Studio delle prestazioni di un sistema a fosfori per mammografia ...

UNIVERSITÀ DEGLI STUDI DI FIRENZE
FACOLTA’ DI MEDICINA E CHIRURGIA
SCUOLA DI SPECIALIZZAZIONE IN FISICA SANITARIA
Studio delle prestazioni di un sistema
a fosfori per mammografia digitale
Tesi di Specializzazione in Fisica Sanitaria del
Dr. Simone Busoni
Relatore Dott. Giacomo Belli
Direttore della scuola Prof. Salvatore Romano
Firenze, 29 Ottobre 2002 Anno Accademico 2001/2002

Contents
Introduction 1
1 Computed Radiography Principles 3
1.1 Projection radiography principles . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 The imaging plate IP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 The single and dual-side reading . . . . . . . . . . . . . . . . . . . . . 11
1.3 The digitization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Characterization of a digital radiography system 15
2.1 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 The Modulation Transfer Function MTF . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Digital MTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 NPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 DQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 DQE operating definition. . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5 NEQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Experimental setup 31
3.1 The mammography unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Dose vs. mAs Relation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.2 The Half Value Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Determination of the X-ray spectrum . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 The FUJI Computed Radiography workstation FCR 5000 MA . . . . . . . . . 38
3.4 IP cassette and photostimulable storage plate . . . . . . . . . . . . . . . . . . 40
3.5 Image readout process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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3.6 Histogram analysis and IP sensitometric curve . . . . . . . . . . . . . . . . . . 43
4 Measurement of the physical quantities contributing to the DQE 46
4.1 Experimental characteristic curve . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Pre-sampling MTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3 EMTF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 NPS measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 NEQ measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 DQE measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6.1 Determination of q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6.2 Measured DQE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Conclusions 72
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Ringraziamenti 77
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Introduction
Digital X-ray imaging devices are increasingly used in medical diagnosis and will widely re-
place conventional (analogic) imaging systems in the future. The benefits of acquisition of
radiological images in digital form became quickly obvious following the introduction of com-
puted tomography by Hounsfield in 1973. These advantages include an increased flexibility in
recording images and display characteristics as well as the possibility of retrieving and trans-
mitting data through communications networks (PACS). It was only with the development of
improved detector technologies, more powerful computers, high-resolution digital displays and
commercial and affordable laser devices, that digital radiography obtained a big diffusion in
other fields, like the standard radiographic projection imaging. Initially, it was thought that
digital radiography would have to match the very demanding limiting spatial resolution perfor-
mance of film-based imaging. However, film imaging is often limited by a lack of exposure
latitude due to the film characteristic curve and by the noise associated with film granularity
and inefficient use of the incident radiation. Further experience has suggested that a high value
of limiting spatial resolution is not as important as the ability to provide excellent image con-
trast over a wide latitude of x-ray exposures for all spatial frequencies up to a more modest
limiting resolution [1]. A digital radiographic system can provide such performances as well as
allowing the implementation of computer image processing techniques [2] for the extraction of
useful medical quantitative information.
Two main digital image acquisition systems have been developed for radiography clinical
applications: the Computed Radiography (CR), currently more diffused, and the Direct Digital
Radiography. The first technology employs an image storage system that only in a second step
is read and digitized, the latter uses a solid state detector which converts immediately the image
information in digital form.
The Computed Radiography (CR) system, which is an X-ray imaging system employing
photostimulable phosphor, is the first digital radiography system of practical use. In the mid
1980s, the CR system was released to the market, and since then great improvements have been
1

made, allowing the diffusion of these technologies in highly specialized fields, as in mammo-
graphical screening where particular performances are required in terms of spatial resolution.
One of the challenges with digital projection radiography, and in mammography field in partic-
ular, is to improve the detection of abnormalities. This last goal can be obtained acting on two
different stages: 1) enhancing the image quality of the acquisition, for instance improving the
S/N ratio and dynamic range, 2) optimizing the display parameters through image processing
so that as much diagnostic information as possible can be extracted from an image read by a
radiologist.
The work and results described in this thesis have the main goal of fully characterize a
mammographical imaging storage system from a physical point of view, leaving to a further
study the optimization of the display parameters of clinical images from a reading radiologist
point of view. The definition of the parameters that describe the specific imaging properties of
these digital X-ray imaging devices is therefore necessary together with a standardization of the
measurement procedures employed. To this end a set of measurements have been performed on
the X-ray equipment and on the mammographical station (Fuji FCR 5000 MA). A set of soft-
ware routines was written to analyze the data and to calculate all the main physical parameters
involved in imaging performances of the system and to compare them with result previously
obtained and published in literature.
In Chapter 1 the working principles of a CR system are described, together with a small
summary of the theory of radiographic imaging. In Chapter 2 the main parameters describ-
ing the performances of a mammographic system are defined and the theory of linear imaging
systems is briefly summarized. The hardware and experimental setup used during this work,
their characterization and the X-ray spectrum determination are described in Chapter 3. Finally,
the measured performances of the CR system in terms of the physical parameters Modulation
Transfer Function MTF, Noise Equivalent Quanta NEQ, Noise Power Spectrum NPS and De-
tective Quantum Efficiency DQE are reported in Chapter 4, together with the description of the
procedures and software codes used to evaluate them.
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Chapter 1
Computed Radiography Principles
A Computed Radiography (CR) system is based on a detector (imaging plate) which allows a
delayed photostimulable light emission after an X-ray exposure.
The image formation process in a CR system can be thought as the sum of several steps in-
volving both the image information storage in a detecting device and the subsequent acquisition
and storage of the image as digital data. The block diagram of the image formation process is
shown in Fig. 1.1 together with the noise sources dominating each step.
The main steps involved in the PSP image acquisition process are summarized in a pictorial
view in Fig. 1.2.
The image information is retrieved in a dedicated system by raster scanning the plate with
a laser to stimulate the luminescence. The light emitted, which is proportional to the absorbed
dose, is then collected by a photomultiplier tube (PMT), digitized and finally stored on a work-
station for subsequent analysis. One of the main advantages of this technology, without forget-
ting those common to all digital radiography systems, is the possibility to expose the imaging
plate (IP) in the same projection-radiography facility usually used with conventional screen-film
based cassettes, since the detector holder has the same geometrical characteristics.
In this chapter the main physical processes and CR principles are summarized. At first,
projection radiography principles and the mechanism of radiation interaction with matters will
be briefly reviewed, together with the process of image storage in a photo-stimulable phosphor
detector. Finally, the reading and digitization steps will be described, with particular emphasis
to the elements and characteristics peculiar to the system under investigation. From now on, if
3

Figure 1.1: Block diagram of the image formation process and main noise sources
Figure 1.2: PSP image acquisition and processing.
not otherwise stated, a set of procedures and a diagnostic mammographical setup is considered.
1.1 Projection radiography principles
The radiographic image is formed by the interaction of X-ray photons with a photon detector
and is therefore a distribution of those photons which are transmitted through the patient and
recorded by the detector. These photons can either be primary photons, which have passed
4

the patient without interacting, or secondary photons, which results from an interaction. The
secondary photons will in general be deflected from their original direction and carry little
useful information. The primary photons, on the other side, give a measure of the probability
that a photon will not interact, and this probability will itself depend upon the sum of the X-ray
attenuating properties of all the tissues the photon traverses. The image is therefore a projection
of the tissue attenuating properties. The scattered photons create a background signal which
degrades contrast. In most cases the majority of the scattered photons can be removed by
placing an anti-scatter device between the patient and the image receptor. This device can be a
grid formed of a series of lead strips that are parallel to the X-ray direction.
A simple mathematical model of the formation of the radiographic image can be derived in
the hypothesis of monochromatic X-ray source. Let us consider all the incident photons to have
energy E and to be parallel to the Þ direction. The detector lies in the ÜÝ plane. We assume that
each photon interacting in the receptor is locally absorbed and that the response of the detector
is linear, so that the image can be considered as a distribution of absorbed energy. If there are
Æ photons per unit area incident on the patient and Á Ü� Ý �Ü�Ý is the energy absorbed in the
area �Ü�Ý of the detector, then:
Á Ü� Ý �Ư �� �� Ê � Ü�Ý�Þ �Þ
� ¯ �×�� �×Ë Ü� Ý� �×� ª �ª��×
(1.1)
where the first term is the primary photons contribution and the second term depends from
the secondary photons. The line integral is over all tissues along the path of the primary pho-
tons reaching the point Ü� Ý and � Ü� Ý� Þ is the linear attenuation coefficient. The scatter
distribution function Ë is defined so that Ë Ü� Ý� �� ª �ª���Ü�Ý gives the number of scattered
photons in the energy range � to � �� and the solid angle ª to ª �ª which pass through
the area �Ü�Ý of the detector. The energy absorption efficiency ¯ of the detector is a function
of both the photon energy and the angle � between the photon direction and the Þ axis. �× is
the energy of the scattered photons. It is worth noting that if the detector has an efficiency not
equal to one, the path length of the photon inside the detecting material will have an important
effect on the global efficiency. In fact scattered photons will be absorbed more efficiently than
primary photons, so that inefficient receptors will enhance the effects of scatter on the image.
The scatter function Ë has a complicated dependence on position and on the distribution of
tissues within the patient. For many applications, it is sufficient to treat it a slowly varying
5

function and to replace the very general integral in Eq. 1.1 with the value at the center of the
image. As the scatter will decrease away from the center, this will give a maximal estimate of
the contrast-degrading effects of the scatter. Eq. 1.1 then simplifies to:
Á Ü� Ý �Ư �� �� Ê � Ü�Ý�Þ �Þ
˯ � � (1.2)
where
�
Ë � Ë � � ª��× �ª��× (1.3)
and
�
¯ � � � ¯ �×�� �×Ë � ��×� ª �ª��×�Ë (1.4)
In practice, it is the ratio of the scattered to primary radiation that is either measured or
calculated, and an appropriate form of Eq. 1.2 is then:
Á Ü� Ý �Ư �� �� Ê � Ü�Ý�Þ �Þ
Ê (1.5)
The parameters used by the radiologist to evaluate the image quality are the radiographic
contrast, the spatial resolution and the noise. The last two terms will be examined more in
detail in the following chapters. The contrast is defined with respect to the target tissue to be
examined. Given the energy ÁÓ�� absorbed by the detector in correspondence of the target and
the energy Á�� � of the surrounding area, the contrast C is defined as:
� � Á�� �
Á�� �
ÁÓ��
(1.6)
The term R is one of the responsible factors of the degradation in contrast in the final image.
Other factors that affects the degradation in contrast are the target thickness and the difference in
linear attenuation coefficients. While the former factor is not under the radiologist control, the
former can be optimized by an appropriate beam quality choice. The contrast decreases rapidly
with increasing photon energy, so that for the best contrast a low photon energy is better. This is
particular important in mammography, where small targets, for example calcifications, must be
detected. A problem arises if we consider the mechanism of interaction of photons with matter.
If the transmission of photons is very low, as is the case with low energy beams, then very few
6

photons will reach the detector and the radiation dose to the tissue will be very high in order
to obtain a signal to noise ratio similar to the one obtained with a more energetic beam. The
choice of energy will therefore be a compromise between the requirements of low dose and high
contrast [3].
At the energy values typical of radiography, the most important photons interactions are the
photoelectric effect and scattering. In soft tissue the photoelectric cross section is larger than
the scatter cross section for energies up to about 25 keV. The photoelectric cross section varies
approximately as the fourth power of the atomic number and inversely as the third power of
the photon energy and it shows discontinuities at the absorption edges. The photon scattering
cross section varies more slowly with energy than does the photoelectric cross section and is
approximately proportional to atomic number [4].
The voltage applied to a X-ray tube dedicated to mammography is usually between 20 and
30 kVp (Vmax), depending on the clinical diagnosis and patient morphology. The X-ray ra-
diation is generated by Bremmstrahlung and by an X-ray emission characteristic of the anode
material. The anode material used in mammography X-ray units is molybdenum, since, with
respect to tungsten targets, it produces lower energy X-rays, which are more appropriate for
imaging thinner body sections at high contrast. Usually the spectrum is attenuated by an added
molybdenum filter (see Chapter 3.1 for the system under investigation) which reduces both the
X-ray component above its K-edge of about 20 keV and the lower energy components. This
is important since it reduces the soft component of the spectrum, which contributes only to the
patient dose and not to image contrast.
1.2 The imaging plate IP
The Imaging Plate is the detecting element of a CR system. It is a multilayer film composed
of a support layer and a sensible layer made of photo-stimulable phosphors. Photo-stimulable
phosphors, also known as storage phosphor, is one of the most successful detectors for digital
radiography to date.
The photo-stimulable phosphor (PSP) stores the absorbed X-ray energy in crystal structure
traps. The trapped energy can be released in a later moment if stimulated by additional light
7

energy of the proper wavelength by the process of photo-stimulated luminescence (PSL here-
after). The output signal shows a linear dependence to the absorbed dose. Furthermore the IP
maintains this linearity over a wide range of exposures (four orders of exposure magnitude).
The PSP is placed in a protective, light-tight cassette with the same geometrical characteris-
tics of a film-based radiographic cassette, so that it can be exposed in a radiographic equipment.
Using X-ray imaging techniques similar to screen film imaging, a latent image, in the form
of trapped electrons is imprinted on the PSP receptor by absorption of the photons transmit-
ted through the object. At this point, the unobservable latent image is processed by inserting
the IP cassette into an image reader, where the PSP plate is extracted from the cassette and
raster-scanned with a highly focused red laser light. A higher energy, low intensity blue photo-
stimulated luminescence signal is emitted, the intensity of which is proportional to the num-
ber of x-ray photons that were absorbed in the local area of the receptor. The PSL signal is
channeled to a photomultiplier tube, converted to a voltage, digitized with an analog to digital
converter, and stored in a digital image matrix. After PSP detector is totally scanned, analysis
of the raw digital data locates the pertinent areas of the useful image. Scaling of the data with
well-defined computer algorithms creates a greyscale image that mimics the analog film image.
Finally, the image is recorded on film, or viewed on a digital image monitor. The described
steps of the PSP reading process are summarized in Fig. 1.3.
PSP devices are based on the principles of photo-stimulated luminescence. When an X-
ray photon deposits energy in the PSP material, the energy can be released by three different
physical processes. Fluorescence is the prompt release of energy in the form of light. This
process is the basis of conventional radiographic intensification screens. PSP imaging plates
also emit fluorescence in sufficient quantity to expose conventional radiographic film [5] [6],
however this is not the intended method of imaging. PSP materials store some of the deposited
energy in defects in their crystal structure, thus they are sometimes called storage phosphors.
This stored energy constitutes the latent image. Over time, the latent image fades spontaneously
by the process of phosphorescence. If stimulated to light of the proper wavelength, the process
of stimulated luminescence can release the trapped energy. The emitted light constitutes the
signal for creating the digital image [7].
Many compounds possess the property of PSL. Few of these materials have characteristics
8

Figure 1.3: Main components of a PSP reader.
desirable for radiography, i.e. a stimulation-absorption peak at a wavelength produced by com-
mon lasers, a stimulated emission peak readily absorbed by common photomultiplier tube input
phosphors, and retention of the latent image without significant signal loss due to phosphores-
cence. The compounds that most closely meet these requirements are alkali-earth halides. Com-
mercial products have been introduced based on Ê��Ð, ����Ö � �Ù , ��� �ÖÁ � �Ù ,
��ËÖ��Ö � �Ù [27]. Trace amounts of impurities, such as �Ù , are added the PSP to
alter its structure and physical properties. The trace impurity is also called an activator. �Ù
replaces the alkali earth in the crystal, forming a luminescence center. Ionization by absorption
of X-rays forms electron/hole pairs in the PSP crystal. An electron/hole pair raises �Ù to
an excited state, �Ù . �Ù produces visible light when it returns to the ground state, �Ù .
Stored energy (in the form of trapped electrons) forms the latent image.
There are currently two major theories for the PSP energy absorption process and subse-
quent formation of luminescence centers: a bimolecular recombination model [8], and a pho-
tostimulable luminescence complex (PSLC) model [9]. The energy levels and the interaction
mechanisms proposed by the two theories are shown in Fig. 1.4.
Physical processes occurring in ����Ö � �Ù using the latter theory appears to closely
9

Figure 1.4: Energy levels of the PSP.
approximate the experimental findings. In this model, the PSLC is a metastable complex at
higher energy (F-center) in close proximity to an �Ù �Ù recombination center. X-rays
absorbed in the PSP induce the formation of ”holes” and ”electrons”, which either activate an
”inactive PSLC” by being captured by an F-center, or form an active PSLC via formation and
recombination of ”excitons” explained by ”F-center physics”.
In either situation, the number of active PSLC’s created (number of electrons trapped in the
metastable site) are proportional to the x-ray dose to the phosphor, critical to the success of the
phosphor as an image receptor.
Fading of the trapped signal will occur exponentially over time, through spontaneous phos-
phorescence. A typical imaging plate will lose about 25% of the stored signal between 10
minutes to 8 hours after an exposure, and more slowly afterwards. Fading introduces uncertain-
ties in output signal that can be controlled by introducing a fixed delay between exposure and
readout to allow decay of the fast component phosphorescence of the stored signal.
The latent image imprinted on the exposed BaFBr:Eu phosphor corresponds to the activated
PLSC’s (F-centers), whose local population of electrons is directly proportional to the incident
x-ray flux for a wide range of exposures, typically exceeding 10,000 to 1 (four orders of expo-
sure magnitude). Stimulation of the �Ù F-center complex and release of the stored electrons
requires a minimum energy of � eV, most easily deposited by a highly focused laser light
10

source of a given wavelength. Laser light produced by He-Ne (� � � nm) and solid state
(� � �� nm) sources are most often used. The incident laser energy excites electrons in the
local F-center sites of the phosphor. According to von Seggern [9], two subsequent energy
pathways within the phosphor matrix are possible-to return to the F-center site without escape,
or to tunnel to an adjacent Eu3+ complex. The latter event is more probable, where the electron
cascades to an intermediate energy state with the release of a non-light emitting ”phonon”. A
light photon of 3 eV energy immediately follows as the electron continues to drop through the
electron orbitals of the �Ù complex to the more stable �Ù energy level.
The readout hardware will be described in Chapter 3.
1.2.1 The single and dual-side reading
One of the most important factors in determining the image quality in an X-ray imaging system
is the X-ray utilization efficiency. In fact the noise components can be roughly classified into
two categories: quantum noise, dependent upon X-ray exposure, and fixed noise, independent
from exposure. Quantum noise includes X-ray photon noise which is caused by X-ray spatial
fluctuations, and “light photon noise”, caused by temporal fluctuations of the photo-electrons in
the photomultiplier tube (PMT). Fixed noise includes, for example, IP structural noise, electri-
cal system noise and laser noise. Especially at low spatial frequencies the X-ray photon noise
is dominant, therefore, in order to achieve an improvement in image quality of the CR sys-
tem, reduction of this noise source would be most effective [11]. In order to optimize X-ray
utilization efficiency, X-ray absorption should be increased, together with the photostimulated
luminescence detected by the photomultiplier tube. With the same image reading system, in-
creasing the thickness of the IP system would be effective, with the drawback of an acceptable
worsening of the spatial resolution performances due to multiple scattering.
In the past the IP scanning system was based on a single-side reading system. However,
in the single-side reading method (see Fig. 1.5), as the thickness of the IP is increased, the
light emission in the deep phosphor layers is less likely to be detected by the PMT, because the
photons must pass through thicker layers of material before they can reach the photodetector
located on the front side. Thus the amount of photostimulated luminescence that can be detected
by the PMT increases but tends to saturate as the phosphor layer is any thicker.
11

Figure 1.5: Single-side reading method and cross section of a conventional IP
To overcome this problem the FUJI mammographic system under investigation employs a
particular IP that consists of a thicker photostimulable phosphor layer (320 �m instead of the
usual 230 �m thickness) combined with a transparent support and a dual-side reading [10]. The
reading system, described in Fig. 1.6, is equipped with a photodetector on the support side as
well, in order to detect light that emission from both sides of the IP.
Figure 1.6: Dual-side reading method and cross section of a transparent support IP
The overall result is that the emitted photons corresponding to the X-ray absorbed in the deep
inside of the phosphor layer can be detected more efficiently by the PMT on the back side. Thus
the X-ray absorption can be increased substantially by adopting a thicker phosphor layer.
The image data detected by the respective PMT are added together with an appropriate
frequency-dependent addition ratio and then used as the final image data. The improvement in
terms of NEQ or DQE described in literature is about 30-40 % [11]. The characteristics of the
12

eading system and of the IP under investigation are describe more in detail in Chapter 3.
1.3 The digitization process
Digitization is a two step process of converting an analog signal into a discrete digital value.
The signal must be sampled and quantized. Sampling determines the location and size of the
PSL signal from a specific area of the PSP receptor, and quantizing determines the average
value of the signal amplitude within the sample area. The output of the PMT is measured at
a specific temporal frequency coordinated with the laser scan rate, and quantized to a discrete
integer value dependent on the amplitude of the signal and the total number of possible digital
values by the Analog to Digital Converter (ADC). The ADC converts the PMT signals at a rate
much faster than the fast scan rate of the laser (on the order of 2000 times faster, corresponding
to the number of pixels in the scan direction). A pixel clock coordinates the time at which a
particular signal is encoded to a physical position on the scan line. Therefore, the ratio between
the ADC sampling rate and the fast scan (line) rate determines the pixel dimension in the scan
direction. The translation speed of the phosphor plate in the sub scan direction coordinates with
the fast scan pixel dimension so that the width of the line is equal to the length of the pixel (i.e.
the pixels are ”square”).
ADC
sample
Figure 1.7: Schematic diagram of the digitization process.
The pixel size is typically between 100 and 200 �m but in mammographical applications a
pixel size of 50 �m is preferred. Typically the ADC converts the signal to a 10,12 or 16 bits
13

digital levels. The system under investigation has a 12 bit digitization ADC prior to implement a
software transformation to 10 bit/pixel image (i.e. 1024 grey levels). Furthermore the FUJI FCR
5000 MA reader uses a logarithmic amplifier on the pre-digitized data. Analog amplification
prior to the final digital conversion reduces quantization errors in the signal estimate. The spatial
sampling process is described in Fig. 1.7.
The distance between two consequent laser beam positions gives the sampling distance,
while the pixel size gives the sampling aperture. In the FCR5000 MA sampling distance and
sampling aperture are equal and correspond to the pixel size. The inverse of twice the sampling
distance corresponds to the Nyquist frequency.
14

Chapter 2
Characterization of a digital radiography
system
In this chapter the main physical quantities involved in the CR characterization will be defined
and the mathematical formalism required will be developed in the Fourier framework. The first
section is devoted to the definition of linear systems and to the Fourier methods for the image
transfer functions. In the following sections the system response function will be described as
well as the concepts of Modulation Transfer Function (MTF), Pre-sampled MTF, Noise Equiv-
alent Quanta and Detective Quantum Efficiency. This last quantity is the one that will be used
as final descriptor of the efficiency and of the global performances of the system under investi-
gation.
2.1 Linear systems
The CR system performances will be analyzed in the spatial frequency domain in order to evalu-
ate the capability to detect structures in clinical images. To this end, Fourier techniques provide
a particularly elegant framework from which we can evolve a description of the formation of
images. A key point in this analysis is the possibility to treat our system as linear. Before
developing the required formalism, the definition of linear system will be recalled. Let suppose
that a bi-dimensional input signal � Ü� Ý passing through some optical and electrical system
results in an output � �� � . The system is linear if:
15

¯ multiplying � Ü� Ý by a constant � produces an output �� �� � .
¯ when the input is a weighted sum of two (or more) functions, �� Ü� Ý �� Ü� Ý , the
output will similarly have the form �� �� � �� �� � , where �� �� � is the image
of �� Ü� Ý .
Furthermore, a linear system will be space invariant if it possesses the property of stationarity.
A system is stationary if changing the position of the input merely changes the location of the
output without altering its functional form. The idea behind much of this is that the output
produced by an optical system can be treated as a linear superposition of the outputs arising
from each of the individual points of the object. Indeed, if we symbolically represent the linear
transformation of the system as ��, the input and output can be written as
� �� � �Ä�� Ü� Ý � (2.1)
Using the shifting property of the Æ-function this becomes
� �� � �Ä
� ��
� Ü �Ý Æ Ü Ü Æ Ý Ý �Ü �Ý
The integral expresses � Ü� Ý as a linear combination of elementary delta functions, each
weighted by a quantity � Ü �Ý . It follows from the second linearity condition that the sys-
�
(2.2)
tem operator can equivalently act on each of the elementary functions, thus obtaining:
� �� � �
�� �
� Ü �Ý Ä Æ Ü Ü Æ Ý
�
Ý �Ü �Ý (2.3)
�
��
�
� Ü Ü �Ý Ý Ä Æ Ü Æ Ý
�
The quantity � Ü �Ý���� � Ä Æ Ü Æ Ý
� �Ü �Ý
� is the response of the linear system to a delta
function located at the point Ü �Ý in the input space, the so-called impulse response. In
general it will depend both on the input and output space coordinates. We can define the Point
Spread Function (PSF) as the normalized impulse response:
ÈË� Ü� Ý� �� � �
ÊÊ
� � � �� �
� Ü� Ý� �� � �Ü�Ý
16
�
� � � �� �
£
(2.4)

where £ is the system gain.
The PSF has a functional form identical to that of the image generated by a Æ-pulse input. If
the system is space invariant, a point-source input can be moved along the object plane without
any effect other than changing the location of its image, the function being the same for any
point Ü� Ý . So the dependence of the PSF on space variables can only be related to Ü� Ý as
far as the location of its center is concerned. The value of the PSF on the image plane merely
depends on the displacement at that location from the particular image point at which the PSF
is centered (see Fig. 2.1).
Figure 2.1: Convolution of a source composed of Æ functions with the PSF. The resulting pattern
is the sum of all the spread out contributions.
In other words it can be stated that:
ÈË� Ü� Ý� �� � �ÈË� � Ü� � Ý (2.5)
then, under the hypotheses of space invariance and linearity,
� �� � �£
��
� Ü� Ý ÈË� � Ü� � Ý �Ü�Ý (2.6)
It is also worth defining the Line Spread Function (LSF) as the integral of the PSF along
one direction, since this is the quantity that will be used to experimentally evaluate the spatial
17

frequency properties of our system.
ÄË� Ü� �� � �
ÊÊ
Ê
� Ü� Ý� �� � �Ý
� Ü� Ý� �� � �Ü�Ý
The LSF carries information about the system response to a linear input.
as:
(2.7)
The expression in Eq. 2.6 is a bi-dimensional convolution integral and is usually expressed
� �� � �£� Ü �Ý ª ÈË� � Ü� � Ý (2.8)
At this point, Fourier theory provides an elegant and powerful framework to deal with the
system analysis, starting from the convolution integral. Let us define the Fourier transforms
� of the functions � Ü� Ý ,� �� � and � � Ü� � Ý (� is the non-normalized PSF):
��� Ü� Ý � � � Ù� Ú , ��� �� � � � � Í� Î and ��� Ü� Ý � � À Ù� Ú . The convolu-
tion theorem states that:
or
���� � ��� ª �� � ��� �¡���� (2.9)
� Í� Î �� Ù� Ú ¡ À Ù� Ú (2.10)
In our subsequent analysis a linear and space invariant system will be considered, unless
otherwise stated. In case the relation between the input mean value of the quantities under
investigation is not linearly related to the output mean value, the linearity condition can be
recovered by using the sensitometric curve.
2.2 The Modulation Transfer Function MTF
In order to estimate the capability of an imaging system to map the amplitudes of input fre-
quency components to the amplitudes of output frequency components, the concept of Modula-
tion Transfer Function is usually introduced for analog systems. Let us consider a bidimensional
18

wave object � �� � , oscillating with spatial frequencies �� and �� and amplitude � around a
constant value �:
� �� � �� ¡ � � � � �� ���
� (2.11)
A highly useful parameter in evaluating the performances of a system is the contrast or modu-
lation Å, defined by:
Å � �Ñ�Ü �Ñ�Ò
�Ñ�Ü �Ñ�Ò
� ��� (2.12)
The image of the object trough the system can be calculated, in the linear space invariant system
approximation, as:
� Ü� Ý �
� �
� �£
� �£
�£
� �
� Ü �� Ý � ¡ � �� � ���� (2.13)
� �
� �
ÈË� Ü �� Ý � � � � � �� ��� ����
ÈË� Ü �� Ý � ����
ÈË� Ü �� Ý � � � � � � � Ü �� � Ý � � � � �Ü ��Ý ���� �£
where £ is the gain of the system. After a change in the integration variables � � Ü � and
� � Ý �, the same expression is:
� Ü� Ý ��£� � � � �Ü ��Ü
� �
ÈË� �� � � � � � �� ��� ���� �£ (2.14)
The integral is the Fourier transform of the PSF and is usually referred to as OTF (Optical
Transfer Function). The OTF is thus defined as the Fourier transform of the output of a system
which has a delta function as its input. The OTF is two-dimensional for a two-dimensional
image, but for simplicity of nomenclature only the one-dimensional case will be considered
here. With a last change of variables �Ü � �� and �Ý � ��, the image � Ü� Ý can be written as:
� � �ÜÜ �ÝÝ
� Ü� Ý ��£ÇÌ� �Ü��Ý �
�£ (2.15)
It appears that the image of a sinusoid is still a sinusoid and, as it could be expected from the
system linearity, with the same spatial frequency of the object; but the amplitude depends both
19

on the gain £ and on the function OTF. The OTF is a complex, frequency dependent function
described by a module and a phase �:
ÇÌ� � �ÇÌ� �� �� � ÅÌ�� ��
(2.16)
where the MTF (Modulation Transfer Function) is the OTF module, and � is also known as
Phase Transfer Function. If the input image is real (as will be always assumed to be the case),
then the real component of the OTF is symmetric about the zero frequency and the imaginary
component is antisymmetric, yielding a symmetric MTF. Therefore only one-half of the MTF
is typically reported (the positive frequency). We stress the fact that the MTF accounts for the
transfer of the modulation between the object and the image, at each spatial frequency. In fact,
the ratio of the object modulation ÅÓ�� and the image modulation Å�Ñ is the MTF:
ÅÓÙØ
Å�Ò
�
�¡£¡ÅÌ�
�£
�
�
� ÅÌ� (2.17)
Furthermore, one of the advantages of using MTF as a performance index of a system is that if
the MTFs for the individual independent components in a system are known, the total MTF is
often simply their product.
The more direct method for the MTF measurements is to expose the detector to individual
monochromatic signals of known amplitude, thus evaluating the MTF for each point in the
spatial frequency domain. The MTF can be measured using filters with a sinusoidal attenuation
profile. This poses several technological constraints on the realization of the filters and of the
measurement, even if a square wave profile (easier to realize), with the proper correction factor,
is used instead of the sinusoidal profile. A second method relies on the property of signals
having a very steep gradient in their structure. A function with an infinite gradient, such as a
Dirac delta or a step function or a narrow slit, has a Fourier spectrum covering all the frequency
domain, with zero values at most on a discrete sample of points. It is difficult to realize a
delta-like input but it is easier to produce a sharp edge or a narrow slit simulating a line, thus
modeling the input with a function whose Fourier transform is known. From the output image,
which contains all the spatial frequency information, the OTF can be obtained. In this work a
slit has been used as mono-dimensional input filter, to evaluate the frequency response of the
system. In fact the Fourier transform of an ideal, zero width, slit in a dimension orthogonal to
20

its direction is a constant function covering all the spatial frequency range. The image of a slit
is the so-called Line Spread Function.
2.2.1 Digital MTF
In a digital system, the sampling procedure usually leads to complication due to undersampling
in the MTF analysis since, as will be explained later, the system response depends on the spatial
frequency content of the images being evaluated. Undersampling in digital systems occurs
when the image is not sampled finely enough to record all spatial frequencies without aliasing.
Undersampling is almost always present to some degree in any real digital imaging device,
and not only makes the physical measurement of MTF more difficult but it also complicates the
proper understanding and interpretation of these quantitative measures. The main complications
are related to the fact that, when applying classical analysis to undersampled digital systems,
the MTF do not behave as transfer amplitude of a single sinusoid and the response of a digital
system to a delta function is not spatially invariant. There are two main elements distinguishing
the MTF of a digital system from that of an analog system: replication of FTs in frequency
space and the overlapping of FT segments from aliasing 1 when the system is undersampled.
While replication and aliasing overlap are certainly related, they are different in the effects they
have on the interpretation of MTF in undersampled digital systems. The replication of FTs
is a result of the infinite sum of sinusoids required to produce a signal comprised of a string
of infinitely sharp delta function, i.e. from multiplying the original function by the sampling
function � ¡ ÁÁÁ Ü� � . The overlap of these FT replications, on the other hand, is the result of
undersampling, by which aliasing causes sampled frequencies above the Nyquist frequency to
contaminate their counterpart frequencies below the Nyquist frequency (see Fig. 2.2).
In order to understand how these effects reflect quantitatively on the MTF, the terms con-
tributing to the system OTF must be considered. The OTF of a digital system is comprised
of a presampling component and a sampled component. The presampled OTF is the result of
image blurring from geometric considerations (for instance focal spot blurring), analog input
1aliasing is so named because a sampled sinusoid of frequency¡Ù above the Nyquist frequency Ù Æ is identical
in every regard to a sampled sinusoid of frequency ¡Ù below Ù Æ (if at the same phase); thus the higher-frequency
sinusoid takes the alias of the lower frequency.
21

(a)
(b)
(c)
-u N
MTF pre
MTF pre
-u N
MTF
Figure 2.2: System response to a single sinusoids. (a) MTF of an analog system. (b) MTF
of a digital system without undersampling. (c) MTF of a digital system with undersampling.
The solid line indicates the total digital MTF including the aliasing overlap from adjacent FT
replications.
subsystems (e.g. the IP response) and the aperture function of the acquisition device (e.g. the
effective laser beam shape):
ÇÌ�ÔÖ� � ÇÌ���ÓÑ ¡ ÇÌ��Ò�ÐÓ�
u 1
MTF
u 1
MTF
u 1
u 2
u 2
u 2
u N
u N
u
u
u
(2.18)
The image is digitally sampled using the sampling theorem, giving rise to the digital OTF
(ÇÌ����):
ÇÌ���� Ù ��ÇÌ�ÔÖ� Ù £ Û ¡ ×�Ò ÙÛ ℄ £ ÁÁÁ Ù� �
(2.19)
where Û is the width of the image, � is the sampling interval, and £ denotes convolution. The
sampling function �¡ÁÁÁ Ü� � is a string of delta functions separated by the pixel spacing � in the
Cartesian space; its Fourier transform ÁÁÁ Ù� � gives rise to replication in frequency space.
The factor Û ¡ ×�Ò ÙÛ in frequency space corresponds to the final image width in Cartesian
22

space. Since Û is almost always much greater than the width of the presampling LSF, one can
usually approximate Û×�Ò ÙÛ as Æ Ù . The ÇÌ���� then simplifies to:
ÇÌ���� Ù � � ÇÌ�ÔÖ� Ù £ ÁÁÁ Ù� �
(2.20)
The MTF is defined for analog systems as �ÇÌ� �. As long as there is no aliasing from
undersampling, the same definition can also be applied to digital systems without alteration.
However, when a digital system is undersampled, two conceptual difficulties arise: the digital
MTF no longer describes the amplitude of a single frequency passed by the system (see Fig. 2.2)
and it is phase dependent, and therefore not spatially invariant as required for the stationarity
property of the linear system definition of MTF.
The difficulties arising from the first item can be explained considering that two working
definition of MTF exist. The MTF can be measured as the response of a system to a delta
function:
ÅÌ���� Ù � �ÇÌ���� Ù �
�ÇÌ����
�
(2.21)
i.e. the MTF is defined as the frequency output of a system when an input consisting of uniform
frequency content is present. On the other side the MTF can be measured as the amplitude
modification of a single sinusoid passed by a system:
ÅÌ� ��� Ù � ��Ì��� Ù �
��Ì�Ò Ù �
(2.22)
where �Ì�Ò and �Ì��� are the amplitudes of the sinusoid before and after sampling. These
two working definitions of MTF are equivalent for analog systems and for digital systems at
frequencies unaffected by aliasing (see for instance Fig. 2.2 (a) and (b)). However the two def-
initions do not agree in undersampled digital systems for frequencies where aliasing causes an
overlap of adjacent MTF replications. In summary, the amplitude response of an undersampled
digital system to a single sinusoid contains replicated but not overlapped values, and is equal to
ÅÌ�ÔÖ�. The standard definition of ÅÌ����, on the other hand, is the response of a system to a
delta function input, and contains overlapped values if undersampled. The second item relates to
the fact that ÅÌ���� depends on the phase relation of the sampling comb function with respect
to the function being digitized, when aliasing occurs [12]. This effect is described in Fig. 2.3,
where the ÅÌ�ÔÖ� response of the system to a Æ function is shown. If the sampling comb is
23

(a)
(b)
(c)
-u N
-u N
-u N
MTF p re
|OTF |
d ig
|OTF |
d ig
Figure 2.3: The effects of phase dependence on digital MTF.
aligned exactly with the delta function, the resulting sampled �ÇÌ����� is given in Fig. 2.3(b).
However, if the phase of the input delta function is shifted by �� with respect to the sampling
comb, the sampled �ÇÌ����� takes the form in Fig. 2.3(c). Values of the shift between and ��
gives intermediate curves. The phase dependence of �ÇÌ����� can be derived mathematically.
If the delta input function is at a distance � relative to the origin and the sampling comb function
is centered on the origin, the ÇÌ���� is given by:
ÇÌ���� Ù� � ��� � ��Ù ÇÌ�ÔÖ� Ù ℄ £ ÁÁÁ Ù� �
u N
u N
u N
u
u
u
(2.23)
where the factor � � ��Ù represents the phase shift � relative to the origin. The value of ÇÌ����
at any frequency is just the sum of all frequency components that overlap the frequency of
interest due to aliasing:
24

ÇÌ���� Ù� � �
�
�
�
� Ó× �Ù�� Ê Ù� ×�Ò �Ù�� Å Ù� ℄ (2.24)
�
�
�×�Ò �Ù�� Ê Ù� Ó× �Ù�� Å Ù� ℄ ¢
where Ê � Ê��ÇÌ�ÔÖ��,Å � ÁÑ�ÇÌ�ÔÖ��, and the term
�
� accounts for the anti-
symmetry of the replications of the imaginary part of ÇÌ����. After computing the amount of
overlap, ÇÌ���� will be the same at each replication; therefore one needs only to perform the
calculations in the frequency range ÙÆ to ÙÆ. The amplitude of ÇÌ����, that is ÅÌ����,
is symmetric about zero frequency, so its value only needs to be calculated in the frequency
range 0 to ÙÆ. For each frequency Ù in this range, one can evaluate the sum of all overlapping
frequencies components in Eq. 2.25 and take the amplitude [12]:
�ÇÌ���� Ù� � � � �
�
��
ÅÌ� ÔÖ� Ù�
�
��
�
�� �����
�Ê Ù� Ê Ù�
È��Å Ù� Å Ù� ℄ Ó×� �� Ù� È��Ù� ℄
�
��
�
�� �����
�Å Ù� Ê Ù�
È��Ê Ù� Ê Ù� ℄
¢ ×�Ò� �� Ù� È��Ù� ℄� � Ù � ÙÆ
(2.25)
where �� � if � and � are both odd or both even, and �� � if � and � have opposite
parities. The frequencies in the sum are denoted Ù �Ù �Ù etc., all positive, and are in the ranges
�Ù �ÙÆ, ÙÆ �Ù � ÙÆ etc. These frequencies are defined as:
Ù� � Ù � ÙÆ for i odd (2.26)
Ù� � ÙÆ Ù � ÙÆ for i even (2.27)
Typically only a few terms of each sum will be needed due to the limited bandwidth of
ÇÌ�ÔÖ�. The diagram of the terms contributing to the real part of ÇÌ���� is shown in Fig. 2.4.
Finally, the value of the ÅÌ���� is given by:
ÅÌ���� Ù � � � �ÇÌ�ÔÖ� Ù � � �
�ÇÌ�ÔÖ� � � �
25
� Ù � ÙÆ
(2.28)

-2u N
Re { OTF }
dig
u1 u2 u3 uN 2uN Figure 2.4: Definition of overlapping frequencies in an undersampled system.
The value of ÅÌ���� outside the range � Ù � ÙÆ can be found by reflecting ÅÌ���� Ù � �
about zero frequency and replicating it every ÙÆ in frequency space. The phase dependence
of ÅÌ���� is demonstrated since ÅÌ���� is explicitly a function of the phase �. If the digital
system were not undersampled, then there would be no overlap of aliased frequency components
and the sum in Eq. 2.26 would have only one term, leaving �ÇÌ�ÔÖ� Ù � � � � ÅÌ�ÔÖ� Ù in
the range � Ù � ÙÆ, which is independent from phase �. In section 4.3 a method will be
described to overcome this problem by defining a new function that takes into account the MTF
average over all possible phase values.
2.3 NPS
The noise contribution can be introduced in this analysis by adding a statistical term Ò �� �
to the expression for the image � Ü� Ý in Eq. 2.6. We obtain:
� �� � �£
��
� Ü� Ý ÈË� � Ü� � Ý �Ü�Ý Ò �� � (2.29)
A first quantitative analysis can be made by considering Ò �� � as a statistical fluctuation
around the average value g(X,Y). The variance � for the statistical process on the image plane
is given by:
� � Ð�Ñ
����� � ��
� �� ��
��
� ��
��
Ò �� � ���� (2.30)
26
u

where � � and � � are the image dimensions. The integration can be extended to the entire
plane if we consider Ò �� � � outside the image. Due to the Parseval theorem [13], the
same relation is valid in the spatial frequency domain for the Fourier components �Ò ����� of
Ò �� � :
� � Ð�Ñ
��� �� � � � � �
�
�
��Ò ����� � ������ (2.31)
Starting from Eq. 2.31, the noise spectral density Ï ����� is defined as:
Ï ����� � Ð�Ñ
��� � � � � � � � ��Ò ����� � (2.32)
Finally, in order to take into account the fact that the process statistics can depend on multiple
realizations, the average of the noise spectral density is computed over a large ensemble of
realizations, thus obtaining the Noise Power Spectrum (NPS):
ÆÈË ����� � Ð�Ñ
��� � � �
� �� � � ��Ò ����� � � (2.33)
The NPS (or Wiener spectrum) describes the variance in amplitude of each frequency com-
ponent of a system.
2.4 DQE
The Detective Quantum Efficiency DQE is a physical quantity that takes into account the imag-
ing system performances both from the noise and from the spatial resolution point of view. In
order to understand the underlying reasons that lead to the definition of the operative DQE def-
inition, let us consider the expected value �� ℄and the variance Î �℄of an image signal. Given
the relation � between input and output we obtain the system characteristic curve in absence
of noise:
��ÇÍÌ ℄�� ��ÁÆ℄ (2.34)
The noise can be added as a term Æ�, with a zero expectation value ��Æ�℄ � , such that the
output is:
ÇÍÌ � � ÁÆ Æ� (2.35)
27

for every exposition. The output variance reflects the dependence from the input signal variance
and from the system intrinsic, not eliminable, variance. In order to quantify the worsening of the
image information due to the system, the simple ratio between the output and output variance
is not significant, since it usually compares different quantities and does not take into account
the functional relation between input and output.
Instead a more efficient quantity is usually used, the DQE, defined as [14]:
�É� � ËÆÊ ÓÙØ
ËÆÊ �Ò
where the signal to noise ratio SNR, for the input and output components respectively, is:
ËÆÊ�Ò � �ÁÆ℄
�ÁÆ
�ÁÆ℄
ËÆÊÓÙØ � ¬ ��ÁÆ℄
¬
¬ � ¬
��ÇÍÌ ¬ �ÓÙØ ℄
��ÇÍÌ ℄¬
��ÁÆ℄
�ÓÙØ
¬
¬
¬ �ÁÆ℄
In the ËÆÊÓÙØ definition, the output signal is projected back to the input.
(2.36)
(2.37)
(2.38)
A further interpretation of the DQE concept can be given if we consider a detector work-
ing as counter, i.e. when input and output are expressed as events number. In the Poisson
distribution validity conditions, it is valid:
and
ËÆÊ ÓÙØ �
¬
�ÁÆ℄
¬ ��ÁÆ℄
¬ �ÓÙØ ¬ ��ÁÆ℄
��ÇÍÌ ℄
� �Ò ��ÁÆ℄ (2.39)
ËÆÊ �Ò � �ÁÆ℄
�ÁÆ℄
¬
¬ ��ÁÆ℄
��ÇÍÌ ℄
��ÁÆ℄ (2.40)
��Ò¬ ��ÁÆ ℄ (2.41)
¬ �ÓÙØ where ËÆÊ ÓÙØ has the dimensions of a quanta flux. In this case the DQE can be written as:
�É� � ËÆÊ ÓÙØ
ËÆÊ �Ò
��Ò � ¬ ��ÁÆ℄
¬
��ÇÍÌ ℄¬
�ÓÙØ 28
(2.42)

2.4.1 DQE operating definition.
This definition can be extended to take into account the spatial modulation of the input signals if
we consider the spatial frequency dependence. From this point of view, the DQE is the transfer
coefficient, for each spatial frequency, of the signal to noise ratio through the system. It can be
compared to the MTF, which describes the modulation transfer. In order to obtain an efficient
operating definition of the DQE, it is necessary to rely on the linearity property of the system
under investigation (see Section 2.1). Let us define the input two dimensional signal � Ü� Ý (in
this case a dose distribution) and the output of the system � Ü� Ý (which is the dose distribution
map obtained applying the inverse sensitometric curve to the image data). The corresponding
input and output fluctuations are ¡� Ü� Ý and ¡� Ü� Ý and their respective power spectra
Ï ¡� Ù� Ú and Ï ¡� Ù� Ú . The power spectrum of an ideal system, which does not generate an
intrinsic noise, is related to the input power spectrum through the MTF:
Ï ¡� Ù� Ú �ÅÌ� Ù� Ú ¡ Ï ¡� Ù� Ú (2.43)
The DQE definition based on the average signals:
¬
¬
¬
�� �
��ÇÍÌ ℄
��ÁÆ℄ ¬ ��Ò �ÓÙØ can be extended so to obtain:
�É� Ù� Ú � ÅÌ� Ù� Ú Ï ¡� Ù� Ú
Ï ¡� Ù� Ú
(2.44)
(2.45)
where the generic average signal has been replaced by a monochromatic wave, with spatial
frequencies Ù� Ú . The input output relation is ruled by the ÅÌ� Ù� Ú coefficient, while the
variance is given by the Ù� Ú component of the corresponding power spectra.
The whole analysis is consistent since, if we introduce a signal of amplitude � Ù� Ú ,we
obtain once more that the DQE is the transfer function of the signal to noise ratio:
ËÆÊ ÓÙØ �
ËÆÊ �Ò �
ÅÌ� Ù� Ú � Ù� Ú
Ï ¡� Ù� Ú
� Ù� Ú
Ï ¡� Ù� Ú
�É� Ù� Ú � ËÆÊ ÓÙØ
ËÆÊ �Ò
� ÅÌ� Ù� Ú Ï ¡� Ù� Ú
Ï ¡� Ù� Ú
29
(2.46)
(2.47)
(2.48)

The experimental data will be analyzed following Eq. 2.45 , as described in Chapter 4.
2.5 NEQ
The quantities described in the previous section lead to the definition of the Noise Equivalent
Quanta, frequently used to evaluate the performances of imaging systems.
Since an ideal detector has the same SNR in the output as in the input, the quantity �ÁÆ ℄
in Eq. 2.41 represents the flux that makes the variance of the output equal to the one of the
real system in an ideal detector. The ideal system in fact works with a bigger flux in input, and
consequently with a better input signal. It can be also written:
�ÁÆ ℄�� �Ò
(2.49)
with �ÁÆ ℄ � �ÁÆ℄ due to the second, adimensional factor smaller than one appearing in
Eq. 2.41. The quantity �ÁÆ ℄ is thus the signal that makes the ideal system equivalent to the
real system from a noise point of view, and it is the so-called Noise Equivalent Quanta NEQ.
From Eq. 2.42 it is straightforward to derive the relation between NEQ and DQE:
�É� � Æ�É
�ÁÆ℄
(2.50)
The physical meaning of the NEQ can be also described from a slightly different point of
view. The detector is exposed to a flux �ÁÆ℄ and a ËÆÊÓÙØ is obtained. If the detector were
ideal, the same ËÆÊÓÙØ would be obtained with a flux given by NEQ (� �ÁÆ℄); the ratio of the
two flux gives an operating definition of the detector efficiency.
30

Chapter 3
Experimental setup
The entire set of measurements performed in order to evaluate the physical performances of a
complete system of CR mammography, based on a FUJI computed radiography system, have
been taken at the CSPO (“Centro Studi Prevenzione Oncologica”) in Florence (Italy) with the
supervision of the Physical Health Staff of the “Azienda Ospedaliera Careggi”.
In this chapter the characteristics of each hardware element will be described, with particular
emphasis devoted to the CR digital acquisition system FUJI FCR 5000 MA and to the X-ray
source, based on a mammography facility manufactured by Instrumentarium, in daily clinical
use at CSPO.
3.1 The mammography unit
The mammographic unit used as X-ray equipment in the present work is an Instrumentarium
model Diamond, with a molybdenum anode and an internal molybdenum filter 25 �m thick (a
25 �m rhodium filter and a 500 �m aluminum filter are also available but have not been used in
this investigation). The exit window isa1mmthick beryllium one. Great care has been taken to
perform all the measurements in a clinical diagnostic-like environment. The X-ray tube voltage
was set to the nominal 28.0 kVp, but a set of measurements have been performed to check the
precision of the control unit. A comparison of the value set on the unit control and of the same
as measured by a High Voltage Voltmeter (Victory Model Nero) is shown in Tab. 3.1.
The other parameter which can be adjusted by the operator is the mAs value, that is strictly
31

Console Value Measured Value
(kVp) (kVp).
24.0 ¦ 0.5 24.0 ¦ 0.1
25.0 ¦ 0.5 25.3 ¦ 0.1
26.0 ¦ 0.5 26.3 ¦ 0.1
27.0 ¦ 0.5 27.2 ¦ 0.1
28.0 ¦ 0.5 28.0 ¦ 0.1
29.0 ¦ 0.5 29.1 ¦ 0.1
30.0 ¦ 0.5 30.2 ¦ 0.1
Table 3.1: Comparison between the mammographic unit tube voltage as shown by the console
and the corresponding values measured by an external Voltmeter.
correlated to the dose impinging on the detector.
3.1.1 Dose vs. mAs Relation
The relation between the mAs value set on the Diamond console and the dose on the plate
has been measured with a calibrated ionization chamber. The ionization chamber Model 10x5-
6M has been manufactured by Radcal Corporation and has a sensible volume of 6 cm . It is
connected to a Radcal Converter Mod.9060 and the dose measured values are read by a Radcal
Monitor Controller Mod.9015. The experimental setup used to obtain the curve dose vs. mAs
is shown in Fig. 3.1.
The X-ray beam has been attenuated by an externally added 30 mm thick PMMA filter
positioned next to the beam beryllium exit window and without the breast compressor. The
source detector distance SDD distance was 62.5 cm and the ionization chamber was positioned
just above the cassette holder. The measured tube voltage was 27.9 kVp with an uncertainty
of 0.1 kVp. The whole setup is as close as possible to the clinical conditions and the data
we obtained allow a subsequent calibration of Dose vs. Pixel values of the acquired digital
images. It should be stressed that the characterization of the spatial resolution and the uniform
dose exposition have been performed with the cassette above the antiscatter grid. Dose values
32

62.8 cm
source-camera
distance
Ionization
chamber
30 mm PMMA
filter
Figure 3.1: Experimental setup used to measure the relation between Dose and mAs.
corresponding to a set of mAs values are shown in Tab. 3.2, where the dose values are the
arithmetic mean of at least three independent measurements. The uncertainties are calculated
as the maximum deviation from the mean.
Just to cross-check, the exposition values for a restricted set of mAs value have been mea-
sured with the Radcal monitor set on the Roengten scale. The exposition and mAs values are
reported in Tab. 3.3. The same values are plotted superimposed to the dose values (in Gray)
in Fig. 3.2, after being rescaled by a factor equal to 8.73 (for photons in Bragg-Gray cavity
approximation 1mR=8.73 �Gy).
3.1.2 The Half Value Layer
The half value layer has been measured positioning aluminum filters close to the X-ray tube
exit. So the X-ray beam is attenuated by an internal 25 �m molybdenum filter, by the 1 mm
beryllium window and by the external aluminum filters. The tube voltage is 27.9 kVp and the
mAS value is set to 12. A second HVL has been measured with a 40 mm PMMA absorber
positioned before the Al filters. In this case the mAs value is 100 mAs while the tube voltage
in unchanged. The experimental setup is the same as the one described in Fig. 3.1. The dose
33

mAs Dose (�Gy) mAs Dose (�Gy)
2.0 11.1 ¦ 0.1 80.0 540.7
4.0 25.4 ¦ 0.1 100.0 675.7
6.0 37.0 ¦ 0.2 125.0 843.3
12.0 80.0 ¦ 0.4 150.0 1013
16.0 106.0 ¦ 0.2 175.0 1183
20.0 132.0 ¦ 0.4 200.0 1354
25.0 166.1 ¦ 0.2 250.0 1696
32.0 215.9 300.0 2032
40.0 267.6 350.0 2375
50.0 335.3 400.0 2713
63.0 425.8
Table 3.2: Measured dose values corresponding to a set of mAs values set on the Diamond
console.
mAs Exposure (mR)
40.0 30.54 ¦ 0.1
80.0 62.05 ¦ 0.2
100.0 77.53 ¦ 0.2
150.0 116.2 ¦ 0.2
Table 3.3: Measured exposure values corresponding to a set of mAs values set on the Diamond
console.
values measured in both cases are shown in Tab. 3.4 and Tab. 3.5 respectively, and plotted in
Fig. 3.3.
The HVL is 0.276 mm of Aluminum without PMMA, and 0.55 mm Al with the 40 mm
PMMA filter.
34

Figure 3.2: Dose vs. X-ray tube mAs values. Black triangles are experimental data. The
exposition values measured as cross-check of the functionality of the ionization chamber are
superimposed (blue squares).
3.2 Determination of the X-ray spectrum
The X-ray spectrum is obtained using a computer model completely based on measured mam-
mography x-ray spectra. The technique does not require any physical assumption concerning
x-ray, simplifying the experimental determination of this quantity. The software routine, de-
veloped in IDL framework, allows the user to calculate the realistic polyenergetic spectra at
any voltage between 18 and 40 kV for a molybdenum anode. Additional spectral shaping by
elemental filters such as molybdenum, which is routine in mammography, as well additional ex-
ternal filters such as PMMA phantoms, can be applied to the raw spectra produced by the model
using the energy dependent Lambert-Beers law with appropriate attenuation coefficients.
Let � ��Πrepresent the photon fluence (photons/mm ) at energy � when a voltage Πis ap-
plied to the x-ray tube. At each energy “bin” (0.5 keV intervals are used), a polynomial function
35

Al thickness Dose
(mm) (�Gy)
0.0 1760
0.1 1314
0.2 1031
0.3 834.7
0.4 680.8
0.5 551.0
1.0 241.3
Table 3.4: Dose attenuation values of the beam after the internal Mo and Be filter. Aluminum
foils are externally added.
Al thickness Dose
(mm) (�Gy)
0.0 319.3
0.1 281.3
0.2 248.6
0.3 219.5
0.4 194.6
0.5 169.2
0.6 150.5
0.7 133.3
1.0 93.6
Table 3.5: Dose attenuation values of the beam after the internal Mo and Be filter and 40 mm
PMMA external filter. Aluminum foils are externally added.
was defined as in Eq. 3.1.
¨ ��Î �� ��℄ � ��℄Î � ��℄Î � ��℄Î (3.1)
36

(a) (b)
Figure 3.3: (a) Attenuation curve with Aluminum filters. (b) Attenuation curve with an added
40 mm PMMA filter.
where ����℄ define the polynomial coefficients and are tabulated in literature [15].
The mammographical system under investigation had an additional 25 �Ñ molybdenum
internal filter and a 1mm thick beryllium window. Finally, the clinical setup was simulated
adding an additional 40 mm PMMA (poly-methil-metacrylate) filter near the focal spot of the x-
ray tube. So the spectrum obtained for the molybdenum target without filter has been corrected
following the attenuation law in Eq. 3.2:
¨��ÐØ ��Î �� �� � ¡�¡��ÐØ ¨ ��Î (3.2)
where �� � is the energy dependent mass attenuation coefficient, � is the filter material
density, and ��ÐØ is the thickness of the filter interposed. ¨��ÐØ ��Î is the photon fluence of
the filtered spectrum.
All spectra have been calculated considering V=27.9 kVp. The computed spectrum after
the Mo anode is shown in Fig. 3.4 , while the spectra filtered with the internal 25 �m Mo filter
and Be 1 mm window, and with an externally added 40 mm PMMA, are shown in Fig. 3.4
37

(a) (b)
Figure 3.4: X-ray spectrum computed from experimental data as described in [15]. (a) No
filtration, 25 �m Mo filter and the final 40 mm PMMA filtered spectrum are considered. (b)
The internal 25 �m Mo filter, a 1 mm Be window and an externally added 40 mm PMMA are
considered.
The spectra obtained agree with the curves found in literature. An independent way to test the
validity of the model used to calculate the spectra is based on the measurement of the half-value
layers (HVLs) for aluminum (see also [18] for another numerical approach).
3.3 The FUJI Computed Radiography workstation FCR 5000
MA
The FCR 5000 MA system is expressly dedicated to mammography applications. It is capable
of reading 50 �m pixel size imaging plates because it employs a dual light collection reading
technology to improve the S/N ratio. Furthermore it incorporates high-density reading and high
speed data processing. The dual light collection IP reading is expected to give a deep contribu-
tion to the improvement of diagnostic performance of mammography, in particular with regard
to the diagnosis of the shapes of micro-calcifications [10]. The FCR 5000 MA is composed of
38

a cassette reader and a touch screen display for easy adjust of the readout process parameters
and operation selection (see Fig. 3.5).
Figure 3.5: The FUJI FCR5000.
The IP handling unit furnishes the cassette with shock absorbing clothes to prevent scratch-
ing of the IP. In addition to IP cleaning brush rollers positioned on both sides of the IP, it has
also cleaning guides to prevent dirtying of the plate and minimize the possibility of image im-
pairment by dust. In the configuration used during this work, the image reader was connected
to an external controller (based on a personal computer) equipped with an image processor.
The controller was connected also to a film printer and to a graphical workstation by means of
a network system. The graphical workstation was based on a further PC, dedicated to image
storage, and a pair of high definition clinical diagnosis monitors manufactured by Barco. The
controller has further features, regarding the image ID registrations and the image acquisition
and processing setup.
The hardware dimensions of the FCR5000 MA are 730 mm width, 700 mm depth and 1565
mm height, for a total weight of 280 Kg. The feed/load time specified for the high resolution
39

Figure 3.6: The graphical workstation equipped with a pair of Barco high resolution monitors
and, on the right, the controller.
18cm¢24cm cassette is about 72 sec, corresponding to a processing capability of about 48
IPs/hour. This reading rate is one of the items that need to be improved in order to provide good
clinical use performances, where higher readout rates are desired.
3.4 IP cassette and photostimulable storage plate
The cassette holding the imaging plate has been designed to cope with the specifications of a
dual light collection mammographic system. The IP is not easily accessible from the exterior
during normal operation and the body is a tight light enclosure. Anyway a fully-openable
design is employed to facilitate interior cleaning. For the protection of both IP surfaces, shock
absorbing clothes are affixed on the back surface as well as the front surface. The cassette is
available in two sizes ( � ¢ �cm and � ¢ cm ), with the same dimensional standards of
the conventional film plate cassettes, so to allow their use in pre-existing X-ray units. A picture
of the IP cassette used in this work is shown in Fig. 3.7.
A bar-code label is attached to the cassette body for IP identification purposes. When used
with an external controller, the whole system allows a bar code identification procedures that
prevents the user from possible mistakes and makes easier the input of the patient parameters
and the association of the image information with the readout data.
The photostimulable storage plate is comprised of a barium fluorobromide/iodide (BaFBr ���I � �)
compound with an activator (Europium) [16]. The high resolution dual side (HR-BD) reading
plate thickness is about 30 % greater with respect to the single side reading plate (� �m),
40

Figure 3.7: The IP cassette.
both manufactured by FUJI for computed radiography. This in order to increase the amount
of X-ray absorption and to reduce the photon noise. In general, increasing the thickness of the
phosphor layer reduces sharpness. However the FUJI IP compensates this effect by minimizing
the phosphor grain size, which enhances sharpness [10]. A comparison between the single side
reading and dual side reading IP layer structure is shown in Fig. 3.8.
Figure 3.8: The IP layer structure for the FUJI High Resolution single and dual side reading
plates.
For all the measurements done, described in Chapter 4, the same IP has been used. Never-
41

theless a set of cassettes has been exposed to the same dose and the results have been compared
in order to be sure of the uniformity of the response over a larger set of IPs. Prior of every ex-
position, the IP is inserted in the cassette reader and submitted to a complete cancellation cycle,
so to minimize possible ghosts effects. Great care has been put in reading the IP with the same
delay (approximately 30 seconds) after each exposure session so to reduce the decay effect of
the stored information.
3.5 Image readout process
The IP readout process starts when the cassette is inserted in the FCR5000 MA reader. The IP
is internally extracted from the cassette box. A laser beam is focused on the sensitive part of the
IP and scan the detector surface in a direction parallel to the longer side (scan direction). When
the first row has been read, the plate moves a step corresponding to a pixel size in a direction
parallel to the shorter side (sub-scan direction) so to start a new row acquisition. The laser
source is a semiconductor device with a spectral emission centered on 660 nm and a power of
60 mW. The light beam is separated in two components by a beam splitter just outside the laser
output window. The less intense components is directed onto a photo-detector that provides
the reference signal necessary to deal with possible intensity fluctuations during the scanning
operation. This is a key feature since the IP response is a function of the laser intensity arriving
on the photostimulable phosphor layer. The main beam is directed towards an optical system
(polygonal mirror, collimating lens, deflection mirror) that has the main goal to steer the bean
on the desired portion of the IP and to maintain an uniform speed and focus on the plate. The
laser spot size is about 50 �m, corresponding to the pixel size dimensions. The scan speed has
an upper limit set by the phosphor decay time constant, equal to ���s. This poses a minimum
limit to the time required for reading an IP with 50 �m pixel size and a 18cm¢24cm size
(3540 ¢4740 pixels), that is about 12 sec. The total read-out time is in any case dominated
by the feed-load-eject mechanical procedure. When the laser reaches the end of the scan line,
it is repositioned to the beginning of a new line by the polygonal mirror. In the meantime the
plate moves in an orthogonal direction with a 50 �m step. This process ends when the entire
detector surface has been exposed to the laser light. The light emitted is proportional to the PSL
42

centers activated in the illuminated area and thus to the dose exposure on the plate, with a linear
dependence. The laser power determines the ratio of the information released by the F-centers
and consequently, the reading time and the remaining unused information.
The radiation emitted by the F-center from both sides of the IP is isotropic. In order to
collect most of the emitted radiation from the upper part of the plate, a light guide receives the
direct light and the light deflected by a mirror that has the goal of cover most of the solid angle
as possible. The lower side of the IP is coupled to a second light guide and a second PMT. The
signal from the two sides, properly weighted, are added together. The photostimulated emis-
sion is centered at about 400 nm, a wavelength which differs deeply from the laser stimulating
radiation, so to avoid interference between the stimulating and detection process. In fact the
photomultiplier tubes have their sensitivity peak at 400 nm.
An analog amplifier stage that applies a logarithmic conversion follows the PMT. This is
necessary to recover the linearity between the photostimulated light and the electrical signal
which had been lost in the PMT photo-electron amplification process [17]. Then the analog to
digital conversion takes place.
The system under investigation digitize the total X-ray induced signal over a range of in-
cident exposures four orders of magnitude wide (from 0.01 mR to 100 mR) keeping a linear
relation between exposure dose and ADC counts. The 12 bit ADC produces 11 bits of effec-
tive quantization levels prior to normalization of the image 10 bit depth. The output data for a
typical high resolution image used during this analysis are stored in 32 MBytes files.
In order to perform a complete cancellation of the information from the IP after the reading
process, the plate is exposed to an intense light that removes the remaining image causing the
decay of the metastable sites still present. Due to the dual side reading technology, the erasure
lamps act on both the surfaces of the imaging plate.
3.6 Histogram analysis and IP sensitometric curve
The digital output represents the pre-processed data. Histogram analysis is applied to the pre-
processed data to define the wanted versus unwanted signals in a scanned image plate for a
particular incident exposure and examination type. As the linear exposure latitude for the imag-
43

ing plate is very wide, a variable reading sensitivity (sensitivity number S) is necessary to map
the stimulated luminescence of the imaging plate to a range of output digital numbers within a
10 bit range. A second parameter, the latitude L, indicates the range of stimulated luminescence
(minimum to maximum) that will be included in the output digital number range.
The histogram of the digital values for each pixel, corresponding to the stimulated lumines-
cence, is created in order to perform an optimization of the data. The method used to assign
one of the 1024 grey scale level to the digital value of each pixel is based on the analysis of the
histogram having on the x axis the dose value and on the y axis the final pixel value number.
The elements used by the FUJI histogram analysis are shown in Fig. 3.9.
PV
Q
mRem
[X-ray dose]
Figure 3.9: Histogram analysis and main parameters involved.
The FUJY FCR5000MA system introduces a quantity × that is declared to be related to the
44

dose by the relation [17]:
× � ÐÓ� ¡ �Ó×� ÑÊ�Ñ (3.3)
× is the parameter linearly connected to the PV. The × values distribution histogram is shown
in Fig. 3.9, where the number of image pixels having the same × value is reported on the left y
axis.
Two parameters are used in order to determine linear relation between × and the grey scale
values. The sensitivity index S is defined as
Ë � ¡
� ×
(3.4)
where × is the × value corresponding to the central point of the grey scale (PV=511 in our
case). The latitude index Ä is defined as:
Ä � ×� ×� (3.5)
where ×� is the minimum × value that corresponds to PV=1023 (response saturation at high
dose) and ×� is the maximum × value that corresponds to PV=0 (response saturation at low dose).
The sensitometric curve is completely determined by the parameters S and L, both shifting the
useful range of the curve along the dose values and the latter alone changing the slope of the
straight line. In fact it can be written for the PV - dose relation:
ÈÎ �
Ä
¡ × �
Ä
� ÐÓ�
Ë
(3.6)
The system under investigation provides three user selectable modes that can be applied to
the histogram: automatic, semiautomatic and fixed mode. In the automatic mode the system
adjust, by means of dedicated threshold and differential algorithms, the L and S values so to
select the useful region of the histogram. In fixed mode the user can select both the L and S
parameter. All the image used for the measurement described in this thesis have been acquired
using the Fixed sensitivity mode, so to keep under control all the parameters necessary to the
off-line analysis. Furthermore a linear transformation curve was used between the input digital
number and the output final pixel value. In other words, in fixed sensitivity mode the user
defines the system speed and the latitude to be used for processing the exposed imaging plate.
Thus the response in this mode is similar to a screen-film cassette. The user can change the
image quality by acting n the mAs and kVp values.
45

Chapter 4
Measurement of the physical quantities
contributing to the DQE
In this chapter the image quality characteristics of the CR system under investigation are re-
ported. Physical criteria, such as modulation transfer function (MTF), noise power spectrum
(NPS), noise equivalent quanta (NEQ) and detective quantum efficiency (DQE), were employed
for this evaluation. In section 4.1 the relation between pixel values and dose is described. In
the following sections all the quantities needed for the evaluation of the DQE are calculated,
in order to determine the DQE and to compare it to other systems in use at present. The dose
effects on these quantities are also shown, together with their spatial frequency dependence.
Unless otherwise stated, a 50 �m pixel resolution (High Resolution HR) and a fixed linearity
acquisition mode have been employed for all the data acquired for the analysis and measure-
ments described in this work. Each image was stored in a DICOM format on the PC controlling
the acquisition procedure, and later transferred to a more powerful calculator for subsequent
raw data analysis.
A reference system is chosen where the X and Y axes are coincident with the IP sides, with
the former directed in a direction orthogonal to the thorax side of the breast (shorter side, called
also sub-scan direction) , and the latter parallel to the thorax side of the breast (longer side - scan
direction). The origin is in the right corner close to the breast, with increasing values moving
far away from the patient.
46

4.1 Experimental characteristic curve
The response curve of the imaging system, in terms of PV as a function of exposure dose, has
been evaluated by exposing the IP at several dose levels with a fixed tube voltage of 28 kVp.
The experimental setup is the same as the one described in Fig. 3.1 since we want to use the
mAs vs. the air kerma calibration curve previously measured. The only difference is that the
ionization chamber was replaced by the IP cassette, while the 30 mm thick PMMA external
filter, close to the beam exit window, remained the same. The cassette was placed just above
the cassette holder, and therefore it was exposed without the anti-scatter grid in between.
The IP has been exposed to the uniform X-ray beam for a set of dose values, the operator
being careful to maintain the reproducibility of the process. After the exposure, the IP was
immediately read, in order to minimize and to keep the temporal decay effect constant for every
measure, and then cancelled. The acquired data were processed in the linear fix mode, allowed
by the FUJI system software, in order to fully control every step of the image processing mode.
The system was operating in High Resolution mode, with a pixel size of 50 �m. The sensitivity
S was set to 200 and the latitude L to 4.0, corresponding to a linear processing curve over the
entire dynamic range of the system. Each DICOM image was read by an IDL software code in
order to access the raw data. The IP is exposed so that its shorter side is parallel to the direction
where the heel effect is bigger. In Fig. 4.1 a row parallel to the direction where the heel effect is
present is shown. In this case the IP was exposed to a uniform beam with a dose of about 676
�Gy.
In order to analyze a uniformly exposed region (ROI) of the image, and to be allowed
to forget the heel effect, a strip 400 pixels wide (2 cm) and 2000 pixels height (10 cm) was
selected. Furthermore the ROI was centered around a point distant 4 cm from the thorax side
laterally centered. The ionization chamber was positioned in the same point during dose-mAs
calibration measurements. The selected area is shown in black in the typical uniformly exposed
image reported in Fig. 4.2.
For each dose, the mean PV has been computed as the average value of the PVs of the ROI
previously described. The corresponding standard deviation SD has been calculated on the ROI
after a plane subtraction, in order to eliminate the first order contribution. If we consider the
plane P that best fits the ROI under investigation and we address each ROI pixel by the indexes
47

Figure 4.1: Section along the shorter side of a uniformly exposed IP. The heel effect in this
direction is clear.
Thorax side
Lateral side
Figure 4.2: Uniformly exposed IP. The ROI used for PV-dose calibration is shown in black. The
left longer side is the one closer to the breast.
�� � , the following equations hold:
ÈÎ � Æ ¡ Å
�
��
�
��
ÈÎ���
(4.1)
ÊÇÁÐ�Ò �� � �ÊÇÁ �� � È �� � (4.2)
where N is the number of pixels (400 in this case) along the X direction and M (2000 in this
case) the number of pixels along the Y direction . SD is the standard deviation of ROIÐ�Ò, with
a value decreasing from 5 PV to 1 PV with increasing dose.
48

The PV curve as a function of dose together with the least square fit performed with a
function f(x) of the type:
� Ü �� ¡ ÐÓ� � ¡ Ü � (4.3)
is shown in Fig. 4.3 (see also 3.6).
Figure 4.3: Pixel Value vs. Dose (�Gy).
The A parameter is connected to the latitude L, since Ä � �� holds. From the fit it
appears that L= ��� ¦ � , in perfect agreement with the predictions. In Fig. 4.4 the same data
set is shown, with the PV as a function of log (Dose), with the goal to stress the linearity of
the system over three orders of magnitude.
From the characteristic curve, it is possible to linearize the PV vs. dose relation and con-
sequently to take benefit from the properties of linear systems in the analysis as described in
chapter 2. In this case the linearized PV (hereafter PVÐ�Ò) are calculated as:
ÈÎÐ�Ò � à ¡
ÈÎ
� (4.4)
where K is a normalization parameter non fundamental in the subsequent analysis.
49

Figure 4.4: Pixel Value plotted as a function of the Log Dose (�Gy).
4.2 Pre-sampling MTF
The overall two dimensional MTF of a digital system as described in Chapter 1 can be expressed
by [19]:
ÅÌ� Ù� Ú � ��ÅÌ�� Ù� Ú ¡ ÅÌ�Ë Ù� Ú ℄ £
�
�
�
Ò�
Æ Ù Ñ�¡Ü� Ú Ò¡Ý �
¡ÅÌ�� Ù� Ú ¡ ÅÌ�� Ù� Ú (4.5)
where Ù� Ú are the spatial frequencies and £ denotes the convolution operator. MTF�(u,v)
,MTFË(u,v), MTF� (u,v) and MTF�(u,v) are the MTFs of analog input, sampling aperture, filter
and display aperture respectively. The Ñ and Ò are integers, and ¡Ü and ¡Ý are the sampling
distances in the Ü and Ý directions.
The product of MTF�(u,v) and MTFË(u,v) is referred to as pre-sampling modulation trans-
fer function MTFÈÊË(u,v) of a digital imaging system [20], which includes the geometric un-
sharpness, if not negligible, the detector unsharpness, and the sampling aperture unsharpness.
The digital MTF (MTF�Á�) of the system can be obtained by the convolution of MTFÈÊË(u,v)
with the comb function in the frequency domain. In this work the MTF due to the filter and the
50

display will be neglected since they are dependent on the operator choice and hardware setup,
and are in any case easily controlled. It should be noted that the MTF�Á� and the overall MTF
may incorrectly indicate the resolution capability of a digital system because both MTFs might
include a false response due to aliasing, while MTFÈÊ� can characterize inherent resolution
properties of a digital imaging system. The effects of aliasing will be described in the next
section, together with the problems deriving from this lack of spatial invariance.
In this work MTFÈÊË is measured by the Fourier transform of a “finely sampled” Line
Spread Function (LSF), which is obtained with a slit slightly tilted with respect to the IP pixel
grid. The sampling distances used for data acquisition were 0.05 mm (20 pixels/mm), with
a standard HR IP having the size of 18x24 cm (3540x4740 pixels). The signal from each
pixel was digitized with a final 10 bit grey scale. The exposure level and the image acquisition
parameters were chosen in order to provide a good quality slit profile, spanning as much as
possible the ADC range. In the examined cases, the LSF images were acquired with a ¦ �m
wide, �� ¦ � mm long slit camera manufactured by Nuclear Associates, model 07-624 (see
Fig 4.5).
Figure 4.5: The slit used for the LSF acquisition.
The camera was positioned by means of a custom, angle graduated, lead holder , which has
the double function of shielding the IP from scattered radiation and of allowing the repeatability
of the measurement and of the slit orientation (see Fig 4.6). The slit was positioned on the
cassette, orthogonally to the central axis the X-ray beam; the experimental setup is analogue to
the one used for the uniform dose expositions; in particular the SDD was 62.5cm, the dose was
about 680 �Gy, the latitude parameter L was 2.5 and the sensitivity S was set to 63. The beam
51

was filtered by a 30 mm block of PMMA, positioned close to the target, in order to approximate
the energy dependent attenuation of breast in a scatter reduced geometry. The distance of the
slit from the IP has been taken into account in all the subsequent analysis.
Figure 4.6: The setup used for the LSF acquisition.
The LSF was measured both in the scan and in the subscan direction. In Fig. 4.7 the two
dimensional image of the slit is shown. The three dimensional profile of the same image is
shown in Fig. 4.8, with the PV on the z axes. The off line linearization process has not been
performed yet.
The slight angle between the slit and the scan or subscan direction allows us to obtain a LSF
more finely sampled than the one that would be reconstructed from a slit parallel to the x and
y axes; in fact, in this last case the sampling distance would be given by the pixel size. The
method to obtain a smaller effective sampling distance is described below. Assume that a slit is
positioned at a slight angle � (usually � Æ ) in the direction perpendicular to the scanning (sub-
scanning) direction in deriving the MTF× �Ò (MTF×Ù�× �Ò). In Fig. 4.9 the schematic diagram
showing this situation is reported, with the IP pixel grid and the slit projection superimposed.
Because of the slight angulation of the slit, a series of different LSFs can be extracted with
respect to the alignment of the slit relative to the sampling coordinate (in the diagram, four
different LSFs superposition of the slit on the pixel grid are shown, since line A is equal to
line E). The LSF of each row is sampled at a distance ¡Ü. From the combined data of each
52

Figure 4.7: Image of the slit used for the LSF and MTF calculation.
(a) (b)
Figure 4.8: Typical slit profile. The PV (z axes) are represented as a function of the x-y IP
coordinate.
row, each representing a LSF sampled at a slightly different point (different phase), a composite
finely sampled LSF can be generated with a smaller effective sampling distance. The number of
points in the LSF is chosen large enough to obtain a sufficient number of values, in the frequency
53

17 13 9 5 1
18 14 10 6 2
19 15 11 7 3
Pixel Grid
Slit Image
20 16 12 8 4
13 9
17 14
10
5 1
18
15
19
16
11
12
6
7
8
3
2
20
4
Figure 4.9: Schematic diagram showing the generation of a composite finely sampled LSF. The
LSFs corresponding to the various alignments of the slit relative to the sampling coordinate are
composed, after a spatial shift, to give the final LSF.
domain, to reconstruct the MTF finely along all the interesting range of spatial frequencies.
So for every slit image it is necessary to evaluate when the LSFs on each rows reproduce
the same pattern. In this work a numerical approach has been chosen but similar results can
be obtained by graphical methods. As it can be inferred from Fig. 4.9, in the hypothesis of
constant slit width and of uniform exposition, the maximum digital value M1 of all the rows is
obtained when the slit is just above a pixel center since the light is not distributed between two
adjacent pixels. We select this row as reference line for the angle calculation. Moving from
this point along a direction orthogonal to the slit, the maximum value of each row is lower than
M, reaching a minimum when the slit center is above the line separating two pixels, and then
increases until a new maximum M2 is reached. The distribution of maxima as a function of the
column number allows to retrieve the slit angle value. A more efficient method has emerged to
be the search, in each row, of the column number corresponding to the maximum. The number
Ò of rows having the maximum in the same column is the number of lines that contribute to the
finely sampled LSF without reproducing the same pattern. Furthermore, more than one LSF
can be obtained from a slit image since this method allows an easy reconstruction of the start
and end point of each composite LSF. The slit angle �, i.e. the angle between the slit and the
54

vertical direction in Fig. 4.9, is given by:
� � �Ö Ø�Ò �Ò (4.6)
while the effective sampling distance ¡Ü is:
¡Ü �¡Ü�Ò �¡Ü ¡ Ø�Ò � (4.7)
In the slit image used for the LSF reconstruction the angle was � ¦ � Æ for the scan
direction and � ¦ � � Æ for the sub-scan direction. To independently check the validity of
the analysis, the maximum value of each row has been plotted as a function of the row number.
Each PV has been normalized to the area under the slit profile at each row to take into account
possible non-uniformity effects of the IP and imperfections of the slit. The distance from two
local mimina is the number Ò (see Fig. 4.10).
Figure 4.10: Maxima of the PV slit values.
We can observe that the graphical method agrees with the numerical one, even if it is less
efficient in the classification of the critical points.
The angle amplitude is a tradeoff between a small angle, which allows to obtain a very
finely sampled LSF, and a big angle that allows to reconstruct more LSFs from the same image
and to average them reducing possible local effect. In this work the minimum angle is given
55

y the pixel size (� �m) and the slit length useful to the analysis (quantified in 6 mm). In
order to fully reconstruct a finely sampled LSF from a single acquisition �Ñ�Ò is about �� Æ ,
but in the analysis an angle between Æ and Æ has always been used. The useful (to the LSF
reconstruction) column width is chosen so to obtain long enough tails without truncating before
reaching the plateau value.
At this point the IP characteristic curves, relating the PV to the relative exposure, were used
for the linearization. The signal can now be expressed in terms of exposure levels instead of
grey levels. Each finely sampled LSF, one for each Ò rows, is then reconstructed by aligning
each row value with the appropriate shift (a ¡Ü integer multiple). The final LSF is the average
of three LSF. The effect of the finite slit width will be taken into account in the MTF calculation.
A typical example of composite LSF is shown in Fig. 4.11; it is clear the effect of fine sampling.
(a) (b)
Figure 4.11: Typical LSF plot. Points deriving from several LSF, extracted from a single im-
age, are superimposed to show the good agreement of the experimental data. (a) LSF in scan
direction. (b) LSF in subscan direction.
In case of failures in the LSF reconstruction, it has emerged that each error in the choice of
the parameters needed by the reconstruction algorithm has macroscopic consequences on the
LSF quality, so to allow an easy rejection of the case.
56

The reconstructed LSF may include in the tails errors due to the glare and quantization ef-
fects. So an exponential extrapolation was employed for the tails, with a truncation level of
approximately 0.01 with respect to the maximum of the LSF. In literature experimental evi-
dences can be found that the exponential extrapolation gives better results.
We will see that the tail corrections affect only the low spatial frequencies and that the
overall effect is not significant because the CR is a low noise system. In Fig. 4.12 the LSF with
tail extrapolation is shown.
Figure 4.12: LSF�ÜØÖ.
The mean composite LSF�ÜØÖ is then Fourier transformed to obtain the presampling MTF.
The smaller effective sampling distance allows to have more reconstructed points for the MTFÔÖ�.
The Optical Transfer Function OTF is calculated as FFT of the LSF. The OTF is then divided
by the factor ×�Ò ��Ï (� is the frequency) to take into account the finite slit witdh W. The
MTF is the modulus of the complex function OTF.
A comparison between the MTFs obtained from the same slit image, before and after the
tales exponential extrapolation, shows that the two curves perfectly agree at the higher frequen-
cies, while the MTF�ÜØÖ has a smoother behavior, as expected, at the lower frequencies (see
Fig. 4.13)
The final MTF× �Ò and MTF×Ù�× �Ò are shown in Fig. 4.14 and Fig. 4.15 .
57

Figure 4.13: Comparison between MTF and MTF�ÜØÖ.
Figure 4.14: MTF× �Ò.
From a comparison between the two curves it emerges that MTF behaves essentially in the
same way for the scan and sub-scan direction, even if they are slightly higher for the sub-scan
slit image. (see Fig. 4.16).
58

Figure 4.15: MTF×Ù�× �Ò.
Figure 4.16: MTF× �Ò and MTF×Ù�× �Ò.
These are the curves that will be used for the subsequent analysis.
59

4.3 EMTF
The phase dependence of ÅÌ���� resulting from undersampling, described in Section 2.2.1,
poses a conceptual problem in that it violates the desired stationarity property of MTF. The
solution adopted in this work is to consider the expectation value of ÅÌ���� averaged over
all phases, which shall be called the expectation MTF (EMTF). The EMTF seems a suitable
descriptor of digital MTF since it satisfies the requirements of being defined approximately as
�ÇÌ����� and being spatially invariant. EMTF is calculated by averaging MTF��� Ù � � over
all possible phase values. For the image of a slit, the phase factor is just the displacement of the
slit center from the origin of the image since all cosinusoids in a slit image have their origins
at the slit center. Furthermore it can be shown that MTF��� ٠� � is the same if � is shifted
by multiples of one pixel [12]. Therefore, the average over all possible phases needs only to
include the range 0 to �. The relation used to evaluate EMTF is then:
�ÅÌ� Ù�
� �
�ÇÌ���� Ù�� � �
��ÅÌ�Ù�� � �� ��
� �ÇÌ���� � � �
(4.8)
where Ù� is the generic spatial frequency, � the phase and �ÇÌ���� Ù�� � � is calculated as
described in Eq. 2.26.
A software code developed in IDL language performs the described calculations with the
sampling interval divided in 10000 steps (i.e. �� � � pixel size). The EMTF curves are
shown in Fig. 4.17 for the scan (a) and sub-scan direction (b). Superimposed are the MTFÔÖ�
curves.
The EMTF was found to be very close to the value of ÅÌ�ÔÖ�, with a difference increasing
with spatial frequencies, as shown in Fig. 4.18, but less than 0.5% at the Nyquist frequency.
Conceptually, EMTF is a reasonable compromise to the problem of phase dependance of
digital MTF. Since MTFÔÖ� is the response of a digital system to a delta function (slit in this
case), EMTF is the response of the system to a slit averaged over all locations in an image.
EMTF is therefore a more accurate reproduction of the response of the total system to a slit
than is MTFÔÖ�, although EMTF will not necessarily represent the delta-function response at
any given location. In the subsequent evaluations of the system NEQ and DQE, the EMTF will
be used. In Fig 4.19 the two EMTF, in scan and subscan directions, are compared.
60

(a) (b)
Figure 4.17: EMTF and MTFÔÖ� (a) scan direction. (b) subscan direction.
4.4 NPS measurement
The Noise Power Spectrum has been measured starting from uniformly exposed images. A large
set of exposure levels has been used. Initially the two-dimensional NPS has been evaluated in
order to study the presence or absence of any off-axis noise structure.
Initially the image data was linearized to the exposure using the characteristic transforma-
tion curve. A big, square ROI, centered on the point of clinical interest (4 cm away from the
chest), was selected for the subsequent analysis for each image corresponding to a different
exposure level. The dimensions of this ROI are such to contain £ smaller ROIs, with 128
pixel side, each shifted from its neighbor by 64 pixels step [21] . A planar trend is subtracted
from each ROI and the Fast Fourier Transform is performed on the flat field data. The NPS is
then calculated as:
ÆÈËÖ�Û Ù� Ú �
� ��Ì��Ð�Ø���Ð� Ü� Ý ℄� �
ÆÜÆÝ
¡Ü¡Ý
(4.9)
where � ��Ì��Ð�Ø���Ð� Ü� Ý ℄� � represents the ensemble average of the Fast Fourier
transformed 128*128 ROIs. ÆÜ and ÆÝ are the number of elements in the Ü and Ý directions
(which are equal to 128 in this case) and ¡Ü and ¡Ý are the pixel pitch (equal to 50 �m).
61

Figure 4.18: Differences between EMTF and MTF.
The NPS used in subsequent calculations is the normalized NPS (ÆÈËÒÓÖÑ�Ð�Þ��), obtained by
scaling the ÆÈËÖ�Û for the mean signal of the £ ROIs (large area signal).
ÆÈËÒÓÖÑ�Ð�Þ�� Ù� Ú �
ÆÈËÖ�Û
large area signal
(4.10)
The two dimensional normalized NPS distribution for an image exposed to ���Gy is
shown in Fig. 4.20.
An evaluation of the two-dimensional NPS image reveals the presence of an apodization
filter that significantly reduces the noise power spectrum above 8 lp/mm in the scan direction.
This filter is likely to be necessary to reduce the high frequency components of the decay lag
characteristics of the photostimulated luminescence that occurs during the readout in the scan
direction. This apodization filter lowers the NPS response, but has little impact on the NEQ and
DQE because the dominant factor above this cut-off frequency is the MTF response, which is
close to zero.
From the 2D NPS, the NPS along the scan and sub-scan direction has been calculated. The
1D NPS along the scan and sub-scan direction is obtained considering a thick slice of four lines
62

Figure 4.19: EMTF scan Vs. EMTF sub-scan.
of the 2D NPS on either side of both scan and sub-scan axis (excluding the axis). For each data
value of coordinates (u,v) in the 2D NPS space, the new 1D frequency value was computed as
Ô Ù Ú . After a rebinning of the frequency axis values, the 1D NPS is obtained [22]. The
assumptions for using this technique for estimating the 1D NPS are that the 2D NPS exhibits
moderate radial symmetry and that the noise data are basically uniform within the small annuli
of spatial frequencies used for regrouping the noise data. The measured NPS in the scan and
sub-scan direction are shown in Fig. 4.21 and Fig. 4.22 respectively for the entire set of exposure
levels used in this analysis.
A comparison between the NPS in the scan and sub-scan direction is performed in Fig. 4.23
4.5 NEQ measurement
If the MTF describes the signal response of a system and the NPS describes the amplitude
variance, both at a given frequency Ù, then the ratio of these two quantities, properly normalized,
63

Figure 4.20: 2 dimensional NPS The dose values is 37 �Gy.
Figure 4.21: NPS as a function of exposure dose values in the scan direction.
64

Figure 4.22: NPS as a function of exposure dose values in the sub-scan direction.
Figure 4.23: Comparison of the NPSs in the scan and -subscan directions for a set of exposure
values.
gives information about the maximum available signal-to-noise ratio as a function of frequency.
65

The Noise Equivalent Quanta was computed, for each exposure level, as [23] [22]:
Æ�É � �
large area signal �ÅÌ� �
ÆÈË �
�ÅÌ� �
�
ÆÈËÒÓÖÑ�Ð�Þ�� �
(4.11)
where � is the spatial frequency and assuming that the measured large area signal is lin-
early related to the input signal. EMTF is the digital expectation MTF, computed numerically
from the pre-sampling MTFs [12] as described in section 4.3. The large area signal is the av-
erage digital value in units of exposure of each ROI after background trend correction. NPS is
the thick-slice one-dimensional NPS near the Ù or Ú axis (see section 4.4) and NPSnormalized is
given by NPS divided the large area signal. The curves describing the NEQ as a function of
spatial frequency are shown in Fig. 4.24 and Fig. 4.25 for each exposure level. The interested
frequencies are in the range between 0 mm and the Nyquist frequency 10 mm .
Figure 4.24: NEQ scan.
In Fig. 4.26 the NEQ for the scan and sub-scan directions are compared. The scan NEQ
is always lower than the corresponding sub-scan curves until the frequency is about 8 mm ,
where it inverts its slope, becoming greater than the sub-scan NEQ after the 9 mm frequency.
This behavior reflects a global characteristic of the system under investigation which has an
evident reduction of the NPS along the scan direction, with respect to the sub-scan direction,
66

Figure 4.25: NEQsubscan.
at higher frequencies. The effect is due to an apodization filter along the scan direction and is
certified in recent works [24] performed on similar systems.
4.6 DQE measurement
The Detective quantum efficiency has been calculated for each exposure level by dividing the
NEQ by the SNR of the incident x-ray beam, as [22]:
�� � �
�ÅÌ� �
ÆÈËÒÓÖÑ�Ð�Þ�� � Õ
� Æ�É �
Õ
(4.12)
where � is the spatial frequency and Õ is the number of x-ray photons incident on the detector
per unit area. The last factor to be determined is Õ.
4.6.1 Determination of q
The total number of incident photons per unit area of the detector at each exposure level, Õ, has
been calculated starting from the X-ray photon fluence per exposure unit at each energy, from
the X-ray spectral distribution and the exposure level (X). The photon fluence per mR � � ,at
67

Figure 4.26: NEQ scan vs NEQ subscan.
energy (�), is best described by the polynomial [23] [25]:
� � � � � �� �� ��� � � ���� � �� � � �� � ¡
� � � (4.13)
In Fig. 4.27 we show the curve obtained fitting X-ray photon fluence per mR between the
range of 5 and 35 keV, as taken from published [23] [25] data.
The X-ray spectrum has been evaluated as described in section 3.2. The exposure value on
the IP surface is evaluated from the calibration curve measurements, described in section 3.1.1
, between mAs and dose on the plate.
The total number of incident photons per unit area of the detector per exposure level X is
calculated as per Eq. 4.14
�� �
Ê Õ � � � ��
Ê Õ � ��
(4.14)
The Õ value for the 40 mm PMMA filtered beam is 52800 photons/mm /mR, while for
the unfiltered beam we obtain 37800. For the 30 mm PMMA case the Õ value is 49700
photons/mm /mR.
68

Figure 4.27: The curve is obtained fitting the X-ray photon fluence per mR (taken from pub-
lished [23] [25] data) between the range of 5 and 35 keV.
4.6.2 Measured DQE
DQE estimates versus spatial frequency as a function of exposure are shown in Fig. 4.28 for the
scan direction and in Fig. 4.29 for the sub-scan direction.
Figure 4.28: DQE scan.
69

Figure 4.29: DQEsubscan.
In Fig. 4.30, a reduced set of the DQE measurements, as a function of exposure level, is
shown in order to allow a comparison between the scan and sub-scan directions.
Figure 4.30: DQE scan vs DQE subscan.
The DQE of this dedicated mammography CR system is significantly better than the one
70

of earlier conventional CR system with a 100 �m sampling pitch when acquired under similar
conditions [24] [26].
71

Conclusions
The work performed in the context of this thesis has concerned the study of the performances
of the Fuji FCR5000 MA CR system, with particular emphasis devoted to the evaluation of the
physical quantities involved in the image acquisition process. The fulfilled measurements are
particular interesting in the context of the expansion of digital mammography to clinical use.
Several technologies are currently under test with the goal to provide the benefits of digital ac-
quisition, image processing, electronic display and storage while, at the same time, producing
an image with as good as or better image quality than the state of the art screen-film detectors.
The results obtained here show that a dedicated digital mammography system based on pho-
tostimulable storage phosphor technology (CR) has good quantitative capabilities in terms of
MTF, NPS, NEQ and DQE, particularly when compared to previous CR mammographic im-
plementations. The system included new features with respect to previous CR equipments, as a
dual-side imaging plate detector, which allowed photostimulated luminescence to be extracted
from both sides of the imaging plate, a thicker phosphor layer and 50 �m sampling pitch.
The X-ray unit has been fully characterized in order to measure the parameters required for
the subsequent analysis (Dose-mAs curve, energy spectrum, voltage calibration). The IP reader
unit has been used in fixed sensitivity mode, with a set of latitude and speed parameters decided
by the user and in linearity mode. Uniform dose expositions at different dose levels have been
used to measure the linearity response of the system, for the pixel value-dose calibration and for
the NPS evaluation. The raw data have been analyzed off line with a set of software routines,
base on IDL platform, completely developed in the framework of this thesis. The software
also allows the user to calculate the X-ray spectrum in function of the attenuation filter and the
voltage applied to the tube.
The MTF has been measured with the slit method. The MTF curve reaches the 5% value of
its maximum at 8 lp/mm, with an improvement with respect to earlier systems [24]. However
this resolution is 30 % less than would be expected from an ideal detector with 10 mm Nyquist
frequency, basically due to the thicker IP layer. Furthermore, in the spatial frequency diagnostic
72

ange (i.e. up to 2.5 lp/mm), the MTF of this system is comparable to the MTF of screen-film
based mammography systems [28].
The evaluation of the two-dimensional NPS image has revealed, as expected from literature,
the presence of an apodization filter that reduces the noise power spectrum significantly above 8
lp/mm in the scan direction. This filter is necessary to reduce the high frequency components of
the decay lag characteristics of the photostimulated luminescence that occurs during the readout
in scan direction.
Finally the NEQ and DQE have been measured since, with widespread consensus in the
international community, they are considered to be the main parameters in order to evaluate
the global performances of an imaging system. The NEQ depends both on the noise properties
and on the spatial resolution performances of the detector. It is therefore a good candidate to
compare different systems, in particular in mammography. In fact in this field the main goals
are often to detect relatively large objects (with dimension about 1 mm) with low contrast,
(low noise required), or small objects (with dimensions of the order of 200 �m) with high
contrast. An accurate evaluation of the NEQ and its dependence on the exposure has been
performed. The DQE of this dedicated mammography CR system is significantly better than
the one of earlier conventional CR system with a 100 �m sampling pitch when acquired under
similar conditions [24] [26]. The measured quantities have revealed to be reproducible and
are in perfect agreement with the ones found in recent literature works, performed on similar
systems [24].
The low throughput of the system, limited by the scan time of about 90 sec for each IP, is still
a limiting factor in the use of this CR equipment in the service of two or more mammography
rooms during clinical use.
The analysis performed here allows to start a new series of studies, involving the clinical
performances of the system with the aid of dedicated phantoms (such as the CD-MAM or sim-
ilar) and of the first clinical cases under the supervision of specialized medical staff.
73

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Ringraziamenti
Ipiù sinceri ringraziamenti vanno al Dr. Giacomo Belli e alla Dott.ssa Barbara Lazzari per
la disponibilità, la professionalità e la simpatia con cui hanno reso meno pesanti le corse su e
giù per Firenze. Tanta gratitudine alla Prof.ssa Marta Bucciolini e al Dr. Cesare Gori per la
pazienza e la gentilezza nel darmi consigli. Un grazie al Prof. Salvatore Romano, anche per
aver semplificato quando possibile la burocrazia e i formalismi. Infine un sorriso a Francesco
R., del quale è il merito se sono arrivato a scrivere questo lavoro.
77