Post on 21-May-2020
PREFACE
During the past 50 years, advanced developments in physics and engineering technology have played anincreasingly important role in the development of modern biomedical sciences. The present set of ten volumeson Biomedical Physics was written to cover the key areas of developments and to illustrate the new approachesas well as methods and theories employed.
Following the general structure of medical care, the new fundamental imaging methods are first coveredstarting from the most sensitive molecular approach with positron emission and single photon emissiontomography (PET and SPECT, Volume 1) via computed tomography and ultrasound (CT and US, Volume2), to magnetic resonance imaging and spectroscopy (MRI and MRSI, Volume 3), and optical molecularimaging (OMI, Volume 4). Each of these volumes is describing the important contributions to the four imagingrevolutions in biomedical imaging since the early 1970s. This part is then followed by an overview of moregeneral biomedical measurement techniques (Volume 5), modern bioinformatics covering the important areasof genomics, proteomics, lipidomics, and metabolomics (Volume 6), and molecular radiation biology andradiation safety (Volume 7). The last set of volumes cover the therapeuticly important field of biomedicalphysics treatment methods starting from Volume 8 on accelerators and radiation sources and detectors, toVolume 9 on radiation interaction with matter, radiation transport theory, and absorbed dose, followed bytreatment planning and treatment optimization. Finally, there is Volume 10 on the wide range of otherphysically based treatment and rehabilitation methods and their biological effects.
It is hoped that the present series will stay alive not least in its Internet version, so new, important areas canbe covered and incorporated as they mature and comprehensive chapters get written. It is hoped that thepresent set of volumes will be useful in biomedical research as well as education where the wide spectrum ofimaging, diagnostics, and therapeutic approaches based on biophysical processes have not been so extensivelycovered in recent years. They may allow an efficient transfer of knowledge from the wide range of methodsavailable that may become of increasing interest in your own area of expertise. I therefore hope that thisextensive reference work on biomedical physics will be of interest for the whole biomedical and applied physicscommunities for years to come.
A. BrahmeKarolinska Institute, Stockholm, Sweden
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INTRODUCTION TO VOLUME 2: X-RAY AND ULTRASOUNDIMAGING
This Volume of Comprehensive Biomedical Physics focuses on the technology and applications of the two mostwidely available categories of morphological imaging techniques, x-ray and ultrasound (US) imaging. Forsure, these two classes of methods dominate the Medical Imaging arena in terms of local availability andfacilities, the number of devices installed worldwide, and the number of exams performed routinely inimaging departments. The word ‘morphological’ used above is, indeed, misleading. When talking aboutRadiology, we suddenly start thinking about chest radiographs, dental panoramic images, fetal USs, oranything else that has to do with the ‘shapes’ of internal organs. The very first radiograph in history (thevery famous Mrs. Roentgen’s hand with ring in 1895) was so impressive that it was immediately clear whatthe main practical application of the x-ray would be after its discovery. Unlike the development of radionu-clide imaging techniques, which have always focused on organ functionality, it took almost a decade for thescientific and medical community to figure out how to get functional information from US images (intro-duced in the 1940s with the first applications in echocardiography in the ’50s and ’60s) and more than acentury to do the same with x-rays.
When Prof. Brahme and Elsevier contacted us about this project, we accepted to collaborate willingly sincewe immediately realized that it would be a great opportunity for us to improve our knowledge in our respectiveareas of research. Given our scientific interests, we were confident in our knowledge of the physics of US andx-rays, but not to the extent that we could consider ourselves true experts in the field. We have certainly learneda lot during the preparation of this Volume, and we must thank all the authors for their patience and for havingfulfilled almost all our requests, even though some of them probably went beyond what they considered to be‘acceptable.’ We spent a lot of time asking the authors to make every single sentence and every mathematicalstep in their Chapters clear. The reader will appreciate the effort devoted to ensuring that there is a logical threadrunning through the book and that the necessary integrations between the chapters were made so as toguarantee a thorough description, not only of the main topics but also of niche and advanced topics, whichwill hopefully help the book find a distinctive collocation within the wide range of US and x-ray imagingliterature. As regards this last point, we have chosen to reserve space for new ideas and new perspectives as far aspossible and to devote entire chapters to the description of important research tools.
The first Section of Volume 2 covers the technology of Medical Imaging based on x-rays. The series of articlespresented in this Section attempts not only to give the reader a clear picture of the state-of-art of current x-rayimaging physics and technology, which are primarily based on the selective attenuation of photons in tissueswith a different atomic number and density, but also of the emerging techniques which are, conversely, basedon the differences of phase shifts of the x-ray wavefront on tissues with different complex refractive indices.Although still experimental, it is of primary importance for physicists, engineers, and scientists to keepup-to-date with the current developments of such emerging techniques which may have a prominent role inthe diagnostic scenario in the near future.
The first Section is therefore structured as follows. Chapters 2.01–2.03 provide the foundations for under-standing the physical and technological aspects of planar and tomographic x-ray imaging. Due to the severalpeculiarities of each diagnostic application of x-ray imaging, Chapters 2.04–2.07 are structured in such a waythat the reader can have a clearer understanding of how each diagnostic task has led to different technologicalevolutions of imaging devices starting from the same physical principles of x-ray emission, interaction withtissues, and detection. Chapters 2.08–2.11 are, on the other hand, more focused on advanced or niche topics inradiology and emerging applications such as phase-contrast imaging, micro-CT, radiation protection, and anin-depth overview of the mathematical foundations of tomographic reconstruction in two and threedimensions.
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xiv Introduction to Volume 2: X-Ray and Ultrasound Imaging
More specifically, this is what the reader will find in each Chapter.Chapter 2.01 introduces the basic aspects of x-ray imaging for diagnostic radiology, the description of
photon interaction processes in the diagnostic energy range, and the introductory physical aspects of x-rayimage formation mechanisms (image contrast based on x-ray attenuation or phase change). It comprises athorough discussion on the radiation quality, from the most common type of source for x-ray imaging, that is,the x-ray tube, and a description of x-ray tube spectra and its shaping through filtration.
Chapter 2.02 describes the primary physical parameters of image quality and their relationship, as well asthe derived parameters obtained from them. In particular, it explains fundamental objective and subjectiveimage quality metrics such as modulation transfer function, noise power spectrum, contrast, contrast transferfunction, signal-to-noise ratio, contrast-to-noise ratio, and low-contrast detectability.
Chapter 2.03 provides an introductory overview of Computed Tomography physics and technology, with astrong focus on the technological evolution that has led to the current configuration of modern multislice CTand cone-beam CT scanners, along with the most important dose reduction techniques as well as a concisereview of common image artifacts and the possible strategies for their correction.
Chapter 2.04 opens the application-driven part of this Section. This chapter addresses the technologies andthe applications of radiology used in the field of oral (or dental) and maxillofacial imaging, describingdedicated x-ray sources for dental intraoral radiology, intraoral detectors, equipment for panoramic and forcephalometric extraoral radiology and cone-beam volumetric imaging of the head.
Chapter 2.05 focuses on mammography. The various stages of the breast imaging chain and the specificrequirements to optimize image quality and patient dose in mammography are reviewed and discussed, andthe latest information on advanced applications such as x-ray tomosynthesis, breast CT, and dual-energymammography are presented.
Chapter 2.06 introduces the reader to the principles and techniques of x-ray imaging based on photonenergy discrimination. This field of study has important applications in quantitative imaging. The theory andapplications of dual-energy and multi-energy (or spectral) imaging are presented and discussed.
Chapter 2.07 deals with quality control (QC) in x-ray imaging. This chapter is intended mainly for medicalphysics experts whose role is to organize and accomplish adequate programs of Quality Assurance within theirimaging departments. QCs for computed and digital radiography, mammography systems, dental systems,digital angiography systems, and to conclude computed tomography systems are presented. Procedures,reference values, and typical periodicity of QCs are reported for each type of equipment.
The wave nature of x-rays, which is commonly disregarded in conventional biomedical imaging applica-tions, plays a fundamental role when coherent sources of x-rays and phase-sensitive techniques are employed.This important and emerging field of application is the subject of Chapter 2.08. The five main different phase-sensitive techniques, namely propagation-based phase-contrast imaging, analyzer-based imaging, coded aper-tures phase-contrast x-ray imaging, interferometry, and grating interferometry are introduced in this Chapter,providing the basic physical principles and presenting selected biomedical applications.
The downscaling of tomographic x-ray imaging technology to the size of small laboratory animals isdiscussed in Chapter 2.09. Due to the growing role of preclinical imaging in phenotyping, drug discovery,and in providing understanding of the mechanisms of disease, we thought it important that the readerunderstand how challenging it might be to obtain high performance in terms of image quality and temporalresolution in subjects that are more than one order of magnitude smaller than humans (and with heart rates upto ten times faster!).
Chapter 2.10 deals with the important and often controversial, delicate topic of the risks associated with themedical employment of x-rays for patients and workers, which however can never be stressed enough. Besidesdebates on the issue, which are mostly about the interpretation of radio-epidemiological data (totally beyondthe scope of this Section), the physical dosimetry of ionizing radiation is a mature science and its application onRadiation Protection in diagnostic radiology is the main focus of this Chapter.
At last, for all those who do not like to look upon the reconstruction software packages of tomographicinstrumentation as just ‘black boxes’ (as it should always be for a serious medical physicist or biomedicalengineer), Chapter 2.11 provides an in-depth description of the mathematical foundations of image recon-struction from projections, a fascinating field of research that has never stopped growing since the invention ofthe CT in the 1970s. This Chapter covers the fundamentals of 2D and 3D image reconstruction algorithms, withan emphasis on the analytical methods for x-ray CT imaging, but with a useful overview of iterative methods fortransmission tomography that are continuously gaining ground on today’s CT instrumentation.
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Introduction to Volume 2: X-Ray and Ultrasound Imaging xv
The second Section of Volume 2 covers the technology of Medical Imaging based on US. US is a multifacetedtechnique which offers innumerable advantages. It can provide information regarding the location of anyacoustic discontinuity and an estimate of the reflection coefficient. It can also provide maps of the propagationspeed and maps of the attenuation coefficient. Besides, it is harmless, simple to use, does not require a speciallyequipped room, and can hence be used effectively at the bedside. Moreover, it provides a good time resolution,it can estimate the speed of moving objects and as far as equipment goes, US equipment is economical andextremely popular.
Chapter 2.12 introduces the reader to the basic principles of US propagation and to the well-known B-modeimaging technique. Chapter 2.13 describes the basic principles which govern the probe operation. The maintopic of this chapter is beamforming by multielement array transducers. Chapter 2.14 deals with the topic ofDoppler US where the Doppler shift is utilized to measure blood flow velocities and direction. Chapter 2.15presents and discusses currently available and emerging US imaging modalities ranging from the standardmodes to nonlinear imaging, quantitative imaging, and perfusion imaging quantification. Chapter 2.16introduces the reader to the basics of nonlinear acoustics and its application to medical US imaging. Thecoefficient of nonlinearity is introduced, together with a variety of equations, which develop step by step fromsimple to more sophisticated models. Chapter 2.17 describes the main medical applications of US and devotesspace to their role as bedside diagnostic completions of the physical examination of the patient. The biologicaleffects occurring in diagnostic ultrasound (DUS) represent an important field of inquiry in non-ionizingradiation biology and Chapter 2.18 is devoted to this topic. Chapters 2.19 and 2.20 are entirely devoted tothe description of two important research tools: the simulation of US fields and the US research platforms(highly flexible scanners with wide access to raw echo data).
Below, the authors themselves introduce their respective chapters to the readers by means of short abstractsand personal messages.
Marcello Demi: Chapter 2.12wants to introduce the reader to the basic principles of US propagation. Firstly,an ideal medium is considered to simplify the approach to the equations that govern US propagation in abiological medium. Secondly, the simplifying assumptions are progressively removed in order to come close tothe work conditions within which physicians usually operate. The most popular US imaging technique, thewell-known echo-pulse or B-mode imaging technique, is also introduced. The main assumptions, which are atthe basis of the reconstruction of the US image sequences, are analyzed separately and the image artifactsgenerated by such assumptions are illustrated.
Han Thijssen: In my opinion, Chapter 2.13 is meant to expose the basic physics of (medical) US to thereader. Furthermore, we extensively introduced the various methods and techniques of imaging with arraytransducers and reviewed the principles and backgrounds of image quality assessment and assurance. Recentdevelopments are introduced and explained.
Massimo Mischi: I am convinced that Chapter 2.13 provides the reader with the essence of array beamform-ing through a few selected and well-connected equations and will help them when they approach relatedproblems, before digging into hundreds of pages without any prior understanding of the topic.
Hans Torp: Chapter 2.14 deals with the topic of Doppler US. In Doppler US, the Doppler shift frommovingblood is utilized to measure blood flow velocities and direction and to extract the weak scattering from bloodfrom much stronger echoes from the vessel wall and other larger tissue structures in the human body. The twoclassical modalities of continuous-wave and pulsed-wave Doppler analysis are introduced and discussed. Thecolor Doppler and vector Doppler imaging techniques are also explored.
Massimo Mischi: Quantitative US imaging is the ultimate instrument we can provide doctors with. In thepast few years, we have experienced a tremendous growth in this field, which is paving the way for newdiagnostic options that we could not have even imagined a few years back. Providing a concise and compre-hensive overview of all the imaging modalities, both the traditional ones, such as B-mode or Doppler, as well asthe emerging ones, such as molecular imaging, elastography, and contrast-enhanced US, has therefore been avery exciting challenge. Hopefully, this work (Chapter 2.15) will set a solid basis and lead to furtherdevelopments.
Libertario Demi: Chapter 2.16 introduces the reader to the basics of nonlinear acoustics and its applicationto medical US imaging. The coefficient of nonlinearity is introduced, together with a variety of equations,which develop step by step from simple to more sophisticated models. Starting from sinusoidal plane waves inlossless nonlinear media, the models are gradually expanded to include the effect of absorption, pulsed planewaves, quasiplanar waves, diffraction, and local nonlinearity.
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xvi Introduction to Volume 2: X-Ray and Ultrasound Imaging
Gino Soldati: Chapter 2.17 describes some medical applications of US and devotes space to their role asbedside diagnostic completions of the physical examination of the patient. The chapter shows how, in thissetting, medical sonography represents a powerful tool for integrating imaging and functional data in thecritically ill. Here, some space is also reserved for unusual applications of sonography and its role in diagnosingdiseases of aerated organs such as the lung.
Douglas Miller: Chapter 2.18 Bioeffects in DUS are a little known and poorly understood biomedical topic.Although the energy radiated from US probes is non-ionizing, the interaction of US with living tissue is ascientifically interesting and complex biophysical problem. There is no distinct boundary between therapeuticand DUS, and on-screen dosimetric indexes therefore are provided on DUS machines. As with all medicalprocedures, the practitioner should be aware of the potential interactions with a biological significance andconsider the risks as well as the benefits to the patient.
Martin Verweij: Chapter 2.19 There are numerous conference proceedings and journal articles that describeindividual simulation methods for medical ultrasound fields. However, there exist very few texts that provide ageneral overview of these methods. In my opinion, what is largely missing is a combined description of thefundamentals and taxonomy of this field, preferably in the form of a review article or book chapter. The editorsgave me the opportunity to write such a text for Comprehensive Biomedical Physics, and some very capable co-authors agreed to join me in this task. Under those circumstances it was not hard to accept the invitation forwriting this book chapter. Knowing the main correspondences and distinctions between simulation methods,you can more efficiently and confidently choose the right method for solving a particular problem.
Piero Tortoli: Chapter 2.20 US research platforms have recently gained unexpected success which isongoing. In a few years, they have gone from being cumbersome pieces of equipment, with their use restrictedto the lab of origin, to portable but sophisticated systems that are distributed worldwide. Due to their nature ofbeing open platforms dedicated to research, they are implicitly dynamic, that is, they must evolve continuouslyso as to be always on the boundary between those needs that are already mature and emerging needs associatedwith brand new ‘hot’ ideas.
Daniele PanettaInstitute of Clinical Physiology (IFC-CNR), National Research Council, Pisa, Italy
Marcello DemiFondazione Toscana Gabriele Monasterio, Pisa, ItalyEVIE
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2.01 Physical Basis of x-Ray ImagingP Russo, Universita di Napoli Federico II, Napoli, Italy; INFN Sezione di Napoli, Napoli, Italy
ã 2014 Elsevier B.V. All rights reserved.
2.01.1 Introductory Concepts 22.01.1.1 Imaging Basics 22.01.1.2 Absorption, Scattering, Refraction of x-Rays 32.01.1.3 Tissue Substitute Materials 82.01.1.4 Absorption Contrast and Phase Contrast 82.01.2 Interaction Processes 142.01.2.1 Photoelectric Absorption 142.01.2.2 Rayleigh (Coherent) Scattering 152.01.2.3 Compton (Incoherent) Scattering 192.01.2.4 Mass Attenuation Coefficients and Dosimetry 232.01.2.4.1 Mass energy transfer coefficient 232.01.2.4.2 Mass energy absorption coefficient 242.01.2.4.3 Exposure and absorbed dose 252.01.3 x-Ray Tubes and Beam Quality in Diagnostic Radiology 262.01.3.1 Beam Attenuation and Beam Shape Descriptors 302.01.3.2 Effect of Varying the Kilovoltage at Fixed Beam Filtration 342.01.3.3 Effect of Varying the Added Filtration at a Fixed Kilovoltage 342.01.3.4 Effect of Varying the Tube Current and Exposure Time at Fixed Kilovoltage 352.01.3.5 Effect of Filtration by Air in the Beam Line 362.01.3.6 Beam Output at Varying Kilovoltages 362.01.3.7 Effect of Voltage Ripple and of Target Angle 372.01.3.8 Attenuation and Beam Hardening 372.01.3.9 Focal Spot Size 392.01.4 Examples of x-Ray Image Formation and Contrast Mechanisms 442.01.4.1 Attenuation Contrast (Absorption and Scattering) 442.01.4.2 Attenuation Plus Phase Contrast 46Acknowledgments 47References 47EVIE
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GlossaryBeam hardening The phenomenon of shift toward higher
energies of the spectral distribution of a polychromatic x-ray
beam, upon transmission of the beam through an
attenuating material, as a consequence of the decreasing
x-ray attenuation coefficient of materials with increasing
photon energy.
K-edge filtering A technique of x-ray beam filtration in
which the spectral intensity distribution of the x-ray beam is
heavily attenuated at energies above a threshold given by the
K-edge absorption energy, characteristic of the filter material,
via photoelectric interaction of the incident photons with
K-shell electrons of atoms in the filter material.
x-Ray absorption imaging A technique for producing
projected shadow images of opaque objects using
penetrating radiation as x-rays, revealing the internal
structure of the objects, based on the measurement with an
image detector of the spatially varying transmitted intensity
through the object determined by the varying amount of
absorption of x-rays.
x-Ray beam filtration Technique for changing the
shape of the spectral distribution of the photon beam from
an x-ray source via transmission through a thin (metal)
sheet in front of the x-ray beam. Spectral shaping occurs
by means of energy-selective x-ray absorption in the sheet,
due to the energy dependence of the absorption
coefficient of the filter material at varying x-ray
energies.
x-Ray phase-contrast imaging A technique for
producing a projected image of opaque objects using
penetrating radiation as x-rays, by measuring how the
electromagnetic waves associated with the propagation of
x-rays are phase-shifted in propagation through matter, as a
result of the spatially varying refractive index at x-ray
wavelengths.
x-Ray photon cross section A quantity with the dimension
of an area, which is proportional to the probability of
occurrence of a particular interaction process (e.g.,
photoelectric effect, Compton effect, and Rayleigh
scattering) of x-ray photons with matter.
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mprehensive Biomedical Physics http://dx.doi.org/10.1016/B978-0-444-53632-7
.00201-X 1Source
x-Rays
R1 R2
Wavefronts
x
y z = 0 z = z 0 z = z 0+ D z
D x
z = z l
z
Object
A
B
a
b
a�
b�
c�c
d�
d
e�
e�
e
Detector
Figure 1 Geometry of x-ray planar imaging. x-Rays diverge from thesource and their direction is deviated by scattering (rays e, e´, e) or leftundeviated (rays b, b´) by transmission through the object, producing a‘shadow’ radiography of the object. An area detector with elements ofsize Dx records a signal in each detector element of size Dx, produced byabsorption of x-rays in the sensitive volume of the detector. Thetransmitted x-ray intensity is dependent on the local properties ofabsorption, scattering, and refraction in the object. In terms ofpropagation of the electromagnetic wave associated with the x-rays, theobject introduces a deformation of the incident waveform due to changeof the phase of the wave, also dependent on the local variation of theobject composition. If the source is of finite size, an image blur(geometric unsharpness) is produced at the edges of the projected objectshadow (the light-gray shaded penumbra regions of rays a and a´). Forsuitably small source size, the phenomenon of x-ray refraction can beobserved: rays at the boundary of denser object details are deviated tothe side (diverge) (e.g., rays c, c´) while at the boundary of lighter objectdetails rays are deviated (converge) toward the denser medium (e.g.,rays d, d´).
2 Physical Basis of x-Ray Imaging
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2.01.1 Introductory Concepts
2.01.1.1 Imaging Basics
x-Ray imaging is adopted in many fields, including diagnostic
radiology, in order to visualize the three-dimensional (3D)
internal structure of ‘objects,’ as biological tissues, which are
opaque to visible radiation. The short wavelengths of x-rays
adopted in diagnostic radiology (in the order of 0.01–0.1 nm)
allow one to penetrate the tissues so that a part of the incident
radiation is transmitted through the sample and can be ana-
lyzed with an area detector for the visualization purpose; the
remaining part is either transmitted while remaining unde-
tected (back- or side-scattered radiation, radiation penetrating
the detector) or absorbed in the object. The fraction of radia-
tion that interacts with the atoms in the body by releasing
energy to them produces absorbed dose of ionizing radiation.
Research into stochastic biological effects of ionizing radiation
showed that adverse effects might result from the absorption of
low doses of ionizing radiation in living matter, so that
hypotheses of a risk hazard for the human body might be
postulated, on a statistical base, for diagnostic medical imaging
procedures employing ionizing radiation producing exposure
to as low effective doses as 100 mSv or less (see Chapters 2.10
and 7.12). These hypotheses of risk factors have to be com-
pared with the beneficial effects on individuals, and on the
population, of such diagnostic procedures in the assessment of
body tissue lesions and medical therapies. In any case, any
diagnostic imaging procedure employing ionizing radiation
should aim at reducing to as low as reasonably possible levels
the radiation dose to the patient, so that it is fundamental to
consider image quality in conjunction with the associated
radiation dose, when discussing the relative merits of any
technique for x-ray imaging of biological organisms.
The purpose of this chapter is to introduce basic aspects of
the physics of x-ray imaging in terms of interaction processes,
spectral properties of the radiation generated by an x-ray tube,
and image contrast mechanism.
The various x-ray imaging techniques require a source of
continuous or pulsed radiation; an object to be imaged, at a
distance R1 from the source; and an area detector, at a distance
R2 from the object (Figure 1). This figure intends to summarize
various aspects of x-ray imaging. First, one can consider
the imaging radiation as rays (of energy� tens of keV) emanat-
ing from a point-like or extended source, traveling in straight
lines in vacuum (or air), or as traveling electromagnetic (e.m.)
waves (of wavelength in the order of 0.01–0.1 nm) and corre-
sponding wave fronts. Some of the rays are unaffected by
passage through the sample (e.g., rays b, b0 in Figure 1), while
some others are deviated (scattered) in the forward or in the
backward direction (e.g., rays e, e´). Rays that are absorbed (to
varying extent) in the different regions of the sample generate a
geometrical shadow of the object contour and of its internal
details (indicated with varying gray levels in Figure 1 on the
detector elements). In the case of a point-like source, rays
passing through any point of discontinuity of the sample (as
at its external contour or at the border of an internal detail)
may pass through it and generate a point image on the detec-
tor. However, in the case of a finite-size source (as with all x-ray
tube sources used in diagnostic radiography units) the rays
originating from various points of the source generate a
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finite-size image of any such object point, when reaching a
distant detector (as for rays a, a´). This generates a penumbra
region (e.g., the region between rays a and a´), which intro-
duces an image blur (‘geometric unsharpness’ effect) increasing
as the distance between the object and the detector increases.
In umbra and penumbra regions, the e.m. wave associated
with x-ray propagation behind the object presents a modula-
tion of its amplitude which determines intensity variations on
the detector, with less intense waves corresponding to propa-
gation through more absorbing regions in the sample. On the
other hand, at the same time, wave fronts (of spherical shape,
for a point-like source at finite distance from the object) are
distorted after traversing the sample because of change of the
wave phase. The deformation of the wave fronts corresponds to
refraction of x-rays, with deviation toward the side of less
material density. This is shown, for example, by rays (c, c´)
and (d, d´) in Figure 1 at the border regions of internal details
of higher density or lower density regions than the background
(details A and B, respectively). This refraction produces an
effect of converging or diverging rays, measurable as x-ray
intensity modulations when placing the detector at suitably
large distances R2 from the object, as indicated in Figure 1
(so-called propagation-based phase contrast imaging; Zhou
and Brahme, 2008; Bravin et al., 2013).
Physical Basis of x-Ray Imaging 3
The relative positions of the source, object, and detector
determine the geometry for projection radiography (with the
detector in front of the source acquiring planar view of the
transmitted x-ray beam through the object) or for scatter imag-
ing (with the detector placed at an angle, e.g., 90�, with respect
to the beam axis, acquiring Compton and Rayleigh scattered
photons in the object). Computed tomography (CT) x-ray
transmission imaging requires the acquisition of hundred
views with a linear or area detector array, by relative rotation
of the source–detector assembly around the object, in planar
geometry (Kalender, 2011); if the detector is placed laterally,
scatter CT is performed (Cesareo et al., 2002). In projection
imaging, R2ffi0 and a contact radiography is executed, for
which R1þR2ffiR1 and no image magnification takes place. If
a divergent x-ray beam is used and if R2 6¼0, then magnification
imaging occurs, with magnification factor M¼(image size/
object size)¼(R1þR2)/R1>1.
The characteristics of the radiographic image acquired
depend on many variables, including
(a) the size, shape, and angular intensity distribution of the
x-ray source;
(b) the spectral distribution of the radiation emitted by the
source (e.g., a monochromatic or a polychromatic beam,
and its spectrum of energies/wavelengths);
(c) the beam divergence angle;
(d) the dose to the irradiated object;
(e) the polarization state, coherence, and the temporal struc-
ture of the emitted radiation;
(f) the 3D absorption, scattering, and refraction properties of
the object, which may include details with a 3D spatial
frequency distribution up to a limit frequency dependent
on the size of the finest details;
(g) the distances R1 and R2;
(h) the two-dimensional (2D) spatial resolution, the intrin-
sic detection efficiency, the 2D noise properties (power
spectrum), and the temporal resolution properties of the
detector;
(i) the angle between the beam axis and the imaging plane;
(j) the energy-dependent attenuation properties of the
medium between the source and the object, and between
the object and the detector.
In principle, all of the above conditions affect the quality of
the x-ray image. Quite generally, one could state that the ‘ideal’
imaging conditions are obtained when using a point-like,
intense x-ray source of monochromatic radiation, delivering
low radiation dose to the sample, at practically short imaging
distances from a detector having a spatial resolution permitting
the visibility of the finest details in the irradiated object and
high detection efficiency, and with negligible noise. In practical
situations in biomedical imaging, an unpolarized, incoherent,
finite-sized, and polyenergetic source of x-rays is adopted
(commonly given by the radiation beam emanating from the
focal spot of an x-ray tube); the detector has a finite pixel size in
the order of 0.1 mm, and a quantum detection efficiency and
an energy absorption efficiency less than 100%. Moreover, the
source has a size in the order of 1 mm, the distance R1þR2 is in
the order of 1 m, and R2ffi0. Owing to fluctuations in the
number of x-ray photons impinging on each detector element
around its mean level (related to the statistics of x-ray photon
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generation), which introduce ‘noise’ in the image, the signal-
to-noise ratio (SNR) for detection has a value dependent on the
intensity of the x-ray field and, thus, on the radiation dose to
the object. Owing to nonideal detector characteristics, the SNR
intrinsic to the incident x-ray field on the area of a detector
element is decreased in the output signal of that element, so
that a degradation of SNR characteristic is introduced, and a
dose higher than in the ideal case must be adopted for imaging.
2.01.1.2 Absorption, Scattering, Refraction of x-Rays
In the diagnostic energy range (10–150 keV), the atomic total
interaction cross section stot (cm2 per atom) can be expressed
as the sum over the individual photon interaction cross sec-
tions of the relevant interaction processes in this range (i.e.,
excluding pair and triplet production and photonuclear inter-
actions which occur at higher energies):
stot ¼ sR þ sC þ tPE [1]
where sR is the cross section for Rayleigh (coherent) scattering,
sC that for Compton (incoherent) scattering, and tPE is the
cross section for photoelectric absorption. In this range, for
photon energies E away from the atomic binding energies
where absorption threshold regions are present, both the
photoabsorption and coherent scattering interactions inmatter
can be described by using complex atomic scattering factors (or
form factors), f¼ f1þ if2. The atomic photoabsorption cross sec-
tion, tPE(E), is related to the imaginary part of the atomic
complex form factor and can be obtained from the values of
f2 using the relation
tPE ¼ 2hcreE
f2 ¼ 2relf2 [2]
where re is the classical electron radius¼(e2/4pe0mec2)¼
2.819380�10�15 m, h the Planck’s constant¼6.62606957
�10�34 J s, c the velocity of light in vacuum¼2.99792458�108 m s�1, E¼hv is the photon energy, and
l¼hc/E the corresponding wavelength, with l (nm)¼1.2398520/E (keV). The atomic coherent scattering cross section
is related to the real part f1 of the complex form factor. Using the
complex form factor f (electrons per atom), one can derive
expressions for the material refractive index and for scattering
and absorption coefficients. The (semiempirical) atomic scatter-
ing factors are based upon photoabsorption measurements of
elements in their elemental state, modeling condensed matter as
a collection of noninteracting atoms: this is true for energies
sufficiently far from absorption thresholds.
The atomic photoelectric cross section tPE is related to the
corresponding cross section per unit mass, or photoelectric mass
attenuation coefficient ma/r (cm2g�1), through
mar¼ NA
AtPE [3]
where NA¼Avogadro’s number (¼6.02214129�1023 mol�1)and A¼atomic mass (number of grams per mole of material).
With Z the atomic number (number of electrons per atom
of an element), in the diagnostic energy range, the cross section
tPE for photoelectric absorption varies approximately as Z4,
while the atomic cross section sC for Compton scattering varies
as Z1 and the atomic cross section sR for Rayleigh scattering
VIER
Photon energy (keV)
m/r
(cm
2 g-1
)
mR/r mtr/r
tPE/r
mC/r
m/r
Carbon (Z = 6)
1
104
103
102
101
100
10-1
10-2
10 100
Figure 2 Mass attenuation coefficients and mass energy transfercoefficient for carbon (graphite) are plotted in the diagnostic energyrange. Data are calculated with the code XMuDat (Nowotny, 1998) withinteraction cross section data from Boone and Chavez (1996).
Photon energy (keV)
Phosphorus (Z = 15)
1 10
104
103
102
101
100
10-1
10-2
10-3
100
m/r
(cm
2 g-1
)
mR/r
mtr/r
tPE/r
mC/r
m/r
Figure 3 Mass attenuation coefficients and mass energy transfercoefficient for phosphorus are plotted in the diagnostic energy range. TheK-edge is at 2.15 keV. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).
4 Physical Basis of x-Ray Imaging
VIER
varies approximately as Z2. For all elements but hydrogen (for
which Z/A¼1), the value of Z/A is between 0.4 and 0.5, with
low-Z elements having Z/Affi0.5. Hence, one can assume that
NA/A∝Z�1: this gives the mass attenuation coefficients a Z
dependence of one power of Z less than for the atomic atten-
uation coefficients.
The atomic cross section for Compton scattering sC is
related to the corresponding cross section per unit mass, or
Compton mass attenuation coefficient mC/r (cm2g�1), through
mCr¼ NA
AsC [4]
while the Rayleigh mass attenuation coefficient mR/r (cm2g�1), isdefined as
mRr¼ NA
AsR [5]
The total mass attenuation coefficient m/r (cm2g�1) can be
obtained as the sum of the mass attenuation coefficients for
each interaction process; without considering pair and triplet
production and neglecting photonuclear interactions, it is
given by
mr
� �¼ mR
r
� �þ mC
r
� �þ ma
r
� �[6]
where mR/r indicates the contribution of the Rayleigh scatter-
ing, mC/r indicates the contribution of the Compton scattering,
and ma/r that of the photoelectric effect. For compounds and
mixtures, the mass attenuation coefficient (m/r)comp can be
expressed as the weighted average of the mass attenuation
coefficients (m/r)i of their elemental constituents, with weights
given by their weight fractions wi :
mr
� �comp
¼Xi
wimr
� �i
[7]
For a compound of chemical formula (X1)a1(X2)a2� � �(Xn)an,
one has
wi ¼ aiAiXi
aiAi
[8]
where Ai is the atomic weight of element Xi in the composition.
The linear attenuation coefficient m (cm�1) for a compound
material of mass density rcomp is m¼(m/r)comprcomp.
The linear attenuation coefficient m (cm�1) for an elemental
material of mass density r, is given by
m ¼ rmr
� �¼ r
marþ mR
rþ mC
r
� �[9]
can be expressed in terms of atomic cross sections as
m ¼ rmarþ mR
rþ mC
r
0@
1A
¼ rNA
AtPE þ sR þ sCð Þ
¼ N tPE þ sR þ sCð Þ
[10]
where N is the number of atoms per unit volume (cm�3).While the linear attenuation coefficient m of a material varies
ELSE
linearly with the density r of the material, the mass attenuation
coefficient m/r is independent of density. The mass density r(g cm�3) can be expressed as r¼(A/NA)N, where NA/A is the
number of atoms per gram of material and the quantity NAZ/A
is the number of electrons per gram of material. The mass
attenuation coefficients (m/r, tPE/r, sR/r, sC/r) [and the mass
energy transfer coefficient (mtr/r) (defined in Section
2.01.2.4)] for some elements (C, Ca, P, I, Pb) and for soft
tissue are shown in Figures 2–6, from attenuation data of
Hubbell and Seltzer (1995) or Boone and Chavez (1996), as
tabulated by the computer code XmuDat (Nowotny, 1998).
Photon attenuation data are also available via the computer
code XCOM by Berger and Hubbell (1987).
For the above-mentioned elements, Figure 7 shows the
ratio of Compton to photoelectric mass attenuation coeffi-
cients, indicating that for low-Z elements Compton scattering
may dominate over photoabsorption at energies of interest for
radiography. On the other hand, for high-Z materials photo-
electric interaction is dominant. Similarly, the coefficients for
soft tissue are plotted versus photon energy in Figure 8, while
Figure 9 shows that Compton scattering in soft tissue is dom-
inant at energies above ffi30 keV where the highest output of a
100-kVp spectrum is contained. For water (Figure 10), the
Photon energy (keV)
m/r
(cm
2 g-1
)
mR/r
mtr/r
tPE/rmC/r
m/r
Calcium (Z = 20)
1
104
103
102
101
100
10-1
10-2
10 100
Figure 4 Mass attenuation coefficients and mass energy transfercoefficient for calcium are plotted in the diagnostic energy range. Notethe K-edge at 4.0 keV. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).
Photon energy (keV)
m/r
(cm
2 g-1
)
mR/r mtr/r
tPE/rmC/r
m/r
Iodine (Z = 53)
1
104
103
102
101
100
10-1
10-2
10 100
Figure 5 Mass attenuation coefficients and mass energy transfercoefficient for iodine are plotted in the diagnostic energy range. Note theK-edge at 33.2 keV and the L-edges at about 5 keV. Iodinatedcompounds are used as injected contrast agents in some contrast-enhanced radiological diagnostic procedures in order to enhance thelabeled tissue contrast with respect to surrounding tissues, due to thelarge increase in attenuation coefficient around the K-edge of iodine. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).
Photon energy (keV)
m/r
(cm
2 g-1
)
mR/r
mtr/r
tPE/r
mC/r
m/r Lead (Z = 82)
1
104
103
102
101
100
10-1
10-2
10-3
10 100
Figure 6 Mass attenuation coefficients and mass energy transfercoefficient for lead are plotted in the diagnostic energy range. Note theK-edge at 88.0 keV, the L-edges between 13.0 and 15.9 keV, and theM-edges between 2.48 and 3.85 keV. Data are calculated with the codeXMuDat (Nowotny, 1998) with interaction cross section data from Booneand Chavez (1996).
Photon energy (keV)
Rat
io (m
C/r
)/(t
PE/r
)
Compton to photoelectric ratio
C
Ca
P
Pb
I
10
102
101
100
10-1
10-2
100
Figure 7 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range, for some low-Z and high-Zelements (C, P, Ca, I, Pb). The horizontal line corresponds to mC/r¼tPE/r : the zone above this line indicates prevalence of Compton scattering,and below the line photoelectric interaction dominates over scattering.
Photon energy (keV)
mC/r tPE/r
mR/r
mtr/r
m/r
1 10
104
103
102
101
100
10-1
10-2
10-3
100
Soft tissue (ICRU-44)
m/r
(cm
2 g-1
)
Figure 8 Mass attenuation coefficients and mass energy transfercoefficient for soft tissue are plotted in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).
Photon energy (keV)
Rat
io (m
C/r
) / (t
PE/r
)
Compton to photoelectric ratio
Nor
mal
ized
sp
ectr
al in
tens
ity
Soft tissue (ICRU-44)
100 kVp x-ray spectrum
10
102
101
100
10–1
10–2
10010–2
10–1
100
101
102
Figure 9 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range (thick line), for soft tissue. Thehorizontal line corresponds to mC/r¼tPE/r. Also shown is the energyspectrum at 100 kVp from an x-ray tube (thin line), normalized to 100%at 34 keV where the maximum of the bremsstrahlung output occurs: thecomparison of the two trends shows that in general radiography, in mostof the energy spectrum, soft tissue scattering through Comptoninteractions is dominant over photoelectric absorption, apart from alow-energy tail in the 15–30 keV range.
Physical Basis of x-Ray Imaging 5
ELSEVIE
R
Photon energy (keV)
(mC/r
) / (t
PE/r
)
Water
Compton to photoelectric ratio
10
102
101
100
10-1
10-2
100
Figure 10 Ratio of Compton to photoelectric mass attenuationcoefficients in the diagnostic energy range, for water.
6 Physical Basis of x-Ray Imaging
Compton mass attenuation coefficient equals the photoelectric
mass attenuation coefficient at about 28 keV, and at 100 keV it
is about 60 times greater.
In what follows, the relation between index of refraction
and attenuation coefficient will be illustrated.
For low x-ray energies or forward scattering (small scatter-
ing angle), one can assume that atoms in a liquid or solid
material scatter the incident radiation as electric dipoles, so
that x-ray interaction with the material can be described using
optical constants like the complex refractive index.
Let us consider atoms as damped, isotropic harmonic oscil-
lators driven by an incident e.m. wave (for simplicity assumed
linearly polarized) exerting on each bound atomic electron of
charge e and mass me an oscillating driving force �eE0eiot atangular frequency o¼2pn¼2pc/l. The equation of motion in
a direction x normal to the propagation direction z, for an
electron having a natural frequency of oscillation o0¼2pn0(resonance frequency), can be written as
med2x
dt2þmeg
dx
dtþmeo2
0x ¼ �eE0eiot [11]
where g is the damping coefficient for that electronic state,
representing a dissipation constant associated with each natu-
ral frequency of oscillation of the atom, so that for each mode,
i, of oscillation we have the three quantities o0i, gi, and f0i, with
f0i being the oscillator strength factor (see Feynman et al.,
1964). The solution of this equation is
x tð Þ ¼ e=m
o2 � o20 � igo
E0eiot [12]
from which the module of the induced dipole moment
p¼�ex is
p oð Þ ¼ e2=me
�o2 þ o20 þ igo
E0 [13]
and the atomic polarizability a(o) can be defined as
a oð Þ ¼ e2=e0me
�o2 þ o20 þ igo
[14]
with e0 being the permittivity of vacuum. Then, in order to take
into account the action exerted by the incident e.m. wave on
each atomic electron and all modes of atomic oscillation char-
acterized by the values of (o0i, gi, fi), one takes the sum over
ELSE
those modes, thus arriving at the expression for the atomic
polarizability
a oð Þ ¼ e2
e0me
Xi
f0i
o20i � o2
� �þ igio[15]
It can be shown that this model leads to an expression for
the complex index of refraction which, for frequencies o�o0i,
can be written as
n ¼ 1� Ne2
2e0me
Xi
f0i
o20i � o2
� �2 þ g2i o2
264
375 o2 � o2
0i
� �þ igio� �
¼ 1�Nrel2o2
2p
Xi
f0i
o20i � o2
� �2 þ g2i o2
264
375 o2 � o2
0i
� �þ igio� �
[16]
where N is the number of atoms of the element per unit
volume, and where the sum is extended over all electronic
states of the atom.
Under the assumption of weak damping and for oscillation
frequencies o�o0i, the complex index of refraction, n, for an
elemental material can be calculated by the simplified
expression
n ¼ 1�Nrel2
2pf1 þ if2ð Þ [17]
with the atomic form factor, f, having a real part f1 and an
imaginary part if2. The index of refraction is then written as
n ¼ 1� d� ib [18]
having defined the dimensionless real numbers refractive index
decrement, d, and the absorption index (or linear absorption coef-
ficient) b as
d rel2
2pNf1 [19]
b rel2
2pNf2 [20]
For a compound or mixture of density rcomp, with Ni the
atom number density of the atomic species i, each having
atomic number Zi, a weight fraction wi, atomic weight Ai, and
complex atomic form factor fi, one has
Ni ¼ wiNA
Arcomp [21]
and the complex refractive index, in the approximation of
atoms scattering as dipoles, can then be written as
n ffi 1� rel2
2p
Xi
Nifi [22]
From this expression, the refractive index decrement of the
compound or mixture, dcomp, can be calculated as
dcomp ¼ rel2
2p
Xi
Nif1,i [23]
VIER
Photon energy (keV)(a)
(b)
n = 1-d-ib
d
bdel
ta o
r b
eta
Breast tissue (ICRU-44)
0
10-4
10-6
10-8
10-10
10-12
10-14
20 40 60 80 100 120 140
Photon energy (keV)
d / bRat
io d /
b
Breast tissue (ICRU-44)107
106
105
104
103
20 40 60 80 100 120 140
Figure 11 (a) The refractive index decrement, d, and the absorptionindex, b, of breast tissue (whose composition is shown in Table 1)versus photon energy in the diagnostic range. The ratio d/b is shown in(b). Data are calculated with the code XOP v2.3 (Sanchez del Rio andDejus, 2003).
Photon energy (keV)
Water, liquid
d
bdel
ta o
r b
eta
1
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10 100
Figure 12 The refractive index decrement, d, and the absorption index,b, of liquid water versus photon energy in the diagnostic range. Data arecalculated with the code XOP (X-ray Oriented Programs) (Sanchez delRio and Dejus, 2003).
Physical Basis of x-Ray Imaging 7
VIER
f1,i being the real part of the atomic form factor for forward
scattering; at the same time, the absorption coefficient, bcomp,
can be calculated as
bcomp ¼rel
2
2p
Xi
Nif2i [24]
f2,i being the coefficient of the imaginary part of the atomic
form factor for forward scattering. Away from absorption
edges, f1,iffiZi electrons per atom so that Nif1,i is the number
of electrons of the atomic species i in the compound or mixture
and SiNif1,i is the electron density re (electrons cm�3). For an
elemental material, the electron density is
re ¼ rNA
AZ [25]
so that by combining eqns [19] and [25], at energies well above
the absorption energies, the refractive index decrement is line-
arly proportional to the electron density via
d ¼ rel2
2pre [26]
For a compound (or mixture), one defines the effective
electron density (re)eff
reð Þeff ¼ rcomp
NA
AeffZeff [27]
where Zeff is the effective atomic number of the compound
(number of electrons per molecule of the compound) and Aeff
is the effective atomic weight, defined as the ratio of the molec-
ular weight of the sample divided by the total number of the
atoms in the compound. Hence, in analogy with eqn [23], for
energies significantly above the absorption edges, the refractive
index decrement is proportional to the effective electron den-
sity of the material:
dcomp ¼ rel2
2preð Þeff ¼
reh2c2
2pE2reð Þeff [28]
and in terms of mass density:
dcomp ¼ reh2c2NA
2p
� �Zeff
Aeff
� �1
E2rcomp [29]
The values of d(E) and b(E) are largely different, with dbeing much larger than b for biological tissues; they show a
generally decreasing trend as a function of energy, away from
threshold energies where discontinuities in b are present.
Figure 11(a) shows the refractive index decrement and the
absorption index for breast tissue as a function of photon
energy, in the diagnostic energy range; their ratio is plotted in
Figure 11(b), where it is seen that d is three to six orders of
magnitude higher than b in the range interesting for radiol-
ogy. The data for d and b for water are shown in Figure 12,
where in the semilog scale the 1/E2 dependence of d is
evident (eqn [28]).
ELSE
The absorption index, b, is related to the cross section for
photoelectric absorption tPE and to the mass attenuation coef-
ficient ma/r by
tPE ¼ 4plN
b [30]
mar¼ 4p
lrb [31]
The transmission T of a collimated x-ray beam through a
slab of thickness d is then given by
8 Physical Basis of x-Ray Imaging
T ¼ exp �NtPEdð Þ ¼ exp � 4plbd
� �¼ exp �madð Þ [32]
where ma is the photoelectric attenuation coefficient (cm�1).The elemental composition of some soft tissues is shown
in Table 1; the total mass attenuation coefficient in the diag-
nostic energy range for some tissues in this table is shown in
Figure 13, while Figure 14(a) shows corresponding values of
the linear attenuation coefficient.
It is seen that in that energy range, m/r and m are a decreas-
ing function of energy, with differences between linear attenu-
ation coefficients of tissues (hence, tissue contrast) decreasing
as energy increases. In Figure 14(b), the plot of ma versus
energy for soft tissue, blood, and lung tissue shows that they
practically have the same values of attenuation coefficient in
the diagnostic range, so that for increasing the radiopacity of,
for example, blood vessels, a suitable contrast agent can be
injected, that is, a solution containing a highly attenuating
medium like iodine-containing compounds (see Figure 5).
2.01.1.3 Tissue Substitute Materials
A number of compound materials are commonly used as tissue
substitutes in radiological imaging, for quality control and for
laboratory studies: for attenuation-based imaging, the require-
ment is to show a linear attenuation coefficient similar to
that of the substituted tissue in the diagnostic energy range.
Soft tissue substitutes include water, polymethylmethacrylate
(PMMA), and, for breast tissues (schematically comprising
skin, fibroglandular, adipose tissues, and microcalcifications),
several epoxy resins like BR10, BR12, CB1, CB2, CB3, CB4,
in addition to paraffin wax, polyethylene as adipose tissue
substitutes, as well as CaCO3 or Al2O3 or Al or Au as possible
substitutes for microcalcifications. Some parameters of tissue
substitute material are reported in Table 2. Figure 15 shows
plots of the attenuation coefficient versus energy in the range of
10–150 keV, for some tissue substitutes.
PMMA is a common soft tissue substitute material for
imaging diagnostics, being cheap and easily machinable. In
attenuation-based imaging, PMMA has attenuation properties
similar to those of a soft tissue; in Figure 16(a) its linear
attenuation coefficient m is plotted versus photon energy
in the diagnostic range together with that of liquid water.
The differences in m are within �25% and þ15% between
PMMA and water in the diagnostic range, with the deviation
(mPMMA�mwater) vanishing at about 36 keV (Figure 16(b)).
In phase contrast imaging, PMMA shows analogous
good properties as a tissue substitute material: Figure 17
shows its values of d and b for energies in the diagnostic
range. In Figure 18, a comparison is shown between the values
of the refractive index decrement d for PMMA and breast tissue
in that energy range, showing a relative deviation of about 12%
between PMMA and tissue. The d/b ratio versus photon energy
is shown in Figure 19 for PMMA and, for comparison, also for
water: the two ratios assume close values at high energies,
differing by less than 1%, whereas they differ by as much as
60% at 10 keV.
Breast tissue microcalcifications (i.e., deposit of a fraction
of a millimeter in size composed of calcium phosphate
or calcium oxalate dehydrate, known as type II and type I
ELSE
microcalcifications, respectively) are important findings in
the diagnosis of breast cancer through mammography (see
Chapter 2.05). As a tissue substitute for such a tissue, fine
grains of calcium carbonate (CaCO3) can be adopted. Figure
20(a) shows values of d and b for CaCO3 in the low-energy
range up to 30 keV, typical of mammography;
microcalcifications (as simulated by CaCO3 grains) generate
contrast in an attenuation-based or phase-based image because
of the differences in their values of b and d, respectively, in the
photon energy range around 20 keV, as shown in Figure
20(b). Note that in this case, contrast is higher in terms of bvalues rather than d values.
2.01.1.4 Absorption Contrast and Phase Contrast
In general, the refractive index n(x,y,z) is a function of the
three-dimensional spatial coordinates of the point of observa-
tion in a nonhomogeneous sample (let us neglect any depen-
dence on time of the refractive index for simplicity). Hence,
both d¼d(x,y,z) and b¼b(x,y,z) depend on the spatial position
as well. With o¼2pc/l being the angular frequency, let us
consider the field at a point P(x,y,z) of an e.m. wave E¼E0exp[i(ot�oz/c)] propagating in free space along the direction
zwith amplitude E0. If the propagation between the source and
P occurs in a homogeneous medium of refractive index n, then
the field of the wave is E´¼E0 exp[i(ot�onz/c)]. By substitut-
ing the complex index of refraction n¼1�d� ib, the wave canbe written as
E0 ¼ E0 exp i ot � on
z
c
0@
1A
24
35
¼ E0 exp i ot � o 1� dð Þ zcþ iob
z
c
24
35
8<:
9=;
¼ E0 exp i ot � oz
c
0@
1A
24
35 exp iod
z
c
0@
1A exp �ob z
c
0@
1A
[33]
In eqn [33], we can see that with respect to propagation in
free space where E¼E0 exp[i(ot�oz/c)], the net effect of con-sidering the propagation in a material slab of complex index of
refraction n is a change (decrease) in amplitude E´/E0 given by
E0
E¼ exp iod
z
c
0@
1A exp �ob z
c
0@
1A
¼ exp i2pldz
0@
1A exp � 2p
lbz
0@
1A
[34]
The above expression shows the presence of a change in the
phase of the wave (given by the first factor on the right-hand
side, related to the refractive index decrement d) and a change
in its amplitude (attenuation) (given by the second factor on
the right-hand side, related to the absorption index b). In other
words, if we interpose a slab of thickness Dz of refractive indexn, between the source and the point P in free space, the result-
ing field at P is E0 exp[i2pdDz/l] exp[�2pbDz/l] and the
changes in phase (Df) and wave amplitude (A/A0) with respect
to propagation in free space are given, respectively, by
VIER
Table 1 Composition of some soft and hard tissues according to ICRU Report 44 (1989), as tabulated in the code XMuDat (Nowotny, 1998) and ascan be found also on the NIST website (http://physics.nist.gov/PhysRefData/XrayMassCoef/tab2.html)
Breast tissue (ICRU-44) Adipose tissue (ICRU-44)
Z Element Weight fraction Z Element Weight fraction
1 H (hydrogen) 0.106 1 H (hydrogen) 0.1146 C (carbon) 0.332 6 C (carbon) 0.5987 N (nitrogen) 0.030 7 N (nitrogen) 0.0078 O (oxygen) 0.527 8 O (oxygen) 0.27811 Na (sodium) 0.001 11 Na (sodium) 0.00115 P (phosphorus) 0.001 16 S (sulfur) 0.00116 S (sulfur) 0.002 17 Cl (chlorine) 0.00117 Cl (chlorine) 0.001Electrons (g�1) 3.32Eþ23 Electrons (g�1) 3.35Eþ23Calc. electrons (cm�3) 3.39Eþ23 Calc. electrons (cm�3) 3.18Eþ23Effective Z 7.07 Effective Z 6.47Mean ratio hZ/Ai 0.55196 Mean ratio hZ/Ai 0.55579Density r (g cm�3) 1.020Eþ00 Density r (g cm�3) 9.500E�01
Bone, cortical (ICRU-44) Brain, gray/white matter (ICRU-44)
Z Element Weight fraction Z Element Weight fraction
1 H (hydrogen) 0.034 1 H (hydrogen) 0.1076 C (carbon) 0.155 6 C (carbon) 0.1457 N (nitrogen) 0.042 7 N (nitrogen) 0.0228 O (oxygen) 0.435 8 O (oxygen) 0.71211 Na (sodium) 0.001 11 Na (sodium) 0.00212 Mg (magnesium) 0.002 15 P (phosphorus) 0.00415 P (phosphorus) 0.103 16 S (sulfur) 0.00216 S (sulfur) 0.003 17 Cl (chlorine) 0.00320 Ca (calcium) 0.225 19 K (potassium) 0.003Electrons (g�1) 3.10Eþ23 Electrons (g�1) 3.33Eþ23Calc. electrons (cm�3) 5.95Eþ23 Calc. electrons (cm�3) 3.46Eþ23Effective Z 13.84 Effective Z 7.65Mean ratio hZ/Ai 0.51478 Mean ratio hZ/Ai 0.55239Density r (g cm�3) 1.920Eþ00 Density r (g cm�3) 1.040Eþ00
Tissue, soft (ICRU-44) Blood, whole (ICRU-44)
Z Element Weight fraction Z Element Weight fraction
1 H (hydrogen) 0.102 1 H (hydrogen) 0.1026 C (carbon) 0.143 6 C (carbon) 0.117 N (nitrogen) 0.034 7 N (nitrogen) 0.0338 O (oxygen) 0.708 8 O (oxygen) 0.74511 Na (sodium) 0.002 11 Na (sodium) 0.00115 P (phosphorus) 0.003 15 P (phosphorus) 0.00116 S (sulfur) 0.003 16 S (sulfur) 0.00217 Cl (chlorine) 0.002 17 Cl (chlorine) 0.00319 K (potassium) 0.003 19 K (potassium) 0.002
26 Fe (iron) 0.001Electrons (g�1) 3.31Eþ23 Electrons (g�1) 3.31Eþ23Calc. electrons (cm�3) 3.51Eþ23 Calc. electrons (cm�3) 3.51Eþ23Effective Z 7.64 Effective Z 7.74Mean ratio hZ/Ai 0.54996 Mean ratio hZ/Ai 0.54999Density r (g cm�3) 1.060Eþ00 Density r (g cm�3) 1.060Eþ00
Lung tissue (ICRU-44) Muscle, skeletal (ICRU-44)
Z Element Weight fraction Z Element Weight fraction
1 H (hydrogen) 0.103 1 H (hydrogen) 0.1026 C (carbon) 0.105 6 C (carbon) 0.1437 N (nitrogen) 0.031 7 N (nitrogen) 0.034
(Continued)
Physical Basis of x-Ray Imaging 9
ELSEVIE
R
Photon energy (keV)
m/r
(cm
2 g-1
)
Tissues:Bone, corticalBrainAdiposeBreast
Mass attenuation coefficients
10
101
100
10-1
100
Figure 13 Total mass attenuation coefficient for some soft and hardtissues (cortical bone, brain (gray/white matter), and adipose and breasttissues) as a function of energy, in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996), for tissues in Table 1.
Photon energy (keV)(a)
(b) Photon energy (keV)10 100
m (c
m-1
)m
(cm
-1)
Tissues:Bone, corticalBrainAdiposeBreast
Tissues (ICRU-44):SoftBloodLung
Linear attenuation coefficients
Linear attenuation coefficients
10
102
101
100
10-1
10-1
100
100
Figure 14 (a) Linear mass attenuation coefficient for some soft andhard tissues (cortical bone, brain (gray/white matter), and adipose andbreast tissues) as a function of energy, in the diagnostic energy range.(b) The almost coincident values of the linear attenuation coefficient for asoft tissue, blood, and lung tissue. Data are calculated with the codeXMuDat (Nowotny, 1998) with interaction cross section data from Booneand Chavez (1996).
Lung tissue (ICRU-44) Muscle, skeletal (ICRU-44)
Z Element Weight fraction Z Element Weight fraction
8 O (oxygen) 0.749 8 O (oxygen) 0.7111 Na (sodium) 0.002 11 Na (sodium) 0.00115 P (phosphorus) 0.002 15 P (phosphorus) 0.00216 S (sulfur) 0.003 16 S (sulfur) 0.00317 Cl (chlorine) 0.003 17 Cl (chlorine) 0.00119 K (potassium) 0.002 19 K (potassium) 0.004Electrons (g�1) 3.32Eþ23 Electrons (g�1) 3.31Eþ23Calc. electrons (cm�3) 3.48Eþ23 Calc. electrons (cm�3) 3.48Eþ23Effective Z 7.66 Effective Z 7.63Mean ratio hZ/Ai 0.55048 Mean ratio hZ/Ai 0.55000Density r (g cm�3) 1.050Eþ00 Density r (g cm�3) 1.050Eþ00
Table 1 (Continued)
10 Physical Basis of x-Ray Imaging
EVIER
Df ¼ � 2pldDz [35]
A
A0¼ exp � 2p
lbDz
� �¼ exp � m
2Dz
� (36)
The intensity I of the wave field is related to the square of
the amplitude A, so that
I
I0¼ exp �mDzð Þ [37]
where
m ¼ 4plb [38]
For a homogeneous material (spatially constant d and b),eqn [37] (known as the Lambert–Beer exponential attenuation
law) can be written as
� lnI
I0
� �¼ 4p
lbDz [39]
while eqn [35] gives the rate of change of the phase
(rad cm�1) as
�dfdz¼ 2p
ld ¼ relre [40]
ELS
We note that for a homogeneous material slab of given dand b, assumed constant in the slab of thickness Dz, at a givenl, eqns [35]–[38] can be combined to give
Df ¼ 1
2
dbln
I
I0
� �[41]
thus showing that in this hypothesis, the phase change Df can
be recovered from an attenuation measurement (I/I0) and
from prior knowledge of the ratio d/b for that material. When
Table 2 Composition of some tissue substitute materials, as calculated with the code XMuDat (Nowotny, 1998)
PMMA (C5H8O2)n BR12a
Z Element Weight fraction Z Element Weight fraction
1 H (hydrogen) 0.080541 1 H (hydrogen) 0.0966 C (carbon) 0.599846 6 C (carbon) 0.7048 O (oxygen) 0.319613 7 N (nitrogen) 0.019
8 O (oxygen) 0.16917 Cl (chlorine) 0.00220 Ca (calcium) 0.009
Electrons (g�1) 3.25Eþ23 Electrons (g�1) 3.267Eþ23Calc. electrons (cm�3) 3.87Eþ23 Calc. electrons (cm�3) 3.17Eþ23Effective Z 6.56 Effective Z 6.73Density r (g cm�3) 1.19Eþ00 Density r (g cm�3) 9.70E�01
Calcium carbonate (CaCO3) Paraffin wax (C25H52)
Z Element Weight fraction Z Element Weight fraction
6 C (carbon) 0.120003 1 H (hydrogen) 0.1486058 O (oxygen) 0.479554 6 C (carbon) 0.85139520 Ca (calcium) 0.400443Electrons (g�1) 3.01Eþ23 Electrons (g�1) 3.45Eþ23Calc. electrons (cm�3) 8.42Eþ23 Calc. electrons (cm�3) 3.21Eþ23Effective Z 15.62 Effective Z 5.51Density r (g cm�3) 2.80Eþ00 Density r (g cm�3) 9.30E�01
Alumina (Al2O3) Polyethylene (C2H4)n
Z Element Weight fraction Z Element Weight fraction
8 O (oxygen) 0.470749 1 H (hydrogen) 0.14371613 Al (aluminum) 0.529251 6 C (carbon) 0.856284Electrons (g�1) 2.95Eþ23 Electrons (g�1) 3.43Eþ23Calc. electrons (cm�3) 1.17Eþ24 Calc. electrons (cm�3) 3.19Eþ23Effective Z 11.28 Effective Z 5.53Density r (g cm�3) 3.97Eþ00 Density r (g cm�3) 9.30E�01
Air, dry Water, liquid (H2O)
Z Element Weight fraction Z Element Weight fraction
6 C (carbon) 0.000124 1 H (hydrogen) 0.1118987 N (nitrogen) 0.755268 8 O (oxygen) 0.8881028 O (oxygen) 0.23178118 Ar (argon) 0.012827Electrons (g�1) 3.01Eþ23 Electrons (g�1) 3.34Eþ23Calc. electrons (cm�3) 3.62Eþ20 Calc. electrons (cm�3) 3.34Eþ23Effective Z 7.77 Effective Z 7.51Density r (g cm�3) 1.20E�03 Density r (g cm�3) 1.00Eþ00
Data for dry air and for liquid water are also included.aData from White (1978).
Physical Basis of x-Ray Imaging 11
ELSEVIE
R
the material is not homogeneous, this simple result is no
longer valid and the measurements of phase change and inten-
sity attenuation are disentangled. From eqns [25] and [38], the
ratio d/b for a material at a given l and the corresponding
energy E can be expressed as
db¼ 2rel
mre ¼
2rehc
m Eð ÞEre [42]
In Figure 21, the rate of phase shift�df/dz for breast tissueand for PMMA (also known as Plexiglas, or lucite), a common
soft tissue substitute material, is shown in the diagnostic
range. In this figure, the relative phase shift versus photon
energy E follows the 1/E trend implied by eqn [40]; more-
over, the closeness of the two curves reflects the similarity of
the values of the effective electron density of breast tissue and
PMMA (3.39�1023 and 3.87�1023cm�3, respectively, see
Tables 1 and 2). For example, at the energy of 17.5 keV of
the Ka characteristic line of Mo of mammography x-ray tube,
a 50-mm compressed breast thickness introduces a phase
advance of about 1000p rad with respect to propagation in
50-mm air.
Photon energy (keV)
PEBR12PMMA
CaCO3
Al2O3Li
near
att
enua
tion
coef
ficie
nt
(cm
-1)
1010-1
100
101
100
Figure 15 Linear attenuation coefficient of some common tissuesubstitute materials (Table 2) in the diagnostic energy range. Data arecalculated with the code XMuDat (Nowotny, 1998) with interaction crosssection data from Boone and Chavez (1996). For BR12, data arecalculated with the XCOM database of NIST.
Photon energy (keV)
Photon energy (keV)(a)
(b)
PMMAWater
36 keV
mPMMA / mwater
Rat
io o
f att
enua
tion
coef
ficie
nts
Line
ar a
tten
uatio
n co
effic
ient
, m
(cm
-1)
00.7
0.8
0.9
1.0
1.1
1.2
10-1
100
101
10 100
20 40 60 80 100 120 140
Figure 16 (a) Linear attenuation coefficient m ofpolymethylmethacrylate (PMMA) and of liquid water in the diagnosticrange. (b) The ratio mPMMA/mwater versus energy: it equals 1 at about36 keV. The data are from the XCOM database of NIST.
PMMA
Photon energy (keV)
n = 1 - d- ib
d
bDel
ta o
r b
eta
110-11
10-9
10-7
10-5
10 100
Figure 17 The refractive index decrement, d, and the absorption indexb, of PMMA versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003).
PMMA
Breast tissue(ICRU-44)
Photon energy (keV)
Ref
ract
ive
ind
ex d
ecre
men
t d
1010-8
10-7
10-6
10-5
d
100
Figure 18 The refractive index decrement, d, of PMMA and ICRU-44breast tissue versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003).
Photon energy (keV)
PMMAWater
d / b
Rat
io d
elta
/ b
eta
00
500
1000
1500
2000
2500
20 40 60 80 100 120 140
Figure 19 The ratio of the refractive index decrement, d, to theabsorption index, b, for PMMA and water versus photon energy in thediagnostic range. Data are calculated with the code XOP (Sanchez del Rioand Dejus, 2003).
12 Physical Basis of x-Ray Imaging
ELSEVIE
R
If the refractive index n of a material varies only along the
direction z, then the phase f of the propagating e.m. wave,
relative to free-space propagation, changes along z by a quan-
tity Df¼f(x,y,z0;l)�f(x,y, z0þDz;l) after traveling a distanceDz, given by
Df ¼ � 2pl
ðz0þDz0
d zð Þdz [43]
At the same time, the intensity of the e.m. wave is
reduced as
I
I0¼ exp
ðz0þDz0
� 4plb zð Þ
�dz
�
¼ exp
ðz0þDz0
�mð Þdz � [44]
Photon energy (keV)(a)
(b) Photon energy (keV)
CaCO3
d
d -CaCO3
b-CaCO3
d -breast tissue
b-breast tissue
d,b
d or
b
b
1 10
10
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-5
10-6
10-7
10-8
10-9
10-10
15 20 25 30
Figure 20 (a) The refractive index decrement, d, and absorptioncoefficient, b, of CaCO3 used as a substitute material formicrocalcifications in breast tissue versus photon energy in the range upto 30 keV typical of mammography. The discontinuity due to the K-edgeof Ca (4.04 keV) is visible. (b) The comparison of d and b values forCaCO3 and ICRU-44 breast tissue shows the origin of absorption andphase contrast of microcalcifications in the energy range typical ofmammography. Data are calculated with the code XOP (Sanchez del Rioand Dejus, 2003).
PMMABreast tissue(ICRU-44)
10
101
102
Photon energy (keV)
Rat
e of
pha
se s
hift
,-d
f/d
z (r
ad m
m-1
)
100
Figure 21 The rate of change of phase �df/dz of breast tissue andof PMMA versus photon energy in the diagnostic range. Data arecalculated with the code XOP (Sanchez del Rio and Dejus, 2003). Materialdata are in Tables 1 and 2.
Physical Basis of x-Ray Imaging 13
ELSE
In general, if d ¼d(x,y,z;l) and b¼b(x,y,z;l), then the pro-
jected phase f ¼f(x,y;z,l) and the projected linear attenuation
coefficient m¼m(x,y;z,l) along the direction z are given by
f x; y; z; lð Þ ¼ � 2pl
ðz�1
d x; y; z0; l
� dz0
[45]
m x; y; z; lð Þ ¼ � 4pl
ðz�1
b x; y; z0; l
� dz0
[46]
Given the relation (see eqn [28]) between the (local) refrac-
tive index decrement of a material and its (local) electron
density re(x,y,z), eqn [44] can also be written as
f x; y; z; lð Þ ¼ �relðz�1
re x; y; z0
� dz0
[47]
For a compound or mixture, one has
f x; y; z; lð Þ ¼ �relðz�1
reð Þeff x; y; z0
� dz0
[48]
showing that the projected phase map is essentially a map of
the projected electron density re(x,y)¼Ðre(x,y,z)dz in the sam-
ple, at a given wavelength.
In the limit of 2D projection radiography, the above formal-
ism provides the description of the interaction of an x-ray wave
with a sample (e.g., a biological tissue) as introducing phase
shift and amplitude attenuation effects in the propagation of the
e.m. wave associated with an x-ray beam. For an absorbing and
scattering thick object illuminated by a given incident wave
field, extending from z¼0 to z¼Dz and containing spatial
inhomogeneities in d (i.e., in electron density) and in m(i.e., in atomic number Z and in mass density r, via photoelec-tric and Compton interactions) over a range of spatial frequen-
cies u (cm�1), the transmitted field T(x,y;zl) at distance zl>Dz ina plane normal to the projection direction z depends through
volume integrals on the contribution at each point (x,y;zl) of all
absorption and scattering events in the whole object volume.
However, if we limit our analysis to the case of a thin object,
then one can consider line integrals of phase f and linear
attenuation coefficient m along the projection direction z, as
given by eqns [45] and [46]. The expression for the transmitted
field at distance zl on a plane perpendicular to the propagation
direction z can be simplified to give the projected transmission
function T(x,y;zl) as (Wu and Liu, 2004):
T x; y; zlð Þ ¼ A x; y; zlð Þeif x;y;zlð Þ [49]
T x; y; zlð Þ ¼ exp if x; y; zlð Þ � m x; y; zlð Þ2
�[50]
Here, f(x,y;zl) and m(x,y;zl) are the projected phase and
projected attenuation coefficients, respectively. In eqn [49], A
(x,y;zl) is the x-ray-transmitted amplitude at the plane z¼zl,
and its intensity A2(x,y;zl) gives the attenuation-based image (or
attenuation image) of conventional projection radiography.
Two-dimensional spatial variations in the transmitted intensity
A2(x,y;zl) depend on the 3D spatial variation of the linear
absorption coefficient b(x,y,z) in the object (as given by eqn
[46]), so that image contrast in the attenuation image is
based on the subject contrast produced by variations in
the projected attenuation coefficient (so-called attenuation
contrast). On the other hand, the projected phase f(x,y;zl) of
the complex-transmitted field at the plane z¼zl represents a
phase image whose contrast arises from spatial variations in the
phase shift produced by x-ray propagation inside the object
(so-called phase contrast), as given by eqn [45]. Contrast in
the projected phase map f(x,y;zl) is related to the spatial
VIER
14 Physical Basis of x-Ray Imaging
inhomogeneities of the (effective) electron density in the
object, as given by eqns [49] and [50]. The linear attenuation
coefficient m for a material scales linearly with the density of
the material and the mass density increases with increasing
atomic number Z for most naturally occurring solid or liquid
elements (i.e., excluding gases). In the diagnostic energy range,
it contains contributions to attenuation arising from photo-
electric interactions and from elastic and inelastic scattering
interactions (eqn [1]). In this range, the photoelectric inter-
action cross section tPE depends on the atomic number and
on the photon energy E approximately as Z4/E3 (see Section
2.01.2.1); correspondingly, the Rayleigh interaction cross sec-
tion sR varies as Z2/E2 while the Compton cross section sCvaries as Z and is approximately independent of E. Hence, eqns
[49] and [50] show that attenuation contrast and phase con-
trast convey information on the spatial distribution of the
(effective) atomic number and of electronic density in the
object. At a given photon energy E and the corresponding
wavelength l, the condition of x-ray thin objects for the deri-
vation of eqns [49] and [50] imposes constraints on the x-ray
transmission T and on the size of the finest detail which can be
imaged at that energy: in the diagnostic range, it has been
indicated that ‘human body parts can be treated as thin objects
for resolutions as high as 10 mm’ (Wu and Liu, 2003).
In projection radiography, 2D maps of object properties
like d(x,y) or m(x,y), integrated over the projection direction z,
can be derived in principle by measurements of the phase
change maps (phase imaging) and by intensity attenuation
maps (absorption-based imaging).
2.01.2 Interaction Processes
The interaction processes of x-ray photons with atoms and elec-
trons in matter in the diagnostic energy range are described in
this section, including the absorption and scattering processes.
Absorption of x-rays occurs through the photoelectric effect,
whereas inelastic and elastic photon scattering occur via the
Compton effect and the Rayleigh scattering effect, respectively.
SE
Atomic number, Z
Ca
Na
S
W
YL
YK
I
00.0
Fluo
resc
ence
yie
ld, Y
k
0.2
0.4
0.6
0.8
1.0
10 20 30 40 50 60 70 80 90 100110
Figure 22 Fluorescence yield for K-shell (YK) and for L-shell (YL)electrons for elements with atomic number between 3 and 110 (YK) andbetween 3 and 100 (YL) (data from Hubbel et al., 1994).
2.01.2.1 Photoelectric Absorption
In the (internal) photoelectric effect, a photon of suitably high
energy E¼hn interacts with a tightly bound atomic electron
(i.e., an orbital electron of the inner atomic shells K, L, M, . . .,
with binding energy Eb), transferring a part of its energy and of
its momentum (q¼hn/c) to the electron, which receives a
kinetic energy Ek and a momentum pe and is ejected from the
atom. The remaining part of photon energy and momentum is
given to the atom, which recoils with kinetic energy Ek,a and a
momentum pa: in the interaction, the incident photon van-
ishes (i.e., it is totally absorbed). This process can take place
with the given bound electron only if hn>Eb and also hn ffi Eb,
or in other words, the total absorption of the incident photon
occurs only in the presence of the binding atom and for pho-
ton energies close to (but higher than) the given electron
binding energy, in order to conserve both energy and momen-
tum in the process: indeed, momentum conservation is reached
only upon considering as nonnegligible the atom recoil
momentum pa. This implies that in the kinematics of the
EL
photoelectric interaction, the direction of the photoelectron
makes an angle different from zero with the direction of the
incident photon, according to an angular distribution which
(by momentum and energy conservation) is dependent on the
photon energy. As long as the photon energy is increasingly
greater than the binding energy (EK or EL or EM) for a given
bound electron (belonging to the shell K, or L, or M, respec-
tively), the interaction becomes less probable with that elec-
tron; for photon energies below a given binding energy,
photoelectric interaction with a lower orbit electron may
occur. Owing to its largely higher rest mass with respect to
that of the bound electron, the atom recoils with a negligible
kinetic energy with respect to the kinetic energy acquired by
the (photo)electron, so that the fraction of the incident photon
energy given to the atom is equal to the potential energy Eb with
which the electron is bound to the atom. Under this descrip-
tion, conservation of energy for this effect can be written as
hn ¼ Ek þ Eb [51]
In the photoelectric effect, in which the ejection of the
electron from an atomic shell leaves a corresponding vacancy
in that shell, promptly filled by a higher shell electron, the
atomic excitation corresponding to the absorption of an
amount of energy given by Eb is followed by a de-excitation
process in which at least a part of this energy can be released
by the emission of a fluorescence (‘characteristic’) x-ray. The
energy of this emitted photon is given by the difference
between the binding energies of the two potential energy levels
between which the electronic transition occurs. This mecha-
nism of atomic de-excitation is relevant for vacancies that are
produced in a K-shell or in an L-shell, during a photoelectric
interaction. The probability of atomic emission of a fluores-
cence x-ray is called the fluorescence yield Y : in particular, if the
emission is determined by a vacancy in the K-shell, it is indi-
cated by YK, and it is YL in the case of a vacancy in the L-shell
(see also Section 2.01.2.4). Given the total number of photo-
electric interaction events in the whole atom, a fraction PKoccurs with the K-shell electrons, and a fraction PL occurs
with L-shell electrons: for values of Z less than 20, PK is 0.9 or
higher, and it is about 0.8 for a high-Z element as tungsten
(Z¼74). Figure 22 shows the fluorescence yield for shells K
and L for varying atomic number Z. It can be seen that the
curve for YL is largely below that for YK; for low-Z elements up
VIER
Photoelectron polar angle q (deg)with respect to the photon direction
20 keV100 keV
00
2
4
6
30 60 90 120 150 180
Pol
ar a
ngle
dep
end
ent
term
of d
s/dW
Figure 23 The term in square bracket in eqn [52] in the text is plottedversus the angle the photoelectron makes with the direction of theincoming photon, at 20 and 100 keV kinetic energy.
Physical Basis of x-Ray Imaging 15
to Z¼20, YK is less than 0.2. Hence, an alternative de-
excitation process must exist so that the atom can release the
fraction of the binding energy not emitted by x-ray fluores-
cence. This process is the Auger effect, in which the excess
atomic energy is transferred to one or several orbital electrons
as kinetic energy, large enough to eject those electrons from the
atom: then, such Auger electrons release their kinetic energy
locally in the surrounding medium. The mechanism of non-
radiative de-excitation through emission of the Auger electron
(s) is also alternative (with probability 1�Y) to the radiative
emission of fluorescent photons for K-shell or L-shell vacancy,
so that radiative and nonradiative de-excitation can coexist in
the same photoelectric interaction event or be totally alterna-
tive to each other.
The differential angular distribution (cm2 sr�1) of the emit-
ted photoelectron in the nonrelativistic regime of the photo-
electric interaction of a photon with energy hn with an atom of
atomic number Z can be expressed with the following formula
(Heitler, 1954):
dsdO¼ 4
ffiffiffi2p
r2eZ5
1374
� �mec
2
hn
� �7=2sin 2y cos 2’
1� b cos yð Þ4" #
[52]
where y is the angle between the direction of the incident
photon and the ejected electrons, ’ is the azimuth angle of
the photoelectron with respect to the x-ray polarization vector,
mec2 is the photoelectron rest mass energy, and b is the ratio of
the photoelectron velocity and the speed of light c in vacuum.
As regards the dependence of ’ on a plane normal to
the direction of propagation of the incoming photon, it fol-
lows from eqn [52] that ds/dO∝cos2’, a condition that is at
the basis of methods for x-ray polarimetry of nonrelativistic
photoelectrons.
As regards the dependence on y for a given ’, ds/dO∝ffi sin2y/(1�bcosy)4¼ f(y). The above formula gives ds/dO¼0 for y¼0� and for y¼180�; thus the above semi-positive
function has an absolute maximum for a photoelectron angle
ymax, depending on the b value and hence on the kinetic
energy of the photoelectron. In particular, owing to the bpower dependence in the denominator of the angular term
in eqn [52], ymax is shifted toward narrower angles as the
kinetic energy of the photoelectron increases. By taking the
two terms in parentheses in eqn [52] as constant in a given
medium and for a given photon energy, the angular depen-
dence on y expressed by f(y) in the last term in square
brackets of eqn [52] is shown in Figure 23 at photoelectron
energies of 20 and 100 keV. At such low kinetic energies (and
corresponding low b values), the term (1�bcosy) is close to
unity so that f(y)∝ sin2y and the photoelectron is preferen-
tially ejected in the direction of the vector electric field of the
incident wave.
By integrating the differential cross section per atom over all
photoelectron emission angles y, one obtains the atomic cross
section for the photoelectric effect tPE (cm2 per atom), whose
dependence on the photon energy hn and on the atomic num-
ber Z in the diagnostic energy range can be approximated by
the following expression:
tPE∝Z4
hnð Þ3 [53]
ELSE
and then, the photoelectric mass attenuation coefficient ma/r(cm2 g�1) (eqn [3]) in the diagnostic range has the following
dependence on Z and on photon energy:
mar∝
Z3
hnð Þ3 [54]
The photoelectric mass attenuation coefficient ma/r of var-
ious elements versus photon energy in the diagnostic range is
plotted in Figures 2–6, while Figures 8 and 13 show the plots
for some soft biological tissues; the plot of ma/r for water is in
Figure 25(b). All these plots indicate that at low energy in low
effective Z materials like soft tissues and water, the photoelec-
tric effect (via the hn3 dependence) dominates over scattering
interactions, with photoelectric mass attenuation coefficient
equal to Compton mass attenuation coefficient at about
30 keV for water (see also Figure 25(c)).
VIER
2.01.2.2 Rayleigh (Coherent) Scattering
In Rayleigh (coherent) scattering, a photon is scattered by
bound electrons without production of excitation or ioniza-
tion of the atom; the interaction involves the entire atom
rather than only its bound electrons so that it can be consid-
ered as a process of elastic scattering since the photon loses
only a negligible fraction of its energy (Hubbell, 1999). Lord
Rayleigh studied this type of scattering of light by air mole-
cules in the last decades of nineteenth century; the term
‘coherent’ reflects the cooperative process of all atomic elec-
trons in producing destructive and constructive interference
effects. In amorphous materials like biological tissues, this
gives an oscillating behavior of the angular distribution of
the Rayleigh cross section.
For sufficiently low frequencies o of light (and correspond-
ing low energies of photons hv¼ho/2p), elastic scattering can
be described by the scattering cross section
ss ¼ 8pr2e3
o4
o2 � o20ð Þ2
[55]
where o0 is the natural frequency of oscillation of the bound
electron which scatters the incoming radiation. In the limit of
scatter from an unbound (‘free’) electron – for which o0!0
and the low frequency v�v0, but with v such that it can be still
16 Physical Basis of x-Ray Imaging
considered low – one obtains the formula for the cross section
sT for classical Thomson scattering from an electron:
ssffisT¼(8/3)pre2¼0.6652448�10�24 cm2 independent of
energy. In the other limit of vv0, and for all atomic natural
frequencies in the atom energy levels for which this relation
holds, scattering occurs by the atom as a whole and atomic
electrons contribute ‘coherently’ to this so-called Rayleigh scatter-
ing process, for which ss�o4. If light scattering occurs coherently
from a whole atomic plane in a crystal, this is called Bragg
scattering.
In all these processes of elastic scattering, no energy is
transferred to the scatter center (electron or atom or molecule),
the atom recoil occurring only to the extent of momentum
conservation; upon interaction, the incident radiation is devi-
ated from its original direction, for any scatter event, according
to the geometry shown in Figure 24 in terms of the scattering
angle 0�y<180� (where 0� is forward scattering) and of the
azimuthal angle 0�f< 360�.In a given material, the probability density function of
scattering at any given angle y, p(y), is a function of energy
and can be highly peaked, while p(f) is constant over f. Thepeak of the p(y) distribution tends to shift to lower scattering
angles as the photon energy increases (see Figure 28).
When traversing any given material thickness, electronic
scattering events may occur for any atom (or groups of
atoms) encountered along the path of the incident radiation,
and the net effect is an angular redistribution of the incoming
beam energy, as described by the differential cross section per
unit solid angle ds(y)/dO versus the scatter angle y. In calcu-
lating the scattered energy fluence out of a given material
volume irradiated by x-ray photons in the energy range of
diagnostic radiology, the single scattering (or first order)
approximation considers the resulting contribution from only
the first scatter event, without consideration for possible suc-
cessive (second or multiple order) Rayleigh scatter processes.
In other words, in the commonly adopted single scatter
approximation, photons reaching the detector after being
Rayleigh scattered in a thickness of a given material have a
history of just one scatter event.
At x-ray energies in the diagnostic energy range, in terms of
angular distribution of the scattered radiation, Rayleigh
(coherent) scattering is highly forward peaked (yffi0). This
LSE
Incident photondirection
Scattered photondirection q
f
Figure 24 Illustration of the scattering angle y (between 0� and 180�)and the azimuthal angle f (between 0� and 360�) that define thegeometry of the scattering process.
E
implies that both primary and coherently (single) scattered
radiations reach the imaging detector at relatively close loca-
tions which depend, among other factors, on the distance
between the scatter center and the image plane. This difference,
due to scatter beam divergence, can be observed, for example,
by using a very narrow primary beam and observing at a
suitable distance from the irradiated object (Johns and Yaffe,
1983). Hence, the shorter the object-to-detector distance,
and the thinner the irradiated object is, the less the influence
(through any structured pattern) of coherent scatter on the
energy fluence distribution on the detector. This occurs at
variance with incoherent (Compton) scattered radiation,
which tends to distribute energy fluence over larger scattering
angles. On the other hand, the probability per unit path length
of being Rayleigh scattered is typically one order of magnitude
lower than for the total interaction probability, so that the
combined effects of high forward scatter and small interaction
coefficients tend to compensate each other in part to produce
a net result of a coherent scatter contributing to a significant
fraction of the total (RayleighþCompton) scatter reaching the
imaging detector, even more for low-energy x-ray imaging as
in mammography (Johns and Yaffe, 1983).
Mass linear attenuation coefficients m/r (cm2g�1) are
related to total atomic (or molecular) cross sections s (cm2
per atom or cm2 per molecule) via m/r¼s(ΝΑ/Α). The interac-tion cross sections (sR, sC/r, and tPE, and total cross section
stot) for water in the diagnostic energy range are shown in
Figure 25(a); the mass linear attenuation coefficients of water
(for which ΝΑ/Α ¼0.0334271�1024 molecules cm�3) are
shown in Figure 25(b); the corresponding percent ratio of
Rayleigh to total coefficients is shown in Figure 25(c).
The differential Thomson cross section per (free) electron
for elastic scattering (cm2 sr�1 per electron) (for unpolarized
photons) is
dsT yð ÞdO
� �elec
¼ r2e2
1þ cos 2y� �
[56]
and the (total, i.e., angle integrated) cross section for Thomson
scattering is
sT ¼ðy¼py¼0
dsT yð Þdy ¼ 8pr2e3
[57]
(sT¼0.6652448 barns per electron, 1 barn¼10�28 m2). The
differential Thomson cross section per scattering angle y is thengiven by
dsT yð Þdy
� �elec
¼ dsT yð ÞdO
dOdy¼ pr2e 1þ cos 2y
� �sin y [58]
and it is plotted in Figure 26. It is symmetrical about the
y¼90� axis and is zero at 0� and 180� with maxima at about
55� and 125�.The differential cross section per atom for coherent
(Rayleigh) scattering is given by
dsR yð ÞdO
� �atom
¼ dsT yð ÞdO
� �elec
F2 x;Zð Þ [59]
where Z is the atomic number and where
x q
2h¼ E
hcsin
y2
� �¼ 1
lsin
y2
� �[60]
VIER
Photon energy (keV)
Mass linear attenuation coefficients
TotalPhotoelectricComptonRayleigh
m/r
(cm
2 g-1
)
20
10-3
10-2
10-1
100
101
40
Water
60 80 100 120 140
Photon energy (keV)(a)
(b)
(c)
Total
0.0334271x1024
Photoelectric
Compton
Cro
ss s
ectio
n (1
024 cm
2 p
er m
olec
ule)
molecules per cm3
Rayleigh
200
2
4
6
8
10
40
Water
60 80 100 120 140
Photon energy (keV)
Ratio of mass attenuation coefficients
Rayleigh / totalCompton / totalPhotoelectric / total
(m/r
) /
(m/r
) tot (%
)
200
20
40
60
80
100
40
Water
60 80 100 120 140
Figure 25 (a) Interaction cross section and (b) mass linear attenuationcoefficients as a function of energy in water, for Rayleigh, Compton,photoelectric, and total interaction, in the diagnostic energy range. (c)Ratio of mass linear attenuation coefficients for water. Rayleigh scatterinteractions contribute negligibly to the total attenuation coefficientexcept in the range around 20 keV for mammography, where it is in theorder of 10% of (m/r)tot. Data are calculated with the code XMuDat(Nowotny, 1998) with interaction cross section data from Boone andChavez (1996).
Scattering angle, q (deg)
Thom
son
diff
eren
tial c
ross
se
ctio
n (1
0-24
cm2 /
deg
)
00.00
0.05
0.10
0.15
0.20
dsT /dq0.25
0.30
30 60 90 120 150 180
Figure 26 Electron differential cross section for Thomson scattering,as a function of the scattering angle (see eqn [58] in the text).
Physical Basis of x-Ray Imaging 17
ELSE
with q¼change of momentum of the photon¼2hx (for energy
E¼hv¼hc/l and scatter angle y).The term F2(x,Z) takes into account the collective effect of
the interference between the scattering from the various elec-
trons in the target atom of atomic number Z, through the
atomic form factor F(x,Z) for Rayleigh scattering, whose values
have been first tabulated by Hubbell et al. (1975). For energies
close to atomic absorption edges, anomalous coherent scatter-
ing occurs at resonant absorption energies, which could be
dealt with by modifying the formalism of the form factors as
(see, e.g., Hugtenburg et al., 2002)
F x;E;Zð Þ ¼ f0 x;Zð Þ þ f0E;Zð Þ þ if
00E;Zð Þ [61]
by introducing energy-dependent real (f0) and imaginary (f00)anomalous scatter (or dispersion) factors.
For an atom with Z electrons, the atomic form factor F(x,Z)
in eqn [59] takes into account the effect on the amplitude of
the wave scattered by an atom in a given direction, due to
superposition of the scattering fields from the different atomic
electrons, with their amplitude determined by the electronic
cross section. When the change of momentum q of the photon
in the atomic scattering event is negligible, simple addition of
the scattering intensities from each of the Z electrons occurs,
and F(ffi0,Z)ffiZ; for increasing q, F(q,Z) decreases. Equation
[61] and anomalous dispersion factors come into play when
considering that when they radiate in response to the incident
wavefield, atomic electrons cannot be considered free: in par-
ticular, K-shell electrons are tightly bound so that their scatter-
ing behavior is dependent on the energy of the incident
wave. This can be described by introducing the complex form
factor, assuming an angle-dependent factor f0(x,Z) and energy-
dependent factors f´(E,Z) and f00(E,Z). As an example, consider
the case of molybdenum (Z¼42) with the energy of the K-edge
20 keV; Figure 27(a) shows the energy-dependent terms f´(E)
and f00(E), with anomalous dispersion evident at the K-edge,
while Figure 27(b) shows the term f0(x,Z) as a function of
the scattering angle at the energy of the Ka1 characteristic line
of Mo (17.479 keV).
From eqn [58], the Rayleigh (coherent) scattering cross
section per atom is then
sR ¼ðy¼py¼0
dsT yð ÞF2 x;Zð Þ
¼ 3
8sT
ð1�1
1þ cos 2y� �
F2 x;Zð Þd cos yð Þ[62]
The Rayleigh scattering differential cross section is
approximately independent of scattering angle up to a few
kiloelectronvolts, but rapidly falls to zero with increasing
scattering angle, at higher energies. The atomic cross section
for Rayleigh scattering (cm2 per atom) is a function of Z and
photon energy E¼hn, and for high energies this dependence
is expressed by
sR � Zn
En2 < n < 2:5ð Þ [63]
For energy in the diagnostic range,
VIER
Photon energy (keV)(a)
(b)
K-edge
Mo (Z = 42)
f�
f��
Ato
mic
form
fact
ors
0
86420
-2-4-6-8
20 40 60 80 100 120 140
Scattering angle (deg)
E = 17.479 keV (Ka1)
Mo (Z = 42)
Ato
mic
sca
tter
ing
fact
or, f
0
05
10
15
20
25
30
35
40
45
10 20 4030 6050 70 80 90
Figure 27 (a) Anomalous atomic scattering form factors f0 and f00 (seeeqn [61] in the text) for molybdenum, as a function of energy (data fromSasaki, 1989). (b) Atomic scattering form factor f0(x,Z) for Mo, at theenergy of Ka1 characteristic line (calculated with the code SCATFAC v1.0,data constant from Waasmaier and Kirfel, 1995).
Photon energy (keV)
Pb (Z = 82)AI (Z = 13)C (Z = 8)R
ayle
igh
scat
terin
g m
ass
atte
nuat
ion
coef
ficie
nt (c
m2
g-1)
1010-3
10-2
10-1
100
101
100
Figure 28 Mass attenuation coefficient for Rayleigh (coherent)scattering in lead, aluminum, and carbon, in the diagnostic energy range(data from NIST database XCOM).
18 Physical Basis of x-Ray Imaging
sR ffi Z2
En[64]
and for the Rayleigh mass attenuation coefficient,
sRrffi Z
En[65]
The Rayleigh mass attenuation coefficient sR/r for carbon,
aluminum, and lead is shown in the log–log plot of Figure 28.
The power law dependence of sR on the atomic number also
explains the low cross section for Rayleigh scattering in soft
tissues, due to their low effective Z (Zffi7.5).
The interference between the electronic scattering fields
gives rise to the coherent overlapping of the scattering contri-
butions from each electron, thus originating differential coher-
ent cross sections significantly greater than the sum of the
single contributions. For example (Johns and Yaffe, 1983),
for water at 60 keV, if one does not consider the interference
phenomenon by the ten electrons of the H2O molecule, then
one just sums up the cross sections from the free oxygen and
from the two free hydrogen atoms, so obtaining
dsR yð ÞdO
� �water
ffi ZdsT yð ÞdO
� �elec
¼ 10dsT yð ÞdO
� �elec
[66]
while the measured differential Rayleigh scattering cross sec-
tion at 60 keV and at the peak scattering angle can be several
times larger. Also the free H2O molecule (independent
molecule approximation, IMA) shows a Rayleigh differential
cross section significantly greater than for the sum of the free O
and the two free H atoms (independent atom approximation,
ELSE
IAA, see later). On the other hand, in the material bulk, groups
of close atoms may be considered as cooperatively acting for
coherent scattering, and in amorphous materials (as in the case
of tissues in diagnostic radiology), where random orientation
of atomic groups is spatially averaged out, one can assume that
the differential cross section dsR(y)/dO for Rayleigh scattering
can be obtained by multiplication of the atomic cross section
(eqn [59], which includes the atomic form factors F(x,Z)) by
an angle-independent interference function I(x) dependent
only on x, this function describing the effects of interference
between electrons in different atomic groups (Poludniowski
et al., 2009):
dsR yð ÞdO
� �atom
¼ dsT yð ÞdO
� �elec
F2 x;Zð ÞI xð Þ [67]
Disregarding the extra-atomic interference processes
between different electrons represents the IAA (Poludniowski
et al., 2009); for example, for water, assuming the IAA implies
I xð Þ ¼ 2F2H x,Z ¼ 1ð Þ þ F2O x,Z ¼ 8ð Þ ffi F2O x,Z ¼ 8ð Þ [68]
For compounds (or mixtures), containing n elements of
weight fraction wi and atomic weight Ai, one has for the atomic
differential cross section (assuming IMA):
dsR yð ÞdO
0@
1A
comp
atom
¼Xni¼1
widsR yð ÞdO
0@
1A
atom, i
¼Xni¼1
wi ¼Xni¼1
widsT yð ÞdO
0@
1A
elec, i
F2comp x;Zi¼1, ..., nð Þ[69]
with the form factor for compounds given by
F2comp x;Zi¼1, ..., nð Þ ¼ dsRdO
0@
1A
comp
atom
�dsRdO
0@
1A
comp
elec
¼Xni¼1
wi
Ai
0@
1A�1Xn
i¼1
wi
AiF2 x;Zið Þ
[70]
Such form factors for water (from tables of the EPDL97
library of Cullen et al., 1997) are shown in polar plots at 5,
20, and 60 keV photon energies in Figure 29. Atomic differen-
tial cross sections for Rayleigh scattering in water (disregarding
VIER
210
240
270
300
3300q
30
60
901 2 3 4 5 6 7
120
150180
Waterform factor
TotalHO
60 keV
210
240
270
300
3300q
30
60
901 2 3 4 5 6 7
120
150180
TotalHO
20 keV
210
240
270
300
330
FF, SF: From EPDL97Normalization independentSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010
WEIGHT FRACTION
H
O
0
q
30
0.11190
0.88810
60
901 2 3 4 5 6 7
120
150180
TotalHO
5 keV
Physical Basis of x-Ray Imaging 19
ELSE
interatomic and intermolecular interference) at 5, 20, and
60 keV are plotted in Figure 30 (Fernandez et al., 2010,
2011), while Figure 31 shows the corresponding angular dis-
tribution of total transmitted intensity of Rayleigh scatter.
The differential cross section for elastic scattering (Thomson
scattering) of linearly polarized photons interacting with a free
electron is given by
dsT yð ÞdO
� �elec
¼ r2e cos2Y [71]
where Y is the angle between the direction of the incident
polarization vector and the direction of the polarization vector
of the scattered photon.
2.01.2.3 Compton (Incoherent) Scattering
Unlike photoelectric absorption, where the interaction of the
incident photon with an atomic bound electron cannot take
place without a recoiling atom, the scattering of a photon off
a loosely bound (virtually, free) electron is possible, neglecting
the influence of the binding atom. In this process, called
Compton effect, the incident photon with energy E¼hn col-
lides with a stationary electron. Then, it is scattered at an angle
f with respect to the direction of incidence, emerging with an
energy E´¼hn´ and giving the fraction (E�E´)/E (not all) of
its energy to the electron which is scattered at an angle y with
respect to the incident photon direction, with a kinetic energy
Ek¼(E�E´) as given by energy conservation. Taking into
account also the conservation law for the total momentum,
we obtain the kinematic relationships between f, y, E, and E´
of the Compton effect:
E0 ¼ E
1þ Emec2
1� cosfð Þ [72]
cot y ¼ 1þ E
mec2
� �tan
f2
� �[73]
Equation [72] is shown in Figure 32 in the diagnostic
energy range, as E´ versus E plot at varying photon scattering
angle f between 0� and 180�. In this plot, a low-energy behav-
ior can be distinguished from the high-energy trend at varying
f. In the region below�10 keV, E´ is equal to E at all scattering
angles: this is a form of elastic scattering described by the
Thomson scattering, whose differential cross section per elec-
tron (eqn [56]) is shown in Figure 33.
At high energy in the diagnostic range, E´¼E only for the
obvious case f¼0. As the photon scattering angle increases
from 0� to 180�, inelastic scattering occurs with increasing
kinetic energy of the Compton electron up to Ekffi(150
�94.5 keV)ffi55.5 keV at f¼180� and E¼150 keV. Ek is plot-
ted versus the incident photon energy in Figure 34, for varying
photon scattering angle.
VIER
Figure 29 Polar plots of atomic form factors for water (and forcontributions from single elements H and O), at 5 keV (right), 20 keV(center), and at 60 keV (left), plotted using the SAP code developed atthe University of Bologna (Fernandez et al., 2010, 2011). Program usedwith authors’ permission.
210
240
270
300
3300q
30
60
90.05 .10 .15
120
150180
Waterslab thickness = 10 cm
AtomicRayleigh scatteringcross section
TotalHO
60 keV
210
240
270
300
3300q
30
60
90.05 .10 .15
.05 .10 .15
120
150180 Total
HO
20 keV
210
240
270
300
330
FF, SF: From EPDL97Normalization: NoneSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010
WEIGHT FRACTION
H
O
0q
30
0.11190
0.88810
60
90
120
150180 Total
HO
5 keV
20 Physical Basis of x-Ray Imaging
ELSE
In the diagnostic range, the scattering angle y of the Comp-
ton electron is only weakly related to the photon scattering
angle f, at varying incident photon energy (Figure 35): y in the
range 0–90� is a decreasing function of f from 0� to 180�, withy¼90� for f¼0�, as well as y¼0� for f¼180�, at all incidentphoton energies.
The differential cross section (cm2sr�1 per electron) for
(unpolarized) photon scattering at an angle f, as derived by
Klein and Nishina, is given by (see Attix, 1986, Chapter 2.06)
dsC fð ÞdO
0@
1A
elec
¼ r2e2
1
1þ E
mec21� cosfð Þ
0BBBB@
1CCCCA
2
1þ E
mec21� cosfð Þ
0@
1Aþ 1
1þ E
mec21� cosfð Þ
0BBBB@
1CCCCA� sin 2f
266664
377775
¼ r2e2
E0
E
0@
1A2
E0
Eþ E
E0� sin 2f
0@
1A
[74]
The Klein–Nishina differential cross section per electron is
plotted in Figure 36 at three energies in the diagnostic range: it
is seen that at low energies it approaches the Thomson cross
section shown in Figure 33, with almost symmetrical distribu-
tion in the forward and backward directions, while at increas-
ing energy a more forward-peaked scattering distribution
develops. The total (i.e., angle integrated) cross section for
Compton scattering is the Klein–Nishina total cross section
(cm2 per electron) given by
sCð Þelec ¼ 2pðf¼pf¼0
dsC fð ÞdO
0@
1A
elec
sinfdf
¼ 2pr2e1þ k
k2
0@
1A 2 1þ kð Þ
1þ 2k� ln 1þ 2kð Þ
k
24
35
8<:
þ ln 1þ 2kð Þ2k
24
35� 1þ 3k
1þ 2kð Þ2
24
359=;
k E
mec2[75]
This function is shown in Figure 37 in the diagnostic
range, and it can be seen that at energies below �10 keV,
sC converges to the Thomson scattering cross section
sT (ffi66.5�10�26 cm2 per electron), decreasing to
ffi45�10�26 cm2 per electron at 150 keV.
VIER
Figure 30 Polar plots of atomic differential cross sections for Rayleighscattering for water (and for contributions from single elements H andO), at 5 keV (right), 20 keV (center), and at 60 keV (left), plotted using theSAP code developed at the University of Bologna (Fernandez et al., 2010,2011). Program used with permission.
210
240
270
300
3300q
30
60
90.05 .10 .15 .20
.0005 .0010
.00010 .00020
120
150180
Rayleighintensity
Waterslab thickness = 10 cm
60 keV
210
240
270
300
3300q
30
60
90
120
15018020 keV
210
240
270
300
330
FF, SF: From EPDL97Normalization: NoneSAP v2.1 (c) Alma Mater Studiorum University of Bologna, 2010
WEIGHT FRACTION
H
O
0q
30
0.11190
0.88810
60
90
120
1501805 keV
Figure 31 Polar plots of total transmitted intensity of Rayleigh scatterfor water, at 5 keV (right), 20 keV (center), and at 60 keV (left), plottedusing the SAP code developed at the University of Bologna (Fernandezet al., 2010, 2011). Program used with permission.
Energy of incident photon, E (keV)
Compton effect
Photon scattering angle, f
E� = E
Ene
rgy
of s
catt
ered
pho
ton,
E
� (k
eV)
0
0�
30�
60�
75�
90�
120�
180�
0
20
40
60
80
100
120
140
20 40 60 80 100 120 140
Figure 32 Energy of Compton scattered photon E´ versus incidentphoton energy E in the range 1–150 keV, for varying values of the photonscattering angle f.
Photon scattering angle, f (deg)
Thom
son
diff
eren
tial c
ross
se
ctio
n (1
0-26
cm2
sr-1
)
0
4
5
6
7
8dsT /dW
30 60 90 120 150 180
Figure 33 Thomson differential cross section per free electron (eqn[56] in the text). This is the limit cross section for the Compton effect atincident photon energies below �10 keV for all scattering angles.
Energy of incident photon, E (keV)
Photon scattering angle, f
Compton effect
Ene
rgy
of s
catt
ered
ele
ctro
n,
Ek
(keV
)
0
0
0�
30�
60�
75�
90�
120�
180�
10
20
30
40
50
60
20 40 60 80 100 120 140
Figure 34 Energy of Compton scattered electron, Ek, versus incidentphoton energy E in the range 1–150 keV, for varying values of the photonscattering angle f.
Physical Basis of x-Ray Imaging 21
ELSEVIE
R
Photon scattering angle, f (deg)
Ele
ctro
n sc
atte
ring
angl
e, q (
deg
)
Incident photon energy, E
0 keV
50 keV
100 keV
150 keV
00
10
20
30
40
50
60
70
80
90
20 40 60 80 100 120
Compton effect
140 160 180
Figure 35 Electron scattering angle y versus photon scattering angle ffor four incident photon energies E in the diagnostic energy range.
Photon scattering angle, f (deg)
Diff
eren
tial c
omp
ton
cros
s se
ctio
n(1
0-26
cm2
sr-1
) per
ele
ctro
n
02
3
4
5
6
7
8
30 60 90 120 150 180
10 keV
50 keV
150 keV
Figure 36 Differential Klein–Nishina cross section per electron forCompton scattering versus photon scattering angle f, given by eqn [74],at varying incident photon energy E.
Incident photon energy (keV)
Com
pto
n cr
oss
sect
ion
(sC) e
lec
(10-2
6 cm
2 p
er e
lect
ron)
(sC)elec
(sT)elec
140
45
50
55
60
65
70
10 100
Figure 37 Klein–Nishina cross section per electron for Comptonscattering in the diagnostic range. At low energies, sC converges to theThomson scattering cross section sT.
22 Physical Basis of x-Ray Imaging
ELSE
Then, the cross section per atom (sC)atom (cm2 per atom) is
obtained as
sCð Þatom ¼ Z sCð Þelec [76]
and the Compton mass attenuation coefficient mC/r (cm2 g�1)(eqn [4]) for an elemental material of density r can be
expressed as
mCr sCð Þatom
r¼ NA
AsCð Þatom ¼
NAZ
AsCð Þelec [77]
Equations [76] and [77] show that while the Compton
cross section per (free) electron (sC)elec is independent of Z
by definition, the Compton cross section per atom (sC)atomvaries as Z and that the cross section per unit mass, mC/r, isindependent of Z due to the slight variation of Z/A (between
0.4 and 0.5) for most elements.
The Compton mass attenuation coefficient for various ele-
ments is plotted in Figures 2–6. Using eqns [7] and [8], the
material compositions in Tables 1 and 2, and eqns [75] and
[77], the Compton mass attenuation coefficient for compound
materials can be derived; Figures 8 and 25(b) show mC/r for
soft tissue and water, respectively.
Equation [74] is valid under the assumption of a free elec-
tron at rest (i.e., with negligible initial kinetic energy). For low-
Z materials and loosely bound electrons, this assumption can
be considered a good approximation only for sufficiently high
photon energies. Electron binding corrections to the differen-
tial Klein–Nishina cross section can be applied by introducing
the incoherent scattering function S(x,Z) as a function of the
change of momentum of the photon, q¼2hx, and of the
atomic number Z :
dsC ’ð ÞdO
� �binding
elec
¼ dsC ’ð ÞdO
� �elec
S x;Zð Þ [78]
(see reviews by Hubbell, 1992, Hubbell et al., 1994, Hubbell,
1999; see also Hirayama, 2000). Binding corrections decrease
the Klein–Nishina differential cross section, to a higher extent
for lower energies in the diagnostic range and for higher Z.
The polarization state of incident x-ray photons has a role
in the description of x-ray scattering at energies in the diagnos-
tic range, also in relation to the fact that the scattering interac-
tion introduces a linear polarization in the scattered photon.
Currently used imaging detectors for diagnostic radiography
are not sensitive to the polarization state of photons; however,
the angular and energy distributions of scattered photons are
sensibly different from those of unpolarized photons, as spec-
troscopy measurements point out.
The Klein–Nishina Compton differential cross section for
linearly polarized photons interacting with a free electron is
given by
dsC ’ð ÞdO
0@
1Apol
elec
¼ r2e4
1
1þ E
mec21� cos’ð Þ
0BBBB@
1CCCCA
2
1þ E
mec21� cos’ð Þ
0@
1Aþ 1
1þ E
mec21� cos’ð Þ
0BBBB@
1CCCCA� 2þ 4 cos 2f
266664
377775
¼ r2e4
E0
E
0@
1A2
E0
Eþ E
E0 � 2þ 4 cos 2f
0@
1A
[79]
VIER
Physical Basis of x-Ray Imaging 23
where ’ is the angle between the direction of the incident
polarization vector and the direction of the polarization vector
of the scattered photon, which can be either in the plane of the
incident polarization vector or perpendicular to it.
2.01.2.4 Mass Attenuation Coefficients and Dosimetry
2.01.2.4.1 Mass energy transfer coefficientFor indirectly ionizing radiation, like photon radiation, the
dosimetric quantity kerma (J kg�1 or Gy) is defined as the
sum of the initial kinetic energies of all charged particles
released by uncharged particles per unit mass (ICRU Report
33, 1980) in a given material. Let us first consider the case of
an elemental material of atomic number Z and density r(g cm�3).
IfF (photons per m2) is the monoenergetic photon fluence of
energy E (J) and C(E)¼EF is the corresponding photon energy
fluence (J m�2), then the kerma, K (J kg�1) (for a monoener-
getic radiation), at a given site in the absorbing material can be
calculated by defining the mass energy transfer coefficient (mtr/r)E,Z (m2kg�1), the ratio of the linear energy transfer coefficient,
mtr (m�1) to the material density r (kg m�3), as
K C Eð Þ mtrr
� �E,Z¼ F E
mtrr
� �E,Z
[80]
If, as commonly used, the photon fluence is expressed in
photons per cm2, E in keV, the density in g cm�3, and the mass
energy transfer coefficient in cm2g�1, then the kerma K (Gy)
can be calculated as
K ¼ 1:6022� 10�13F Emtrr
� �E,Z
[81]
where the coefficient (1.6022�10�13 Gy gkeV�1) takes into
account the change from SI units of measurement. In the case
of a polyenergetic spectrum, then the photon fluence spectrum
FE(E)¼dF/dE and the energy fluence spectrum CE(E)¼dC/
dE can be introduced, respectively. Then, with the same units
and for a polyenergetic x-ray spectrum, the kerma K can be
calculated as
K ¼ 1:6022� 10�13ðE00
FE Emtr Eð Þr
�Z
dE [82]
x-Ray photons in the diagnostic energy range (which
excludes pair production and photonuclear interactions) trans-
fer energy to charged particles via photoelectric or Compton
processes (Rayleigh scattering is an elastic process which does
not contribute to energy transfer); hence, in analogy with the
total mass attenuation coefficient, the mass energy transfer
coefficient can be written as
mtrr
� �¼ mtr, C
r
� �þ mtr, a
r
� �[83]
where the first term on the right-hand side takes into account
the contribution of Compton scattering and the second term,
that of photoelectric events. If all of the energy of x-ray photons
interacting at a point in a material of mass attenuation coeffi-
cient (m/r)E,Z (or, more rigorously speaking, interacting in a
small volume centered to that point) is transferred to charged
ELSE
particles and is converted into kinetic energy of those particles,
then the kerma K is equal to C(E)(m/r). On the other hand, if
only a fraction of the energy of primary photons is converted
into kinetic energy of charged particles at the interaction site
(the remaining part being due to secondary photons that
escape that site carrying away energy hn0<hn), then the kerma
K is equal to C(E)(mtr/r). Hence, the ratio mtr/m of the mass
energy transfer coefficient mtr/r to the (total) mass attenuation
coefficient m/r is equal to the ratio of the energy transferred as
kinetic energy Ek to charged particles to the incident x-ray
energy E0, and can be written as
mtr, Cr
0@
1A ¼ fC
mCr
0@
1A,
mtr, ar
0@
1A ¼ fa
mar
0@
1A
fa ¼ EkE0
, fC ¼ E0k
E0
mtrr
0@
1A ¼ fC
mCr
0@
1Aþ fa
mar
0@
1A
[84]
where Ek and E0k in eqn [84] for the energy transfer fractions
fa and fC represent the kinetic energy of the photoelectron and
of the Compton scattered electron, respectively; the quantity
(mtr/r) depends on the photon energy E and on the atomic
number Z of the material. Plots of the mass energy transfer
coefficient for some elements and for soft tissue, in the diag-
nostic energy range, have been shown in Figures 2–6, and 8.
To illustrate the meaning of mtr,a/r (for photoelectric effect)
(Attix, 1986), let us consider the energy transfer process for
photons in the diagnostic energy range, taking the photoab-
sorption and scattering processes separately into account. In
the photoelectric effect, the fraction (Ek/E0) of the initial pho-
ton energy E0 transferred as kinetic energy Ek to the photoelec-
tron is given by
EkE0¼ E0 � EK, L
E0[85]
where EK,L is the binding energy of the atom for the electronic
shell (K or L) interested in the interaction. In the atomic
relaxation process following the ionization process, the atom
will dispose of this energy by either radiative (fluorescence)
transition(s), in which characteristic x-ray(s) are emitted (the
probability of occurrence of this event being given by YK, the
fluorescence yield for shell K, or YL, the fluorescence yield for
shell L) or, alternatively, by nonradiative transitions(s) in
which Auger electron(s) are emitted. In the first case, energy
is transported away from the interaction site by characteristic
photons, while in the second case energy is transferred to
charged particles and deposited locally. In other words, if
Auger electrons are produced in the photoelectric interaction,
then all of the initial photon energy is transferred to charged
particles and mtr,a/r¼ma/r, whereas if fluorescent x-rays are
emitted, then mtr,a/r<ma/r. The ratio mtr,a/ma is the energy
transfer fraction given by (1�hEfli)/E0 where hEfli is the aver-
age energy of fluorescence radiation. For a K-shell photoelectric
interaction (i.e., for E0 EK), the difference between these
two terms can be obtained by considering the ratio
(E0�PKYK hEfli)/E0. In this expression, Efl is the photon energy
of the characteristic x-ray emitted ( Efl¼EKa1, or Efl¼EKa2, or
VIER
Photon energy (keV)
Air, dry
Mas
s en
ergy
tra
nsfe
r co
effic
ient
(cm
2 g-1
)
1010-2
10-1
100
101
100
m tr /r
Figure 39 Mass energy transfer coefficient for dry air (near sea level),
24 Physical Basis of x-Ray Imaging
VIE
Efl¼EKb1, etc.) and hEfli defined above is the average value of
these photon energies; PK is the fraction of all photoelectric
interactions occurring in the K-shell. Then, for E0 EK, one has
mtr, ar
� �¼ mtr
r
� �1� PKYK
EKh iE0
� �[86]
correspondingly, for EL�E0<EK, one has, with analogous
terms and definitions:
mtr, ar
� �¼ mtr
r
� �1� PLYL
ELh iE0
� �[87]
The fluorescence yield is an increasing function of the
atomic number, and for a given Z it decreases from the
K-series to the L-series. YK for elements with 3�Z�110 and
YL for elements with 3�Z�100 are shown in Figure 22.
For compounds or mixtures, the rule of the weighted aver-
age (by weight) of single components’ coefficients applies:
mtrr
� �comp
¼Xi
fimtrr
� �i
[88]
Then, for a compound or a mixture, the kerma in eqns
[80]–[82] can be written as
K C Eð Þ mtr Eð Þr
� �comp
¼ FEmtrr
� �comp
[89]
K ¼ 1:6022� 10�13FEmtrr
� �comp
[90]
K ¼ 1:6022� 10�13ðE00
FE Emtrr
� �comp
dE [91]
Equations [89] and [90] apply to a monochromatic beam
of energy E, and eqn [91] represents the kerma for a polychro-
matic beam with a spectrum extending up to a maximum
energy E0. In the above formulae, as in eqns [81] and [82],
the coefficient (1.6022�10�13 Gy gkeV�1) converts from SI
units to more common units (keV, Gy, cm2g�1) for energy E,
kerma K and energy transfer coefficient mtr/r, respectively.The mass energy transfer coefficients for water are plotted in
Figure 38, where the contributions from the photoelectric andLSE
Photon energy (keV)
Water
Compton, m tr,C /r
Photoelectric, m tr,a /r
m tr /r
m /r
Mas
s at
tenu
atio
n co
effic
ient
s (c
m2
g-1)
2010-3
10-2
10-1
100
40 60 80 100 120 140
Figure 38 Mass total attenuation coefficient, energy transfercoefficient, and energy transfer contribution from photoelectric andCompton effects for water, plotted in the diagnostic energy range. Dataare calculated with the code XMuDat (Nowotny, 1998) with interactioncross section data from Boone and Chavez (1996).
E
Compton effects are illustrated, as well as the occurrence of
a mtr/m ratio which decreases to an almost constant value
(ffi0.17) as energy increases, in the diagnostic range.
The mass energy transfer coefficient for dry air is shown in
Figure 39 in the diagnostic range, and from this curve, using
eqn [90], the kerma in air per unit fluence K/F (Gy cm2) is
plotted as a function of energy in Figure 40.
2.01.2.4.2 Mass energy absorption coefficientOnce photon energy is transferred to charged particles in the
form of kinetic energy, the average energy of all secondary
charged particles (electrons, in the diagnostic range) can either
be deposited locally at the site of interaction through colli-
sional losses (excitations and ionizations), or be lost through
bremsstrahlung radiation (which is nonnegligible for energetic
electrons and for high-Z materials along the electron paths)
or other means like fluorescence emission following electron
impact ionization. The occurrence of radiative losses in the
energy deposition process can be taken into account in
the definition of kerma, K, by considering, for the two separate
energy loss processes, a collision kerma, Kc, and a radiative
kerma, Kr, respectively:
K ¼ Kc þ Kr [92]R
Photon energy (keV)
Air, dry
K/F
Air
kerm
a p
er u
nit
pho
ton
fluen
ce (G
y cm
2 )
1001010-13
10-12
10-11
Figure 40 Air kerma per unit photon fluence for monoenergetic x-raysin the diagnostic range.
plotted in the diagnostic energy range. Data are calculated with thecode XMuDat (Nowotny, 1998) with interaction cross section data fromBoone and Chavez (1996).
Physical Basis of x-Ray Imaging 25
At a given monoenergetic photon energy E, the collision
kerma Kc is related to the energy fluence C(E) according to
Kc ¼ Cmenr
� �E,Z¼ FE
menr
� �E,Z
[93]
where men/r is the mass energy absorption coefficient (m2kg�1 orcm2g�1). If, in a photon interaction, hEki is the average energyof all secondary charged particles and hEri is the average energyof all photon radiation produced by those particles, then by
introducing the fraction g of hEri to hEki(0<g<1) through the
quantity
Ekh i � Erh iEkh i ¼ 1� Erh i
Ekh i 1� g [94]
one can express the mass energy absorption coefficient in terms
of the energy transfer coefficient as
menr
� �E,Z¼ mtr
r
� �E,Z
1� gð Þ [95]
Thus, men/r is less than mtr/r and men/rffimtr/r when gffi0,
for example when bremsstrahlung production is negligible as
in low-Z materials and at low energy E. The mass energy
absorption coefficient for water is plotted in Figure 41. For
water and soft tissues, gffi0 in the diagnostic energy range,
while for tungsten (Z¼74) at 100 keV, gffi0.5.
2.01.2.4.3 Exposure and absorbed dosex-Rays produce ionization in matter (i.e., they create electron–
ion pairs), so that the dosimetry of an x-ray beam can be
performed with an ionization chamber, by measuring the
amount of electric charge of one type produced in a material
volume (typically, a few cubic centimeters of air) divided by
the mass of material contained in the volume, a quantity
termed as x-ray exposure, X. The energy deposited in air for
production of one ion pair (or electron) varies little (less
than 2%) in the diagnostic energy range; its mean value is�Wð Þair ffi 33:97 eV per electron, so that with e the electron
charge, the mean value in J C�1 is�We
� �air¼ 33:97JC�1. The
unit for exposure is the roentgen (R), with 1R¼2.580�10�4Ckg�1. The exposure X (C kg�1) at a point in a monoenergetic
(E¼hn) x-ray beam, where the energy fluence is C, is then
LSE
Photon energy (keV)
Water
men/r
m/r
Mas
s at
tenu
atio
n co
effic
ient
(cm
2 g-1
)
20
101
10-2
10-1
100
40 60 80 100 120 140
Figure 41 Mass total attenuation coefficient and energy absorptioncoefficient for liquid water in the diagnostic energy range (data from NISTdatabase XRAYCOEF with data from Hubbell and Seltzer (1995)).
E
X Ckg�1� � ¼ C
menr
� �E, air
�W
e
� ��1air
[96]
and by using eqn [93], the exposure X (C kg�1) in eqn [96] is
related to the collision kerma in air (J kg�1) via
X C kg�1� � ¼ Kcð Þair
�W
e
� ��1air
¼ Kcð Þair J kg�1� �
33:97JC�1[97]
By expressing X in roentgen, R (a historical unit of
measurement defined as the quantity of radiation which
liberates one electrostatic unit of charge per cubic centime-
ters of air at standard temperature and pressure), the rela-
tion between exposure (in R) and collision kerma in air
becomes
X ¼ 1
2:580� 10�4Kcð Þair
�W
e
0@
1A�1
air
¼ 1
2:580� 10�4Kcð Þair33:97
¼ Kcð Þair8:76426� 10�3
[98]
or, equivalently,
Kcð Þair J kg�1� � ¼ 8:76426� 10�3X Rð Þ [99]
X Rð Þ ¼ 1
8:76426� 10�3Kcð Þair ffi 114 Kcð Þair J kg�1
� �[100]
and also,
X Ckg�1� � ¼ 2:580� 10�4X Rð Þ [101]
X Rð Þ ¼ 3875:97X Ckg�1� �
[102]
With absorbed dose D in matter defined in terms of energy
imparted per unit mass (1 Gy¼1 J kg�1), under conditions
of charged particle equilibrium (CPE) in the air volume of
the chamber, the (absorbed) dose in air, Dair, at the point
of measurement, is related to the collisional air kerma Kc via
Dair ¼ Kcð Þair CPEð Þ [103]
and using eqn [97] and eqn [99], one can obtain the expression
relating dose in air and exposure:
Dair Gyð Þ ¼ 33:97X Ckg�1� �
[104]
and also,
Dair mGyð Þ ¼ 8:76426X Rð Þ [105]
In air and within the diagnostic energy range, the radiative
kerma is Krffi0 so that KffiKc and then, using eqns [92] and
[103], DairffiKair.
The dose at a point in a medium with energy absorption
coefficient (men/r)medium, with respect to the dose in air, is
given by
Dmedium ¼ Dair
menr
� medium
menr
� air
264
375 [106]
and in terms of exposure X, using eqn [104],
VIER
26 Physical Basis of x-Ray Imaging
Dmedium Gyð Þ ¼ 33:97
menr
� medium
menr
� air
264
375X Ckg�1
� �[107]
and using eqn [105],
Dmedium mGyð Þ ¼ 8:76426
menr
� medium
menr
� air
264
375X Rð Þ [108]
The term in square brackets (‘F-factor,’ here expressed as
Gy kg C�1 andmGy R�1) converts the exposure in air into dose
in a medium; its value is shown for various materials in the
diagnostic range, in Figure 42.
Using eqns [90], [104], and [105], and by considering
that (mtr/r)airffi(men/r)air in the diagnostic range, the dose in a
medium is related to the air kerma Kair by
Dmedium ffi Kair
menr
0@
1A
medium
menr
0@
1A
air
26666664
37777775
¼ 1:6022� 10�13FEmenr
0@
1A
medium
[109]
As an example of the use of eqn [109], the dose in breast
tissue (ICRU-44) per million photons per cm2 calculated
with this formula is shown in Figure 43. In absorption-based
contrast imaging of the breast (mammography), low-energy
photons around 20 keV are used instead of high-energy pho-
tons as in general radiography, in order to exploit the larger
differences in the values of the linear attenuation coefficient of
breast tissues (adipose, fibroglandular, tumor, and microcalcifi-
cations; see, e.g., Figures 14(a) and 15). However, Figure 43
shows that low-energy photons give higher dose to breast tissue
than high-energy photons in the diagnostic range: the mini-
mum in Kbreast/F is between 50 and 70 keV. On the other
hand, phase-contrast imaging relies on the difference in the
SE
Photon energy (keV)
F-fa
ctor
(mG
yR
-1)
F-fa
ctor
(Gy
kgC
-1)
Bone, cortical
Water
Adipose
Breast
15
6789
10
20
30
40
50
6070
10 10019
2327313539
78
116
155
194
233271
Figure 42 F-factor for various materials, in the range 1–150 keV.The sharp edges correspond to element K-edges in the materialcomposition. Data for mass energy absorption coefficient were derivedwith the code XMuDat (Nowotny, 1998) with interaction cross sectiondata from Boone and Chavez (1996).
EL
refractive index decrement d of tissues, whose values are much
larger than b, so that there is potential of dose reduction in
breast imaging based on phase-contrast techniques in the diag-
nostic range (Lewis, 2004), for example, employing photon
energies at 40–80 keV.
2.01.3 x-Ray Tubes and Beam Quality in DiagnosticRadiology
For an x-ray tube of given anode target material and given type
of applied voltage waveform, the shape of the output photon
energy spectrum (generally referred to as x-ray beam quality)
depends on the applied tube voltage (as characterized by the
kVp value) and on the tube inherent filtration. With respect to
a pure (unfiltered) bremsstrahlung spectrum from a thick tar-
get, actual beam spectra from an x-ray tube show (i) super-
imposed characteristic (fluorescence) x-ray lines (due to
interaction, in the anode target material, of electrons acceler-
ated toward the anode with kinetic energy greater than the
electron K-shell or L-shell binding energies of the target mate-
rial, and also to photoelectric interactions of bremsstrahlung
x-rays with target-bound electrons), in addition to (ii) beam
attenuation at low energies, resulting from photoelectric (self)
absorption of bremsstrahlung radiation in the surface layers of
the target material and photoelectric interactions in the mate-
rials (cooling liquid, glass envelope, tube housing, exit win-
dow) encountered by the photon radiation before exiting the
output window of the x-ray tube. This latter component repre-
sents an intrinsic filtration of the beam produced in the thick
target and as it exits the output window, which can be
expressed, for example, as an equivalent thickness of filtration
in millimeters of Al, typically in the range of 0.5–1 mm Al (at
70–75 kV) for general radiography x-ray tubes, and down to
0.1 mm Al for a mammography tube. It should be noted that
the inherent filtration is a function of x-ray tube voltage and
waveform.
Figure 44 shows schematically the shape of a bremsstrah-
lung x-ray spectrum from a thick tungsten target, the curve
of an actual filtered bremsstrahlung spectrum, and the
VIER
Photon energy (keV)
K/F
(mG
y/1
mill
ion
pho
tons
/cm
2 )
Breast tissue (ICRU-44)
1010-1
100
101
100
Figure 43 Dose in breast tissue per fluence of 106 photons cm�2 in thediagnostic energy range. Data for mass energy absorption coefficientof breast tissue (ICRU-44) were derived with the code XMuDat (Nowotny,1998) with interaction cross section data from Boone and Chavez(1996). For this plot, see also Lewis (2004).
Table 4 Energy and relative intensity of characteristic K-lines oftungsten, molybdenum, and rhodium
Tungsten (Z¼74)a
x-Ray line Energy(keV)
Relativeintensity
Ka2 57.984 57.600
Physical Basis of x-Ray Imaging 27
contribution from K-shell characteristic radiation lines. The
dominant contribution to characteristic radiation (for electron
kinetic energies above the K-edge of W at 69.525 keV, Table 3)
comes (in order of intensity of emission) from the Ka1line (59.321 keV, shell transition K LIII in W), from the Ka2line (57.984 keV, shell transition K LII), from the Kb3 line
(66.950 keV, shell transition K MII), and from the Kb1 line
(67.244 keV, shell transition K MIII) (Table 4). After consid-
ering photon attenuation by a target thickness of 2 mm in
escaping the target, the spectrum assumes its typical attenu-
ated shape at low energies; additional filtration in the beam
determines further attenuation of photons at energies
below �0.01 MeV. L-shell binding energies in tungsten
(LI¼12.098 keV; LII¼11.541 keV; LIII¼10.204 keV) deter-
mine the sharp transitions around 10 keV (Figure 44).
The unfiltered thick-target spectrum (differential radiant-
energy r(E) in J MeV�1 versus photon energy E¼hv in MeV)
has the decreasing linear trend of the Kramers spectrum,
given by
Photon energy (MeV)
Unfiltered bremsstrahlung
Filtered through 2 mm tungsten
Bre
mss
trah
lung
out
put
(J M
eV-1
)
0.000
2
4
6
8
10
12
0.02 0.04 0.06
K-edge at69.5250 keV
Ka1
Ka2
Kb1
Kb2
LIII edge(10.2068 keV)
LI edge (12.0998 keV)LII edge (11.5440 keV)
0.08
Figure 44 The continuous thick line shows the bremsstrahlungdifferential energy spectrum r(E) (J MeV�1) as a function of photonenergy E (MeV) for electron kinetic energy 0.08 MeV incident on a thicktungsten anode target. Also shown (not to scale) is the spectrum ofthe characteristic emission K-lines of tungsten (see Table 3). Thecontinuous thin line is the bremsstrahlung spectrum attenuated by alayer of tungsten of thickness 2 mm, simulating the expected effect oftarget self-attenuation on the shape of the unfiltered spectrum; L-edgesare seen at about 0.01 MeV. Characteristic L-lines are not shown forclarity.
Table 3 Physical data of elemental materials used in x-ray beam filtration
Element Symbol Atomicnumber, Z
Density, r(g cm�3)
Electronic dens(cm�3)
Aluminum Al 13 2.699 7.83Eþ23Barium Ba 56 3.50 8.60Eþ23Copper Cu 29 8.96 2.46Eþ24Iodine I 53 4.93 1.24Eþ24Molybdenum Mo 42 10.22 2.69Eþ24Rhodium Rh 45 12.41 3.27Eþ24Tin Sn 50 7.31 1.85Eþ24Tungsten W 74 19.30 4.68Eþ24Lead Pb 82 11.35 2.71Eþ24aExperimental data from NIST x-ray Transition Energies database http://physics.nist.gov/Phy
Data from the computer code XMuDat based on tabulations from Boone and Chavez (1996)
ELSE
r Eð Þ ¼ const:� I� t � Z � E0 � Eð Þ [110]
where I is the tube current (mA), t is the exposure time (s), and
hn0E0 (MeV) is the maximum photon energy in the spectrum.
In turn, the maximum photon energy is equal to the kinetic
energy Ek of electrons (elementary charge e) incident on the
anode target, accelerated by a tube voltage V (kV)¼Ek/e. By
integrating the triangular Kramers spectrum up to the maximum
photon energy, one obtains the total bremsstrahlung produc-
tion radiant energy R (J) in a thick anode target of atomic
number Z of an x-ray tube (see Attix, 1986, Chapter 2.08):
, as anode target, contrast medium or for shielding
ity Electrons(g�1)
K-shell binding energy(keV)a
Fluorescence K-yield
2.90Eþ23 1.560 0.03102.46Eþ23 37.452 0.90202.75Eþ23 8.980 0.43902.52Eþ23 33.167 0.88102.64Eþ23 20.000 0.75902.63Eþ23 23.222 0.80102.54Eþ23 29.200 0.85602.42Eþ23 69.525 0.97102.38Eþ23 88.006 0.9830
sRefData/XrayTrans/Html/search.html.
.
Ka1 59.321 100.000Kb3þKb1þKb5/1þKb5/2 ffi67.2 32.126Kb2/1þKb2/2þKb4/1þKb4/2þKb2/3þKb2/4
ffi69.1 8.417
Molybdenum (Z¼42)
x-Ray line Energy (keV)b Relative intensityc
Ka2 17.374 50–53Ka1 17.479 100Kb3 19.590 10Kb1 19.608 20
Rhodium (Z¼45)
x-Ray line Energy (keV)b Relative intensityc
Ka2 20.074 50–53Ka1 20.216 100Kb3 22.699 10Kb1 22.724 20
aAs reported in Attix (1986).bExperimental data from NIST x-ray Transition energies database http://physics.nist.
gov/PhysRefData/XrayTrans/Html/search.html.cFrom NPL database of x-ray adsorption edges and characteristic x-ray line energies
http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_2/4_2_1.html.
VIER
28 Physical Basis of x-Ray Imaging
R Eð Þ ¼ const:0 � I� t � Z � E2k
¼ const:00 � I� t � Z � V2 [111]
From eqn [111], it is seen that for an anode of given atomic
number Z, at a fixed tube current and exposure time, R goes as
the square of the x-ray tube voltage V (kVp), and that at fixed
tube voltage, R increases proportionally to the increase of
the product of the tube current I and the exposure time t. The
constant in eqn [110] is about 2 J MeV�2 mA�1 s�1. At fixedkilovoltage, by multiplying the current–time product (mAs) by
a factor k, each ordinate in the unfiltered r(E) bremsstrahlung
x-ray spectrum as well as the slope of the spectrum is multi-
plied by k. On the other hand, by keeping the mAs values
constant and changing the tube voltage, the spectrum shifts
in parallel along the energy axis, without changing its slope.
The quadratic dependence of R(E) on Ek (and on V) is strictly
valid only for the unfiltered bremsstrahlung x-ray spectrum,
and a different power law exponent can be observed in the
output spectrum, for filtered beams.
As regards the efficiency of the production of bremsstrah-
lung x-rays from a thick target x-ray tube, in which electrons
(Ek�0.15 MeV) are stopped in the target, it depends on the
ratio of radiative (dEk/rdx)r to collision (dEk/rdx)c mass stop-
ping power. Figure 45(a) shows this ratio in percent for tung-
sten in the range 10–150 keV of kinetic energy; it is seen that it
Kinetic energy (keV)
Kinetic energy (MeV)
(dE
k/rd
x)/(
dE
k/rd
x)c
(%)
(a)
(b)
W (Z = 74)
W (Z = 74)
Mo (Z = 42)
200.000
0.005
Rad
iatio
n yi
eld 0.010
0.015
0.00.02 0.04 0.06 0.08 0.10 0.12 0.14
0.5
1.0
1.5
2.0
2.5
3.0
40 60 80 100 120 140
Figure 45 (a) Radiative to collision mass stopping power versus kineticenergy Ek for the case Z¼74 (tungsten anode), from NIST databaseESTAR. (b) Electron radiation yield in tungsten and molybdenum in thekinetic energy range 10–150 keV; data after Berger and Seltzer, astabulated in the Radiological Toolbox code (Eckerman and Sjoreen, 1996).
ELSE
is between 0.2% and 3% in the energy range of interest for
radiography with x-ray tubes.
The ratio of radiative to collision stopping power is directly
proportional to the atomic number Z of the target material,
and at varying electron kinetic energy below 0.15 MeV, it can
be empirically approximated as {Z�Ek� [700þ200 log10(Ek/
3)]�1} (Attix, 1986). The ratio of radiative to total (radiative
and collision) mass stopping power {[(dEk/rdx)r]/[(dEk/rdx)rþ(dEk/rdx)c]} is a function of the current kinetic energy
of the slowing down electron; by averaging this ratio across
the energies up to the initial kinetic energy Ek, one obtains an
estimate of the radiation yield Y, that is, the fraction of the
electron initial kinetic energy Ek that is emitted as e.m. radia-
tion during the slowing down process up to thermalizing in
the material. Indeed, the total bremsstrahlung production radi-
ant energy R∝Y�Ek� I� t. The electron radiation yield (as
derived from NIST database ESTAR, http://physics.nist.gov/
PhysRefData/Star/Text/ESTAR.html) is plotted in Figure 45(b)
for W and Mo targets, showing that it is less than 1% for
electron kinetic energies lower than 100 keV. This means that
for tube voltages less than 100 kV, only less than 1% of the
electron beam energy incident on the target is emitted as x-rays
while more than 99% is deposited on the target through colli-
sions and produces the heating of the target: this anode ther-
mal power loading is to be accounted for by the anode cooling
system. In terms of power density at the tube (electronic) focal
spot, hot-cathode rotating solid-metal anode x-ray tubes for
diagnostic radiology (with electrical powers up to ffi100 kW
applied by the high-voltage generator) are capable of handling
up to ffi100 kW mm�2. In terms of apparent focal spot size
(i.e., the size of the focal spot as viewed from the central axis
of the x-ray beam), taking into account the angled anode
configuration in the so-called line-focus technology (see
Section 2.01.3.9) which allows for a reduction up to a factor
1:10 of the focal spot length and area, this means that powers
up to ffi100 kW can be managed with an (apparent) linear
focal spot size of a fraction of a millimeter. Microfocus x-ray
tubes with an apparent focal spot diameter of 10 mm or less
have typically an electron beam power of 10 W or less and
hence can handle a power density of less than 100 kW mm�2,while liquid metal jet anode x-ray tubes can handle in excess
of 1000 kW mm�2 (Larsson et al., 2013).
Figure 46 shows the x-ray spectrum for 80-keV electrons
incident on a thick target tungsten-anode x-ray tube after
filtration through 0.5 mm of Al. The data have been calcul-
ated with the software code SpekCalc (Poludniowski, 2007;
Poludniowski and Evans, 2007) with 0.1 keV resolution and a
photon takeoff angle of 30� from the target, and are intended
to illustrate the effect of typical (inherent) beam filtration
and the contribution of the characteristic lines, which is less
than 0.25% of the total tube output. A model (TASMICS) for
producing X-ray tube spectra from 20 kV to 640 kV has been
introduced recently (Hernandez and Boone, 2014).
Adding thin sheets of a highly pure (>99.9%) material
(e.g., Al, Cu, Mo, Rh, Sn, and others) at the output port of
the x-ray tube, which selectively absorb (via photoelectric inter-
actions) the output photons as a function of their energy, is
the common way of roughly changing the shape of the x-ray
tube spectrum in order to reduce the spectral intensity in low-
energy bands. This practice is adopted for reducing patient
(skin) radiation dose during the diagnostic imaging exam, for
VIER
Photon energy (keV)
K-edge
K-edge
Eb= 69.5 keV
0.5 mm AI added filter
W anode 150 kVp12 deg anode angle
20105
106
107
40 60 80 100 120 140 160
Pho
tons
keV
−1 c
m−2
mA
−1
s−1 @
1m
Photon energy (keV)(a)
(b)
(c)
Photon energy (keV)
0.5 mm AI added filter
W anode, 80 kVp12 deg anode angle
10
66 68 70 72 74
0.0
5.0x106
1.0x107
1.5x107
20 30 40 50 60 70 80
b)
No.
pho
tons
keV
−1 c
m−2
mA
−1
s−1 @
1m
Figure 46 (a) Calculated x-ray spectrum at 80 kVp tube voltage from atungsten anode tube, target angle 12�, after filtration through 0.5 mmAl (typical equivalent inherent filtration due to target self-filtration, tubehousing, cooling oil, exit window). At 1 m distance from the source, thetotal bremsstrahlung tube output is 216.5 mGy mA�1s�1, and the totalcharacteristic tube output is 0.7575 mGy mA�1s�1, so that the totaloutput is 217.3 mGy mA�1s�1. Note the characteristic Ka and Kb lines oftungsten at about 58–69 keV. Note also that no air filtration wasconsidered. (b) Details of the same spectrum around 70 keV, to point outthe hardly visible decrease in intensity at the K-edge of tungsten.The K-edge absorption in the target is more evident in the spectrumshown in a semilog plot in (c) (150 kV, 0.5 mm filter, 1 m air). Data arecalculated by the code SpekCalc (Poludniowski, 2007; Poludniowskiand Evans, 2007).
Table 5 Minimum HVL at various x-ray tube potentials required byFood and Drug Administration (FDA) regulation for all x-ray systems,except dental x-ray systems designed for use with intraoral imagereceptors, manufactured on or after June 10, 2006 (21 CFR, Ch. I, }1020.30, 4-1-12 Edition, U.S. Gov. Printing Office, available at web sitewww.gpo.gov)
Measuredoperatingpotential (kVp)
Minimum HVL(mm Al)
Measuredoperatingpotential (kVp)
MinimumHVL(mm Al)
30 0.3 90 3.240 0.4 100 3.650 0.5 110 3.951 1.3 120 4.360 1.5 130 4.770 1.8 140 571 2.5 150 5.480 2.9
Table 6 Minimum HVL at various x-ray tube potentials required byFDA regulation for dental x-ray systems designed for use with intraoralimage receptors, manufactured after December 1, 1980 (21 CFR, Ch. I, }1020.30, 4-1-12 Edition, U.S. Gov. Printing Office, available at web sitewww.gpo.gov)
Measuredoperatingpotential (kVp)
Minimum HVL(mm Al)
Measuredoperatingpotential (kVp)
MinimumHVL(mm Al)
30 1.5 90 2.540 1.5 100 2.750 1.5 110 3.051 1.3 120 3.260 1.5 130 3.570 1.5 140 3.871 2.1 150 4.180 2.3
Physical Basis of x-Ray Imaging 29
ELSEVIE
R
reducing radiation scatter in the subject, and/or to increase the
average energy of the beam, with respect to the unfiltered (i.e.,
with only inherent filtration) energy spectrum.
x-Ray tube spectral filtration can be expressed as the total
filtration (addedþ inherent), a parameter also used for
radiation protection. Indeed, regulatory boards require or rec-
ommend a limit for the minimum total filtration in the x-ray
beam for diagnostic purposes; for example, the ICRP (Interna-
tional Commission on Radiological Protection) recommends
that the total filtration above 70 kV should be equivalent to not
less than 2.5 mmAl. In the United States, federal regulations by
the Food and Drug Administration (FDA) (CFR, 2012), require
minimum half value layer (HVL) (mm Al) values as in Table 5
(for all x-ray systems, except dental x-ray systems designed
for use with intraoral image receptors, manufactured on or
after June 10, 2006) and in Table 6 (for dental x-ray systems
designed for use with intraoral image receptors, manufactured
after December 1, 1980). Increasing the filtration even further
could be advantageous for additional limitation of patient dose
in general radiography, without sacrificing image quality
(Behrman, 2003). Such a spectral filtration is the result of
the energy-dependent mass attenuation coefficient (m/r) of
the filter material of density r in the low-energy range where
photoelectric effect is dominant (Table 3; Figure 47). An abrupt
increase of the attenuation coefficient occurs at the threshold
energy corresponding to K-shell absorption (see values in the
last column of Table 3 and Figure 48), thus leading to an
increased photoabsorption at energies above such threshold.
Atomic number, Z
Ene
rgy
(keV
)
Cu
01
10
100
10 20 30 40 50 60 70 80 90 100
MoSn
AIBa
K-edge
K-edge electron binding energy
W
IRh
Figure 48 K-edge binding energy of elements with Z¼1 to 100. Someelements, indicated by the arrows, are commonly selected for x-raytube target, beam filtering, or contrast enhancement materials. Datafrom computer code MUCOEFF are from Boone and Chavez (1996).
Photon energy (keV)
Additional filtration: 0.5 mm Be + 0.6 m air
W anode 28 kV1 mm AI filter(Emean= 20.3 keV)
Rh anode 28 kV0 . 0 2 5 mm Rh filter(Emean= 18.21 keV)
Mo anode 28 kV0.030 mm Mo filter(Emean= 16.78 keV)
Rel
ativ
e no
. of p
hoto
ns
100.0
0.2
0.4
0.6
0.8
1.0
15 20 25 30
Rh K-edge
Mo K-edge
Figure 49 Example x-ray spectra for compressed breast imaging witha mammographic unit, calculated with the MASMIP, RASMIP, andTASMIP codes (Boone and Seibert, 1997; Boone et al., 1997) for x-raytubes with Mo, Rh, and W or W/Rh anodes, respectively, with addedfiltration of Mo, Rh, or Al. The spectra are normalized to thecorresponding peak value for 28 kV (Mo/Mo or Rh/Rh anode/filter), aswell as for the 28-kV, W-target spectrum with Al filter. K-edges of Moand Rh are indicated (see Table 3).
Photon energy (keV)
AI
CuMo
Mass attenuation coefficient
Rh
m/r
(cm
2 g−1
)
101
102
101
100
10−1
102
Figure 47 The graph shows the mass attenuation coefficient, in thediagnostic energy range, for elements aluminum, copper, molybdenum,rhodium, and tin, commonly used in the form of thin pure metal sheetsfor x-ray tube added filtration and beam shaping. Sharp transitions in thegraph show elemental K-shell absorption edges. Data from computercode MUCOEFF (Boone and Chavez, 1996) are available at web siteftp://ftp.aip.org/epaps/medical_phys/E-MPHYA-23-1997/.
30 Physical Basis of x-Ray Imaging
ELSE
From the Lambert–Beer exponential attenuation law, the
(energy-dependent) beam attenuation introduced by inserting
a material (metal) layer of thickness Dx downstream in the
beam at the exit window of the x-ray tube is given by exp
[�(m/r)rDx], where m¼m(E). By multiplying the spectral ordi-
nates at each energy in the x-ray tube output spectrum by
the corresponding attenuation factor, the shape of the filtered
beam can be calculated.
x-Ray tube spectra employed in mammography setups
require a low-energy spectrum with photon energies in the
approximate range of 15–25 keV. Indeed, incident photons
with energy lower than �15 keV determine full absorption
and high absorbed dose in the breast tissue, while photons
with energy higher than �25 keV determine a reduction of
tissue radiographic contrast. On the other hand, the limited
penetration of 15–25 keV radiation is well met to the reduced
thickness (from a few to several centimeters) of compressed
breast tissue during a mammographic exam. Indeed, studies
with monochromatic x-ray beams produced at synchrotron
radiation facilities with photon energy in this range (providing
incident fluence rate in the order of 1010 photons cm�2s�1)(e.g., at the SYRMEP beamline dedicated to mammography at
the ELETTRA synchrotron radiation facility in Trieste, Italy)
(Dreossi et al., 2008) provided evidence that the optimal breast
tissue contrast for lesion detection is for monoenergetic photons
in that low-energy range; with this monochromatic source the
optimal photon energy for imaging can be selected also with
regard to the lowest radiation dose, depending on the com-
pressed breast thickness. In order to produce narrow spectral
width beams from x-ray tubes, molybdenum or rhodium target
anodes are commonly used (with Mo or Rh filtration), at tube
voltages approximately in the range 22–33 kV; for digital mam-
mography detectors, tungsten anode tubes operated up to 50 kV
(with Al filtration) are also employed. A comparison of three
spectra of possible use in mammography is shown in Figure 49.
VIER
2.01.3.1 Beam Attenuation and Beam Shape Descriptors
Direct measurements of x-ray photon spectra from an x-ray tube
is not commonly performed either in the lab or in the clinical
environment, since the high photon fluence rate (in the order of
109 photons cm�2s�1 per mA tube current at 1 m distance, see
Figure 46) far exceeds the count rate capabilities of spectroscopic
detectors (which is limited in practice to �105counts s�1) witha low pileup rate. A high-count rate waveform-digitization
technique by Stumbo et al. (2004) and Bottigli et al. (2006)
allows pileup events to be rejected via off-line processing of the
digitized signal from a semiconductor detector. In general, for
the purpose of reducing the count rate, strong beam collima-
tion (e.g., with pinhole of aperture 0.1 mm or less) is usually
employed in order to reduce the photon flux on the detector, but
this may determine spectral artifacts in terms of spectral defor-
mations and/or inaccuracy in the determination of the spectral
intensity, due to uncertainty in the effective aperture size, its
Physical Basis of x-Ray Imaging 31
orientation with respect to the beam central axis, and scatter
from the aperture sides. Spectral artifacts also arise from escape
peaks with detectors based on high-bandgap compound semi-
conductor substrates (e.g., CdTe or CdZnTe). Recovery of spec-
tral artifacts due to escape peaks is usually performed with a
post-processing stripping procedure (e.g.,Matsumoto et al.,
2000; Seelentag and Panzer, 1979). Other spectral artifacts
may be related to the detector response, to the presence of K-
edge(s) of the detectormaterial which alter the spectral shape, to
the presence of escape peaks by interaction with the detector
contacts, to edge effects from interaction events at the border of
the detector volume, to incomplete charge collection in the
detector, and to beam scattering from the setup environment
reaching the detector. Example experimental spectrawith aCdTe
detector are shown in Figure 50.
As a rough yet practical descriptor of x-ray beam quality, the
HVL is considered. HVL is defined as the thickness (e.g., in
millimeters) of a filter material (typically aluminum or copper)
required to reduce the x-ray exposure, X, by one-half. x-Ray
exposure is determined by measuring the electrical charges, in
terms of number of ion pairs, produced by x-ray ionization in a
given mass of air (1R¼2.58�10�4C kg�1 of air), using an
ionization chamber. Indeed, the number of ion pairs created in
the chamber air volume is directly related to the beam intensity at
each energy in the x-ray spectrum. For a given anodematerial, the
HVL is a function of tube voltage (kVp), the generator waveform,
and the shape of the x-ray spectrum, as determined by the total
(inherentþadded) filtration of the beam. The HVL is used as
a rough descriptor of the x-ray spectral shape in order to
Photon energy (keV)(a)
(b)
Cou
nts
mm
-2 m
A-1
s-1
keV
-1
@ 6
0cm
1 mm AI added filtration
Mo anode30 kVp
50
500
1000
1500
10 15 20 25 30
Photon energy (keV)
Cou
nts
mm
-2 m
A-1
s-1
keV
-1
@ 6
0cm
1 mm AI added filtrationMo anode30 kVp
5100
101
102
103
10 15 20 25 30
Figure 50 Spectrum from a molybdenum anode x-ray tube (30 kVp,1 mm Al filtration) measured with a commercial CdTe detector (AMPTEKXR-100T-CdTe, 3�3 mm2) (a) in linear and (b) in log scale.
ELSE
characterize an x-ray beam, with reference to the attenuation
properties of the given x-ray beam in a given material.
Attenuation refers to the decrease of the number of x-rays
incident on a given area of a sample in passing through the
sample, and is caused by both photoabsorption and scatter
in the sample, depending on the photon energy and on
the sample thickness and composition. If the x-ray beam is
monochromatic, with energy E, then the HVL value would
completely characterize the beam energy. Indeed, the (mono-
energetic) Lambert–Beer attenuation law of a pencil beam in
a thin layer of thickness t (cm) of a material with density rand total mass attenuation coefficient m/r (cm2g�1) is given by
N
N0¼ exp � m Eð Þ
rrt
� �[112]
and in terms of logarithmic attenuation,
lnN
N0
� �¼ � m Eð Þ
rrt [113]
where N0 is the number of photons incident on the thin layer
and N the number of photons exiting from the thin layer.
This law is strictly valid for a monoenergetic collimated beam
and in narrow beam geometry (also good geometry), that is when
the setup for exposure measurements with the attenuated
and unattenuated beams is such to exclude any contribution
of radiation scattered from the attenuator material and from
the surroundings.
The attenuation law can be expressed in terms of exposure X
for an incident exposure X0:
X
X0¼ exp � m Eð Þ
rrt
� �[114]
and for logarithmic attenuation,
lnX
X0
� �¼ � m Eð Þ
rrt [115]
For attenuation by an HVL at energy E,
X
X0¼ 1
2¼ exp � m Eð Þ
rrHVL
� �[116]
HVL ¼ ln 2m Eð Þr r¼ 0:693147
m Eð Þ cm [117]
hence the HVL is uniquely determined by the beam energy E,
for a given material with known value of linear attenuation
coefficient m (cm�1) at that energy. On the other hand, x-ray
tube spectra are polychromatic, and then characterized by the
distribution of the photon fluence FΕ(E) (photons per cm2)
versus photon energy E (keV) up to the maximum beam energy
E0. Following (Boone, 2000), we have
N0 ¼ðE00
AFE
x Eð Þ dE [118]
N ¼ðE00
AFE
x Eð Þ exp �m Eð Þr
rt �
dE [119]
(where A is the irradiated area), and the x-ray attenuationN(t)/
N0 in a sheet of material of thickness t, and the mass attenua-
tion coefficient m(E)/r is expressed by
VIER
80 keV
40 keV
1m
per
1 m
As
ion
N/N
0
0.4
0.5
0.60.70.80.9
1 7.33
32 Physical Basis of x-Ray Imaging
N tð ÞN0¼
ðE00
AFE
x Eð Þ exp �m Eð Þr
rt �
dEðE00
AFE
x Eð Þ dE[120]
The function x(E) (photons cm�2R�1) describes the pho-
ton fluence per unit exposure and is given by (Boone, 2000)
x Eð Þ ¼ 5:43� 1010
E men Eð Þr
� air
[121]
where (men/r)air (cm2g�1) is the mass energy absorption coef-
ficient for air and E is in kiloelectronvolt. x(E) is plotted in
Figure 51 in the diagnostic energy range; the marked energy
dependence reflects that of the attenuation coefficient of air,
the common absorption material in ionization chambers used
for exposure measurements.
The total photon fluence per exposure FΤ (photons
cm�2R�1) over the whole x-ray spectrum up to the maxi-
mum photon energy E0 can be obtained by calculating the
quantity (Boone, 2000)
FT ¼
ðE00
FE dEðE00
FE
x Eð Þ dE[122]
If the exposure X (in R) at the distance of interest from the
source is measured, the total photon fluence F can be obtained
simply as the product F¼FTX. The photon fluence per expo-
sureFΤ can be calculated via eqn [122] once the x-ray spectrum
FΕ is known (e.g., by calculation with codes like TASMIP and
SpekCalc or TASMICS); tables of FΤ are provided by Boone
(2000) for various combinations of kilovolt, filtration, andHVL.
Once the x-ray photon fluence is known at each energy in
the beam, using the conversion function provided by eqn
[121] and with known material attenuation data, eqn [120]
can then be used to provide the expected attenuation in any
(homogeneous) material volume, as well as the expected inci-
dent fluence spectrum on the sample and on the downstream
imaging detector. As an example of the above considerations,
consider the attenuation of an 80-kV x-ray beam (filtered by
2.5 mm Al and 1 m air) by a sheet of aluminum of thickness t
LSE
Photon energy, E (keV)
x(E
) (p
hoto
ns c
m-2
R-1
)
00
3x1010
2x1010
1x1010
20 40 60 80 100 120 140
Figure 51 The function x(E) (photon fluence in photons cm�2 per unitexposure in R, in air, at x-ray energy E) in the diagnostic energy range(data are calculated from eqn [121] in the text, with attenuation data forair calculated with the code XMuDat, Nowotny, 1998, with interactioncross section data from Boone and Chavez (1996)).
E
in the range 1–15 mm. Figure 52 shows in a semilogarithmic
plot the attenuationN(t)/N0 calculated by eqn [120], using the
spectral intensity for an 80-kV spectrum calculated with the
computer code SpekCalc (Poludniowski, 2007; Poludniowski
and Evans, 2007) and tables of mass attenuation coefficients
(e.g., from the XCOM database of Berger and Hubbell (1987),
freely available at the website of NIST, National Institute of
Standards and Technology in USA) or from freely available
software codes that make reference to those or other tabulated
data, as from Boone and Chavez (1996).
We note that the calculated attenuation curve for the poly-
energetic beam in Figure 52 shows a marked nonlinear trend,
with increasing curvature at increasing material thickness; on
the other hand, the monoenergetic beam does show a linear
trend in this semilog plot, as obvious from eqn [113]. The
nonlinearity in the plot of exposure attenuation versus attenu-
ator thickness reveals a phenomenon known as beam harden-
ing, whereby the progressive removal of low-energy photons
from the beam transmitted through increasing thicknesses of a
material determines a modification of the beam spectral shape
and a shift of the average (and effective) energy toward higher
energies (whence, a harder x-ray beam).
For the 80-kV beam, the HVL is graphically derived as
2.78 mm Al; the corresponding effective linear attenuation coef-
ficient meff is calculated as
meff mr
� �eff
r 0:693147
HVL cmð Þ cm�1 [123]
and is meff¼2.49 cm�1, or (m/r)eff¼2.49 cm�1/2.7gcm�3¼0.92 cm2g�1. From the m(E)/r versus E curve for alu-
minum (Figure 47), this value corresponds to a beam effective
energy of 32.3 keV (Al) for the polychromatic beam. Increasing
the added filtration of the x-ray beam will cause an increase of
the HVL and a decrease of the effective attenuation coefficient
VIER
Aluminum thickness (mm)
80 kV
32 keVHVL2
HVL1
2.5 mm AI filterE
xpos
ure
(mR
) @
Att
enua
t
00.090.1
0.2
0.3
5 10 15
0.73
Figure 52 Attenuation curves (in terms of normalized exposure as afunction of thickness) in aluminum, calculated for an 80-kVpolychromatic x-ray beam. The (first) half value layer (HVL) is derivedfrom this plot as the abscissa for attenuation¼0.5 and isHVL1¼2.78 mm. Also shown for comparison are attenuations calculatedfor monochromatic x-ray beams of energy 80, 40, and 32 keV, this lastbeing the beam effective energy at which the HVL is the same for thepolychromatic and monochromatic beams. The second HVL is alsoindicated as the additional aluminum thickness for which theattenuation¼0.25 and is HVL2¼4.04 mm, so that the homogeneitycoefficient for the 80-kV beam is 2.78/4.04¼0.69.
Tube voltage (kVp)
Total filtration: 2.5 mm AI + 1 m Air
Linear fits (R2> 0.999, P<0.0001)
Hal
f val
ue la
yer
(mm
AI)
200
1
2
3
4
5
6
7
8
9
40 60 80
HVL2
HVL1
100 120 140 160
Tube voltage (kVp) (b)
(a)
Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m Air
Parabolic fits (R2> 0.999, P<0.0001)
Hal
f val
ue la
yer
(mm
AI)
200123456789
101112
40 60 80
HVL2
HVL1
100 120 140 160
Figure 54 (a) The first and second HVL values as a function ofkilovoltage for a tungsten target x-ray tube with 12� anode angle and2.5 mm Al total filtration. The curve shows a linear trend as indicated bythe linear fits. (b) With respect to the same x-ray tube and filtration as inplot (a), the filtration has been increased by adding 0.25 mm Cu,producing an increase in beam HVL, closer values of HVL1 and HVL2, andthe appearance of a curvature in the trend of HVL versus tube voltage.Data are calculated with the code SpekCalc (Poludniowski, 2007;Poludniowski and Evans, 2007).
Physical Basis of x-Ray Imaging 33
VIER
meff. The relationship between (m/r)eff and HVL for Al and Cu
attenuators is shown in Figure 53.
Equation [116] defines the first HVL, or HVL1, as the thick-
ness required to reduce the exposure (or the exposure rate) by a
factor 0.5; once inserted in the beam a thickness HVL1 of
attenuator, the second HVL, or HVL2, is the additional thickness
required to determine, in the same conditions, an additional
reduction of the exposure by a factor 0.5, that is a total atten-
uation by a factor 0.25. Also used are the quarter value layer
(QVL), the attenuator thickness that reduces the exposure to
one-quarter of the unattenuated exposure (i.e., HVL2¼QVL-
�HVL1), and the tenth value layer, the attenuator thickness
that reduces the unattenuated exposure to 1/10.
The condition HVL1¼HVL2 strictly holds only for mono-
energetic beams, so that the homogeneity coefficient HVL1/HVL2can be used as a rough parameter to indicate how added beam
filtration is effective in reducing the spectral width and increas-
ing the spectral homogeneity of the x-ray beam, with its value
approaching unity as the spectrum approaches monochroma-
ticity. The two values (first and second HVL) provide a more
accurate description of the (polychromatic) x-ray spectrum
than a single value; a more refined characterization of the
beam quality would be provided by indication of the higher-
order HVL, that is, those filter material thicknesses which, once
inserted consecutively in the beam path (usually at the output
port of the x-ray tube, in the collimator/filter unit), determine
corresponding further reduction of the beam exposure by a
fraction 0.5; HVLs exceeding the tenth order (HVL10) have
been proposed for this purpose.
As an example, the HVL as a function of the x-ray tube
voltage for a W anode tube with 12� target angle is shown in
Figure 54(a) (with 2.5 mm Al filtration) and Figure 54(b)
(with 2.5 mm Alþ0.25 mm Cu total filtration). It is seen that
adding filtration determines an increase of HVL, a quadratic
trend of HVL versus tube kilovoltage and an increase of the
homogeneity coefficient; the trend of this coefficient versus
tube voltage is shown in Figure 55 for the two example beam
filtrations.
The implicit functional relationship between the effec-
tive energy and the HVL expressed by eqn [123] is shown in
Figure 56, from data in Figure 47 of the total attenuation
coefficient for Al and Cu.
LSE
0.010.1
1
10
100
HVL (mm)
(m/r
) eff
(cm
2 g-1
)
CuAI
0.1 1 10
Figure 53 The relationship between effective mass attenuationcoefficient and first HVL in aluminum or copper, given by eqn [123]in the text.
Tube voltage (kVp)
2.5 mm AI + 0.25 mm Cu2.5 mm AI
12 deg W anode
Hom
ogen
eity
coe
ffici
ent
HV
L 1/H
VL 2
200.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
40 60 80 100 120 140 160
Figure 55 From the data in Figure 54, the homogeneity coefficient hasbeen calculated and shown as a function of tube voltage. Adding filtrationreduces the width of the x-ray beam energy spectrum (i.e., high HVL1/HVL2 ratio), which is even more for high kilovolt spectra.
E
HVL1 (mm)
First half value layer
Cu
AI
Effe
ctiv
e en
ergy
(keV
)
0.0110
20
30
40
50
60708090
100
0.1 1 10
QVL (mm)(b)
(a)
Quarter value layer
Cu
AI
Effe
ctiv
e en
ergy
(keV
)
0.1
10
20
30
40
50
60708090
100
1 10
Figure 56 The effective energy versus HVL (a) or quarter value layer (b)for Al and Cu, from data in Figure 53.
Photon energy (keV)
No.
pho
tons
keV
-1 c
m-2
mA
-1s-1
@ 1
m
200
8x106
2x1074x107
6x106
4x106
2x106
40 60 80 100 120 140
50 kVp
Filtration: 2.5 mm AI + 1 m Air
70 kVp90 kVp110 kVp130 kVp150 kVp
160
Figure 57 x-Ray spectra at 50–150 kVp in 20 kVp steps, at 1 mdistance from the focal spot of a tungsten anode tube with target angle30�, after filtration through 2.5 mm Al and considering the filtrationintroduced by 1 m air between the source and the detector. Data arecalculated with the code SpekCalc.
34 Physical Basis of x-Ray Imaging
Fulfillment of the requirement of narrow beam geometry is
important in assuring correctness of data interpretation in
attenuation measurements; on the contrary, operation under
broad beam (or bad) geometry, with a nonnegligible scatter
component present in the measurement, produces inaccurate
estimates of beam quality, for example, HVL and effective
energy. Since inclusion of scattered radiation in the exposure
measurement increases the detected signal, for any given atten-
uator thickness attenuation tends to be lower in bad geometry
than in good geometry, with corresponding increase of the
(apparent) HVL.
ELSE
2.01.3.2 Effect of Varying the Kilovoltage at FixedBeam Filtration
Figure 57 shows the effect of varying the accelerating potential
on the shape of the x-ray spectrum from a tungsten anode x-ray
tube, at fixed beam filtration. Bremsstrahlung continuum is
evident from about 15 keV up to the maximum photon energy
corresponding to the given kilovoltage (50–150 kV), as well as
characteristic K-shell lines for electron kinetic energies above
the K-edge of tungsten. The spectra up to 90 kV show a linear
high-energy continuum resembling the unfiltered bremsstrah-
lung (Figure 44), while for higher kilovoltages a curvature of
the continuum spectrum is evident.
The spectra in Figure 57 are for a tungsten anode tube with
12� anode angle, with 2.5 mm Al added filtration and includ-
ing air filtration in the beam. The beam effective energy at
varying kilovoltage, for two filtrations (0.5 mm Al, equivalent
to a typical inherent filtration, and 2.5 mm Al) is shown in
Figure 58(a) and 58(b), respectively; the mean energy shows
an almost linear trend at low kilovoltages, with evident curved
trend at higher tube voltages and higher filtration. This trend is
more evident by increasing even further the beam filtration
(0.25 mm Cu added to the 2.5 mm Al filter) as shown in
Figure 58(c).VIER
2.01.3.3 Effect of Varying the Added Filtration at a FixedKilovoltage
Once a given kilovoltage for a patient exposure has been
selected, the beam filtration should be such that the HVL is
greater than a minimum value, as prescribed by approved pro-
tocols and norms, in order to keep the radiation dose to the
patient as low as possible. For example, in the United States
under FDA regulations, for all x-ray systems except dental
systems designed for use in intraoral image receptors, a mini-
mum HVL of 2.5 mm Al is required for all tube voltages of
71 kVp or higher (Table 5). Figure 59 shows the modifications
of the spectral shape of the beam from a tungsten-anode x-ray
tube, at 80 kV, upon increasing in steps the added filtration
from 0.5 to 4 mm Al. The corresponding changes in the effec-
tive energy (Al) and in the HVL of the beam are shown in
Figure 60, indicating the spectral shift toward higher energies
(beam hardening).
A specific application of beam filtration is in x-ray imaging
of the compressed breast with a mammography setup. In this
case, with anode target materials as Mo or Rh often employed
in mammography tubes, filtration tends to decrease the brems-
strahlung continuum intensity at too-low and too-high ener-
gies without affecting much the intensity of target characteristic
lines, centered at photon energies (between 17 and 23 keV)
where the tissue contrast is optimal. For this purpose, K-edge
filtering is applied, namely, attenuation foils are inserted in
Photon energy (keV)
Added filtration mm AI
Pho
tons
keV
−1 c
m−2
mA
−1 s
−1
@ 1
m
100.0
1.2x107
8.0x106
1.0x107
4.0x106
6.0x106
2.0x106
20 30 40
4.03.02.01.00.5
50 60
W anode12 deg
70 80
Figure 59 x-Ray spectra at 80 kV tube voltage, with increasing amountof added filtration. Data are calculated with the code SpekCalc.
Tube voltage (kVp)
Mean energyEffective energy (AI)
Filtration: 2.5 mm AI + 1 m air
Mea
n or
effe
ctiv
e en
ergy
(keV
)
12 deg anode angle
2020
30
40
50
60
40 60 80 100 120 140 160
Tube voltage (kVp)(a)
(b)
(c)
Mean energyEffective energy (AI)
Filtration: 0.5 mm AI + 1 m airM
ean
or e
ffect
ive
ener
gy (k
eV)
12 deg anode angle
20
20
30
40
50
60
40 60 80 100 120 140 160
Tube voltage (kVp)
Mean energy
Effective energy (AI)
Parabolic fit(R2=0.99921, P<0.0001)
Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m air
Mea
n or
effe
ctiv
e en
ergy
(keV
)
2020
30
40
50
60
70
40 60 80 100 120 140 160
Figure 58 Mean energy and effective energy (in Al) for beams from anx-ray tube with tungsten anode, 12� anode angle, and filtration of 0.5 mmAl (a) or 2.5 mm Al (b), in addition to 1 m air filtration; the data pointshave been connected with a straight line for better showing the trend. (c)The filtration increased to 2.5 mm Alþ0.25 mm Cu, where the fitindicated by the continuous line shows the quadratic trend of the meanenergy versus kilovoltage. Data are calculated with the code SpekCalc.
Added filtration, mm AI0
01234
20
30
40
50
1 2 3 4
Effective energy (keV)X Mean energy (keV)
01234
50
40
30
20
80 kV
HV
L (m
m A
I)
Figure 60 Effective energy (Al), mean energy, and HVL for the x-rayspectra shown in Figure 59, showing beam hardening upon increasedadded filtration (mm Al).
Physical Basis of x-Ray Imaging 35
ELSEVIE
R
the beam path at the tube output port, made of materials with
a K-shell threshold energy in the spectral region of interest:
for example, a molybdenum filter is used for a Mo anode
(so-called Mo/Mo anode/filter combination), or a rhodium
filter is used with a molybdenum (Mo/Rh) or with a rhodium
anode (Rh/Rh) (Figure 61(a) and 61(b)). In general radio-
graphy, or for breast imaging techniques for the uncompressed
breast, the use of K-edge filtration of tungsten spectra can be
investigated for selection of the optimal kilovoltage/filter
material for maximization of tissue contrast (e.g., Prionas
et al., 2011) (Figure 61(c)).
2.01.3.4 Effect of Varying the Tube Current and ExposureTime at Fixed Kilovoltage
At fixed tube potential and at a given filtration, the x-ray beam
quantity (the integral of the spectral intensity over the whole
photon energy range in the x-ray spectrum) varies with the tube
current I and exposure time t as predicted by eqn [111], that is,
the spectral intensity at each energy is multiplied by the value
of the product I∙t (mAs): this is shown in the example plot of
Figure 62, where an 80-kVp x-ray spectrum with 0.5 mm Al
filtration is calculated for I∙t¼50, 100, and 150 mAs. The area
under the curve gives the energy fluence at 1 m distance from
the source, for a corresponding calculated air kerma of
266 mGy (@50 mA s), 531 mGy (@100 mA s), and 797 mGy(@150 mA s).
Photon energy (keV)
Pho
tons
keV
−1 c
m−2
mA
−1 s
−1
@ 1
m
No air
1 m air
W anode80 kVp0.5 mm AI
00
2x106
4x106
6x106
8x106
1x107
10 20 30 40 50 60 70 80 90
Figure 63 The effect of air attenuation in the beam path between thesource and detector: spectra for an 80-kVp tube voltage, 12� anodeangle, 0.5 mm Al filtration, without or with 1 m air filtration. Data arecalculated with the code SpekCalc.
Photon energy (keV)
Filtration: 0.6 m air + 0.5 mm Be
Filtration: 0.6 m air + 0.5 mm Be
Added filtration:0 mm Rh (Emean= 15.19 keV)
0.025 mm Rh (Emean=18.21 keV)
Added filtration:
0 mm Mo(Emean= 15.23 keV)
0.030 mm Mo
(Emean= 16.78 keV)
Rh anode28 kVp
Mo anode28 kVp
Rel
ativ
e no
. pho
tons
Photon energy (keV)
Pho
tons
keV
−1 c
m−2
mA
−1
s−1
@ 1
m
Sn K-edge
0.5 mm AI +
0.5 mm AI + 1 m air W anode80 kVp
0.2 mm Sn
0
50.0
0.2
0.4
0.6
0.8
1.0
Rel
ativ
e no
. pho
tons
0.0
0.2
0.4
0.6
0.8
1.0
10 15 20 25 30
Photon energy (keV)(a)
(b)
(c)
5 10 15 20 25 30
0
2x107
1x107
8x106
6x106
4x106
2x106
10 20 30 40 50 60 70 80 90
Figure 61 The effect of K-edge filtration on x-ray tube spectra. (a) x-Rayspectra from a mammography x-ray tube with a molybdenum anode,operated at 28 kVp. The spectrum with no added filtration (thick line) has amean energy of about 15 keV, with significant contribution to beam fluencefrom low-energy photons. Upon filtration with a 0.030 mm molybdenumfoil (hence, the term Mo/Mo for anode/filter combination), the meanenergy increases to about 17 keV. (b) Mammography spectra obtainedwith a Rh/Rh combination at 28 kVp. (c) In general radiography, withtungsten anode tubes, K-edge filtration can be used for tissue contrastoptimization: here, an 80-kV spectrum (12� anode angle) is filtered with0.2 mm tin, with K-edge at about 29 keV (Table 3). Data are calculated withthe MASMIP spectral model (Boone et al., 1997) for Mo and Rh anodes,and with the code SpekCalc for W anode.
Photon energy (keV)
50 mAs100 mAs150 mAs
0.5 mm AI + 1 m air W anode80 kVp
00.0
5.0x108
Pho
tons
keV
−1 c
m−2
@ 1
m
1.0x109
1.5x109
2.0x109
2.5x109
10 20 30 40 50 60 70 80 90
Figure 62 The effect on the x-ray tube spectral shape from a change inthe product of tube current and exposure time (mAs) at fixed tubepotential, for an 80-kVp beam with 12� anode angle and 0.5 mm Alfiltration. Data are calculated with the code SpekCalc.
36 Physical Basis of x-Ray Imaging
ELSEVIE
R
2.01.3.5 Effect of Filtration by Air in the Beam Line
In some spectra shown in the previous paragraphs, attenuation
by air in the beam path between x-ray source and detector was
included in the calculation.
Figure 63 shows the effect of including a 1 m air thick-
ness in the beam, at 80 kVp. Owing to air absorption (see
Figure 51), a slight decrease in spectral intensity is produced
at low energies. In this example, by considering air attenuation,
the calculated total (bremsstrahlungþcharacteristic) tube out-
put at 1 m distance decreases from 217 to 194 mGymA�1 s�1 and the mean energy goes from 36.2 to 36.7 keV,
with the HVL changing from 0.939 to 1.05 mm Al.
2.01.3.6 Beam Output at Varying Kilovoltages
Figure 64 shows how the beam output (mGy mA�1 s�1) variesat varying kilovoltage in the diagnostic energy range, for an
x-ray tube with 30� tungsten anode angle, with 2.5 mm Al
filtration (Figure 64(a)) or 2.5 mm Alþ0.25 mm Cu filtration
(Figure 64(b)). The total output is calculated by summing up
the contribution to exposure due to bremsstrahlung radiation
and that due to characteristic radiation, at 1 m of distance
from the source. The beam output shows a quadratic trend
with kVp, in agreement with eqn [111]. Visual comparison of
plots (a) and (b) in this figure shows that increased filtration
determines a higher curvature in the beam output versus tube
voltage curve.
Tube voltage (kVp)
Total output = (1)+(2)
Total filtration: 2.5 mm AI + 0.25 mm Cu + 1 m Air
Bremsstrahlung (1)
Parabolic fit
(R2=0.99995, P<0.0001)
Characteristic (2)
Bea
m o
utp
ut (µ
Gy
mA
−1
s−1 @
1 m
)
20
0
25
50
75
100
125
40 60 80 100 120 140 160
Tube voltage (kVp)(a)
(b)
Total output = (1)+(2)
Total filtration: 2.5 mm AI + 1 m Air
Bremsstrahlung (1)
Parabolic fit
(R2=0.9999, P<0.0001)
Characteristic (2)
Bea
m o
utp
ut (µ
Gy
mA
−1
s−1 @
1 m
)
20
0
50
100
150
200
250
40 60 80 100 120 140 160
Figure 64 The beam output (from bremsstrahlung, characteristic,and total radiation) from a tungsten-anode x-ray tube as a function oftube voltage. (a) 2.5 mm Al filtration; (b) 2.5 mm Alþ0.25 mm Cufiltration. The total output has been fitted with a quadratic trend(continuous lines), in agreement with eqn [111] in the text. Data arecalculated with the code SpekCalc.
Photon energy (keV)
Filtration: 2.5 mm AI + 1 m air
No.
of p
hoto
ns m
m−2
keV
−1
per
1 m
Gy
air
kerm
a @
1m
0
4x105
6x105
2x105
20 30
100%5%10%1%
1%5%
10%100%
40 50 60
80 kVp
70 80
Figure 65 The effect of varying ripple levels on the shape of an 80-kVpx-ray spectrum, at fixed exposure. Data are calculated with the codeTASMIP (Boone and Seibert, 1997).
Photon energy (keV)
Filtration: 2.5 mm AI + 1 m air
Pho
tons
keV
−1 c
m−2
mA
−1 s
−1
@ 1
m
100
4x106
3x106
2x106
1x106
20 30 40
7 deg10 deg12 deg14 deg16 deg
50 60
80 kVp
anode angle
70 80
Figure 66 The effect of varying anode target angle (from 7� to 16�) onthe shape of the spectrum from an x-ray tube, at 80 kVp and 2.5 mmAl filtration. Decreasing the target angle decreases the spectral intensityat low energies in the bremsstrahlung spectrum. The tube output at1 m decreases from 71 to 58 mGy mA�1s�1 in going from 16� to7� anode angle. Data are calculated with the code SpekCalc.
Physical Basis of x-Ray Imaging 37
VIER
2.01.3.7 Effect of Voltage Ripple and of Target Angle
Ripple in the anode tube potential, that is, residual oscillations
in kilovoltage from the x-ray tube high-voltage generator mea-
sured in percent as the factor 100(Vmax�Vmin)/Vmax, indicates
the deviation from a constant-voltage operation of the x-ray
tube, producing a corresponding variation in tube output and
the maximum photon energies in the spectrum. While old-
fashioned single-phase x-ray generators have 100% ripple,
modern x-ray tube high-voltage generators employ high-
frequency inverter circuits that can reduce the ripple to 1% or
less, though ripple factors of several percent are associated with
triple-phase high-voltage generators. Voltage ripple can be eas-
ily introduced in calculations of empirical spectral models like
TASMIP (Boone et al., 1997).
Figure 65 shows an example of spectral changes due to
varying ripple levels, for an 80-kVp spectrum with 2.5 mm Al
filtration, operated at 1%, 5%, 10%, or 100%.
x-Ray tubes with different anode angles produce slightly
different spectra. Figure 66 shows calculated spectra at 80 kVp,
for varying target angles. A decrease (18%) in tube output and
an increase (3%) in beam average energy can be observed in the
calculated spectra, for decreasing target angle from 16� to 7�.
ELSE
2.01.3.8 Attenuation and Beam Hardening
Transmission of a polychromatic x-ray beam through an object
produces a modification of the spectral shape, with increasing
average photon energy for increasing traversed thickness.
This phenomenon of beam hardening is relevant to general
radiography since the spectrum of x-rays incident on the detec-
tor can be slightly yet sensibly changed with respect to the
incident spectrum; the effect is due to the energy-selective
attenuation of the beam introduced by the energy dependence
of the tissue attenuation coefficient. In CT imaging, beam
hardening produces an underestimation of the reconstructed
effective attenuation coefficient of tissues for inner regions in
the object (see Chapter 2.03, Section 5.1). To illustrate this
beam hardening effect with an example, Figure 67 shows
calculated spectra for an 80-kVp beam from an x-ray tube,
after transmission through a water slab of varying thickness,
from 25 to 150 mm. Upon increasing the material thickness in
the beam path, reduced transmission (i.e., decrease of the
spectral intensity at all photon energies in the beam) is evident
as well as an increase in the average energy due to weakening of
the low-energy part of the spectrum. This is also shown quan-
titatively in Figure 68(a), where the ratio of attenuated to
unattenuated beam is plotted against photon energy, and in
Water slab thickness (mm)0
28
32
36
40
Mea
n of
effe
ctiv
e en
ergy
(keV
)
44
48
52 80 kVp
Filtration: 2.5 mm AI + 1 m air56
25 50 75 100
Parabolic fits
Emean
Eeff (AI)
125 150
Photon energy (keV)10
(a)
(b)
0.0
0.1
0.2
0.3
Rat
io a
tten
uate
d/u
natt
enua
ted
spec
tral
inte
nsity
0.4
0.5
0.6 80 kVp
Filtration: 2.5 mm AI + 1 m air0.7
20 30 40 50
Water slabthickness
150 mm
100 mm
50 mm
25 mm
807060 90 100
Figure 68 From calculated spectra as in Figure 67, the following werederived: (a) the intensity ratio at each energy and the mean energy of thepolychromatic beam and (b) the corresponding effective energy as afunction of the thickness of the water slab through which the beam istransmitted. A quadratic least-squares fit is also shown for bothquantities. Increased mean (and effective) energy at increased slabthickness shows hardening of the polychromatic beam, due to anincreasingly efficient phenomenon of removal of low-energy photonsfrom the primary beam, as indicated by decreasing transmission at lowenergies.
Photon energy (keV)
Pho
tons
keV
−1 c
m−2
mA
−1 s
−1 @
1m
100
150
100
50
25
0
mm H2O
Filtration: 2.5 mm AI + 1 m air
80 kVp
14 deganode angle
1x106
2x106
3x106
4x106
20 30 40 50 60 70 80
Figure 67 Spectral shape modification by transmission of an 80-kVpbeam (curve ‘0 mm H2O’) through a water slab of varying thickness, from25 to 150 mm. Data are calculated with the code SpekCalc.
38 Physical Basis of x-Ray Imaging
ELSE
Figure 68(b), where the mean energy and the effective energy
(in Al) corresponding to the spectra in Figure 67 are plotted as
a function of the slab thickness; quadratic fits in this plot show
the approximate trend of this beam hardening effect, at 80 kVp
in water, for the range of thicknesses explored. Another way of
looking at beam hardening effects with polychromatic beams
in passing through thick objects is related to the behavior
previously shown in Figure 52.
When performing attenuation measurements via exposure
X(t) measurements with an ionization chamber and a homo-
geneous slab of thickness t made of a given material, the trend
of the logarithmic attenuation � ln[X(t)/X(t¼0)] as a function
of t is linear for thin objects (i.e., mt1, where m is the linear
attenuation coefficient of the material) and it deviates from
linearity at large thicknesses, with the slope of the curve
decreasing as the thickness increases. A calculation, following
eqn [115], was performed for the previous example of water
and an 80-kVp beam (Figure 69): the data points up to 15 mm
water thickness are well fitted by a linear trend (in the semilog
scale of this figure) whose slope is higher than the one
obtained by a linear fit to data points at slab thicknesses of
100 mm or larger.
In order to show the effect of such a phenomenon in digital
radiography imaging, Figure 70(a) shows the image (taken
with a CMOS flat panel detector with CsI:Tl scintillator layer)
of an aluminum alloy (Al-100) wedge in contact with the
detector, taken at 80 kVp; analogous images were recorded at
70, 60, and 50 kVp. The wedge introduces a spatially decreas-
ing absorption-based attenuation in the beam in the horizon-
tal direction in this Figure 70(a), due to decreasing absorber
thickness. The pixel values I were normalized to the corre-
sponding average value I0 in the absence of the object in the
beam, and a vertically-averaged horizontal line profile along
the direction of decreasing attenuation was drawn on images
at each kilovoltage; the horizontal axis was then scaled in order
to show Al-100 absorber thicknesses in the direction of the
beam. These profiles (Figure 70(b)), shown in a semilog plot,
are nonlinear and attain curvature at large thicknesses, thus
VIER
Water slab thickness, t (mm)
Linear fit, small t
Linear fit, large t
Filtration: 2.5 mm AI + 1 m air
80 kVp
00.01
0.1
Att
enua
tion
X/X
0
1
20 40 60 80 100 120 140 160
Figure 69 For the 80-kVp x-ray spectrum and varying thickness t of awater slab, attenuation data X/X0 (calculated via eqn [114] in the text)versus t provide another graphical evidence of beam hardening: the slopeof the curve (in the semilog plot) decreases as the thickness increases. Aconstant slope over the data points would indicate absence of beamhardening, as in the case of a monoenergetic beam.
Photon energy (keV)
Cou
nts
mm
−2 m
A−1
s−1
keV
−1
@60
cm
5100
101
102
103
10
30 kVp
No PMMA
10 mm PMMA
20 mm PMMA
Mo anode1 mm AI
15 20 25 30 35
Figure 71 Spectra from a molybdenum anode x-ray tube (30 kVp,1 mm Al filtration) measured with a commercial CdTe detector (AMPTEKXR-100T-CdTe, 3�3 mm2) in air with and without insertion in thebeam path of a PMMA slab of thickness 10 or 20 mm. Few scatteredevents (at energies below �12 keV) and pileup events at energies above30 keV are detected. The three vertical arrows point to the value of themean energy of the corresponding spectrum (at 20.76, 21.49, and21.91 keV, respectively), increasing with increasing attenuation in thebeam.Aluminum thickness (mm)
Ave
rage
pix
el v
alue
I/I 0
00.09
0.1
0.2
0.3
0.4
0.5
0.60.70.80.9
1
(a)
(b)
1.00.80.60.40.2
10 20 30 40 50 60
50 kVp
60 kVp
70 kVp
80 kVp
70 80
Figure 70 (a) Digital radiography of a wedge made of an aluminumalloy (Al-100) at 80 kVp, with pixel values normalized to 1 at regionswhere there is no aluminum in the beam path. (b) Horizontal average lineprofile of the images at 80, 70, 60, and 50 kVp, plotted versus thecorresponding aluminum thickness in the beam. Curved trends in thissemilog plot show the effect of beam hardening.
Physical Basis of x-Ray Imaging 39
showing the effect of beam hardening, less pronounced at low
kilovoltage (smaller spectral width) than at higher kilovoltage.
The analysis of these profiles, however, is complicated by the
presence of an energy-dependent absorption in the scintillator
layer of the detector.
Beam hardening is more pronounced for wider range of
photon energies in the incident spectrum (and is not present
for a purely monochromatic beam), so that reducing the x-ray
spectral width (e.g., by beam filtration) helps in reducing this
effect. In mammography, with spectral widths extending only
over a range of about 20 keV, a less evident beam hardening
effect is expected for soft-tissue attenuation, though breast
tissue exhibits a significant decrease in attenuation coefficient
from 10 to 30 keV (Figure 14(a)). Figure 71 shows measured
spectra from a Mo anode x-ray tube at 30 kVp, with additional
filtration of 1 mm Al, with and without insertion in the beam
path of a PMMA slab of 10 or 20 mm thickness; after traversing
20 mm of PMMA, the measured variation in the mean spectral
energy is just 1.2 keV (from 20.76 to 21.91 keV). An even
weaker effect can be observed with a Mo/Mo anode/filter
combination, for compressed breast thicknesses of several
centimeters.
ELSE
2.01.3.9 Focal Spot Size
The size, shape, and intensity distribution of the focal spot of
an x-ray tube have important effects on the quality of x-ray
images, in terms of 2D spatial resolution and level of spatial
coherence of the emitted radiation. In particular, image
blurring due to geometric unsharpness introduced by the finite
size of the x-ray source plays an important role in magnifica-
tion imaging (see, for instance, its impact in micro-CT imaging
as described in Chapter 2.09). x-Rays from an x-ray tube are
generated upon directing energetic electrons on a small portion
(typically less than a few square millimeters) of the target anode.
While the area of the surface of the target of the tube where x-rays
are emerging has an obvious interpretation as the x-ray source
area, indeed the ‘focal spot’ is definedas the area of x-ray emission
on the anode of the x-ray tube, as seen from the measuring device.
The focal spot properties are dependent on the x-ray tube voltage
and tube current, and its assessment implies careful indication of
the many details of the measurement setup. In principle, the
measurement of the focal spot can be performed on any plane
intercepting the x-ray beam, using ad hoc detectors andmeasure-
ment devices or employing the imaging receptor itself; in this last
case, the plane is coincident with the image plane.
In a radiographic system, the image receptor is an area
detector positioned at a certain distance from the x-ray source;
by considering various arbitrary positions (x, y) on the detector
surface in the x-ray field, one has from the above definition
that the apparent focal spot has properties (e.g., as size, shape,
and intensity distribution) which vary over the detector plane,
both in the x and in the y direction, these being perpendicular
to the anode–cathode axis and parallel to it, respectively. In
other words, at a given source-to-detector distance, the way in
which different portions of an extended object are irradiated by
the x-ray beam from the focal spot of an x-ray tube varies in
dependence of their relative position within the beam. This is
true also for the spectral properties of the incident beam, since
also the spectral shape of the x-ray beam varies (though in
limited quantitative terms) in dependence of the take-off
angle of bremsstrahlung and characteristic photons from the
surface of the x-ray tube. In particular, there is a slight beam
hardening effect in the cathode to anode direction, in addition
to a pronounced decrease of the angular intensity of the x-ray
VIER
Cathode Anode
336
349
362
375
388
Figure 72 Image of a flat field (i.e., a flood illumination of thedetector without any object in the beam path) of the 80-kVp beam froman x-ray tube obtained with a scintillator-based digital radiographydetector. The black-line overlay is the profile along the horizontal whiteline, showing decreased intensity of the beam from the cathode tothe anode direction (heel effect).
40 Physical Basis of x-Ray Imaging
beam in the direction from the cathode to the anode due
to self-absorption in the target, a phenomenon known as
heel effect (Figure 72). Thus, the complex phenomenon of
x-ray generation from an x-ray tube introduces a great com-
plexity in the description of the spatially variant image quality
on the detector plane, due to the properties of the focal spot
and to the imaging geometry.
Generalized theory of geometric unsharpness in x-ray image
formation with an x-ray tube shows that the optical transfer
function, OTF(u,v), of the focal spot fully determines the so-
called field characteristics at any point (x,y) in the image plane
in terms of spatial frequencies u and v; specifically, it explains
the geometrical unsharpness effect.
As explained in the next chapter, the OTF is the Fourier
transform of the point spread function, PSF(x,y), which char-
acterizes the (spatially variant) intensity distribution of the
focal spot on the image plane as projected from a point in
the object plane; the modulus of the (complex) OTF is the
modulation transfer function, MTF(u,v). For simplicity, mea-
surement of the 2D PSF is often replaced by measurement of
1D line spread function, LSF(x) and LSF(y), as an approximate
representation of the PSF(x,y) as PSF(x,0) and PSF(0,y),
obtained, for example, by using a slit in the object plane and
orienting it along the anode–cathode direction (for focal spot
width assessment) and then perpendicular to it (focal spot
length), and then measuring the projected line intensities at a
position in the image plane. This implies that the 1D MTF(u)
and MTF(v) describe the geometric unsharpness at the mea-
surement point, in the spatial frequency domain. It can be
shown that once the PSF(x,y) is measured at the central beam
position in the image plane, the PSF at any field position can
be analytically derived from the central PSF, essentially from
geometrical considerations.
ELSE
The complexity of the geometric unsharpness determina-
tions is usually reduced by defining suitable metrics and con-
ventional rules that ease the task (and the related complexity
and cost of the measurement devices) for the full determina-
tion of the focal spot properties via the PSF(x,y). Among these,
a fundamental quantity is the size of the focal spot; as a crude
yet practical way of assigning the shape and size of the focal
spot, one assigns to it just two dimensions (focal spot length
and width, along the anode–cathode axis and along
the corresponding orthogonal direction, respectively). With
respect to the physical size of the source area on the target,
the effective focal spot length can be quite reduced (by a factor
siny) due to the introduction of angled anodes, where the
target surface is angled by 90þy� with respect to the anode–
cathode direction (line-focus principle). This implies a reduc-
tion of the focal spot length in the angular direction from the
cathode to the anode, while the focal spot width is left unaf-
fected by the target angle. As regards the imaging geometry,
focal spot measurement protocols usually fix the position on
the image plane where the measurements of the focal spot size
should be performed. By adopting a positioning scheme with
the x-ray tube anode-to-cathode axis parallel to the imaging
receptor plane (as is common in general radiography), the
cone-beam of x-ray emerging from the focal spot area on the
target (having an angular width limited by the target angle and
by the beam collimators) is characterized by a reference axis
(which bisects the beam cone angle) and by a central ray,
which is defined by the axis from the focal spot normal to
the receptor plane (Figure 73).
Measurement of the focal spot size can be performed along
the reference axis, which usually is made to intersect the imag-
ing receptor at the center of the detector-sensitive area. Note
that in case (as in many mammography setups) where the
anode-to-cathode axis has a tilt angle with respect to the imag-
ing plane, the central ray is not anymore along the reference
axis, and values of focal spot size from measurements per-
formed on the imaging plane along the reference axis should
be processed to take into account the tube tilt angle. As x-ray
optics, either a slit (e.g., 0.01 mm wide and a few millimeters
long) or a pinhole (e.g., 0.03 mm aperture diameter) is used in
the object plane, at a distance from the image plane in order to
exploit some image magnification (e.g., between 2� and 5�).Figure 74 shows the image of a focal spot from a mammo-
graphic x-ray tube evaluated at the central beam position, as
obtained using a pinhole with 0.025 mm aperture. The figure
shows how the length and width of the focal spot (nominal
size, 0.3 mm) are determined from the full width at half max-
imum (FWHM) value of the line profiles, with measurements
in practical agreement with nominal values. The complex
shape and intensity distribution of the focal spot in this
image are also evident in Figure 75, which refers to the focal
spot of the x-ray tube in a general radiography unit with a
nominal focal spot size of 0.6 mm.
A special case of focal spot sizemeasurements concernsmini-
focus andmicrofocus x-ray tubes. In the case of such fixed-anode
low-power tubes,multiple-aperture optics can be employed. The
shape, size, and intensity distribution of the focal spot of a 40-W
tube operated at constant voltage, with a nominal spot of
0.040 mm, determined at 35 kV with a multiple-pinhole optics,
are shown in Figure 76(a). The shape of the focal spot reveals a
VIER
Field of view(a) (b)
Image plane Image plane
Target angleTube tilt angle
Target angle
Central ray andreference axis
Referenceaxis
X-ray tube
q
q
Field of view
Figure 73 (a) Geometry of general radiography imaging with an x-ray tube; (b) the tube is tilted with respect to the image plane, so that the referenceaxis is not anymore normal to the image plane. Tube tilting coupled to small anode angles in tubes with biangular targets is commonly employed inmammography.
Vertical line profile
Horizontal line profile
Figure 74 Magnified (4.5�) pinhole image of the focal spot of a mammographic Mo–Mo x-ray tube, and corresponding vertical and horizontal centralline profiles. Nominal focal spot size 0.3 mm, technique factors 28 kV, 50 mAs; photostimulable imaging plate detector with 0.05 mm�0.05 mmpixel size. The full width at half maximum (FWHM) spot size was measured as 0.24 mm (width)�0.32 mm (length).
Physical Basis of x-Ray Imaging 41
ELSEVIE
R
double-Gaussian intensity distribution with a central Gaussian
round spot with a 1D halo, which could be due to off-focal
radiation (Figure 77). The same focal spot imaged with a slit in
the anode–cathode axis and in the perpendicular direction pro-
duced LSF(x) and LSF(y) whose corresponding MTF(u) andMTF
(v) curves are shown in Figure 76(b), confirming the greater
geometrical unsharpness introduced in the direction of the spot
length for this tube.
Figure 78 shows the effect of the tube voltage on the shape
of the focal spot, at fixed tube current, for this 0.040-mm focal
spot. The focal spot size appears to vary with tube voltage in
one direction and to keep constant in the other direction, with
details of the tube electron optics and off-focal radiation gen-
eration possibly explaining the development of a focal spot
halo around a round central spot.
The finite size of focal spots has large effects on the spatial
resolution properties of the x-ray imaging system. As shown in
Figure 1, attenuation-based projection radiography with a
finite-size source introduces an image blur (geometrical
unsharpness) producing a penumbra region at the borders of
Figure 75 (a) Magnified (3�) pinhole image of the focal spot of a W anode x-ray tube, and (b) corresponding 3D representation of the intensitydistribution of the x-ray beam. Nominal focal spot size 0.6 mm, technique factors 80 kV, 50 mAs; photostimulable imaging plate detector with 0.1 mmpixel size. The FWHM spot size was measured as 0.74 mm (width)�0.75 mm (length).
Spatial frequency (mm-1)(b)
(a)
MTF
00.00.10.20.30.40.50.60.70.80.91.0
2 4 6 8 10
Focal spot lengthFocal spot width
80 kVp
12 14
0.1 mm
Figure 76 (a) Magnified (6�) image of the focal spot shown withhorizontal and vertical line profiles across its center, and (b)corresponding MTF curves evaluated in the direction perpendicular(width) and parallel (length) to the cathode–anode axis of the x-ray tube.The image of the focal spot was acquired with a multiple-pinholeoptics (coded aperture) and shows a bright round core with 0.099 mmFWHM diameter and a halo in the direction of the cathode–anode axis,which extends 0.2 mm across. The x-ray tube is a W target, fixed anodex-ray tube P/N 97007 by Oxford Instruments X-Ray Technology, Inc.(Scotts Valley, CA, USA) with a carbon fiber window and inherentfiltration of 1.8 mm Al, operated at 35 kV and 0.125 mA. The focal spotsize of 40 mm (manufacturer’s nominal value according to the IEC 336standard) was compatible with the above estimates given the differencesin the measurement methods. The MTF was derived from the modulusof the Fourier transform of the line spread function (LSF) of the focalspot, measured in projection imaging with a slit oriented along the lengthand width of the focal spot, respectively.
42 Physical Basis of x-Ray Imaging
VIER
objects and internal details. By geometrical considerations, one
can easily derive that the image of a point object P at a distance
R1 from a uniform-intensity source of linear size F, projected
on the detector and a distance R2 from that point, is no longer a
point but has a fine size f given by
f ¼ FR2
R1¼ F M� 1ð Þ [124]
where M is the magnification factor for point P. Similarly, the
image of a sharp absorbing edge at P is blurred by a width f. If
the intensity profile of the (linear) source has a nonuniform
shape, for example, a Gaussian-like shape as in Figure 78, then
the image of a point has a Gaussian shape too, with a FWHM
value approximated by the value of f. The image of a sharp edge
would appear as a sigmoidal curve (edge spread function, ESF),
whose characteristic width is given by the FWHM of the deriv-
ative of the ESF, equal to the line spread function (LSF) of the
edge. This represents a degradation of the spatial resolution of
the imaging system (geometric unsharpness), whose amount
increases (i.e., f increases) with increasing focal spot size and
increasing magnificationM. Since x-ray tube focal spot is in the
order of a fraction of a millimeter, image magnification intro-
duces an undesirable image blur effect which is minimized by
placing the object as close as possible to the detector, so that
Mffi1 and fffi0 in eqn [124].
The finite size of focal spots has also important conse-
quences in phase-contrast x-ray imaging employing polychro-
matic beams from an x-ray tube. Most techniques for retrieving
the phase change map f(x,y,z) of objects irradiated with x-ray
beams require the use of partially coherent illumination of the
sample. Indeed, the incident waves should be at least partially
coherent, in order for interference effects to occur between e.m.
waves diffracted by the object contour or by spatial nonhomo-
geneities in its refractive index decrement d(x,y,z). Coherence at
ELSE
Lateral distance (nm)
vacuumtungsten
e− 80 keV12�
4200
4300 Dep
th in
tun
gste
n (n
m)
3100
1900
700
0
21000.0−2100−4200
Figure 77 Production of off-focal radiation. The plot shows the lateral view of 200 tracks of an 80-keV electron beam incident (with 12� angle and 1 mmlateral beam size) from vacuum on the surface of a tungsten target. While penetrating as much as a few micrometers in the tungsten target, afraction of the electrons are backscattered in the vacuum emerging backward in regions distant from the incidence beam. In an x-ray tube, theseelectrons would be directed again on the surface of the tungsten target by the high electric field between the cathode and anode, originating an off-focalradiation (simulation was performed with the code CASINO v. 2.42, Drouin et al., 2007).
Position (mm)
Focal spot length
80 kV50 kV35 kV
0.0(b)
(a)
0.0
0.2
0.4
Nor
mal
ized
inte
nsity
(a.u
.)
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
Position (mm)
Focal spot width
80 kV50 kV35 kV
0.00.0
0.2
0.4
Nor
mal
ized
inte
nsity
(a.u
.)
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
Figure 78 LSF of the focal spot for the width (a) and length (b) of the0.040-mm focal spot shown in Figure 76, showing the change in thefocal spot shape and intensity upon varying the x-ray tube voltage (35,50, or 80 kV) at a fixed current of 0.125 mA. The curves are normalized tothe central height of the 80-kV curve. Magnification factor, 3.15�.
Physical Basis of x-Ray Imaging 43
ELSE
a point in a radiation field refers both to the angular distribu-
tion of the wave field at that point (so-called spatial or lateral
coherence) and to the temporal structure of the radiation field
(referred to as temporal or longitudinal coherence), related to the
spectrum of frequencies (photon energies) of the source. High
temporal coherence implies high monochromaticity of the
source radiation. The degree of spatial coherence for a mono-
chromatic wave of wavelength l from an extended (angularly
incoherent) source, when the wavefront reaches a given point
P in space, is given by the Fourier transform of the angular
intensity distribution I(a) of the source, so it is related to its size
s and intensity distribution. Quantitatively, it can be calculated
by defining the complex quantity
lcoh ¼
ðI að Þei2pl sa daðI að Þda
[125]
Its modulus (the ‘fringe visibility’ in interference experi-
ments) has values between 0 and 1 (with value 1 assumed for
the case of monochromatic plane waves, and value 0 assumed
for the case of no visible interference effect due to lack of
coherence). lcoh is related to a quantity called lateral coherence
length Lcoh, the linear size of a region in space where the wave
field is strongly correlated, which can be calculated as
Lcoh ffi la¼ l
Dls
[126]
where a is the (small) angle subtended by the source of linear
size s (e.g., the focal spot size of the x-ray tube) from the point P
at a distance Dl from the source. Hence, using an x-ray tube as
the source of radiation, its coherence properties are highly
dependent on the size F of the focal spot; in clinical settings,
the focal spot (linear) size is in the order of 1 mm, and the
source-to-object distance, R1, is in the order of 1 m; the above
formula also holds in the case of polychromatic x-ray beams as
from an x-ray tube, and characterizing its spectrum by its
average wavelength �l, one has
Lcoh �lR1
F[127]
or, in terms of average spectral energy �E,
Lcoh ¼ hc�E
� �R1
F[128]
As a numerical example, at 50 keV average photon energy �E
(corresponding to an x-ray wavelength of 2.48�10�11m), one
VIER
Position (mm)
Nor
mal
ized
pix
el v
alue
0
A
(b)
(a)
A B C D E
B CD
E
0.94
0.96
0.98
1.00
10 20 30 40 50 60 70 80
Figure 80 (a) Contact digital radiography (i.e., with the detector planealmost coincident with the back surface of the object, R1þR2¼72 cm,magnification factor Mffi1) at 40 kV of a U-shaped PMMA frame,1.95-mm thick, on which thin (25 mm diameter) gold wires have beenwrapped up. Image regions marked with letters A–E indicate areascorresponding to PMMA, a wire, another wire, PMMA, and air,respectively. Pixel (50-mm pitch) values have been normalized to theaverage value in areas corresponding to air absorption. (b) Horizontal lineprofile along the white line indicated in the image in (a): absorption-basedintensity attenuation is evident in regions corresponding to areas A–Din the contact radiography. Note the regions, indicated by the arrows,where an increased signal is present at PMMA/air edges. The focal spotsize of the x-ray tube was 5 mm.
Focal spot size F:
0.01 mm
0.05 mm
0.1 mm
10
10−5
10−4
L coh
(mm
) 10−3
10−2
100Average photon energy (keV)
Lateral coherence length at R1= 1 m and focal spot size F
1 mm
Figure 79 Lateral coherence length Lcoh as a function of the averageenergy of the polychromatic beam from an x-ray tube, as calculated witheqn [126] using a source–object distance R1¼1 m and a varying focalspot size between 1 and 0.01 mm.
44 Physical Basis of x-Ray Imaging
VIER
has Lcohffi2.48�10�5mm. Thus, the x-ray e.m. wave at 1 mdistance from a diagnostic x-ray tube may lose spatial coher-
ence over regions whose linear size is only about 25 nm. With
the above figures, in order to gain lateral coherence, even
impractically large source–object distances would be relatively
ineffective, since Lcoh would reach 0.5 mm only after the e.m.
wave travels a distance R1¼20 m. On the other hand, reducing
the focal spot size may increase the spatial coherence of the
radiation from x-ray tube sources.
Figure 79 shows the value of Lcoh at a distance of 1 m from
the focal spot of an x-ray tube having a varying size from
1 mm (as in a general radiography x-ray tube) to 0.1 mm
(minifocus source) down to 0.01 mm (microfocus source),
as a function of the average energy of the x-ray beam �E, as
given by eqn [128]. From Figure 79, it can be seen that
coherence length is higher at lower energy and that in the
diagnostic energy range, it is less than 1 mm. In this context,
breast imaging would potentially provide an important appli-
cation of phase-contrast imaging techniques employing par-
tially coherent sources, since in that case low-energy spectra
are used with mean energies at about 17–18 keV, and focal
spot sizes are typically as small as 0.3 mm (or 0.1 mm, as
employed in magnification mammography and in phase-
contrast mammography).ELS
E
2.01.4 Examples of x-Ray Image Formationand Contrast Mechanisms
In Section 2.01.4.1, a few examples of attenuation-contrast
and phase-contrast images are illustrated. These examples,
relying on simple physical objects of known materials, will
help the reader to better understand the practical aspects of
image formation according to different contrast mechanisms.
2.01.4.1 Attenuation Contrast (Absorption and Scattering)
Figure 80(a) shows the radiography of a sample test, acquired
in contact with a 50-mm-pitch flat panel digital detector at
80 kVp. The sample consists of a PMMA frame made out of a
Dz¼1.95-mm thick PMMA slab with highly polished edges
and surfaces. Thin gold wires (25 mm diameter) have been
wrapped up on the U-shaped PMMA frame, as thin absorbing
details.
The normalized, horizontal line profile shown in Figure
80(b) reveals that the PMMA slab introduces an average atten-
uation I/I0ffi0.945, from which the effective attenuation coef-
ficient of PMMA can be calculated as m¼�(1/Dz)ln(I/I0)ffi0.290 cm�1 corresponding to an effective beam energy
of 38 keV in PMMA. Such attenuation corresponds to an
image contrast between PMMA and air C¼(I�I0)/I0ffi5.5%,
and to an attenuation contrast between the wires and PMMA
Cffi1%. Owing to the limited spatial resolution of the detector
(about 0.16 mm FWHM, measured), the thin wires have a
width in this profile that is close to 0.16 mm, so that a signal
averaging effect due to detector undersampling produces an
apparent attenuation in the gold wire (I/I0ffi0.955 at the center
of the wire profile) which is much less than the one calculated
at 38 keV (I/I0¼0.555). By considering the rate of phase
change in PMMA, �df/dz¼35.237 rad mm�1 at 38 keV (see
Figure 21), the phase shift Df in the 1.95-mm thick PMMA
slab with respect to surrounding air can be calculated as
21.87p rad.On the other hand, an estimate of Df can be obtained
by using eqn [41]. From the measured attenuation � ln(I/I0¼0.945)ffi0.046 and from the value (¼2457.17) of the
ratio of d/b for PMMA at 38 keV (Figure 19), one obtains
Dfffi22.12p rad, in close agreement with the calculated value.
Physical Basis of x-Ray Imaging 45
Figure 81(a) shows the phase map (i.e., the x,y distribution of
the phase advance Df introduced by the homogeneous slab
with respect to the corresponding propagation in free space):
the assumption of uniform d/b ratio in the object (i.e., assum-
ing that the object is made only of PMMA) allows one to apply
eqn [41] and to derive this image from the logarithm of the
corresponding attenuation map I/I0 shown in Figure 80(a).
Line profiles (along PMMA and air only, Figure 81(b), and
along PMMA, PMMAþwire, and air, Figure 81(c)) show that
the average phase shift (with respect to free space) is ffi70 rad
in PMMA with an additional ffi70 rad attributed to the gold
wires. Note that the image contrast between PMMA and air and
between the wires and PMMA in this reconstructed phase
image is increased with respect to the case of the attenuation
image, since correspondingly, d is about 2500 times higher
than b. The phase contrast between the gold wires and
PMMA is not the true one corresponding to phase change in
the wires, since the whole object was assumed a homogeneous
PMMA slab. Since at the average energy of 38 keV the ratio of
attenuation index between Au and PMMA isffi50:1, in terms of
PMMA attenuation thickness the 25-mm thick Au wires can be
thought of as being equivalent to 1.25 mm of PMMA, which
gives them an estimated phase change close to the value for the
20
0 20
A
B C
D
E
20
A
F G H I L
B C D E
M
0
-20
-40
-60
-80
Position (mm)40 60 80
0-20-40-60-80
-100-120-140-160
Position (mm)(c)
(b)
(a)
Pha
se s
hift
(rad
)P
hase
shi
ft (r
ad)
0 10
F G H I LM
20 30 40 50 60 70 80
Figure 81 (a) Retrieved phase map corresponding to the contactradiography in Figure 80(a), obtained using eqn [41] in the text. (b)Horizontal profile along the upper white line and (c) horizontal profile alongthe lower white line in (a). Letters (A–M) label regions indicated in the imagein (a). The large step in phase shift (about �70 rad) corresponds to thePMMA–air edge, numerically similar to the phase shift introduced by thegold wires as difference between��70 and ��140 rad.
ELSE
1.95-mm thick PMMA frame. This shows that even under the
rough approximation of homogeneity for a nonhomogeneous
sample, image contrast can be generated in phase imaging,
though the quantitative relationship between Df and the
material density r is lost (eqn [41]).
Figure 80(a) shows also other interesting features: the atten-
uation in air is not constant in regions close to the edges of the
slab, as pointed by the arrows in Figure 80(b); a slight increase
in the signal is present there, which can be attributed to the
contribution from Compton scattering in the PMMA slab.
Scattering at large angles, as represented by inelastic scatter-
ing at variance with elastic scattering, redirects incident photons
to lateral directions, and at the transition region between two
different materials (PMMA and air, in this case) this produces an
enhancement of the signal toward the side of the border region,
where the scattering attenuation mCDz is smaller, mC being the
Compton scattering attenuation coefficient at the average beam
energy. In the present case (with mC¼0.213 cm�1 at 38 keV in
the PMMA slab and mCDzffi0.0414), this side corresponds to
‘air’ regions with respect to ‘PMMA’ regions, hence the increased
signal visible in Figure 80. This redirection of rays from
the forward direction toward the less scattering side should
be accompanied by a decrease of the x-ray field intensity
on the more scattering side. Indeed, this is not apparent in
Figure 80 due to the image noise and to the thin (1.95 mm)
sample thickness, but imaging of a thicker (23.52 mm) PMMA
sample in exactly the same irradiation conditions as in that
figure does reveal this condition (Figure 82).
The large extent of the zone across the plate edge which is
seen in the radiography of Figure 82 as decreased or increased
intensity (about 4 cm across the sharp edge of the PMMA slab)
allows to attribute this effect to Compton scattering rather than
Rayleigh scattering in the slab, due to the low divergence of the
coherent scatter forward peaked intensity and the short dis-
tance (a few millimeters) between the rear slab surface and the
detector active surface. Moreover, the higher beam energy
(80 kV tube voltage rather than 40 kV as in Figure 81)
VIER
PMMA
10 mm
Air
Figure 82 Composite image showing the overlay, on the contactradiography of the sharp edge of a thick (23.52 mm) PMMA slabacquired at 80 kVp, of a horizontal profile drawn along the white line atthe center of the figure. The profile evidences the effect of scatteringin the slab, which produces a reduced intensity recorded for the slab (lefthalf of the figure) and a corresponding increased intensity recorded onthe side of air (right half of the figure). The dashed lines indicate theaverage intensity on the two sides of the slab away from the edge: thedifference between these intensity levels is due to absorption in thePMMA plate.
46 Physical Basis of x-Ray Imaging
determines a higher contribution of Compton scattering in the
sample. This thick PMMA slab introduces an attenuation
(between the average values of the signal in the slab and
those in the air, away from the edge, indicated by the black
dashed lines in Figure 82) of I/I0ffi0.69. This attenuation
comes from absorption contrast between the PMMA plate
and the same thickness of air in the beam path. When sub-
tracting these average attenuation levels from the correspond-
ing intensity on the two sides of the edge, two half Gaussian
profiles are obtained with the same width of 26.2 mm FWHM,
which can be taken as an estimate of the width of the LSF for
scattering in the sample with the given x-ray spectrum.
2.01.4.2 Attenuation Plus Phase Contrast
The above examples of projection radiography were produced
with the detector in contact with the back object surface, that is,
with the image plane at a distance zl¼z0þDz from the source,
where Dz is the object thickness (Figure 1). Upon increasing
the distance R2 from the object to the detector (i.e., with
zl>z0þDz) while keeping the source–object distance fixed –
hence introducing magnification in the imaging technique if a
divergent x-ray source is adopted, as for x-ray tube focal spots –
one expects that Compton scattering intensity reaching the
detector reduces because of divergence of the beam in the air
gap beyond the object, and that Rayleigh scattering could
reveal some effects. Indeed, the angular distribution of the
differential cross section for Rayleigh scattering (Section
2.01.2.2) in an amorphous material like liquid water (and
biological tissues) shows an oscillating behavior with an abso-
lute maximum at a scattering angle of a few degrees (Johns and
Yaffe, 1983) and a minimum in the forward direction: at
60 keV photon energy, this maximum occurs at 3.8�, and at
20 keV it occurs at 11.3�. This oscillating behavior is related to
the interference of coherent scatter from electrons belonging to
different molecules, and it introduces a small-amplitude, low-
frequency modulation of the transmitted intensity distribution
on the image plane, at a distance R2>0 from the object.
Figure 1 schematizes the presence of refraction effects in
projection radiography, related to the distortion of x-ray wave-
fronts when traversing a nonhomogeneous object, irradiated
with monochromatic or polychromatic x-ray beams. Variation
in object thickness (i.e., in the direction of propagation z) and
the nonhomogeneity of the absorption index b in the sample
are responsible for attenuation contrast in the transmitted field
intensity and the phase shift effects introduced by the propa-
gation in the object with respect to propagation in air (or
coupling medium). On the other hand, in addition to varia-
tion in object thickness, local variations in the (x,y) spatial
distribution of the refractive index decrement d(x,y,z) due to,
for example, details with a lower or higher density than the
background density of the object produce perturbation of the
wavefronts and deviation (by a few arc seconds, in biological
tissues) in the direction of x-rays at the border of those non-
homogeneous regions. Slightly refracted waves propagate
through slightly different path lengths in the sample, thus
acquiring a phase difference Df(x,y) while propagating in the
direction z (Figure 1). This phenomenon introduces a phase
gradient (�@f/@x, �@f/@y) in the plane transverse to the
direction of propagation, whose modulus is proportional to
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the angular deviation of the x-rays undergoing refraction. The
phase differences of diffracted waves which propagate in the
region behind the object and downstream the x-ray beam
produce via interference effects a redistribution of the wave
intensity in a (transverse) image plane at a distance from the
object, which shows up as sharp (x,y) variations in the trans-
mitted intensity corresponding to the projected position of
nonhomogeneities in the object or variation in the thickness
of the object. In the mathematical description of this distribu-
tion (see Chapter 2.08), the intensity of the transmitted wave
exiting the object (eqn [50]) is ‘transported’ from the output
object plane to the image plane giving a lateral distribution T
(x,y) which contains dependences on both the projected atten-
uation profile and the Laplacian (@2f/@x2þ@2f/@y2) of the
projected phase change f(x,y) (eqn [47]).These effects can be
observed with monochromatic x-ray sources (e.g., from a syn-
chrotron radiation source) as well as with polychromatic
sources as from an x-ray tube, even in the near field (Fresnel
diffraction in the free space) (Wilkins et al., 1996). For a given
spatial resolution Dx of the imaging detector, one expects that
the extent of these effects depends, among other parameters,
on the distance R2 between the object and detector as a result of
the low refraction angles (a few arc seconds), on the average
wavelength �l, and on the spatial frequency 1/D introduced by
spatial nonuniformities of size D in the object. Wu and Liu
(2003) have introduced the parameter shear length, Lshear,
defined as
Lshear �lR1R2
R1 þ R2
1
D¼ �l
R2
M
1
D[129]
whereM is the magnification factor, for introducing a visibility
parameter for phase-contrast effect in the propagation-based
geometry. This figure of merit for phase-contrast visibility,
related to the coherence degree in the incident and
transmitted wavefields, is defined as (Wu and Liu, 2003):
LshearLcoh
¼�l R1R2
R1þR2
1D
�l R1
F
¼ R2F
R1 þ R2
1
D¼ M� 1ð ÞF
M
1
D[130]
For phase contrast of a nonhomogeneity of size D to be
visible, one should have Lshear/Lcoh1, and for Lshear/Lcoh<1,
one has a partially coherent wavefield.
Figure 83(a) shows details of the radiography (40 kV,
5 mm focal spot size, R1ffi7 cm, R1ffi65 cm, Mffi10,�l ffi 0:04592nm, �E ffi 27keV, detector pixel Dx¼50 mm) of
the sharp edge (transition from region D to region E) of
the PMMA frame whose contact radiography is shown in
Figure 80(a). In this case, Lcohffi0.6 mm and Lshearffi0.03 mm,
so that phase-contrast effect could be visible. Sharp dark and
white fringes are visible at the two sides of the PMMA edge
(as shown by the profile in Figure 83(b)), which reveal the
dependence of the recorded intensity on the second derivative
@2f/@y2 of the projected phase across the PMMA–air transi-
tion. However, the attenuation in the PMMA frame being not
negligible (as seen in the contact radiography of Figure 80(a)),
both attenuation and phase contrast are present in this radiog-
raphy. In principle, scaling and ‘subtraction’ of the contact
image from the phase-contrast image allow one to derive the
two separate information of the (projected) attenuation map
m(x,y) and phase map f(x,y) of the object. So-called phase
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Distance (pixels)
0
(b)
(a)
0.74
0.76
0.78
0.80
100
Gra
y va
lue
200
A
300 400
Figure 83 (a) Detail of a magnification radiography (Mffi10) of thesharp edge of the PMMA edge shown in Figure 80(a), taken at 40 kV andwith a focal spot size of 5 mm. (b) The horizontal line profile (averagedover all lines in the image) reveals the presence of sharp variations in theintensity across the edge, as black and white fringes, which are thesignature of phase-contrast effects in this propagation-based geometry.The difference A between the average signal levels reflects the absorptionin the PMMA sample with respect to free space.
Physical Basis of x-Ray Imaging 47
retrieval methods exist, as explained in Chapter 2.08 of this
volume, which under suitable approximations allow deriving
separately the absorption image and the phase image of the
sample, exploiting image acquisitions at two or even at just
one distance R2 from the detector.
Acknowledgments
The author wishes to thank Dr. G. Mettivier, S. Curion, M.
Quattrocchi, V. Capano for their assistance in the acquisition
of radiographies, and Prof. E. Perillo for his comments on parts
of the manuscript.SE
L
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Relevant Websites
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