider the glandular fraction of the tissue: a 0%

glandular tissue has a glandular fraction next to 0 and this means that the

mass which dose is inversely proportional to is very small. The result is a

higher dose with respect to a 50% and a 100% glandular tissue.

8 10 12 140.6

0.7

0.8

0.9

E= 38 keV

Dg

N (G

y/

Gy)


0

50

100

Fig. 3.16: DgN values for the studied phantoms as a function their diameter at the energy of 38 keV. The three curves refer (from the top to the bottom) to the 0%, 50% and 100% glandularity phantoms. They are expressed in µGy/µGy as a result of the ratio between glandular dose and air kerma.

Tab. 3.1: DgN values for the 0%, 50% and 100% glandularity phantoms and 8, 10, 12 and 14 cm diameters at the energy of 38 keV.


DgN (µGy/µGy) [fg=0%]



8 0.8939 ± 0.0011 0.8537 ± 0.0008 0.8048 ± 0.0007

10 0.852 ± 0.003 0.801 ± 0.005 0.742 ± 0.004

12 0.807 ± 0.004 0.745 ± 0.006 0.680 ± 0.005

14 0.768 ± 0.003 0.699 ± 0.003 0.629 ± 0.002

49

Actually, there are some issues. Does the cumulative dose found by

convolving the DSF coincide with the dose absorbed when the phantom is

completely scanned, i.e. when a photon beam of the same dimensions of the

breast is used? Moreover, it is necessary to have a pratical way to measure

air kerma, since it is not possible to use an air phantom. The most common

way to do it is to use a ionization chamber. So, do the air kerma values

measured with the ionization chamber coincide with the values found with

the air phantom? For this purpose, a series of simulations has been

implemented for a 50% glandularity, 12 cm diameter and 9 cm height

phantom, at the energy of 38 keV. Three beams of identical photon fluence

of 40 mm-2 have been simulated in order to compare the doses absorbed by

glandular tissue, as shown in table 3.2. The first, of 1 mm height, is the same

used to compute the dose spread functions for the 50 cm height phantoms;

the second, 4 mm height, has approximately the same dimensions of the

Elettra beam and the third, 90 mm height, simulates the total scan of the

phantom. The width of the phantom is always 150 mm, the same dimension

of the Elettra beam. In the first case, it is necessary to calculate the

cumulative dose; at the opposite, for the 90 mm beam it is not necessary,

while for the 4 mm beam there are two possibilities. Even if a total scan of

the phantom is the most realistic situation, one could decide to scan only a

slice of 4 mm: it is indispensable to know the glandular dose also for this

case.

Tab. 3.2: The three photon beams simulated in order to compare the results in DgN values. The table lists the height and the width in mm of the photon beams used and the corresponding number of photons to be simulated in order to have the same photon fluence. The simulations have been repeated 10 times.

Height (mm) Width (mm) Photon number per run

1 150 4870

4 150 19200

90 150 432000

50

The resulting glandular dose profiles for the total scan are shown in fig. 3.17.

From the top to the bottom, the three curves correspond to the total scan

with a beam of 90 mm size, the cumulative dose obtained by the

convolution of DSF(z) obtained with the beam of 1 mm size and finally the

cumulative dose obtained from the dose profile due to the 4 mm beam. The

profiles have the same trend: they decrease from the center to the borders

of the phantom due to the scattering contribution, which are prominent at

the center because dose tails do not influence in a considerable way the

profile in periphery. The profile obtained with the 90 mm beam results

slightly higher than the other two. The difference is greater for the 4 mm

beam with respect to the 1 mm beam, in particular at the borders, but it is

always smaller than 5%. This is probably due to the finite size of the dose

profile used to compute the cumulative dose: while proceeding with the

convolution, dose tails are not summed correctly, and this leads to

underestimate the dose at the borders of the phantom, and probably, this

effect is enhanced by the beam size.

As for the air kerma measurements, the same beams described in tab.

3.1 have been used to compute air dose in a 12 cm diameter, 9 cm height

phantom and using a ionization chamber. Table 3.2 shows the results found

for the mean value. The columns describe, in order, beam size, air kerma

obtained with the air phantom and air kerma obtained using the ionization

chamber. The second row has been splitted in two because of the possibility

to scan the whole phantom (1) or only a slice (2) during the tomographic

exam.

The trend for the mean values for the three cases in which the

phantom is totally scanned is the same observed for breast tissue: the

convolution of dose profiles underestimates air kerma of some percent

because of the influence of the tails of the distributions and their smaller

contribute in the sum at the borders of the phantom. In particular, even in

this case the convolution performed with the 1 mm beam appears better

than that done with the 4 mm beam. Air kerma measured with the

51

ionization chamber results slightly smaller than the values found with the

air phantom (2); probably, this is due to the different amount of events

scored in the two cases.

-4 -2 0 2 40.1

0.2

0.3

0.4

0.5

Gla

nd

ula

r d

ose

(G

y)

z position (cm)

90 mm

1 mm

4 mm

E= 38 keV

fg=50%

d=12 cm

h=9 cm

Fig. 3.17: The three dose profiles resulting from the scans with the three different beam as a function of z position along phantom axis with photon fluence of 40 mm-2. From the top to the bottom, the three curves correspond to the total scan with a beam of 90 mm size, the cumulative dose obtained by the convolution of DSF(z) obtained with the beam of 1 mm size and finally the cumulative dose obtained from the dose profile due to the the 4 mm beam.

Tab. 3.3: For the three different beams, air kerma values obtained by means of the simulations. The second column is referred to the values calculated with the 12 cm diameter, 9 cm height phantom while the third column is referred to air kerma measured with a ionization chamber. The second row is splitted in two because of the possibility to scan the whole phantom (1) or only a slice (2) during the tomographic exam.

Beam size Air kerma (air phantom)

(µGy)

Air kerma (ionization chamber)

(µGy)

1 mm 0.626 ± 0.005 ---

4 mm (1) 0.5973 ± 0.0004 ---

(2) 0.03084 ± 0.00002 0.02618 ± 0.00011

9 cm 0.627 ± 0.004 0.610 ± 0.002

Tab. 3.4 reports the different DgN values found for the different cases. The

first column reports the sizes of the three beam implemented in the

simulations; the second column (*) reports the DgN values when air kerma

measurement is simulated by means of the air phantom, while the third

column (**) reports DgN obtained when the air kerma measurement is

simulated with the ionization chamber. In the first (*) case, the values range

52

between 0.745 and 0.751. There are small differences between the values

found by means of the cumulative dose and the one found for the total scan;

these differences are smaller than 1%. In particular, the value labeled with

(2), referred to a single slice scan, are compatible with the value found for

the phantom totally irradiated. The third column (**) reports the DgN

values when air kerma simulations are performed with the ionization

chamber; these values are greater of about 10% with respect to the former

because ionization chamber underestimates the air kerma values found

using the air phantom.

Tab. 3.4: DgN values obtained with the three different beams in the case of the air kerma simulated with the air phantom (*) or with the ionization chamber (**). (1) refers to the phantom totally scanned while (2) to the single slice scan.

Beam size

(mm)

DgN (*)

(µGy/ µGy)

DgN (**)

(µGy/ µGy)

1 mm 0.745 ± 0.006 ---

3.94 mm (1) 0.7468 ± 0.0006 ---

(2) 0.7510 ± 0.0007 0.880 ± 0.004

9 cm 0.7496 ± 0.0005 0.770 ± 0.003

Finally, tab. 3.5 reports DgN values obtained evaluating air kerma

following eq. (1):

(1) 𝐾 = Φ ⋅ 𝐸 ⋅ (𝜇𝑡𝑟

𝜌)

𝑎𝑖𝑟

where Φ is photon fluence in mm-2 , E is photon energy and (𝜇𝑡𝑟

𝜌)

𝑎𝑖𝑟 is the

mass energy transfer coefficient of air.

Tab. 3.5: DgN values obtained with the three different beams with air kerma calculated with the analytic formula (1).

Phantom

diameter

(cm)

DgN (µGy/µGy)

[fg=0%]

DgN (µGy/µGy)

[fg=50%]

DgN (µGy/µGy)

[fg=100%]

8 0.6804 ± 0.0014 0.6497 ± 0.0015 0.6125 ± 0.0008

10 0.6541 ± 0.0009 0.6146 ± 0.0012 0.5698 ± 0.0004

12 0.630 ± 0.002 0.5814 ± 0.0009 0.5304 ± 0.0010

14 0.602 ± 0.001 0.5470 ± 0.0014 0.4921 ± 0.0008

53

3.2.5 Compton multiplicity

Compton scattering has a primary importance in dose diffusion within the

phantom and in image quality. Actually, scattered radiation causes dose

depositions far from the point where the photons had their first interaction:

in the case of the human body, it means that a great attention has to be paid

to radiosensitive organs, since they can be accumulate dose even if they are

not directly irradiated. While analyzing the DSF, we already noticed that

they assume very small values far from the center of the phantom, but these

values are never zero. Thus, it is interesting to study how many times a

photon can have Compton interactions, because it may help to better

understand dose distributions. Fig. 3.18 shows a Compton multiplicity

histogram for a 50% glandularity phantom, 14 cm diameter at the energy of

38 keV. The histogram shows the fraction of photons entered in the

phantom which had undergone a certain number of Compton interactions

within it versus the number of Compton events. The photons which had no

Compton interactions are included in the first bin of the histogram and

classified with three criteria: the first subcolumn is for those photons which

had a photoelectric interaction, the second for Rayleigh events and the third

is for transmitted photons. m value stands for the histogram average

calculated for those photons who had at least a Compton interaction while

passing through the phantom. n value means the percent of photons which

had undergone more than 5 Compton interactions: the sum of the columns

of the histogram and n is 1. The graph shows that less than 20% of primary

photons had a photoelectric interaction, while about 10% of them had a

Rayleigh and only about 12% of all the photons passes “undisturbed”

through the phantom. About 70% of the photons had at least a Compton

event. The probability of multiple scattering decreases rapidly with the

number of interactions. Actually, a Compton event means a loss of energy,

and every time a photon is scattered due to a Compton event, the

probability of a photoelectric interaction increases, thus only about 2% of

all the photons is scattered more than 5 times. These events are pretty rare

54

and are favored by phantom height: 50 cm of tissue can contribute

significantly to radiation spreading. This fact influences the average values,

which are showed in fig. 3.19 (a); these values increase with increasing

diameter, because a greater volume of interaction favors Compton

scattering. Also glandularity has an important influence, since Compton

cross section is higher for adipose tissue than for glandular; 100% glandular

tissue absorbs more radiation, thus limiting the number of Compton events

a photon can undergo. Fig. 13.19 (b) shows the cumulative distribution

function of Compton events versus the number of Compton events for the

14 cm diameter phantoms, 0%, 50% and 100% glandularities, at the energy

of 38 keV. It was computed only for the photons that had at least a Compton

interaction; so, it can be seen that about a half of all the scattered photons

underwent only one scattering event. The trends depend only weekly from

glandularity: 0% curve is only a little higher than 50 % and 100% curves.

The functions asymptote to 1 and reach 90% of their value before 4: almost

all the photons are absorbed by the phantom or escape from it after 4

Compton interactions, which could change significantly their direction,

contributing to dose spreading in the whole phantom.

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

E=38 keV

f_g=50%

d=14 cm

m=2.05

n=0.02

ph

oto

n fra

ctio

n

# of Compton events

Compton

Photoelectric

Rayleigh

Transmitted

Fig. 3.18: Compton multiplicity histogram for a 14 cm diameter phantom of 50% glandularity at the energy of 38 keV. The histogram shows the fraction of photons entered in the phantom which had undergone a certain number of Compton interactions versus the number of Compton events. The photons which did not undergo Compton interactions are included in the first bin of the histogram: the first subcolumn is for those photons which had a photoelectric interaction, the second for Rayleigh events and the third is for transmitted photons. m value stands for the histogram average calculated for the photons who had at least a Compton interaction. n value means the percent of photons which had undergone more than 5 Compton interactions.

55

8 10 12 141.7

1.8

1.9

2.0

2.1

2.2

Ave

rag

e o

f C

om

pto

n e

ve

nts


0 %

50 %

100 %

E=38 keV

(b)

1 2 3 4 5 6 7 8 9

0.4

0.5

0.6

0.7

0.8

0.9

1.0

E= 38 keV

d=14 cm

Cu

mu

lative

fu

nctio

n

# of Compton events

0 %

50 %

100 %

(c) Fig. 3.19: (a) Average of the Compton multiplicity histograms versus phantom diameter. The values are computed only for those photons which had at least a Compton event; from the top to the bottom, the curves represent the averages for the 0%, 50% and 100% glandularity phantoms. (b) Cumulative distribution function relative to the 14 cm diameter phantoms, for all the three glandularities studied, versus the number of Compton events, at the energy of 38 keV.

3.2.6 Fluence at detector surface

Fig. 3.20 shows beam transmittance as a function of phantom diameter for

all the phantoms studied, evaluated in the detector area shadowed by the

phantom.

8 10 12 140

5

10

15

20

25

E=38 keV

Tra

nsm

itta

nce (

%)


0

50

100

0% th.

Fig. 3.20: From the top to the bottom: beam transmittance evaluated in the phantom shadow (1 mm x phantom diameter) for the 0%, 50% and 100% breast phantoms versus phantom diameter at the energy of 38 keV. Bottom curve represents beam transmittance calculated for slabs of 0% glandular tissue whose thicknesses coincide with phantom diameters.

From the top to the bottom, the curves are referred to 0%, 50% and 100%

glandularity phantoms, while the last curve shows theoretical

transmittance for a 0% glandularity slab evaluated with Lambert-Beer

attenuation law:

(1) 𝐼 = 𝐼0𝑒−𝜇𝑥

56

where I is beam intensity at detector plane, I0 is initial beam intensity, µ is

the linear attenuation coefficient and x takes in account the different

thicknesses of the slabs, from 8 to 14 cm, the same range of phantom

diameters. As the graph shows, simulated values range from about 7% for

a 100% glandularity 14 cm diameter phantom, to almost 25% for a 0%

glandularity, 8 cm diameter phantom. Thus, 100% glandular phantoms,

which have an higher density and absorb more radiation, transmit less

photons than 0% phantoms. Phantom diameter influences dramatically

these values, which fall off approximately with an exponential trend for

greater diameters, because the thickness that photon have to pass through

before reaching the detector increases. It can be seen from the lower curve

in fig. 3.20 that the values found by simulations are about 10% higher than

the theoretical values calculated for a slab whose thickness coincide with

phantom diameter. This happens because phantom profile is circular and

not flat: at the border of the phantom, attenuation is small because photons

have to pass through a smaller thickness, and this contributes to increment

transmittance with respect a uniform slab. A uniform slab attenuates all the

photons in the same way because the thickness the photons have to pass

through is constant. Thus, the differences between the upper and the lower

curve are due to geometry, as if the phantom would be a slab with an

effective diameter smaller than its real value.

The beam profile on the detector plane, normalized to the photons

fluence, is showed in semi-log scale in fig. 3.21 for four different 50%

glandularity phantoms; from the top to the bottom, the profiles for 8, 10, 12

and 14 cm diameters are showed. At the borders, these profiles appear flat

because of all the primary photons that do not pass through the phantom

and reach the detector. Actually, a little fraction of these photons is

attenuated by the great air gap which separates the phantom and the

detector: 2 m of air cause an attenuation of about 7% at the energy of 38 keV.

Proceeding towards the center of the profiles, from the top to the bottom,

the curves show the effect of phantom diameter on attenuation: an 8 cm

57

diameter phantom attenuates about 8% less than a 14% diameter. The trend

of these curves is decreasing from the border to the center due to phantom

shape: at the center, photons have to pass through the whole diameter, so

there is a minimum, while at the border only a little fraction of them is

attenuated.

0 3 6 9 12 15

0.1

1

10 cm

12 cm

14 cm

o

ut/

in

Horizontal position (cm)

8 cm

E=38 keV

fg=50%

Fig. 3.21: Relative intensity profile on the detector along beam width (from 0 to 15 cm) for a 50% glandularity phantom and four different diameters (from the top to the bottom: 8, 10, 12 and 14 cm). The plot is represented in log scale and shows photon fluence at detector plane normalized to photon fluence entering the phantom.

Fig. 3.22 shows the effect of different phantom compositions.

2 4 6 8 10 12

0.1

1

fg=50% (th.)

fg=100%

fg=0%

E=38 keV

d=12 cm

o

ut/

in

Horizontal position (cm)

Fig. 3.22: Photon fluence at detector plane normalized to photon fluence entering the phantom for a 12 cm diameter breast phantom of three different glandularities (from the top to the bottom: 0%, 50% and 100%) versus horizontal position. The horizontal plot labeled with fg=50% (th.) is the theoretical profile of a slab made of 50% glandular tissue and 12 cm thickness.

The graph shows in log scale the intensity profiles in correspondence of the

zone shadowed by the phantom for three 12 cm diameter phantoms of three

different glandularities (0%, 50% and 100% from the top to the bottom). As

58

expected, an adipose tissue phantom attenuates less than a 50% and a 100%

glandular phantom. This effect is more visible at the center, where the

difference is about 5%, and becomes less evident at the border. The

horizontal line shows the attenuation for a 50% glandular tissue uniform

slab of 12 cm thickness calculated with eq. (1). With respect to the cylindrical

phantom, the values are always smaller; the differences diminish

proceeding from the borders to the center, where photons have to pass

through greater and greater thickness. In particular, in correspondence of

the diameter, the two values assume almost the same value. Actually, the

circular profile is higher also due to the little amount of scatter that reach

that zone, which is not taken in account in eq. (1). These considerations

explain why phantom transmittance values showed in fig. 3.20 are always

higher than the values calculated for a uniform slab. Fig. 3.23 (a) shows the

energy spectrum of the photons reaching the whole detector area (2.5x25

cm2) for a 50% glandularity, 12 cm diameter phantom. The plot, in semi-log

scale, shows the number of photons of a given energy, normalized to the

total number of photons of the beam versus photon energy, for energies

greater than 30 keV. Actually, it is more useful studying the energy

spectrum only in that part of the image which is used for image

reconstruction, i.e. the part shadowed by the phantom. It is showed in fig.

3.23 (b) for the same phantom as above; the plot shows the number of

photons reaching an area of 0.1x12 cm2 normalized to the integral of the

curve. The trend is the same showed in fig. 3.23 (a); scatter contribution does

not exceed 10-5 of the total. Low energy photons, resulting from multiple

scattering, are attenuated with greater probability with respect higher

energy photons and their amount is negligible (less of 10-6 of the total). It is

to say that these values are so small also due to the laminar profile of the

photon beam and its small size (1 mm): such characteristics reduce the

amount of scatter radiation which reaches the detector. Moreover, also the

great air gap enhances scatter rejection: even photons with a little scattering

angle diverge sufficiently not to reach the detector.

59

30 32 34 36 3810

-7

10-6

10-5

10-4

10-3

10-2

10-1

Energy spectrum

E=38 keV

fg=50 %

d=12 cm

area=2.5x25 cm2

Re

lative #

of p

hoto

ns

Energy (keV) (a)

30 32 34 36 3810

-7

10-6

10-5

10-4

10-3

10-2

10-1

Energy spectrum

E=38 keV

fg=50 %

d=12 cm

area=0.1x12 cm2

Re

lative

# o

f ph

oto

ns

Energy (keV) (b)

30 32 34 36 381E-5

1E-4

1E-30.6

0.81

E=38 keV

d=12 cm

area=0.1x12 cm2

fg=0%

Cu

mula

tive

fun

ctio

n

Energy (keV)

fg=100%

(c)

Fig. 3.23: (a) Energy spectrum (in semi-log scale) for a phantom of 12 cm diameter and 50% glandularity versus photon energy; beam energy is 38 keV. This spectrum has been evaluated in the whole detector area (2.5x25 cm2). Data are normalized to the total number of photons; only data for energies beyond 30 keV are shown. (b) Same spectrum as above, evaluated only in the phantom shadow on detector plane (0.1x12 cm2). Data are normalized to the number of photons detected. (c) Cumulative distribution function (in semi-log scale) for a 12 cm diameter phantom and three different glandularities (0%, 50% and 100%) versus photon energy. Data are referred to the graph showed in (b).

Actually, if d is the distance between phantom and detector (2 m) and l1 and

l2 are sides of the detector (2.5 cm and 25 cm, respectively), the angle

covered by the detector is given by

60

tan−1 (𝑙1

𝑑) = tan−1 (

25 𝑐𝑚

200 𝑐𝑚) ≅ 7°

in the horizontal direction and

tan−1 (𝑙2

𝑑) = tan−1 (

2.5 𝑐𝑚

200 𝑐𝑚) ≅ 0.7°

in the transverse direction: only photons scattered at very small angles are

accepted. Increasing energy, there is an important contribute due to

Compton scattering. Actually, a Compton shoulder of the first order can be

noticed in the graph, which extends from 33 to 38 keV. Its width represents

the maximum energy transferred to the electron after a Compton

interaction and is given by the formula

Δ𝐸𝑚𝑎𝑥 =2𝐸0

2

𝑚0𝑐2 + 2𝐸0

where E0 is the initial photon energy and m0c2 is the mass at rest of the

electron. The difference between E0 and ΔEmax is then photon energy after

the interaction. If E0 is 38 keV, ΔEmax is 4.92 keV: the minimum of energy for

the photons reaching the detector after a single Compton scattering is then

33.08 keV. The peak of the spectrum is at 38 keV, whose value is about 4

orders of magnitude higher than the Compton shoulder because of the

influence of the primary photons at the border of the beam which do not

interact with the phantom.

The graph in fig. 3.23 (c) shows the cumulative distribution function

for the energy spectrum for three phantoms of 12 cm diameter and 0%, 50%

and 100% glandularities. Using Pixirad detector [18], there is the possibility

to set two different energy thresholds, and so it is important to know how

much signal one loses varying the threshold. The plot evidences that there

is no substantial difference due to phantom composition; anyway, due to

the very little amount of scatter photons that reach the detector, this loss is

minimal (always less than 10-4 of the total) for every value of the threshold.

61

However, it is to remembered that these data are evaluated for a 1 mm size

photon beam and so, for a 3.94 mm size beam, as the one used in SYRMA-

CT project, the results could be different.

3.2.7 Comparison with the literature

SR BCT is a novel technique in the field of biomedical imaging; most of the

literature is dedicated to cone beam BCT. Anyway, it is useful to compare

the two geometries in order to find similarities and differences: the trend in

the values should be approximately the same. Scatter dose spread functions

can be compared to those found by Boone [16] using 120 kV photons in fan

beam geometry and 30-40 cm diameter water phantoms. Actually, water

phantoms are used because they mimic soft tissue absorbing properties. The

mean energy of a 120 kV photons is greater than 38 keV, the energy used in

the simulations for this thesis; moreover, phantom diameters are almost

three times bigger than the ones simulated for this work. Water sDSF have

the same trend found for breast tissue, with a great peak at z=0 and an

exponential decrement at the borders of the phantoms. As found for breast

tissue phantoms, dose in peripheral zone decreases more rapidly than in

the central zone. However, due to the greater phantom diameter and higher

energy employed, scatter contribution is more evident even at large

distances from the center of the phantoms. The values found for

equilibrium dose for water phantoms are about 45% greater than the values

of DSF(0). The same values computed for breast phantoms shows

equilibrium dose values about 40% greater than DSF(0). This can be

attributed again to the different conditions (phantom materials and

diameter, different mean energies of the beam): at lower energies and

diameters, scatter contribute is minor and this means that it contributes less

to the cumulative dose in z=0. Dose SPR assumes in all the cases values

centered around 1, which means that scatter photons contribute to the dose

distribution as much as primary photons, even if scatter contribute is

slightly higher for bigger phantoms. This is not surprising, since

62

photoelectric and Compton cross sections are about the same at 38 keV for

soft tissues. Water phantoms used in Boone’s simulations have greater

diameters and then scatter photons can spread more in the phantoms, thus

increasing the dose SPR, which can reach values of about 2, even if the

energy is the same. Dose radial profiles can be compared to those found by

Thacker and Glick [6] for breast phantoms surrounded by a layer of 2 mm

(instead of 1.45 as assumed in this thesis) skin, with photons of 40 keV

energy. The trend is the same already described in par. 3.2.4: dose has a

maximum in the periphery and a minimum in the center of the phantom,

and differences between center and periphery in the order of about 25-30%.

The difference between these two values calculated for the simulations of

this thesis range between 30% and 70%, passing from a 8 cm to a 14 cm

diameter phantom. Actually, it is to consider that Thacker and Glick

simulations were performed with breast size of 1.5 times the radius. Thus

dose homogeneity is greater, because of a smaller contribution of scattered

radiation from great distance, which is not negligible for extremely long

phantoms as the ones simulated in this thesis. These values can be also

compared to the ones found by Sechopoulos [22] at the center of a hemi-

ellipsoidal breast phantom with a 49 kV photon beam, whose mean energy

is 30.3 keV. In this case, the difference between maximum and minimum is

about 60%, and it can be justified by means of the minor effective energy

employed. Actually, at 30.3 keV the linear attenuation coefficient is greater

than at 38 keV; a great fraction of the photons is absorbed before reaching

the center of the phantom, thus the dose distribution is less uniform than at

higher energies. DgN values can be compared with these found by C. Fedon

[55] during his PhD research for the SYRMA-CT project. He simulated

phantoms of 8, 10, 12 and 14 cm diameter and 0%, 50% and 100%

glandularity, as the ones of this thesis, and several energies, including 38

keV. Values found have the same trend: they decrease increasing

glandularity and increasing breast diameter, but there are small differences,

always smaller than 4%, probably due to the different way to score dose

63

deposition: glandular dose has not been computed as cumulative dose from

a DSF but irradiating the whole phantom. As seen before, while talking

about the differences in computing dose by means of the DSF or with a

photon beam which irradiates the whole phantom at the same time, the first

method causes an underestimate of dose.

0.45 0.50 0.55 0.60 0.65 0.700.45

0.50

0.55

0.60

0.65

0.70

E= 38 keV

Dg

N (

C.

Fe

do

n, U

Trie

ste

)

DgN this work

Equation y = a + b*

Adj. R-Squar 0.99564

Value Standard Err

Christian Intercept -0.0109 0.01235

Christian Slope 1.03269 0.0206

(a)

8 10 12 14

0.5

0.6

0.7

E= 38 keV

fg=100%

fg=50%

Dg

N (G

y/

Gy)


fg=0%

(b)

Fig. 3.24: (a) Comparison between all the DgN values found in this work and the ones found by C. Fedon [24] at the energy of 38 keV. (b) Comparison between DgN values found in this work (full points, straight line) and the ones found by C.Fedon (empty point, dashed line) versus phantom diameter at the energy of 38 keV. The three curves, from the top to the bottom, are referred to 0%, 50% and 100% glandularity phantoms.

The graph in fig. 3.24 shows the DgN values found in this work versus those

found by C. Fedon; DgN values of this work, found by means of the

convolution of the DSF, are slightly smaller than the values found

irradiating the whole phantom, and the difference, always smaller than 2%,

diminishes increasing glandularity. This is due to the underestimate of dose

with the DSF, which is greater for phantom of smaller glandularities: scatter

dose tails have greater importance in those phantoms, since Compton

64

scattering cross section make dose spread in the whole phantom, even at

great distances from the center. Thus, due to the underestimation of the

scatter dose tails, cumulative dose in adipose tissue phantoms results

smaller than the value found irradiating the whole phantom at the same

time.

The great air gap contributes to the rejection of scattered radiation

on the detector, whose contribute diminishes with increasing the gap

dimension. It is interesting to compare transmittance found with the one by

Boone [23], for breast phantoms in cone beam geometry. For a phantom of

14 cm diameter, at the mean energy of 38.2 keV, the value found is 3.28%

versus 8.19% in parallel beam geometry. This is due to the different paths

that a photon experiences in these two geometries: actually, in cone beam

geometry the probability that a photon is absorbed is greater than in parallel

beam geometry because their mean path while passing through the

phantom is longer. On the opposite, in parallel beam geometry, in

particular, at the border of the phantoms, the photons pass through a

smaller thickness of material, thus reducing absorption and increasing the

amount of radiation on the detector even of 5-10%.

65

Conclusions

This thesis has been realized within the SYRMA-CT project, promoted by

INFN. A Monte Carlo simulation code with Geant4 has been developed.

This code simulates SYRMA-CT experimental setup, in particular the

photon beam, breast phantoms and the detector. Geant4 Livermore sub-

package was used to simulate electromagnetic interactions. The scoring of

dose and other quantities has been performed with Geant4 class

G4UserSteppingAction. The code has been validated by comparing the

results of CTDI measurements and simulations. For this purpose, a pencil

ionization chamber inside Plexiglas phantoms of 8, 10, 12 and 14 cm

diameter and 15 cm height has been simulated; synchrotron laminar beam

and monochromatic energies in the range 18 − 40 𝑘𝑒𝑉 were employed.

Dose values scored in the chamber have been elaborated to compute

𝐶𝑇𝐷𝐼100,𝑤𝑃𝑀𝑀𝐴 and 𝐶𝑇𝐷𝐼100,𝑐

𝑃𝑀𝑀𝐴normalized to 𝐶𝑇𝐷𝐼100,𝑐𝑎𝑖𝑟 . Simulations replicate

well the measurements trend, in particular if the real beam, obtained with

the exposure of radiochromic films, is employed. Then, dose distributions

in breast phantoms have been simulated. The phantoms had diameters of

8, 10, 12 and 14 cm and fixed height of 50 cm in order to obtain dose

distributions. The phantoms were divided in 1 mm3 voxels and absorbed

dose was registered in each voxel, distinguishing between primary and

scatter interactions. The results were elaborated to compute DgN and the

dose spread functions. Compton multiplicity has been evaluated to obtain

histograms of the number of Compton events that photons undergo while

passing through the phantom. Finally, also energy spectra of photons

reaching the detector and beam transmittance have been evaluated.

From dose spread functions, we obtained important information

about dose distributions. The utility of DSF is the possibility to study dose

distributions in phantom of arbitrary length by running only a simulation

at its center. Dose spread functions have been used to compute cumulative

and equilibrium dose and to evaluate the importance of scatter in dose

distribution along the axis of the phantom. Also the profiles along phantom

66

diameter were analyzed in order to study the dose distribution in the

coronal plane of the phantoms. The difference between phantoms of

different glandularity and diameter has been evaluated. The dose

distributions allow the development of new models for the radiological

risk, which are based on integral values like MGD, at now. Dose spread

functions allowed to evaluate DgN values for the different phantoms,

which are an indispensable information for the dosimetric protocol. This

method for computing DgN has also been compared with the method

consisting in irradiating the whole phantom; there is a slight difference

between them, due to the importance of dose tails, which is underestimated

in computing cumulative dose. Also air kerma measurements with an air

phantom and an ionization chamber have been compared, and the same

underestimation of dose found for breast phantoms was found. Detector

simulations allowed the evaluation of the photon fluence at the entrance of

the detector; this is useful for future evaluations of image quality, since it

depends strongly on the amount of radiation detected.

These simulations may be useful for the future development of

SYRMA-CT project. The code permits the study of the scan protocol, and

DgN for helical scans with pitches greater or lower than 1 will be evaluated.

Also, further study is necessary to evaluate the signal-to-noise ratio (SNR),

the contrast and other important parameters for image quality. New

simulations could be done with the presence of masses or details inside the

phantoms to evaluate these parameters and implementing the real

characteristics of the detector, its substrate material and pixel size in

particular. These aspects are fundamental to study the tradeoff between

dose and image quality, which is necessary to ensure a good diagnosis by

the radiologists and a minimum health risk for the patients.

67

Appendix A. Code user’s guide

This code has been developed within the project SYRMA-CT. The aim is to

evaluate the MGD for breasts of different diameters and compositions in

SR BCT; the breast is implemented as a cylindrical phantom of breast tissue

surrounded by skin. As a second approximation, this code also provides

dose distributions in the phantom, both in axial and in radial direction.

Moreover, it permits to calculate the energy spectrum of the photons

entering the detector, their fluence and the histogram of the Compton

events that happen within the phantom.

The PC used to run the simulations has a quad-core processor (Intel

Core I7 mod. 3770, 4 GHz CPU); Geant4 vers. 10.0 is installed on a virtual

machine with Ubuntu 13.04 operative system, with 8 GB RAM DDR3

dedicated. Ten classes have been implemented, for a total of 3700 rows of

C++ code. As for the computational time, it varies with phantom diameter

and beam energy: increasing these characteristics, more and more events

have to be simulated, thus the time for a simulation grows. At the energy of

38 keV and in the range 8-14 cm for phantom diameter, the time necessary

to process 105 photons is about 5 s. The amount of RAM dedicated to

Ubuntu is sufficient to manage four processes at the same time. Memory

occupied by the results of a simulation varies with phantom height, since

the number of files produced to compute dose maps is proportional to

phantom height in mm; in general, it is about 30 MB.

The phantom can rotate and translate. All the phantom

characteristics can be set by macro: its dimensions (diameter and height), its

position in the reference frame, its material (with also the possibility to

choose skin thickness). It is also possible to insert an ionization chamber at

its center or at the periphery, if one wants to evaluate CTDI, and a spherical

mass, whose diameter and position can be modified by macro, if one wants

to study image quality parameters.

68

As for the detector, its dimensions, position, material and pixel size can be

set via macro. For this thesis, the detector was made of air, so we only

studied the properties of the photons that reached it and not the interactions

within it. Macros also permit to manage beam characteristics: its shape,

energy, the particles. So, a laminar and monochromatic beam has been used

in these simulations, whose size was 15x0.394 cm2.

How to modify, compile and launch the program.

As soon as one opens a terminal, in order to compile or use the code, he has

to write this command:

cd /home/adminlab/geant4.10.00.p01-install/share/Geant4-

10.0.1/geant4make; source geant4make.sh

and then to go to the folder where the code is in by using the command

cd /home/adminlab/g4work/B1copia2

The code, for practical purposes, has been copied in four different folders

(from B1copia2 to B1copia5); I’ll use “B1copia2” as reference. For a not

expert user, macro commands which I’ll describe later will be sufficient to

manage the main characteristics of this program. To compile the code after

to have modified it, the user must write this command on Ubuntu terminal:

make

(if this doesn’t go, try to use make clean before). To launch the program, the

command to use it

exampleB1copia2

Table 1 sums up the commands to use to manage the code.

69

Tab. 1: Commands usable to manage the code within Ubuntu operative system.

Everytime a terminal is

opened to use Geant4

cd /home/adminlab/geant4.10.00.p01-install/share/Geant4-10.0.1/geant4make; source

geant4make.sh

To move into a specific folder

cd /home/adminlab/g4work/B1copia2

Only to compile the code

make

To use the code exampleB1copia2

To stop a simulation

CTRL+z

Macros and commands.

In order not to modify the code every time one has to launch a simulation,

some commands have been implemented to easily manage some

parameters, like phantom dimensions and material. These commands must

be written in a file whose extension is .mac or .g4mac and, when the

program is launched, they are executed by writing the command

/control/execute macro_name.mac

where macro_name is the name of the file above mentioned. The reference

frame used is such that phantom axis coincides with z axis and the beam is

in the x,z plane.

Fig. 1: (a) Reference frame in Geant4 world and (b) breast phantom oriented along z axis.

70

Fig. 2: Image obtained from a 12 cm diameter, 50 cm lenght, 50% glandularity phantom irradiated with 100 photons at the energy of 38 keV.

Table 2 contains all the commands created (UOM stands for Unit Of

Measurement):

Tab. 2: User commands to manage the simulations.

Typology Name Input

parameter

Description

Macro &

loop

/B1/loopmacro Int

Defines how many times a macro is executed in loop

/B1/setmacro String

Chooses what macro is executed via loopmacro command

Rotation

and

Translation

/B1/setRotAngle Double

Defines the rotation angle (in degrees) of the phantom

/B1/setzpos Double+UOM

Sets the position in the reference frame of the center of the phantom

Maps /B1/map String

Chooses one of the three maps (primary, secondary or total)

71

/B1/printmap String Prints one of the three maps

/B1/printhisto String

Prints dose histograms for the selected map

/B1/printprof String

Prints DSF(z) for the primary or secondary dose or their sum

/B1/clearmap String

Empties the selected map in order to free RAM.

Ionization

chamber

/B1/setchamberposition String

Set ionization chamber position by using center or periphery strings.

/B1/setcap Bool

To put or remove build up cap from the ionization chamber

Phantom

/B1/setphantmaterial String Set phantom material

/B1/setdiameter Double+UOM Set phantom diameter

/B1/setbreast Double+UOM Set phantom height

/B1/setskin Double+UOM Set skin thickness

Detector

/B1/setdistance Double+UOM

Set the distance of the detector from the center of the phantom along y axis

/B1/setdetmaterial String Set detector material; air is default.

/B1/setdetector_x Double+UOM Set detector x size

72

/B1/setdetector_z Double+UOM Set detector z size

/B1/setdim_voxel Double+UOM Set detector pixel size

Optional

mass

within the

phantom

B1/setmass Bool

To put in or remove a mass from the phantom; false is default.

/B1/setmassposition Double+UOM

Set mass position in the phantom along x axis

/B1/setmassdiameter Double+UOM Set mass diameter

/B1/setmassmaterial String Set mass material.

As for the materials, the syntax to follow is: G4_(material name with

underscores in place of the spaces); the complete list is at the end of Geant4

user’s guide.

Tab. 3: Geant4 syntax for some materials usable in the simulations.

Name Geant4 syntax

Air G4_AIR

Water G4_WATER

Adipose tissue BREAST_TISSUE_0

Breast tissue 50% glandular BREAST_TISSUE_50

Breast tissue 100% glandular BREAST_TISSUE_100

PMMA G4_PLEXIGLASS

Polyethylene G4_POLYETHYLENE

CdTe G4_CADMIUM_TELLURIDE

BGO G4_BGO

Chemical elements, for example He G4_He

73

Sometimes, some commands have to be used in particular combinations,

for example to make phantom rotate or translate:

/control/alias InitialValue 0

/control/alias FinalValue 359

/control/alias stepSize 1

/control/loop angle_loop.mac angle {InitialValue} {FinalValue} {StepSize}

The user must set initial value, final value and the step. In the macro

mentioned macro at 4th row of the example (angle_loop.mac) the following

command must be set:

/B1/setRotAngle {angle}

Similarly one could set a translation; they can be combined to perform a

helical scan, for example.

As mentioned above, also the photon beam can be set by macro. All the

commands are reported in Geant4 user’s guide, here I’ll report only a part:

Tab. 4: User commands to manage the particle beam.

Name Input parameter Description

/gps/particle String Set the particle of the beam.

/gps/energy Double+UOM Set monochromatic energy of the beam

/gps/direction Double Double

Double

Imposta tramite i tre coseni direttori la direzione del fascio

/gps/pos/type String

Set source distribution: Point, Plane, Beam, Surface, Volume

/gps/pos/shape String

Set source shape: Circle, Annulus, Ellipse, Square, Rectangle; Shpere, Ellipsoid, Cylinder, Para

74

/gps/pos/centre Double Double

Double

Set beam position in the reference frame

/gps/pos/halfx (halfy,

halfz) Double+UOM

Set the half dimensions of the beam

/gps/ang/type String

Set angular distribution of the beam: iso, cos, planar, beam1d, beam2d, focused, user

/gps/ang/rot1 (rot2) Double Double

Double

Rotation axes for the angular distribution of the beam

/gps/ang/mintheta

(minphi, maxtheta,

maxphi)

Double+UOM

Sets theta and phi angles

/gps/ang/sigmax

(sigmay) Double+UOM

Set standard deviation on the position of the beam along x (y) axis

To use a polychromatic beam, the spectrum must be normalized to 1 and

the following commands must be used:

/gps/ene/type User

/gps/hist/type energy

/gps/hist/point (energy in MeV) weight

For example, /gps/hist/point 0.038 0.05 if 5% of the photons has the energy

of 38 keV.

Visualization.

Visualization slows down simulations but it can be useful to check the

correct execution of the simulations. To enable it, the user must give this

command before a simulation starts:

/control/execute vis.mac

75

vis.mac macro, contained in the folder with the code, defines some of the

commands used in visualization; table 3 lists other useful commands.

Tab. 5: Commands usable while visualization is open.

Name Input

parameter

Description

/vis/viewer/set/viewpointThetaPhi Double

Double

Set the angle from which the simulation is visualized

/vis/viewer/zoom

/vis/viewer/zoomTo Double Set the zoom

/vis/viewer/set/style s or w

To visualize surface or wireframes as contours of the volumes

/vis/drawVolume String

To visualize a specific volume, for example Phantom

Output.

Output files will be as much as the times a simulation is repeated. The folder

that contains output file has a name like

(material)_(diameter)mm_(beam energy)keV

For example, for a 12 cm diameter, 50% glandularity and 38 keV

monochromatic energy, file name will be

50_120mm_38keV

This folder contains many folders:

-CTDI (if the ionization chamber is inserted);

-flusso (photon flux on detector plane);

-flussoPrimari (primary photon flux);

76

-flussoSecondari (scatter photon flux);

-info: contains some information:

Compton histograms average

Edep*G1

f(E)2

Maximum dose in a voxel *

Maximum dose in a coronal plane *

Maximum glandular dose in a coronal plane *

(*these results are reported for primary and scatter distributions and their

sum);

-mappe: contains dose maps for all the coronal plane of the phantoms;

-profili: contains DSF(z);

-nCompton: contains Compton multiplicity histograms;

-spettri: contains energy spectra of the photons at detector plane.

1 Edep is the amount of energy deposited after an interaction; G is the factor defined in [ 10], used to compute the amount of dose absorbed by glandular tissue. 2 f(E) is the energy deposited by a photon divided by its initial energy.

77

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