Phase-Contrast X-Ray Imaging of Complex Objects

kth royal institute of technology

Doctoral Thesis in Physics Phase-Contrast X-Ray Imaging of Complex Objects

ILIAN HÄGGMARK

Stockholm, Sweden 2021 Phase-Contrast X-Ray Imaging of Complex Objects

ILIAN HÄGGMARK

Academic Dissertation which, with due permission of the KTH Royal Institute of Technology, is submitted for public defence for the Degree of Doctor of Philosophy on Friday the 28th May 2021, at 1:00 p.m. in Zoom and the BioX Library, AlbaNova University Center, Roslagstullsbacken 21, Stockholm.

Doctoral Thesis in Physics KTH Royal Institute of Technology Stockholm, Sweden 2021 © Ilian Häggmark

ISBN 978-91-7873-847-2 TRITA-SCI-FOU 2021:03

Printed by: Universitetsservice US-AB, Sweden 2021 iii

Preface

The doctoral work on phase-contrast X-ray imaging presented in this Thesis was carried out between 2016 and 2021 in the Biomedical and X-ray physics group at the Department of Applied Physics, KTH Royal Institute of Tech- nology, under the supervision of Prof. Hans Hertz and Assoc. Prof. Anna Burvall. iv v

Abstract

X-ray imaging is a group of techniques using electromagnetic radiation of high energy. The ability to quickly visualize internal structures in thick opaque objects has made it an indispensable tool in research, medicine, and industry. Contrast is generally achieved by differential absorption, however, this mechanism has a strong dependence on atomic number. This results in low contrast within materials consisting of mainly elements of low atomic number, such as hydrogen, carbon and oxygen, e.g., soft organic matter. The problem with low contrast is further complicated by limitations in radiation dose. To improve contrast the phase shift of the X-rays can be measured without increasing the dose.

This Thesis concerns one method to harness this phase signal – propagation- based phase-contrast X-ray imaging (PBI). Three aspects on how to image complex objects are addressed: multi-material phase retrieval, simulations of clinical imaging, and small-animal imaging on compact systems. First, the derivation of a previously published method for multi-material phase retrieval is shown. A comparison between this method and another further reveals important differences. Secondly, a strategy to use large digital voxel-based phantoms for clinical imaging is developed. The method is demonstrated on a mammography phantom and in a reader study on clinical lung imaging. Finally, a compact X-ray system is used to demonstrate imaging of vascular canals in rat bone and high-resolution lung imaging on free-breathing mice, i.e., without mechanical ventilation. vi

Sammanfattning

Röntgenavbildning är en samling tekniker som använder elektromagnetisk strålning av hög energi. Förmågan att snabbt åskådliggöra inre strukturer i tjocka ogenomskinliga objekt har gjort den till ett oumbärligt redskap inom forskning, medicin och industri. Kontrast åstadkoms generellt genom skillnader i absorption av röntgenstrålningen, men denna mekanism är starkt beroende av atomnummer. Detta resulterar i låg kontrast för material som huvudsakligen består av grundämnen med lågt atomnummer som väte, kol, och syre – typiskt mjuk biologisk vävnad. Begränsningar i stråldos försvårar ytterligare problemet med låg kontrast. För att förbättra kontrasten kan röntgenstrålningens fasskift mätas utan att dosen ökas.

Denna avhandling behandlar en metod som nyttjar denna fassignal – pro- pagationsbaserad faskontrast (PBI). Tre aspekter av hur komplexa objekt kan avbildas behandlas: fasåterhämtning av objekt med flera material, si- muleringar av klinisk avbildning och smådjursavbildning med kompakta röntgensystem. Först görs en härledning av en tidigare publicerad metod för fasåterhämtning av objekt med flera material. Viktiga skillnader visas i en jämförelse mellan denna metod och en annan. Sedan utvecklas en strategi för hur stora digitala voxelbaserade fantomer kan användas för klinisk avbildning. Den förevisas på en mammografifantom och tillämpas i en studie med radiologer om klinisk lungavbildning. Slutligen demonstreras hur kompakta röntgensystem kan användas för att avbilda nätverket av vaskulära kanaler i råttben och hur lungavbildning kan utföras med hög upplösning på möss som andas naturligt, d.v.s. utan mekanisk ventilering. vii

List of papers

This Thesis is based on the following papers:

Paper A I. Häggmark, W. Vågberg, H. M. Hertz, and A. Burvall, ”Comparison of quantitative multi-material phase-retrieval algorithms in propagation-based phase-contrast X-ray tomography”, Opt. Express 25(26), 33543–33558 (2017).

Paper B I. Häggmark, K. Shaker, and H. M. Hertz, ”In Silico Phase-Contrast X- Ray Imaging of Anthropomorphic Voxel-Based Phantoms”, IEEE T. Med. Imaging 40(2), 539–548 (2021).

Paper C I. Häggmark*, K. Shaker*, S. Nyrén, B. Al-Amiry, E. Abadi, W. Segars, E. Samei, and H. M. Hertz, ”Propagation-based phase-contrast CXR: a virtual clinical study”, manuscript in preparation.

Paper D I. Häggmark, J. Romell, S. Lewin, C. Öhman, and H. M. Hertz, ”Cellular-Resolution Imaging of Microstructures in Rat Bone using Laboratory Propagation-Based Phase-Contrast X-ray Tomography”, Microsc. Microanal. 24(S2), 368–369 (2018).

Paper E K. Shaker*, I. Häggmark*, J. Reichmann, M. Arsenian-Henriksson, and H. M. Hertz, ”Resolving the terminal bronchioles in free-breathing mice: propagation-based phase-contrast CT”, manuscript.

* Shared first authorship. viii

Other publications

The author has contributed to the following publications, which are related to this Thesis but have not been included in it.

I. Häggmark, W. Vågberg, H. M. Hertz, and A. Burvall, ”Biomedical Ap- plications of Multi-Material Phase Retrieval in Propagation-Based Phase- Contrast Imaging”, Microsc. Microanal. 24(S2), 370–371 (2018).

J. Romell, I. Häggmark, W. Twengström, M. Romell, S. Häggman, S. Ikram, and H. M. Hertz, ”Virtual histology of dried and mummified biological samples by laboratory phase-contrast tomography”, Proc. SPIE 11112, 111120S (2019). ix

List of Abbreviations

CCD Charge-coupled device CMOS Complementary metal-oxide-semiconductor CT Computed tomography CTF Contrast transfer function CXR Chest X-ray FBP Filtered back projection FDK Feldkamp-Davis-Kress FOV Field of view FRC Fourier ring correlation FSC Fourier shell correlation FWHM Full width at half maximum ICS Inverse Compton scattering LMJ Liquid-metal-jet LWFA Laser wakefield acceleration MC Monte Carlo PBI Propagation-based imaging PCI Phase-contrast X-ray imaging PSF Point spread function SNR Signal-to-noise ratio TIE Transport-of-intensity equation VCT Virtual clinical trial WP Wave propagation x

Contents

Preface iii

Abstract v

Sammanfattning vi

List of papers vii

Other publications viii

List of Abbreviations ix

0 Historical note 1 0.1 Discovery ...... 1 0.2 Development of X-ray imaging ...... 4 0.3 Imaging with phase ...... 5

1 Introduction 7 1.1 Image quality ...... 7 1.2 Phase-contrast X-ray imaging ...... 8 1.3 Complex objects ...... 9

2 X-ray matter interaction 11 2.1 Fundamental interactions ...... 12 2.1.1 Single-photon interactions ...... 12 2.1.2 Ionizing radiation and dose ...... 15 2.2 Complex refractive index ...... 16

3 Sources and detectors 19 3.1 X-ray sources ...... 19 3.1.1 X-ray tubes ...... 19 3.1.2 Liquid-metal-jet sources ...... 22 xi

3.1.3 Synchrotrons ...... 22 3.1.4 Laser-based sources ...... 23 3.1.5 Comparison of sources ...... 24 3.2 X-ray detectors ...... 25

4 X-ray image formation 31 4.1 Attenuation-based imaging ...... 31 4.2 Phase-contrast imaging ...... 33 4.2.1 Mathematical description ...... 34 4.2.2 Phase-contrast imaging methods ...... 36 4.3 Propagation-based imaging ...... 39 4.3.1 Basic phenomenon ...... 39 4.3.2 Parameters ...... 40 4.4 Tomography ...... 43 4.4.1 Fourier slice theorem ...... 43 4.4.2 Reconstruction ...... 44 4.4.3 Angular sampling ...... 46

5 Phase retrieval 49 5.1 The phase problem ...... 49 5.2 Single-material methods ...... 50 5.3 Multi-material methods ...... 51

6 X-ray imaging simulations 55 6.1 Methods ...... 55 6.2 Object models ...... 57 6.2.1 Sampling ...... 58 6.2.2 Upsampling ...... 59 6.3 Virtual clinical trials ...... 61

7 Small-animal imaging on compact systems 63 7.1 Visualizing vascular canals in bone ...... 63 7.2 In-vivo lung tomography ...... 64

8 Conclusions and outlook 69

A Fourier Transform 71

Summary of papers 75

Acknowledgements 77

References 79

HISTORICAL NOTE | 1

Chapter 0

Historical note

Below follows a short and incomplete description of selected events in the development of X-ray imaging and phase imaging.

0.1 Discovery

Wilhelm Conrad Röntgen’s discovery of X-rays 1895 was immediately recognized as a significant contribution to science. The speed with which the technique was adopted around the world remains to this day unparalleled. On November 8, Röntgen made the discovery of an unknown radiation. Before the end of the year, on December 28, he published a solid report on his findings [1]. On January 5, Die Presse in Vienna published an article about the discovery entitled “Eine sensationelle Entdeckung” [2]. On January 23, Nature published a translation of Röntgen’s original report [3]. The already widespread use of the key hardware components to perform imaging – the discharge tube and film or a fluorescent screen – enabled scientists and enthusiasts to quickly begin their own experiments. On February 14, Science published an article by Edwin B. Frost, who had imaged various objects including a broken arm [4]. The following day New York Times included a news item with the title “X Rays Find a Needle in a Foot” which described how surgeons in Toronto used an X-ray image to locate a needle for removal [5]. In March Walter König at Physikalischer Verein in Frankfurt am Main showed X-ray images of animals, human teeth, and even mummies [6]. In May the first military X-ray was performed in Naples by Guiseppe Alvaro [7].

In Sweden, the discovery was received with great interest. On January 9, newspapers described Röntgen’s work [8]. One month later, on February 9, Knut Ångström and Hjalmar Öhrvall at Uppsala University acquired 2 | HISTORICAL NOTE

images of frogs and a rat [9, 10]. In February Thor Stenbeck also did his first experiments with X-rays on Triewaldsgränd in Stockholm, which were published on the 29th [11]. An early case of brain surgery aided by X-ray imaging was overseen by Salomon E. Henschen1, medical doctor at Uppsala University Hospital. A miller shot through the left eye in August 1895, with the bullet still in his head, had a good recovery, but received severe headache after a year and sought medical attention on September 2nd, 1896. Stenbeck acquired images to locate the bullet and on February 2 the following year, 1897, the surgeon Karl Gustaf Lennander removed the bullet at the University hospital. The patient recovered well and was discharged two weeks later [13–16]. In 1896 Stenbeck started using X-rays not only for imaging but also to treat patients for lupus vulgaris, skin lesions caused by tuberculosis. He later opened a clinic on Mäster Samuelsgatan in Stockholm and performed one of the first successful radiation therapies of cancer (carcinoma) on June 4th 1899 [17]. The patient was alive and well 30 years later and was present at the 2nd International Radiology conference in Stockholm, 1928.

This initial enthusiasm, evident by the hundreds of articles published only in 1896 [18], was not dampened by the reports of the harm of X-rays [19, 20]. Already in August 1896 the term radiation burns was used [21, 22]. Some refuted the dangers of exposing living organisms to radiation, but the risks of radiation dose became painfully evident over time with the death and disfigurement of users [23]. The dentist William Rollins conducted experiments showing the risks of X-rays and advocated careful use, e.g., shielding of sources and limiting the exposed area [22, 24, 25]. Reckless and frivolous use such as in entertainment gradually decreased although adoption of a responsible use took many years. As an imaging technique, especially in the medical domain, X-ray imaging nevertheless grew to become an irreplaceable tool.

Despite the immediate adoption of X-rays as an imaging technique, fundamental understanding of their nature was still lacking. Röntgen hy- pothesized in his first article that X-rays might be longitudinal waves in the ether [1].2 J. J. Thomson among others immediately began to investigate this [27]. G. G. Stokes advocated that “cathode rays” were charged molecules and that each molecule produced a small X-ray pulse [28]. A significant contribution to understanding X-rays was done by Charles Barkla

1Henschen was a famous doctor and neurologist which perhaps is evident from the fact that he was summoned to the sickbed of Vladimir Lenin in 1923 [12] 2Despite Michelson and Morley’s experiment in 1887 [26], the ether theory was still widely supported. HISTORICAL NOTE | 3

in Cambridge, who in 1904 showed that X-rays were polarized, a feature of electromagnetic radiation [29]. He later discovered how X-rays have properties characteristic of the elements from where they are released [30, 31]. Several attempts were made to show diffraction of X-rays, since this was considered a defining quality of electro-magnetic radiation. Herman Haga in Groningen and Cornelis H. Wind in Utrecht published results on diffraction using V-shaped slits and estimated the X-ray wavelength to be 0.05 nm, but they were not considered reliable enough [32–34]. That this was a problem yet to be solved is evident from Arnold Sommerfeld’s words in a letter to Wilhelm Wien in May 1905 [35]:

Es ist eigentlich eine Schmach, daß man 10 Jahre nach der Röntgen’schen Entdeckung immer noch nicht weiß, was in den Röntgenstr. eigentl. los ist.

Bernhard Walter and Robert Pohl performed a diffraction experiment similar to Haga and Wind in 1908 and questioned their results and estimated a shorter wavelength [36, 37]. Peter Koch traced Walter and Pohl’s diffraction curves a few years later using a photometer and Sommerfeld analyzed the result [38, 39]. Sommerfeld’s wavelength, 휆 = 0.04 nm was similar to Haga and Wind’s result. This result also agreed with estimates made by Wien based on Planck’s energy quantization [40]. A wider acceptance for the presence of diffraction was, however, not reached yet.

In 1850 Auguste Bravais formulated a theory about the lattice structure of crystals [41]. Max von Laue realized in 1912 that these structures could serve to show the diffraction of X-rays as the atomic lattice structure should be the same order of magnitude as the wavelength of X-rays [42]. Together with Walter Friedrich and Paul Knipping at Ludwig-Maximillian Universität in München, a crystal was shown to produce a diffraction pattern [43,44]. This finally convinced the scientific community that X-rays were electromagnetic radiation and earned Laue the 1914 Nobel prize in physics [45]. It was also the starting point of the field X-ray crystallography.

In Sweden, X-ray research was quickly picked up, as previously mentioned, but most early contributions concerned clinical imaging. A few fundamental results deserve, however, attention. A property of electromagnetic radiation that Röntgen without success tried to show in X-rays in 1895 was refraction by prisms. His failure was merely a lack of precision, as the short wavelength resulted in an exceedingly small refraction. The first indication of reflection from crystals was shown by Wilhelm Stenström in Lund 1919 [46, 47]. The definitive experiment showing that X-ray could be refracted by prism was 4 | HISTORICAL NOTE

carried out by Axel Larsson, Manne Siegbahn and Ivar Waller in Uppsala 1924 [48,49]. Siegbahn, professor in Physics, developed X-ray spectroscopy which was awarded the Nobel prize in physics in 1924 [50]. Waller developed a theory for scattering of X-rays in lattices under thermal vibrations [51]. The quality of this work is perhaps best illustrated by the words of Max Born: “Waller extended his investigations to the background scattering in his brilliant dissertation (Uppsala, 1925), which contains the complete formulae for the most general type of lattices; nothing essential has been added to these formulae by later theoretical research, apart from improvements in the presentation and practical application.” [52]. Another interesting work was also carried out in the same group by Gunnar Kellström. His 1932 thesis contains images showing X-ray interference patterns [53]. These images are near-field holographic images published 16 years before Gabor published his famous paper on holography in 1948 [54, 55].

0.2 Development of X-ray imaging

The principle and technique by which Röntgen obtained his first images are essentially the same 125 years later. A high voltage is used to accelerate electrons to a target where X-rays are produced. These X-rays pass through an object (patient) and expose photographic film. The contrast originates from the differential absorption of X-rays in the object. The devices that produce and detect X-rays have been developed and refined over time, like with the introduction of Coolidge tubes, rotating anodes and digital detectors. Outside the medical setting, accelerator sources such as synchrotrons have had a huge impact on research and a large number of techniques using X-rays as a probe have been developed. In clinical imaging the only major conceptual change since the beginning is perhaps the introduction of Computed Tomography (CT) in 1971 by Godfrey Hounsfield [56, 57]. Alvarez showed in 1976 the potential benefit of using spectral information, but the clinical implementation of this is still not complete [58].

In recent years, significant development has been carried out on brighter sources, energy resolving detectors, and advanced image post-processing. Another area of development concerns the fundamental mechanism that makes an image, the interaction between the X-rays and the object. This is of high relevance as conventional X-ray imaging is a flawed technique in the sense that the mechanism creating images, absorption of X-rays, is also potentially harmful to the object. This is particularly troubling for imaging objects with small differences in absorption such as soft tissue, and hasled to the development of methods detecting phase. HISTORICAL NOTE | 5

0.3 Imaging with phase

The problem of limited visibility due to low absorption is by no means new, nor restricted to X-ray imaging. Robert Hooke, one of the pioneers of microscopy, showed in 1672 at Royal Society how the air flow from the flame of a candle refracted light [59, 60]. Thomas Young’s interference experiment in 1803, showed wave properties of light and how interference could visualize phase, which was used in many configurations, e.g., the Michelson interferometer and the Mach-Zender interferometer [26, 61–63]. Another famous example is found in the study of shock waves in the 19th century. Schlieren (German: ”streaks”) Photography developed by August Toepler allowed for visualization of wavefronts [64]. Toepler’s method is based on the idea to image shifts in air density by singling out refracted rays. The difference in absorption is negligible, but by removing the background illumination and visualizing only refracted rays the shifts in density become apparent [65]. Parallels to phase imaging can also be found in nature. Caustic lines (Greek: 휅훼휐휎휏 ́휊휍, kaustos, ”burnt”) is one such phenomenon. It refers to the bright moving lines that appear on the bottom of clear, shallow waters due to refraction at the water surface. The difference in absorption between light passing through the crest orthe trough is negligible in this case, but the intensity difference caused by the refraction is substantial. The lensing of the surface results in a positive lens so bright lines are formed. The same phenomenon can be observed if a small insect or branch is floating on a still surface.3

Phase is the property describing the position on a cyclic waveform. This corresponds to the shape of a wavefront. A distortion in this wavefront can be considered as a phase shift and also as a refraction. Techniques aiming to form images from this shift are referred to as phase-contrast techniques. Modern phase-contrast imaging is sometimes considered to have been born with Frits Zernike, who in the 1930s realized the possibilities to image phase when absorption is negligible [66]. He applied this concept in microscopy for which he was awarded the Nobel Prize in 1953 [67]. Zernike’s idea to use phase was not new, but the technique gave a linear relationship between phase shift and intensity which enabled quantitative imaging. The same lenses used for Zernike phase contrast were, however, not possible to use with X-rays. Ulrich Bonse and Michael Hart were the first to realize phase- contrast imaging using X-rays in 1965. With a conventional interferometric approach using crystals, the phase could be directly recorded [68,69]. This approach, achieved with an accelerator-based source (synchrotron), was fol-

3If a drop of oil rests on the surface, however, the wetting of the water creates a negative lens and the border of the drop is dark. 6 | HISTORICAL NOTE

lowed by new techniques and applications, but still limited to these sources for the coming 30 years. In 1995 Anatoly Snigirev et al. at the Euro- pean Synchrotron Radiation Facility (ESRF) in France showed that with sufficient coherence and distance between the sample and detector adiffrac- tion pattern could be recorded [70]. This is near-field diffraction or Fresnel diffraction which gives an image that depends on the Laplacian of thephase shift. A phase signal could in other words be directly captured without any optical elements. In 1996 Stephen Wilkins et al. at the Commonwealth Sci- entific and Industrial Research Organisation (CSIRO) in Australia showed that this method works well also for laboratory sources with low temporal coherence, thus opening up possibilities for a much broader range of applications [71]. 25 years later, phase-contrast X-ray imaging has become a research field with a wide range of applications and several techniques for laboratory imaging, some of which are close to clinical implementation. INTRODUCTION | 7

Chapter 1

Introduction

Imaging is the art of recording a spatial structure using radiation. A good naturally occurring example is the mammalian eye. Visible light reflected or refracted in an object goes through the lens and is captured by the retina. For a long time, the concept of imaging was limited to a few optical systems: the eye, telescopes and later microscopes and cameras. In recent time, many different types of radiation have been used with numerous techniques to perform imaging. X-ray imaging is one of those. In X-ray imaging the radiation is X-rays, electromagnetic radiation with high energy. Its main advantage is the penetrating ability which allows for imaging the interior of thick objects, opaque to visible light (see Fig. 1.1). This means, however, also that reflection and refraction is weak and that an optical system likea camera with a lens is not practical. Instead the object is positioned between the detector and X-ray source and a shadow image is obtained. Despite this apparent simplicity, X-ray imaging is a powerful technique with a broad range of applications in research, medicine, and industry.

1.1 Image quality

In imaging, there is a never-ending struggle to improve the image quality, and to faster and more efficiently record the structure of interest. To assess the quality of an acquired image is not trivial, but three good metrics that broadly characterize an image are resolution, contrast and noise. Resolution is how small features can be seen, contrast how well one can distinguish between different features, and noise how much random intensity variations degrade the image. To complicate matters, all three quality metrics are connected. High resolution is irrelevant if it, for example, is drowned in noise or if the contrast is close to zero. High quality must also be balanced against measurement time and radiation dose, even if a good imaging system 8 | INTRODUCTION

Figure 1.1. A box, opaque to visible light, with complex internal structure imaged using a micro-focus X-ray source. The black bar at the bottom is 150 mm. is used. What is optimal varies greatly with application, from a conveyor belt where speed is critical, to day-long ultra-high resolution scans where stability and noise is critical, to clinical imaging where radiation dose must be kept low.

1.2 Phase-contrast X-ray imaging

For imaging in general, and X-ray imaging specifically, much development has led to better radiation sources and detectors which have improved resolution, speed, and enabled more efficient use of photons. Conventional clinical X-ray imaging is, however, inherently flawed in a sense as the fundamental mechanism for contrast is harmful to living organisms. The absorption of ionizing radiation damages DNA and increases the risk of cancer. The contrast achieved in soft tissue and other materials with low atomic number is also severely limiting image quality. This makes phase-contrast techniques an interesting option. By imaging the phase shift of the X-rays, not just the attenuation, more information is obtained which can provide better contrast, even with reduced X-ray dose. A number of different techniques have been developed to image the phase shift for both research and industrial applications [72]. This doctoral work is about one such technique, propagation-based phase-contrast X-ray imaging, or propagation- based imaging (PBI) in short. It is characterized by the edge enhancement between materials in the image (see Fig. 1.2) and the absence of optics in the experimental arrangement. INTRODUCTION | 9

(a) (b)

Figure 1.2. Wasp leg joint between tibia (lower left) and tarsus (upper middle) with tibial spurs. (a) Attenuation image. (b) Phase-contrast image. Both images are acquired with the same X-ray dose and effective pixel size using a micro-focus X-ray source. Scale bars are 500 µm.

1.3 Complex objects

The topic of this doctoral work has been to improve imaging of complex objects in propagation-based imaging. So what is meant by complex in this context? It is perhaps best understood by considering the opposite: what are simple objects? In this field it is relatively small solid objects made from one material which is not strongly absorbing. Diverging from this narrow focus leads to a host of challenges. Three such challenges are presented in this work. The first is how to perform image post-processing, so-called phase retrieval, on images of objects containing multiple materials, the second deals with how to simulate clinical imaging, and the third how to perform pre-clinical imaging with a compact laboratory system.

The outline of the Thesis is as follows. Chapters 2–4 give a brief background to X-ray imaging and phase-contrast imaging. Chapters 5–7 describe more specific details on the three challenges presented and summarizes the keyre- sults from corresponding papers (A–E). Chapter 8 adds concluding remarks and an outlook.

X-RAY MATTER INTERACTION | 11

Chapter 2

X-ray matter interaction

How X-rays interact with matter is fundamental for understanding X-ray imaging. A salient feature of this description is the strong dependence on X- ray energy. X-rays are electromagnetic radiation and share many properties with visible light, which is more familiar to us. They may in many instances be described as particles (photons) and in others as waves, or as Albert Einstein and Leopold Infeld wrote [73]:

It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either. We are faced with a new kind of difficulty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do.

This chapter explains the most relevant types of interactions, first on the atomic scale, and then on the macroscopic scale using the complex refractive index.

Wavelength 1 um 100 nm 10 nm 1 nm 100 pm 10 pm 1pm

Ultraviolet Hard X-ray Soft X-ray Gamma

1 eV 10 eV 100 eV 1 keV 10 keV 100 keV 1 MeV Photon energy

Figure 2.1. The electromagnetic spectrum between visible light and gamma radiation. 12 | X-RAY MATTER INTERACTION

2.1 Fundamental interactions

In Fig. 2.1 part of the electromagnetic spectrum is shown. Although the boundaries between different types of radiation are not precise, X-rays can be found in the region around 10 nm to 0.01 nm or equivalently 120 eV to 120 keV. Both energy and wavelength are commonly used to describe the electromagnetic radiation in the X-ray region. Wavelength 휆 and energy 퐸 are related by the expression 푐ℎ 퐸 = , (2.1) 휆 where 푐 = 299 792 458 m/s is the speed of light, ℎ = 6.626 070 15 ⋅ 10−34 J s is Planck’s constant and energy is given in Joules [J] [74]. As the Joule is a large unit, the kiloelectronvolt is more suitable in the hard X-ray range: 푐ℎ 1.239 퐸 [keV] = ≈ , (2.2) 푒휆 휆[nm] where 푒 = 1.602 176 634 ⋅ 10−19 C is the elementary charge. The high energy of X-rays compared to visible light results in two significant differences: a much lower interaction with matter which allows X-ray to pass through thick opaque objects, and strong ionization, i.e., creating ions and free electrons from atoms which result in secondary interactions.

2.1.1 Single-photon interactions Below, three fundamental interactions between single photons and atoms or electrons are summarized: photoelectric effect, inelastic, and elastic scattering. These interactions contribute to attenuate X-rays traveling through a medium (see Fig. 2.2). Attenuation is described by the attenuation coefficient, 휇 [1/m]. It is defined such that attenuation of intensity overa distance 1/휇 is given by 1/e ≈ 0.37, i.e., a reduction with 63%. The total attenuation is the sum of all interactions:

휇 = 휇abs + 휇inelastic + 휇elastic. (2.3)

Depending on the energy range, certain interactions can be neglected. Pair production, the creation of an electron and a positron from a photon, will, for example, only occur at much higher photon energies than X-rays.

Photoelectric effect In photoelectric effect, colloquially called absorption, the photon transfers its energy to an electron and this photoelectron is knocked out (see Fig. X-RAY MATTER INTERACTION | 13

103 Elastic scattering

] Inelastic scattering 1 - Photoelectric absorption Total attenuation 102

101

Attenuation coefficient [m coefficient Attenuation 100

0 10 20 30 40 50 60 70 80 90 100 Energy [keV]

Figure 2.2. Attenuation coefficient for soft tissue. Data from NIST [75].

2.3a). This is in other words an ionization. For this to happen, the photon energy has to exceed the electron binding energy. When the atom relaxes and the vacancy is filled by another electron the energy is released either as a characteristic photon, i.e., fluorescence, or as an Auger electron [76, 77]. The fluorescence photon is named according to the Siegbahn notation [78]: first by the capital letter of the shell of the vacancy, K, L, Metc.,and second by a Greek letter and a number depending on which spin-orbitals the electron moves between. Three important lines for X-ray imaging are K훼1, K훼2 and K훽1. The probability of photo-electric effect is strongly dependent on photon energy (퐸) and the atomic number (푍) of the material, 휇abs ∝ 푍4/퐸3 [79]. This expression is, however, only an approximation and does not account for absorption edges. An absorption edge is a steep increase in attenuation, due to more possibilities of interaction when the photon energy passes the binding energy of a new shell.

Inelastic scattering Scattering is an event when a photon interacts with an atom and the direction is changed, but no absorption takes place. Inelastic scattering, also called Compton scattering or incoherent scattering, occurs when the photon loses energy in the interaction (see Fig. 2.3b). The lost energy is transferred to an electron which is ejected from the atom [80]. The energy (퐸′) after the redirection by the angle 휃 is given by

1 1 1 ′ = + 2 (1 − cos 휃), (2.4) 퐸 퐸 푚푒푐 14 | X-RAY MATTER INTERACTION

- - - M K - - - L α - -

- - K + - + - - - - M - - - - L - - K

(a)

------

- - - - + + - + - + ------

(b) (c)

Figure 2.3. Schematic drawing of three types of X-ray matter interaction. (a) Photo-electric effect. On the left a photon knocks out a photo electron, leavinga vacancy. On the right two processes to fill that vacancy are shown, fluorescence (top) and Auger (bottom). (b) Inelastic scattering. (c) Elastic scattering.

2 where 푚푒푐 = 511 keV is the rest energy of the electron. Inelastic scattering is relatively constant in the relevant energy range, 10 keV - 100 keV (see Fig. 2.2) as the attenuation mostly depends on density, 휇inelastic ∝ 휌.

Elastic scattering Elastic or coherent scattering occurs when no energy is lost at the photon redirection. It is coherent as the phase of the photon is not randomized by the interaction. This interaction is also called Rayleigh scattering or Thomson scattering depending on if it occurs with an entire atom or just an electron. The attenuation depends on energy and atomic number as 2 휇elastic ∝ 푍/퐸 . X-RAY MATTER INTERACTION | 15

2.1.2 Ionizing radiation and dose As previously mentioned, a central problem to X-ray imaging is the danger of ionizing radiation. Ionization produces radicals, molecules and ions with an unpaired valence electron. Radicals are highly reactive and can break bonds in contact with molecules, such as proteins and nucleic acids. This causes both deterministic and stochastic damage (see Tab. 2.1). The former is due to cell death and typically associated with high exposures. Stochastic damage is due to DNA damage. This can trigger cell mutation which can lead to cancer. To minimize damage to living organisms the dose of radiation exposure is measured and controlled in X-ray imaging.

Deterministic Stochastic Dependence on dose More severe More probable Time until effect Within hours or days Years Fundamental problem Cell death Cell mutation Symptoms/effects Hair loss, damage to blood, Cancer, genetic effects bone marrow and intestines, sterility

Table 2.1. Difference between deterministic and stochastic radiation damage.

Dose is commonly given in Gray [Gy] or Sievert [Sv] [81]. They share the same SI base unit, J/kg. Gy measures a physical quantity – the energy deposited per mass, also called absorbed dose 퐷. Sievert takes the harmful effect on biological tissue into consideration. By modifying thedose (퐷) with a weighting factor 푊푅 depending on the radiation causing the ionization an equivalent dose 퐻 is obtained. The weighting factor is 1 for photons and 20 for alpha particles. A second weighting factor depending on irradiated tissue (푊푇) is introduced to obtain the effective dose,

퐸 = ∑ 푊푇 ⋅ 푊⏟푅 ⋅ 퐷푇, (2.5) 푇 퐻푇 where the subscript 푇 signifies the quantity for a certain tissue. This isto account for the fact that some parts of the body are much less sensitive than others. Arms, legs, bone, and skin belong to the former while bone marrow, lymphatic tissue, and reproductive organs belong to the latter. Medical standard procedures in X-ray imaging range from 0.001 mSv to 20 mSv. To set this in context, one can compare it to LD50/60 for humans, meaning an X-ray dose resulting in 50% death within 60 day in a group of people. There is no exact number due to limited data and many contributing factors, but 4 Sv is an approximate figure [81, 82]. This in other words 200 times 20 mSv, the dose of a larger CT examination. Another illuminating figure 16 | X-RAY MATTER INTERACTION

is the background radiation all humans experience each day from natural sources. It is a few mSv per year, but varies [83]. This gives an indication that X-ray examinations done occasionally with a sub mSv dose should pose low risk. It is, however, still not known if a very small dose adds to a stochastic damage or if there is a threshold at low doses. The former assumption, the linear no-threshold model, is assumed, by some only as a precaution, but it has been heavily criticized [84].

2.2 Complex refractive index

To describe X-ray interaction with bulk material, the refractive index is more convenient than considering individual interactions. In optics the refractive index 푛 is a dimensionless number describing the speed of light 푣 inside a material compared with the vacuum speed 푐:

푛 = 푐/푣. (2.6)

This also determines the refraction of light between different media. A refractive index of 1 corresponds to vacuum, and many light transmitting materials, such as glass, are found in the range 1 to 2. To describe the X-ray properties of a material a complex refractive index is introduced [85]. As X-ray interaction is small compared to visible light 푛 is close to unity and therefore defined as 푛 = 1 − 훿 + 푖훽, (2.7) where 훿 is the decrement of the real part, and 훽 the absorption. For biological materials in the hard X-ray range, 훿 is around 10−6 − 10−8 and 훽 10−7 − 10−13. The decrement 훿 is given by 푟 휆2 푟 휆2 푍푁 훿 = 푒 휌 = 푒 A 휌, (2.8) 2휋 푒 2휋 푀

−15 where 푟푒 = 2.818 ⋅ 10 m is the classical electron radius, 휌푒 is the electron density, 푁A is the Avogadro’s number, 푀 the molar mass, and 휌 is the mass density.

Waves in matter To understand the use of 훿 and 훽, consider an electromagnetic plane wave propagating in the 푧 direction,

푖푘푧 푈(푧) = 푈0e , (2.9) 2휋 where 푘 = 휆 is the wave number. This expression is, however, only valid in vacuum. In a medium the complex refractive index is added, X-RAY MATTER INTERACTION | 17

∆α

−a U0 U0e

Figure 2.4. Phase shift (휙) and absorption (i.e., amplitude changed by e−푎) of wave passing through a medium.

푖푘푛푧 푖푘푧 −푖푘훿푧 −푘훽푧 푈(푧) = 푈0e = 푈0e e e . (2.10)

We note that the first part of the expanded expression is identical to Eq. (2.9). This corresponds to the original unperturbed wave. The following two exponentials containing 훿 and 훽 change the wave. The first exponential is complex and describes a phase shift. Compared to the original wave the phase is changed by −푘훿푧, which we call 휙 (see Fig. 2.4). In the more general case, 훿 is a function of space:

2휋 휙(푥, 푦) = − ∫ 훿(푥, 푦, 푧)d푧. (2.11) 휆

1 This corresponds to a refraction in the ray picture by an angle Δ훼 = 푘 ∇휙. The second exponential (third in Eq. 2.10) is real and will decrease the amplitude of the wave, i.e., account for absorption (see Fig. 2.4). The change in intensity is given by the square of the amplitude change (e−푘훽푧)2 = −2푘훽푧 4휋 e . The factor 2푘훽 = 휆 훽 is the attenuation coefficient:

4휋 휇 = 훽. (2.12) 휆

This is in other words the well known Beer-Lambert law,

−휇푧 퐼 = 퐼0e , (2.13) 18 | X-RAY MATTER INTERACTION

where 퐼0 is the original intensity and 퐼 is the attenuated intensity. We call the absorption exponential 푎, which in the general case becomes 1 푎(푥, 푦) = ∫ 휇(푥, 푦, 푧)d푧. (2.14) 2

The change that a medium causes to a wave can then be described as

e−푎(푥,푦)+푖휙(푥,푦). (2.15)

We will refer to this as the transmission function. SOURCES AND DETECTORS | 19

Chapter 3

Sources and detectors

This chapter covers the generation and detection of X-rays. This is a critical part of the imaging process, which sets important limits on imaging performance. A few different types of X-ray sources and X-ray detectors are introduced with the focus on advantages and limitations in the technology.

3.1 X-ray sources

X-rays can be generated with a wide range of techniques that have drastically different properties. Compact sources are covered in the firsttwo sections, and large accelerators and laser-based sources in the following two. A brief comment on the comparison of sources is given in the final section.

3.1.1 X-ray tubes Since 1895, the X-ray tube has been the workhorse of X-ray imaging [1]. The basic principle is to direct a beam of electrons onto a metal target which will emit X-rays. X-ray tubes can therefore also be called electron-impact sources. To give the electrons sufficient energy a high voltage is applied between the electron emitter (cathode) and the target (anode).

Generation of X-rays Two different physical effects are responsible for the generation ofX-rays in X-ray tubes: characteristic line emission and bremsstrahlung (see Fig. 3.1) [81]. In the former, core shell electrons are knocked out in the anode materials. The relaxation of the atoms causes fluorescence, i.e., X-ray photons corresponding to the shell transitions are emitted. These transitions are specific to the element and leads to discrete characteristic emission energies. Bremsstrahlung is German for braking radiation. When 20 | SOURCES AND DETECTORS

1012 2

1.5

1 Photons/s/sr/keV 0.5

0 0 10 20 30 40 50 60 70 Photon energy [keV]

Figure 3.1. Spectrum from an electron-impact source (liquid-metal-jet source, see Section 3.1.2) showing characteristic line emission around 10 keV and between 24 keV and 30 keV. The broad baseline is the bremsstrahlung. The acceleration voltage is 70 kV, the power 100 W and the emission spot diameter 10 µm.

electrons reach the anode material they are affected by the positive charge of the atom nuclei. This leads to braking or change in trajectory, i.e., acceleration, which in turn results in emission of photons over a broad spectrum. The probability of bremsstrahlung decreases linearly with photon energy and the highest photon energy occurs when one electron transfers all of its kinetic energy to one photon [86]. The kinetic energy of one electron is determined by the high voltage 푈 [V] in the tube. As the electron energy is given by charge times voltage (퐸 = 푒푈) the unit electronvolt (eV) has been defined, where the numerical value ofthe voltage is the same as the electron energy – an electron accelerated with 100 kV receives 100 keV energy.

The X-rays emitted from an X-ray tube consist of a spectrum of different energies, i.e., the radiation is polychromatic. The lower bound is in practice set by absorption. Additional filtration of the X-ray spectrum is doneby inserting a sheet of metal in the beam path. This will predominantly remove low-energy photons and thus increase the mean energy. Controlling the energy is important in X-ray imaging as the spectrum needs to match the object and detector to maximize interaction and detection while limiting absorption. SOURCES AND DETECTORS | 21

Cu W 15 Mo

10 Power [W] Power

0 0 5 10 15 20 25

Figure 3.2. E-beam power at anode melting as function of e-beam spot diameter for three common anode materials: Cu, W, and Mo. Approximate result from [88].

X-ray emission spot

In the X-ray tube the electrons are focused to a small spot on the anode to generate X-rays. As will be covered in the next chapter, the size of this spot is highly important for imaging as a smaller spot enables higher resolution and spatial coherence. Coulomb repulsion will work against this concentration, but in practice the limiting factor is the conversion efficiency from kinetic electron energy to X-ray energy, which approximately is given by the formula [87]

휂 = 푘푍푈, (3.1) where 푘 ≈ 9 ⋅ 10−10 V−1 is a constant and 푍 the atomic number of the anode material. Using a tungsten (푍 = 74) anode and an acceleration voltage of 120 kV, 휂 is less than 0.01. More than 99% of the energy reaching the anode will in other words become heat, and if the power load is too high, the anode melts. As small spot size and high power is beneficial for imaging a useful quantity is brightness [photons/s/mm2/mrad2], photons generated per second per mm2 spot size per mrad2 divergence angle. For spot sizes in the range 5 − 20 µm the maximum power is only a few watts (see Fig. 3.2). Different source technologies, such as rotating anodes and micro-focus tubes, have been developed to improve the brightness (see Fig. 3.3), but for all techniques there is a trade-off between power and spot size. 22 | SOURCES AND DETECTORS

e e e e

(a) (b) (c) (d)

Figure 3.3. X-ray Tube designs. (a) Static solid anode. (b) Rotating anode. (c) Liquid-metal-jet anode (jet direction perpendicular to the page). (d) Transmission anode. Note that electron impact creates X-ray radiation in all direction, but only the cone defined by the tube aperture is shown.

3.1.2 Liquid-metal-jet sources The limiting factor in modern X-ray tubes is the power load on the anode. The liquid-metal-jet is a source created to push that boundary. It was created by Hans Hertz and Oscar Hemberg in the Biomedical and X-ray physics group at KTH, but is now developed and sold commercially [89]. The basic principle is to replace the solid metal anode with a high-speed jet of liquid metal [90]. This improves the issue with power load in three ways: 1) The fast movement of the jet quickly removes heat. 2) A jet is regenerative, thus removing damage. The power load can thus be closer to vaporizing the jet than a solid anode can be to melting. 3) The jet offers a high stability compared to a rotating anode, which allows for a small electron beam focus.

Liquid-metal-jet sources are typically run with the alloy Galinstan as anode material. This is a eutectic mix of gallium (Ga), indium (In), and tin (Sn), meaning that its melting temperature, −19∘C, is lower than the melting points of all constituents. Galinstan has two main clusters of emission peaks, one with Ga K훼 at 9.2 keV and one with In K훼 at 24 keV (see Fig. 3.1 and Tab. 3.1). Other alloys with heavier elements could also be used to shift the spectrum to higher energies [91]. The e-beam spot can be tuned to a minimum size around 5 µm. With a 10 µm apparent spot the maximum source power is approximately 100 W.

3.1.3 Synchrotrons Synchrotrons are large-scale accelerators that circulate relativistic electrons to emit electromagnetic radiation [85]. The electrons are first accelerated to relativistic speeds and then injected in a storage ring (see Fig. 3.4) SOURCES AND DETECTORS | 23

Z Element K훼1 K훼2 K훽1 31 Ga 9.25 9.22 10.26 49 In 24.21 24.00 27.28 50 Sn 25.27 25.04 28.49

Table 3.1. Emissions lines [keV] of elements in Gallinstan. Values are rounded to two decimal places. Data from X-Ray Data Booklet [92]. where they are kept circulating due to powerful electromagnets. The ring diameter can be up to several hundreds of meters. The ring is actually a regular convex polygon, e.g., an icosagon (20 sides) (an octagon in Fig. 3.4) with bending magnets deflecting electrons to pass between straight sections. The acceleration that the electrons experience results in the emission of electromagnetic radiation. A very strong beam with low divergence is produced. The radiation covers a broad spectrum, i.e., it is polychromatic, but the abundance of light makes it possible to pick out desired energies with crystals to obtain monochromatic radiation. Along straight sections, insertion devices, wigglers and undulators, can be constructed to improve beam properties. A spatially alternating magnetic field forces the electrons to oscillate, which can produce both higher and lower divergence as well as change the spectrum. The high flux, low divergence, and monochromatic radiation make synchrotrons ideal sources to perform fast and demanding experiments. Critical drawbacks are, however, the high cost, large size, and limited access.

3.1.4 Laser-based sources As X-ray tubes have limited capabilities and synchrotrons have limited access, the development of new X-ray sources is still active research. Two notable principles are based on the interaction between electrons and laser light. This removes the limitation of a physical anode while limiting the size of the setup.

The first type uses inverse Compton scattering (ICS). A bunch of electrons are accelerated and fed into a laser cavity where high-power laser pulses move in the opposite direction to the electrons. When an electron bunch meets a laser pulse, X-rays are generated. This process can be described by ICS, but also as electrons oscillating in the electromagnetic field of the laser [93, 94]. ICS sources can produce tunable quasi-monochromatic radiation. The Munich Compact Light Source (muCLS) is one such example. With an energy range of 15–35 keV it is suitable for X-ray imaging of biological samples and small animals [95, 96]. Several ongoing 24 | SOURCES AND DETECTORS

Figure 3.4. Schematic drawing of a synchrotron storage ring. Straight sections form the ”ring” (here: octagon). Bending magnets (red) keep the electrons circulating. In the direction of the red arrows beam lines can be constructed to use the electromagnetic radiation generated by the electrons. In the straight sections insertion devices (black) can be constructed to improve the coherence of the beam lines. projects aim to extend the energy range of these sources to higher energies [97, 98]. Drawbacks of current ICS sources are limited stability, moderate flux, large size, and high cost.

The second type uses laser wakefield acceleration (LWFA). A Terawatt (TW) laser pulse is focused in a gas which generates a plasma. The in- tense laser pulse creates a wake in the plasma with a strong magnetic field which pulls in electrons in a process called self-injection. These electrons oscillate in the electromagnetic field, which generates X-rays, not unlike an undulator. This creates very short pulses of X-rays, down to tens of fem- toseconds (10−15 s) [99]. This has been shown with phase-contrast imaging on biological samples [100] and tomography [101, 102]. It is, however, unclear today if this technique will reach any broader use as the stability, pulse repetition rate, and average flux is low. The field of view is also small, around 1 × 1 cm2 [103]. The lasers needed are furthermore large (>100 m2) and cost several millions of USD, which is many orders of magnitude more costly than an X-ray tube.

3.1.5 Comparison of sources Comparing different source technologies is not simple. Apart from ”hard” parameters such as divergence, spot size and flux, other factors like SOURCES AND DETECTORS | 25

Source LMJ CLS LWFA Flux [ph/s] 1.4 ⋅ 1011 4.3 ⋅ 1010 1.1 ⋅ 109 Divergence [mrad] 183 4 10 Emission spot diameter [µm] 8 100 ∼ 1 Brightness [ph/s/mrad2/mm2] 7 ⋅ 1010 3.4 ⋅ 1011 3.4 ⋅ 1013

Table 3.2. Brightness and key parameters from three source types, LMJ [104], CLS [96] and LWFA [102]. The values are given for the energy 15 − 35 keV for LMJ, 35 keV (5% bandwidth) for CLS, and a broad spectrum (>5 keV) with critical energy at 25 keV for LWFA. stability and reliability are essential for operation. Source footprint (i.e., physical size) and cost are also necessary to consider. To make things more difficult the application and imaging method has a significant impact on what is a suitable source. The brightness measure introduced above is therefore only a first indication of performance, and potentially misleading if different source technologies are compared without context. Recent LWFA experiments have, e.g., reported brightness values surpassing LMJ and CLS sources, but the flux is much lower, which limits actual use (see Table 3.2).

In summary, synchrotrons can provide beams superior in most aspects, with a few exceptions, such as the short pulses achieved with LFWA. LMJ, CLS, and LWFA are three technologies developed to provide more compact al- ternatives. LMJ is the most mature technology, with high stability and reliability as well as relatively low cost and small footprint. CLS and especially LFWA are more unstable and have larger footprints and costs. They have more unique characteristics compared with LMJ that could be useful in certain applications, but if a broader adoption is possible remains today unclear.

3.2 X-ray detectors

Detecting X-rays in imaging has traditionally been done with fluorescent screens and film. Today it is almost exclusively done using solid state detectors with pixel arrays. There are two main principles used in these detectors: indirect and direct detection (see Fig. 3.5).

Indirect detection Indirect detection is when X-ray photons are converted to visible light photons before detection. For a long time, the conversion and detection was 26 | SOURCES AND DETECTORS

Scintillator Photoconductor Optical transfer layer Photodiodes Electrical circuit Electrical circuit

(a) (b)

Figure 3.5. Schematic of the main components in indirect and direct detection. (a) Indirect detection with scintillator (visible photons in red). The optical transfer layer can be a FOP or a lens system (see Fig. 3.6). The photodiodes and electronics (electrons shown as black dots) can be a CCD, a CMOS sensor or a TFT array. (b) Direct detection. The photoconductor can, e.g., be CdTe. The underlying readout electronics can be the same as on a CMOS sensor. done with fluorescent screens and film respectively, but today so-called scintillators and conventional CCD or CMOS sensors are used. A scintillator is a material exhibiting scintillation, a form of luminescence where ionizing radiation, e.g., an X-ray photon, is converted to visible photons. Two common scintillator materials are Tl doped CsI and Tb doped Gadox (Ga2O2S). The visible photon yield is tens of photons per keV [105]. To protect the sensor and to guide the light a fiber optics plate (FOP) or an optical system can be placed between the scintillator and the sensor (see Fig. 3.6). The sensors are commonly 2048×2048 or 4096×4096 with pixel sizes from a few micrometers to 200 µm. The choice of CCD or CMOS sensor depends on the application. In the past CCDs have had lower noise and smaller pixel- to-pixel variations due to a single high-quality analog-to-digital converter (ADC) on the chip. CMOS sensors have on the other hand been faster (read-out speed) and more inexpensive. With recent development of high- quality scientific CMOS sensors (sCMOS) the difference is not as clear-cut, and CMOS sensors are taking over much of the high-end market.

Direct detection Direct detection is when X-ray photons immediately are converted to electron-hole pairs in the sensor matrix. The obvious advantage compared to indirect detection is the removal of visible light as an intermediary and the associated image-quality degradation. A challenge is the high demands on the sensor material: it must both stop the X-rays and work as a semiconductor. Only a few materials have achieved some success: Si, CdTe, CdZnTe (CZT), amorphous Se (a-Se), and GaAs. Fabrication of some of these materials remains a challenge. Most sensors are also built of smaller modules with gaps in between, which leads to loss of information. SOURCES AND DETECTORS | 27

CCD

Lens FOP

X-rays CCD X-rays

mirror

scintillator scintillator light excluding foil light excluding foil

(a) (b)

Figure 3.6. Basic components in a scintillator detector. (a) detector with FOP, (b) detector with optical system. Visible light is shown in red.

Detector characteristics An ideal detector would register the exact location and energy of each incoming photon. Real detectors are unfortunately far from this ideal. Charge summing is currently the most common detection mechanism. The photons hitting a pixel in the sensor are converted to counts, which roughly corresponds to the total energy of the photons. Photon counting, where one photon corresponds to one count, is also an emerging option.

A key characteristic of a detector is its resolution, which is closely coupled to the pixel size. Apart from some special detection schemes, the pixel size sets an upper bound for the sensor resolution. Further degradation, often exceeding the pixel size, is caused by scattering and interaction in the scintillator, FOP and sensor. This degradation is generally described using the point spread function (PSF). The PSF shows how blurred a point source would be if imaged with the detector (see Fig. 3.7). It sets an upper bound for detector resolution. It is not uncommon for indirect detectors to have a PSF significantly larger than the pixel size, while it is more similar for direct detection. As the PSF in most cases can be assumed to be fairly symmetric, its impact is summed up by the Full width at half maximum (FWHM). For a normal distribution with standard deviation 휎 it is

FWHM = 2√2 ln(2) 휎. (3.2) A second key characteristic is the efficiency of the detector. The efficiency is 28 | SOURCES AND DETECTORS

PSF(x) FWHM

Figure 3.7. Schematic of how spread of visible light (red) in the scintillator creates blurring which is described by the PSF. The FWHM of a one-dimensional PSF is shown. essentially how much of the incoming light is contributing to counts in the image. First the detector needs stopping power – the scintillator, or sensor in direct detection, must absorb X-ray photons for them to be detected. This depends on the material and its thickness (see Fig. 3.8). Second, the photons absorbed in the sensor must be converted to digital counts with as little loss as possible. Thermal (dark) noise and electronic noise degrade this conversion. Thermal noise can be reduced with sensor cooling and both thermal and electronic noise eliminated with photon counting detectors. All images will, however, contain Poisson noise, also called shot noise, since this is an inherent property of photons. Considering photon noise only, the signal-to-noise√ ratio (SNR) for 푁 photons, which have the standard deviation 푁, becomes 푁 √ SNR = √ = 푁. (3.3) 푁

Increasing the number of photons to improve SNR is in other words quite√ unfavorable. Double the photons and the SNR increases by 41% ( 2 ≈ 1.41).

The main challenge in detector technology for high-resolution X-ray imaging is the trade-off between resolution and efficiency. A thick scintillator will absorb X-ray photons well, but the thickness will also decrease the resolution and vice versa. This becomes increasingly difficult at high photon energies. Direct detectors do not have such a clear dependence on thickness, which makes them more suitable for certain high-energy applications. Compton scattering and fluorescence in the sensor material will, however, add a significant PSF depending on the sensor material. SOURCES AND DETECTORS | 29

1 1 mm Si 1 mm Gadox 1 mm a-Se 0.8 1 mm CsI 1 mm CdTe

0.6

Attenuation 0.4

0.2

0 20 40 60 80 100 120 140 Energy [keV]

Figure 3.8. Attenuation of common materials used as scintillators and for direct detection sensors. The fraction of radiation attenuated as a function of energy is given for five materials: Si, Gadox, a-Se, CsI, and CdTe. Note that the PSFvaries drastically for these materials if the same thickness is used.

In time-dependent applications the detector speed can be critical. For sensors with a large number of pixels the sensor readout speed is the limiting factor. CMOS sensors outperform CCDs in this area. Another factor is whether a global or rolling shutter is used. With a global shutter all pixels are recording at the same time. A rolling shutter will record sequentially over the array. This can cause artifacts if the object or illumination changes.

X-RAY IMAGE FORMATION | 31

Chapter 4

X-ray image formation

This chapter contains the main description of X-ray image formation. The first three sections describe different techniques for X-ray imaging, conventional attenuation-based imaging, an overview of phase-contrast X-ray imaging techniques not used in this work, and propagation-based imaging in some more detail. The last section describes tomography – 3D imaging, which applies to all techniques.

4.1 Attenuation-based imaging

Attenuation-based, or conventional, X-ray imaging is based only on the contrast mechanism of differential attenuation by absorption and scattering. This is succinctly described by the Beer-Lambert law,

−휇푇 퐼 = 퐼0e , (4.1) where the intensity in the recorded image depends on the thickness of the object (푇) and how strongly it attenuates the radiation (휇). The negligible

detector image

Figure 4.1. Attenuation-based imaging. The attenuation in the object creates a shadow image on the detector. 32 | X-RAY IMAGE FORMATION

source plane object plane image plane

(M − 1)s

s a Ma

R1 R2

Figure 4.2. Magnification of object and source spot is given by congruent triangles in the schematic drawing. The penumbral blurring shown at the image edges is present over the entire image. refraction means that only a shadow image is formed behind the object (see Fig. 4.1). If diverging radiation (fan or cone beam) is used a magnification 푀 can also be introduced as 푅 + 푅 푀 = 1 2 , (4.2) 푅1 where 푅1 is the source-to-object distance and 푅2 the object-to-detector distance (see Fig. 4.2). As the image is magnified in the detector plane, the relative pixel size is demagnified by the same factor. It is therefore convenient to introduce an effective pixel size 푝eff = 푝/푀 given a physical pixel size 푝. An equivalent argument can be made for the detector PSF. These effective values are given for the object plane (see Fig. 4.2). The blurring and reduction of resolution due to detector pixels and PSF can thus be minimized by increasing the magnification. This will, however, also affect the blurring from the source spot. The source magnification inthe detector plane is 푅2/푅1 = 푀 − 1 (see Fig. 4.2). In the object plane the magnification becomes (푀 − 1)/푀. A small 푅2 (compared to 푅1), which is common in attenuation-based imaging, will then give a demagnification1 of the source spot. A large spot can then be used without reducing the system PSF. This is, unfortunately, in direct conflict with using a large 푀 to compensate for detector pixels and PSF. Reducing spot size is difficult due to the power load constraints mentioned in Chapter 3. Reducing pixel

1Technically there is always a demagnification, but it is negligible for large 푀. X-RAY IMAGE FORMATION | 33

104

103

Adipose tissue

2 Soft tissueAluminium / 10 Cortical bone

Iron 101

100 100 101 102 103 Photon energy [keV]

Figure 4.3. Ratio between 훿 and 훽 for two elements (Al, Fe) and three biological tissue types. size and PSF is technically possible, but results in low SNR as the flux per pixel drops. The image intensity changes as 1 퐼 (푀푥, 푀푦) = 퐼 (푥, 푦), (4.3) d 푀 2 o where 퐼d is the detector intensity and 퐼o the intensity in the object plane.

4.2 Phase-contrast imaging

Measuring phase shift in addition to attenuation provides more information about the sample. The phase-contrast signal is also much stronger in many cases. To show this in an intuitive way, the ratio 훿/훽 is often given for different energies and materials (see Fig. 4.3). For soft tissue, the ratio is more than 1000 for energies relevant in pre-clinical and clinical imaging. Improvements in SNR of two orders of magnitude have been reported and assuming Poisson statistics, this implies a dose improvement of four orders of magnitude [106, 107]. It should, however, be noted that phase contrast ultimately depends on differences in electron density, which is proportional to the mass density. If two features have the same electron density, the phase shift is also the same, which results in no contrast. 34 | X-RAY IMAGE FORMATION

4.2.1 Mathematical description Below follows a short and approximate mathematical description of image formation in phase-contrast X-ray imaging (PCI). Several steps will be omitted for brevity. We begin from Maxwell’s equations, the fundamental equations for classical electromagnetism and optics. First we neglect po- larization, anisotropy, and assume time harmonic fields, i.e., 퐸(푥, 푦, 푧, 푡) = 퐸(푥, 푦, 푧)e−푖휔푡. This gives us the inhomogeneous Helmholtz equation, (∇2 + 푛(푟)2푘2) 퐴(푟) = 0, (4.4) where 푛(푟) is a function of space, hence inhomogenous. The differential 2 휕2 휕2 휕2 operator, ∇⟂ = 휕푥2 + 휕푦2 + 휕푧2 , is the Laplacian. The function 퐴(푟) is a complex scalar wave field. Assuming a plane wave 퐴(푟) = 푢(푟)e푖푘푧 and a paraxial field, that transverse changes are small compared to changes inthe 휕2푢 휕푢 propagation direction (| 휕푧2 | ≪ 푘| 휕푧 |), the equation can be simplified: 휕 (∇2 + 2푖푘 + 푘2(푛2(푟) − 1)) 푢(푟) = 0. (4.5) ⟂ 휕푧 This can be called the inhomogeneous paraxial Helmholtz equation, which 2 휕2 휕2 contains ∇⟂ = 휕푥2 + 휕푦2 , the transverse part of the Laplacian. In free space 푛 = 1 and we obtain an even simpler form, 휕 (∇2 + 2푖푘 ) 푢(푟) = 0, (4.6) ⟂ 휕푧 the homogenous paraxial equation [108].√ If we assume that our wave of X- rays can be written on the form 푢(푟) = 퐼e푖휙, where 퐼 is the intensity and 휙 the phase, we get the Transport-of-intensity equation (TIE) [109]: 휕 1 퐼 = − ∇ ⋅ (퐼∇ 휙). (4.7) 휕푧 푘 ⟂ ⟂

What does this say in simple terms? If we have a wave front, 퐼∇⟂휙 describes its intensity 퐼 and direction ∇⟂휙 (the gradient is perpendicular to the wavefront). The operator ∇⟂⋅ is the divergence of this wavefront. If the wavefront is diverging, the right side is negative, if it is converging, it is positive. The left side gives the change in intensity due to a change in propagation distance. In other words, the TIE says something quite trivial: if the illumination diverges the intensity goes down, and if it converges it goes up. Let us now look at an approximate expression for the intensity of the image. We assume the partial derivative can be approximated with the intensity at two 푧-positions. For convenience we set the first one at 푧 = 0. The TIE then becomes 퐼(푥, 푦, 푧) − 퐼(푥, 푦, 푧 = 0) 1 = − ∇ ⋅ (퐼∇ 휙). (4.8) 푧 푘 ⟂ ⟂ X-RAY IMAGE FORMATION | 35

(a) (b) (c)

Figure 4.4. Derivatives. (a) Projected thickness of a cylinder. (b) 1st order derivative of projected thickness. (c) 2nd order derivative of projected thickness.

Reordering the terms yields 푧 퐼(푥, 푦, 푧) = 퐼(푥, 푦, 푧 = 0) − ∇ ⋅ (퐼∇ 휙). (4.9) 푘 ⟂ ⟂ Finally, the divergence expression is expanded: 푧 퐼(푥, 푦, 푧) = 퐼(푥, 푦, 푧 = 0) − [∇ 퐼 ⋅ ∇ 휙 + 퐼∇2 휙)] . (4.10) 푘 ⟂ ⟂ ⟂ The right side has two terms. The intensity at 푧 = 0, unaware of any phase, and the phase term. We can make a few more observations. The phase term is proportional to 푧 and therefore zero at 푧 = 0. Propagation is in other words necessary for this type of phase imaging. A second observation is that we have two separate terms containing 휙, a first and second derivative (see Fig. 4.4). For most samples it is a good approximation that ∇⟂퐼 ≈ 0, i.e., that there are no strong transverse variations in intensity. If 퐼(푥, 푦, 푧 = 0) is the intensity from the Beer-Lambert law, the intensity can be given as

푧 퐼(푥, 푦, 푧) ≈ 퐼(푥, 푦, 푧 = 0) (1 + ∇2 휙(푥, 푦)) 푘 ⟂ 푟 푧 (4.11) = 퐼(푥, 푦, 푧 = 0) (1 + 2휋 푒 ∇2 ∫ 휌 (푥, 푦, 푧)d푧) , 푘2 ⟂ 푒 where 휌푒 is the electron density. This is the intensity for PBI. The ”1” corresponds to the conventional attenuation image and the second term is the intensity change due to the phase shift. Slightly simplified, the transverse dependent part of the second term contains the Laplacian of the electron density. The image is hence a combination of the intensity image and the second order derivative of the projected electron density (see Fig. 4.4). In this approximation we can see that the energy only exists as a factor before the Laplacian of the electron density and polychromatic radiation can thus be used [71]. An alternative to simple propagation is to introduce a 36 | X-RAY IMAGE FORMATION

strong intensity variation, e.g., with a grating, to measure the gradient of the phase, ∇⟂휙. This is done in most other phase-contrast methods.

4.2.2 Phase-contrast imaging methods

There are several different methods for measuring the phase in PCI. This work focuses solely on propagation-based imaging, covered in the following section. A thorough description and comparison of other methods is out of scope, but below five notable examples are briefly explained with some advantages and drawbacks. For more extensive comparisons, see [110–113].

Crystal interferometer

The crystal interferometer is a classic interferometric approach where a beam passing through the object is combined with a reference beam from the same source to form an interference pattern (see Fig. 4.5a). Three Si crystals are used to split, redirect and recombine a parallel beam. The crystal interferometer was developed by Bonse and Hart in 1965 [68]. Later, tomography of biological samples has been demonstrated [114]. The phase (휙) is directly measured with high sensitivity. With intensity measured as 퐼 = |1 + e푖휙|2 = 2(1 + cos 휙) for a pure phase object, a phase-wrapping problem appears, as modulo 2휋 of the phase is measured. Another, more critical drawback is the need for very high intensity due to i) absorption in the monochromator and crystals and ii) the beam split in the second and third crystal (which is not shown in 4.5a). A parallel beam is also necessary. Crystal interferometry is thus practically limited to synchrotrons. The monolithic Si crystals required limit also the object size.

Analyzer-based imaging

In analyzer-based imaging (ABI), a parallel monochromatic X-ray beam goes through the object and is reflected off a crystal onto a detector (see Fig. 4.5b). Introduced in 1979 as X-ray Schlieren [115, 116], it was further developed and applied to biological samples in the 1990s [117–119]. The refracted X-rays will only be reflected by the crystal if the Bragg condition is fulfilled. By slightly shifting the angle of the crystal, an intensity curve can be recorded from which the first derivative of the phase in one direction (∇휙푥) can be retrieved. Critical drawbacks are the need for high stability and parallel, monochromatic radiation. Like the crystal interferometer, ABI is thus practically limited to synchrotrons. X-RAY IMAGE FORMATION | 37

Detector Sample

Sample

Detector

(a) (b)

Detector Detector Sample Sample

Detector

Sample

(e)

Figure 4.5. Schematic drawings of phase-contrast techniques. (a) Crystal interferometer. (b) Analyzer-based imaging. (c) Grating-based imaging. (d) Speckle- based imaging. (e) Edge-illumination. Note that both (a) and (b) require optics to achieve a monochromatic, parallel beam prior to the setup (not shown in the figure). 38 | X-RAY IMAGE FORMATION

Grating-based imaging Grating-based imaging (GBI) uses a number of gratings to record the phase (see Fig. 4.5c). The main grating splits the beam behind the object. In- terference in the beam creates an intensity pattern known as a Talbot self- image [120]. This pattern is then analyzed with a high-resolution detector or a second grating. GBI was pioneered at synchrotrons in the early 2000s [110, 121, 122], but moderate requirement on temporal and spatial coherence makes compact micro-focus sources an option [122]. To enable use with conventional laboratory sources, a third grating can be introduced between the source and the sample to improve the spatial coherence of the source. This so-called Talbot-Lau setup will, however, not affect the resolution which remains low due to the large source spot [123]. GBI can measure three separate signals: absorption, the first derivative of the phase in one direction (∇휙푥), and scattering [124]. With two-dimensional gratings it is possible to record both phase directions (∇휙) [125, 126]. The scattering signal is referred to as dark-field and can provide a complementary contrast. GBI can yield quantitative information, even on laboratory sources [127,128]. It is extensively used on laboratory sources and recently in clinical trials [129].

Speckle-based imaging In speckle-based imaging (SBI) a diffuser, e.g., a sandpaper, is inserted before the object (see Fig. 4.5d). This creates a speckle pattern on the detector. The shift of the speckle pattern over a small area, a window, caused by the object is compared to a reference image, which gives the first derivative of the phase (∇휙) [130]. Resolution depends on speckle and window size. The same three signals as for GBI can be obtained [131]. Stepping, i.e., multiple images acquired at different transverse positions, can be used to improve resolution, but this requires small and precise movements. The advantage of SBI is that it can be used on compact micro-focus sources and that it requires no grating fabrication nor optics between object and detector.

Edge-illumination In edge-illumination (EI), a grid creates separated beamlets which are refracted in the object (see Fig. 4.5e). The pixel detector is partly covered by a second grid. By measuring the intensity of each pixel, the deviation of beamlets, and thus phase shift, can be calculated [132]. EI measures ∇휙푥 and can also produce a dark-field signal [133]. Edge-illumination is also compatible with incoherent lab sources [134, 135]. Schemes to record X-RAY IMAGE FORMATION | 39

detector image

Figure 4.6. PBI. An object with higher 훿 than its surrounding material produces a bright fridge on the outer edge and a dark fridge inside. two-dimensional differential phase is possible [136]. A single shot method with one 2D grid and detector with subpixel resolution has also been proposed [137, 138].

4.3 Propagation-based imaging

Propagation-based imaging uses only the coherence of the radiation and a propagation distance between the sample and the detector. Unlike all other phase-contrast methods, no optics, i.e., crystals, grids, etc. are used. This is a considerable advantage as optics can bring vibration problems, difficult fabrication, limited acceptance angle, and dose increase due to absorption between object and detector. There is also no need for stepping and the phase is recorded in both transverse directions. PBI is finally compatible with laboratory sources [71]. The main drawbacks are the need for a source with spatial coherence, a high-resolution detector, and that phase and absorption signals are mixed in the image.

4.3.1 Basic phenomenon Fundamentally, PBI can be understood as radiation being phase shifted in the object interfering with the unperturbed wave. If the distance to interfere is sufficiently long, constructive and destructive interference occurs. This appears as bright and dark fringes in the images (see Fig. 4.6). A good mathematical description of this phenomenon is Fresnel diffraction. This is a description of diffraction in the near field, unlike Fraunhofer diffraction (colloquially referred to as ”diffraction”), which is in the far field. The regime is determined by the Fresnel number,

푑2 퐹 = , (4.12) 푧휆 40 | X-RAY IMAGE FORMATION

where 푑 could be taken as the smallest feature in the object and 푧 is understood as the propagation distance, from object to image plane. We will see in the following section how this factor changes depending on the setup. The Fraunhofer regime (far field) is when 퐹 ≪ 1 and Fresnel regime (near field) when 퐹 ≫ 1. For a typical wavelength, 0.62 Å (20 keV), 푑 = 10 µm and 푧 = 30 cm 퐹 becomes 5.38 and the Fresnel description is applicable. The Fresnel diffraction is given by the Fresnel diffraction integral [139]:

푖푘푧 e 푖푘 [(푥−푥 )2+(푦−푦 )2] 푈(푥, 푦, 푧) = ∬ 푈(푥 , 푦 , 0)e 2푧 0 0 d푥 d푦 , (4.13) 푖휆푧 0 0 0 0 where 푥0, 푦0 are coordinates in the plane 푧 = 0 and 푥, 푦 in a plane the distance 푧 away. This is also a solution to Eq. (4.6) [140]. In Chapter 6, a more convenient form of the integral for computation is presented. The transverse size of the first Fresnel fringe, i.e., the first bright intensity maximum and first dark intensity minimum, is given by [141] √ 휆푧. (4.14)

This is a useful number as it provides a simple measure of the effect which then can be placed in relation to other parameters such as coherence and sampling.

4.3.2 Parameters Although simple at a first glance, propagation-based imaging depends on several often conflicting factors. In Section 4.1 the choice of magnification was mentioned. In PBI the propagation distance is added. For parallel radiation the propagation distance 푧 is the physical distance 푅2. With a cone beam, however, 푧 depends also on magnification. To make a distinction 푧 is called the effective propagation distance2 for cone beams,

푅2 푅1 ⋅ 푅2 푧eff = = . (4.15) 푀 푅1 + 푅2

The distance 푧 must in other words be substituted for 푧eff in equations, e.g., Eq. (4.11). Formally one can show that the intensity distribution from the Fresnel diffraction from a point source (spherical wave) is equal tothe intensity distribution in the plane wave case at a different distance. This is called the Fresnel scaling theorem [108]: 1 푥 푦 푧 퐼 (푥, 푦, 푧) = 퐼 ( , , ), (4.16) sphere 푀 2 plane 푀 푀 푀 2 The effective propagation distance is here denoted 푧eff, but 푑 and Δ is also common notations. X-RAY IMAGE FORMATION | 41

2 M z eff

1.75 0.9 2

1.5 1.5 0.8

1.25 0.7 2 [m]

1 1 3

R 0.6

1.5 0.75 0.5 3 2 0.4 0.5

0.3 3 6 0.25 0.2 6 1.5 2 0.1 0 6 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 [m] R2

Figure 4.7. How effective propagation distance 푧eff relates to 푅1, 푅2, and 푀. Note the rapid decrease in 푧eff for higher magnifications.

where 푧 is 푅2 in this case. The effective propagation distance leads to problems as a high 푀 might be necessary to resolve the edge enhancement, but this will reduce 푧eff and with it the edge enhancement (see Fig. 4.7). One strategy is to use an inverse geometry, i.e., where 푅1 ≫ 푅2, like in attenuation-based imaging to limit spot impact. This puts, however, higher requirements on the detector which is hard to combine with high energy and dose limited imaging.

Which is the optimal propagation distance? One guide is the contrast- transfer function (CTF) [142, 143]. By assuming a weak object, i.e., that e푖휙−푎 ≈ 1 + 푖휙 − 푎 the ”image” intensity in Fourier space is [142]

퐼(푢, 푣) ≈ 훿푘(푢, 푣) + 2Φ(푢, 푣) sin 휒 − 2퐴(푢, 푣) cos 휒, (4.17) where 훿푘 is a Dirac delta (not to be confused with the decrement of the 42 | X-RAY IMAGE FORMATION

1 Amplitude Phase

0.5

0 CTF

-0.5

-1

0 0.5 1 1.5 2 2.5 3 3.5 4

Figure 4.8. Contrast transfer function (CTF). The CTF is plotted against the √ √ dimensionless coordinate 휒 = 휋휆푧푢2 and 푢 is the spatial frequency in one dimension. refractive index), which corresponds to the incoming X-ray radiation. A dimensionless coordinate is introduced, 휒 = 휋휆푧(푢2 + 푣2), and Φ and 퐴 are the Fourier transforms of 휙 and 푎 respectively. The functions sin 휒 and cos 휒 are then defined as the CTF for phase and amplitude respectively. These two functions are shown in Fig. 4.8. The amplitude CTF is optimal at zero propagation distance, while the phase CTF is optimal at 휒 = 휋/2, which results in the condition 휋/2 = 휋휆푧(푢2 + 푣2), or in one dimension, 1 푧 = . (4.18) 2휆푢2 The optimal propagation distance grows linearly with energy and quadrat- ically with feature size. In practice this means that PBI has an upper limit in energy and feature size as distance grows too large to obtain significant contrast. Where that limit is depends on many factors such as source brightness and detector. As mentioned above, spatial coherence is necessary for 휆푅1 PBI. This is given as the transverse coherence length ℓ푠 ∝ 푠 , where 푠 is the source size. For optimal contrast [142] √ 휆푧 휆푅 √ < 1 . (4.19) 2 푠 If the spot size is 8 µm, the energy 9.2 keV, and propagation distance 0.2 m, the condition would be fulfilled at 푅1 = 0.27 m. The distance increases with energy by the square root so at 24 keV 푅1 should optimally be more than 0.44 m. The distance increases linearly with spot size, so if also the spot is X-RAY IMAGE FORMATION | 43

changed, e.g., to 15 µm, the new 푅1 should be 0.51 m. For higher energies and larger spots Eq. (4.19) is difficult to fulfill with a compact source, but phase contrast can still improve images, even if contrast is not optimal. One should finally note that no general optimum exists [144].

4.4 Tomography

Acquiring a radiogram, a single X-ray image of an object, means that the three-dimensional (3D) structure of the object is mapped to two dimensions (2D) in the image. This mapping is called a projection and the image is sometimes referred to as a projection image. A point in the image is, to a first approximation, determined by the absorption along a straight line going from the source, through the object, and to the image. Radiograms or projection images are useful, but they have a flaw: structures overlap in a projection. This limits the possibility to see features, particularly small and faint ones. To solve this problem tomography was developed.

Tomography, from the Greek 휏 ́휊휇휊휍, tomos, ”slice” or ”section” and 훾휌 ́훼휑휔, grapho, ”to write” is a technique to image an arbitrary section in an object. This avoids overlapping of structures and if multiple sections are recorded, sequentially or simultaneously, a full 3D image of an object can be obtained. The images produced with tomography are known as tomograms. In practice, tomography is accomplished by acquiring many projection images of an object from different rotation angles perpendicular to the normal ofthe sections to be imaged. It is referred to as CT – Computed tomography or CAT – Computerized axial tomography.

4.4.1 Fourier slice theorem The Radon transform published in 1917 is a fundamental part of the mathematical description of tomography [146, 147]. It calculates the projection images of an object from a number of rotation angles. The Radon transform of a single section is an image where one dimension is the spatial coordinate that lies in the plan of the section and is perpendicular to the direction of the radiation. The other coordinate is the rotation angle. The image is called a sinogram as features in the object trance sinusoidal curves (see Fig. 4.9). The main problem in tomography is how to reconstruct the section from the experimentally acquired sinogram.

Tomographic reconstruction can be understood by considering the Fourier slice theorem (see Fig. 4.10). It states that the Fourier transform of a projection of a section and a radial line in the Fourier transform of the same 44 | X-RAY IMAGE FORMATION

θ1

θ2 x

θ θ1 θ2

Figure 4.9. A detector row (푥-coordinate), collects a projection of a section and for multiple angles they form a sinogram. The Radon transform is the mathematical description of going from the section on the left to the sinogram on the right. section are equal. In other words, if projection images from different angles are acquired of an object, each image contributes to a radial line in Fourier space, and when the image formed by the radial lines is converted back to real space, it corresponds to a section through the object. This means that an inverse radon transform can be constructed which calculates the original object from the projection values. This is called filtered back projection (FBP) [148]. A filter step is necessary to include as the data is sampled more densely for low spatial frequencies in Fourier space (see Fig. 4.11). Without it, the dense sampling leads to blurring. Performing the reconstruction in the case above is computationally simple for an idealized situation with uniform parallel monochromatic illumination. To reconstruct objects acquired in non-idealized conditions, which is almost always the case in laboratory imaging, different reconstruction methods have been developed.

4.4.2 Reconstruction There are two main branches of methods for tomographic reconstruction: deterministic and iterative. The former calculates the object in one step while the latter goes back and forth between the projection images and the tomogram to reduce the error in the reconstruction (see Fig. 4.12). Iterative methods are nowadays common in medical CT as they better can reduce X-RAY IMAGE FORMATION | 45

Fourier Transform

Intensity y v

y0 x0

θ x θ u

Inverse Fourier Transform

Figure 4.10. Fourier slice theorem. A parallel beam of X-rays passes through a section in a cylindrical object and registers as an intensity 1D image on the detector. After Fourier transformation this data corresponds to a radial line in Fourier space. The section of the original object is obtained by calculating the inverse Fourier transform of the image in Fourier space. Adapted from [145].

(a) (b)

Figure 4.11. Filtering in the reconstruction process. (a) Equally sized polar regions in Fourier space, which should be sampled to reconstruct a slice. (b) Actual data has no radial dependence. This leads to a gradual oversampling towards the center, i.e., for low frequencies. Without compensation this will lead to blurry images. 46 | X-RAY IMAGE FORMATION

Object Projections Forward projection

Back projection

Figure 4.12. A forward projector creates projection images from an object and the back projector vice versa. Acquiring images is in other words a forward projector. Deterministic reconstruction methods use a single back projector. Iterative methods can use both forward and back projectors. noise and artifacts [149]. The demanding computations typically necessary lead, however, to long reconstruction times in high-resolution X-ray imaging as data size is considerably larger. The significantly faster deterministic methods have therefore been historically preferred. The foremost deterministic method is the FBP method. With parallel radiation all sinograms (sections) are independent which makes it computationally simple and fast. For cone beam illumination the problem becomes harder as the radiation can pass several axial slices which means that sinograms are not independent. Here the most common method is FDK cone-beam reconstruction, which is an acronym of the creators of the method, Feldkamp, Davis, and Kress [150]. This method can produce an accurate reconstruction, but artifacts limit the result in practice. Common artifacts are beam hardening, the change in X-ray spectrum as the radiation pass through the object, photon starvation, streaks that appear due to very high absorption for certain angles, and ring artifacts, concentric rings around the rotation center due to detector calibration errors or faulty pixels [151]. The most fundamental artifact is perhaps undersampling due to few projection images.

4.4.3 Angular sampling A critical factor in tomography is the number of projections recorded, i.e., the angular sampling. If all other conditions are ideal this will limit the resolution. To estimate the impact of angular sampling, a simple sampling criterion can be derived by assuming uniform sampling in the Fourier domain. If the sampling in the section is Δ푥, the field of view (FOV) 푥width, the corresponding sampling in Fourier space is Δ푢 and the maximum frequency 푢 , then the following relations are true: Δ푢 = 1 and max 푥width 1 푢max = 2Δ푥 (see Fig. 4.13). This follows the Nyquist-Shannon sampling theorem [152, 153]. The length of the red line in Fig. 4.13c is 푢maxΔ휃 and the blue Δ푢. If they are equal the reconstructed section is not limited by angular sampling: X-RAY IMAGE FORMATION | 47

real space y freq. space v v

x u u

∆x xwidth ∆u umax

(a) (b) (c)

Figure 4.13. Angular sampling. (a) Real space. (b) Fourier space. (c) Angular sampling in Fourier space. One criterion is to choose the number of angles such that the red line has the same length as the blue, i.e., the angular and radial sampling are the same.

2Δ푥 푢maxΔ휃 = Δ푢 ⇒ Δ휃 = . (4.20) 푥width As the line from a projection image covers frequencies from positive max to negative min, only a half circle needs to be sampled. Over 180∘ the number of angles 푁 then becomes 휋 푥 푁 = = 휋 width . (4.21) Δ휃 2Δ푥

An imaged object can have the maximum size 퐷 = 푥width and the smallest feature size is 푑 = 2Δ푥 given Nyquist-Shannon sampling. The criterion can then be expressed as 퐷 푁 = 휋 . (4.22) 푑 This is the Crowther criterion [154]. From a dose perspective, however, it is important to note that the SNR of the reconstructed slice does depend on the total dose, but not on the number of projections when the sampling criterion is fulfilled [155]. Another important point is that high image quality can be achieved with a lower number of angles [156]. This is partly because Crowther is based on Nyquist sampling which is an upper bound for necessary sampling. It also scales with 퐷, i.e., sampling varies with distance to rotation center. Slices will therefore be oversampled in the center and undersampled (angular sampling limited) towards the edge.

PHASE RETRIEVAL | 49

Chapter 5

Phase retrieval

In this chapter, phase-retrieval methods based on approximations of the Transport-of-intensity equation (TIE) are covered. The first section describes what phase retrieval is. Section 5.2 introduces the standard method in PBI – Pagnin’s method. Section 5.3 covers two TIE-based extensions for multi-material objects and the results from Paper A.

5.1 The phase problem

Propagation-based phase-contrast X-ray imaging has the drawback of recording images with a mix of two signals – phase and attenuation. This necessitates phase retrieval, a process in which the phase shift is calculated from the recorded intensity. This is used in crystallography and many types of imaging. Like tomographic reconstructions, this is a classical type of inverse problem. A number of causal factors produce an observable effect, the intensity distribution, but performing the reversed operation, calculating the cause from the effect is non trivial. Phase retrieval problems are often ill-posed. A well posed problem can be defined as having a solution which exists, is unique, and is stable.

In more simple terms, two unknowns, phase and absorption, cannot be calculated from one observation: intensity. The only option is to add more information. This can be in the form of approximations, e.g., assuming a relationship between the unknowns or by adding more observations, i.e. taking more images. A number of methods have been designed for PBI using both strategies [157, 158]. Acquiring multiple images appears simple, but is experimentally difficult. Two independent images must be acquired, meaning that a well known factor in the setup must be changed to get a new observation. Merely acquiring two identical images save for the noise 50 | PHASE RETRIEVAL

is not enough. Changing distance in the setup is the first option which has been used at synchrotrons [158, 159]. For laboratory setups this is very challenging as magnification and spectrum changes. Imaging of fast processes or living animals also makes this impractical. An alternative to changing setup distances is to use a polychromatic spectrum and an energy resolving detector. This would give independent images for each energy bin recorded. Access to detectors with high energy resolution has limited development in this direction.

5.2 Single-material methods

Paganin’s method, developed 20 years ago, is still the most widely used phase retrieval method in PBI [160]. It assumes that the object consists of a single material with material parameters 훿 and 휇 and is based on the TIE (see Section 4.2) [109]:

2휋 휕 퐼(푥, 푦) = −∇ ⋅ (퐼(푥, 푦)∇ 휙(푥, 푦)) . (5.1) 휆 휕푧 ⟂ ⟂ With a few approximations Paganin and coauthors showed that the TIE could be rewritten as

1 퐼(r ) 휇푇 (r ) = − ln (픉−1 { 픉 { ⟂ }}) , (5.2) ⟂ 2 훿 2 퐼 ( ) 4휋 푧eff 휇 w⟂ + 1 0 r⟂ where r⟂ = 푥, 푦 is transverse spatial coordinates, 푇 (r⟂) is the thickness of the object, 퐼(r⟂) is the intensity on the detector, 퐼0(r⟂) is the intensity 2 2 2 without the object and w⟂ = 푢 +푣 are spatial frequencies. The phase shift can be calculated by multiplying the result with 훿/휇. The solution assumes that the object is homogeneous, i.e., that it consists of a single material, that intensity at propagation distance 0 is given by the Beer-Lambert law and that the object is thin so that the phase shift is given by 휙 = −푘훿푇. These assumption allows us to transform the TIE to

훿 퐼(r ) 2 −휇푇 (r⟂) ⟂ (−푧eff ∇⟂ + 1) e = . (5.3) 휇 퐼0(r⟂)

By using a Fourier transform Eq. (5.2) is obtained. There are several advantages to this solution. There is no explicit use of energy, and even though 훿 and 휇 both are functions of energy, the ratio 훿/휇 is used which weakens the energy dependence. Paganin’s formula therefore works well with polychromatic radiation, which makes it applicable both to synchrotron and laboratory imaging. In practice, the approximate solution to the TIE is a PHASE RETRIEVAL | 51

convolution with a filter which in the Fourier domain is 1 퐻 = . 2 훿 2 (5.4) 4휋 푧eff 휇 w⟂ + 1

This corresponds to a low-pass filter. The highest frequencies (edge enhancement) are therefore removed along with noise of high frequency. Computa- −1 tionally, it is very fast to execute a filtering operation like 픉 {퐻 ⋅ 픉{퐼/퐼0}} using Fast Fourier Transform (FFT) (for more details, see Appendix A). The computational time is therefore negligible compared to tomographic reconstruction. The method is also very robust since the denominator in the filter term 퐻 never is zero. For more details, see e.g. [157].

5.3 Multi-material methods

Despite the success of Paganin’s method, the approximation of single material is too narrow to apply to many samples. This has led to the development of more complex methods. In Paper A two such methods were considered: Beltran’s method [161] and Ullherr’s method [162] (in the paper referred to as the parallel and linear method respectively). The purpose was mainly to compare the methods in terms of qualitative and quantitative results and ease of use. Both methods are limited to tomographic imaging as they rely on segmentation in the reconstructed section.

Beltran’s method Beltran’s method, published in 2010, is an extension of Paganin’s method [161]. In short, acquired images are processed in two different ways and following two parallel tomographic reconstructions, the data sets are combined using segmentation. The method assumes that there is one dominating encasing material (subscript 1) and one or more other materials (subscript 2). The data is first processed with Paganin’s method which gives 휇1푇, and 휇1 after the reconstruction. For other materials, (휇2 − 휇1)푇 is calculated with a new expression (Eq. 5.5) which becomes 휇2 − 휇1 after the reconstruction. In the reconstruction no materials overlap and a segmentation can separate encasing and other materials. The final 3D image is the encasing (value 휇1) from Paganin’s method together with other materials (value 휇2) retrieved with the new expression. To get the right value, 휇1 is added to the calculated 휇2 − 휇1. Beltran et al. derived the following expression from the TIE to calculate (휇2 − 휇1)푇:

−1 퐼(r⟂) (휇2 − 휇1)푇2(r⟂) = − ln (픉 {퐻 픉 { }}) . (5.5) 퐼0(r⟂) exp(−휇1퐴(r⟂)) 52 | PHASE RETRIEVAL

A C E G 1 3 5

I 7

B D F H 2 4 6

Figure 5.1. Schematic drawing of Beltran’s method. 1) The projection image (B) is used in Beltran’s formula together with the projected thickness 퐴(r⟂) (A) to retrieve (휇2 − 휇1)푇2 (C). 2) D is retrieved from B using Eq. (5.2). 3/4) C and D are reconstructed separately with slices shown in E and F, respectively. 5/6) A segmentation is performed with an intensity threshold in F. This is used to obtain material 2 from E, and to remove material 2 from F. 7) The cut out G is combined with H. The value of the encasing (휇1) must be added to G before merging to get 휇2 in the final image I. Adapted from Paper A.

D 3 5

A B C E F G 1 2 3 4 5

Figure 5.2. Schematic drawing of Ullherr’s method. 1) Phase retrieval on projection images (A) using Eq. (5.7), 2) Reconstruction, 3) Segmentation where the highly absorbing part D is removed, 4) Additional phase retrieval on the volume image (E), 5) Merge of D and F to form final image G. Adapted from Paper A.

The filter 퐻 substitutes 훿/휇 for Δ훿/Δ휇 = (훿2 − 훿1)/(휇2 − 휇1) compared to Paganin’s method:

1 퐻 = . 2 Δ훿 2 (5.6) 4휋 푧eff Δ휇 w⟂ + 1

A factor exp(−휇1퐴(r⟂)) is also introduced in Eq. (5.5). 퐴(r⟂) is the projected thickness of the entire object which must be estimated or calculated with Paganin’s method. PHASE RETRIEVAL | 53

Ullherr’s method In 2015, M. Ullherr and S. Zabler published a phase-retrieval methods to improve multi-material phase retrieval like Beltran’s method, but with a simpler implementation. It is designed two work well for 3-material objects with two different material interfaces and with one highly absorbing material. This appears restricting, but is useful for many samples, e.g. a mouse which essentially is bone, soft tissue, and air, or an artery which might consist of hard calcifications, soft tissue and air. The method uses a modified expression to phase retrieve the raw data:

−1 퐼(r⟂) ∫ 휇(r⟂)d푧 = − ln (픉 {퐻 픉 { }}) . (5.7) 퐼0(r⟂)

Δ훿 The filter 퐻 is the same as in Eq. (5.6). The ratio Δ휇 is chosen to work well for the highly absorbing material. This implies that the filter leaves some edge enhancement. After the tomographic reconstruction segmentation is used to remove the highly absorbing materials. The remaining part is then filtered again. The two parts are finally combined.

Comparison In Paper A the derivation of Eq. (5.7) from the TIE was shown. The method should therefore produce accurate numbers within the accuracy of the underlying approximations. It is not obvious that phase retrieval filtering in the tomographic reconstruction is compatible with quantitative imaging, but assuming that the logarithm in Eq. (5.7) is approximately linear, i.e. that ln(푥) ≈ 푥, the filtering operation that is phase retrieval commutes with the tomographic reconstruction [163]. These theoretical results were confirmed with wave-propagation simulations (see chapter 6) on a 3-material object.

One important difference noted in Paper A between the methods is that Ullherr’s method produces better results with typical three materials objects. This is because Beltran’s method introduces inaccuracies by using Paganin’s method in the first step. In practice this means that material interfaces are blurred, something that cannot be amended in the segmentation step. Ullherr’s method splits the filtering, thus making it possible to avoid blurring any edges.

X-RAY IMAGING SIMULATIONS | 55

Chapter 6

X-ray imaging simulations

In all kinds of simulations and modelling a central question is how well the simulation or model predicts reality. This is important also in phase- contrast imaging as fast and accurate simulations can give insight into the techniques and enable quick virtual experiments that would be difficult or impossible to carry out experimentally. In this chapter, work to improve phase-contrast simulations by using new object models is described. This is of particular interest in simulations of clinical imaging. The simulation method is first briefly described, followed by the main problem: howto incorporate sophisticated object models in simulations. In the last section an object model is used to perform virtual clinical study.

6.1 Methods

There are a number of ways to simulate imaging with X-rays (see Fig. 6.1). One of the most well known and reliable is Monte Carlo (MC) based methods. MC methods simulate single photons or particles by tracking their

(a) (b) (c)

Figure 6.1. Simulation types. (a) Monte Carlo. (b) Ray tracing. (c) Wave propagation. The effect of attenuation and phase shift is added in the dashed exit plane. The wave is then propagated to the detector (solid line to the right). 56 | X-RAY IMAGING SIMULATIONS

paths through a simulated environment and the probability and effect of different interactions. They are accurate, but also slow as each photon is simulated individually. Acceleration with a graphics processing unit (GPU) is possible [164], but simulation time can still be long and memory limitation in the GPU restricts what kind of models that can be included. As MC methods take a particle view on reality they are not optimal for simulating wave phenomena. Another option is ray tracing. This approach can also be slow as single rays are simulated and it can take weeks to simulate an image even with powerful GPUs [165]. A third option is wave propagation (WP). WP simulates the evolution of phase and intensity in a plane wave. Individual rays or photons are not simulated which makes it much faster. It does not include effects such as Compton scattering, but the results have shown good correspondence with experimental data [166]. In this work, open-source MC code has been used to calculate dose estimates, but simulations of phase contrast imaging have been done with WP.

Wave propagation In Section 4.3 the Fresnel diffraction integral was given as an approximate description of the intensity pattern formed by PBI. To use this result for simulations the integral can also be interpreted as a convolution. Ignoring the factor e푖푘푧 as it does not affect the final intensity, the integral is

푈(푥, 푦, 푧) = ∬ 푈(푥0, 푦0, 0)ℎ(푥 − 푥0, 푦 − 푦0, 푧)d푥0d푦0, (6.1) where ℎ is known as the Fresnel propagator: 1 푖푘 ℎ(푥, 푦) = exp ( (푥2 + 푦2)) . (6.2) 푖휆푧 2푧

A computationally efficient way of calculating the convolution is by Fourier transform (see Appendix A)

−1 푈(푥, 푦, 푧) = 픉 {픉{푈(푥0, 푦0, 0)} ⋅ 픉{ℎ(푥, 푦)}} (6.3)

To save additional computations, the Fresnel propagator can be analytically transformed once:

퐻(푢, 푣) = 픉{ℎ(푥, 푦)} = exp(−푖휋푧휆(푢2 + 푣2)). (6.4)

The wave exiting the object, 푈(푥0, 푦0), is calculated as a perfect incoming plane wave times the transmission function,

exp(−푎(푥, 푦) + 푖휙(푥, 푦)), (6.5) X-RAY IMAGING SIMULATIONS | 57

where 푎 and 휙 are attenuation and phase shift respectively as described in Section 2.2. The wave perturbation of the entire object is added in the exit plane, i.e., a virtual plane right after the object (see Fig. 6.1c). This is essentially the thin lens approximation, in this context known as the projection approximation. Mathematically it means that the Laplacian in Eq. (4.5) is negligible. This has been extensively investigated by Morgan et al., who state that if the propagation distance is much larger than the object thickness it gives an accurate solution [167, 168]. One intuitive way to check if the projection approximation is reasonable is if the transverse deviation of X-rays due to refraction is larger than the simulated pixels. Deviation angles are on the order of 훿 so this gives the condition

훿푇 < Δ푥, (6.6) where 푇 is the object thickness and Δ푥 the pixel size. For typical values, 훿 = 10−6 and Δ푥 = 1 µm, 푇 < 1 m. Although approximate, this indicates that the projection approximation is compatible with thick, decimeter sized objects.

In the simulation of an X-ray image, the convolution between the transmission function and the Fresnel propagator creates an ideal image. To make it comparable with a real image, source and detector effects and photon noise must be included. The source and detector cause blurring. From the spectrum, geometry, and detector properties, the pixel count for the open beam image is calculated. It is used to scale the ideal intensity image. The spatial sampling in the simulation is changed to the detector pixel size and Poisson noise is added. It is possible to extend this to include more detector effects such as thermal noise, readout noise etc., but as the target ofthe simulations in Paper B and C is future clinical imaging, photon counting was assumed making Poisson noise a good approximation.

6.2 Object models

An important part of realistic simulations is the object model. The model can have different resolution and capture different features of the object. Models can be built from simple geometrical objects or experimental data. The way of storing the model impacts also its use. The most common type is voxel-based models (see Fig. 6.2). They are intuitive, simple to use in MC and WP and share a direct similarity with experimental data. The main drawback is the need for much computational power and large storage if high resolution is used. Another type is surface-based models. They are built from experimental (voxel-based) data, but consists only of surfaces, which 58 | X-RAY IMAGING SIMULATIONS

(a) (b) (c)

Figure 6.2. Object types. (a) Voxel-based. (b) Polygon mesh with vertices, edges, and faces. (c) NURBS. A surface is defined by pulls from points. reduces storage. Among surface models vertex-based or polygon mesh models are common. A vertex is a point where two lines meet. Lines, vertices and faces form polygons, typically triangles (see Fig. 6.2). Another used surface model is non-uniform rational B-spline (NURBS). Here, surfaces are represented by splines, functions that are piecewise polynomials. The advantage of these models is the smooth surface, reduced memory need, and flexibility. For a biological model this could enable simulations of biological motions such as respiration and simplify anatomic modification and variation. For the work on simulations in this Thesis, voxel-based models have been used as they enable a simple calculation of the transmission function. Today there are both voxel-based and NURBS-based models for pre-clinical and clinical imaging [169–172]. Structures smaller than 100 µm are rarely included as it is difficult to create such detailed models and there is little need with commonly used clinical resolution. PBI, which is typically used with one to two orders of magnitude higher resolution would benefit from more detailed models, but even with resolution in the range 50 − 100 µm PBI is possible which means current models have sufficient complexity for relatively low-resolution PBI. Performing wave propagation simulations of PBI in this range is, however, limited by the sampling.

6.2.1 Sampling In Paper B the sampling limit for PBI is addressed. To capture the phenomenon correctly the sampling must roughly obey √ 휆푧 Δ푥 < , (6.7) 2 where 푧 is the propagation distance and Δ푥 sample step size. This sampling is needed to reproduce the Fresnel propagator sufficiently well. For typical X-RAY IMAGING SIMULATIONS | 59

120

100 2 4

80 6

60 2 4 8

40 6 Photon energy [keV] energy Photon

8 12 20 12 20 20 0 0 5 10 15 Propagation distance [m]

Figure 6.3. Sampling required in simulations of phase-contrast imaging. The lines indicate the gradually increasing sampling (in micrometers) for higher energies and shorter propagation distances. For a certain energy and propagation distance, the closest line below gives an upper bound for the sampling. values in a high-resolution laboratory setup this leads to sampling around 1 µm (see Fig. 6.3). This is obviously far higher sampling than what is available in pre-clinical and clinical models. Even NURBS-based models, which do not have a sharp sampling due to smooth surfaces and thus can be converted to voxel-based models of high sampling, are not compatible with micrometer sampling as code generally can not handle the memory requirements associated with micrometer sampling. A way around this problem, presented in Paper B, is to upsample the model.

6.2.2 Upsampling Upsampling is the process of resampling a function or signal. The purpose is to achieve the same result as if the original function had been sampled more densely. In Fig. 6.4 a sine function is shown. If this function is downsampled to only a few discretization points a jagged curve is obtained. Upsampling is the process to recreate the original smooth function. This can be done perfectly if the Nyquist sampling criterion is obeyed. Upsampling is for example done when a digital music signal is converted to an analogue signal in a sound system. A function describing the structures in an organ would, however, be complicated and have a high Nyquist frequency. An 60 | X-RAY IMAGING SIMULATIONS

(a) (b) (c)

Figure 6.4. Sampling of a sine function. (a) Analytic function. (b) Discretization with high sampling. (c) Discretization with low sampling. Sampling points shown in red.

(a) (b) (c) (d)

Figure 6.5. Sampling artifacts in PBI shown on a coarsely sampled sphere. (a) Low propagator sampling. (b) Low object sampling. (c) High object sampling achieved with upsampling in 2D. (d) High object sampling achieved with upsampling in 3D. exact reconstruction is therefore not possible for the type of coarsely sampled data considered in this chapter, but the function can be assumed to be fairly smooth, which can allow for an approximate upsampling.

In Fig. 6.5 the sampling problem is shown in the context of a PBI image of a sphere. Low propagator sampling, i.e., violation of Eq. (6.7) results in reduced or no phase contrast. High propagator sampling in combination with low object sampling creates artifacts in the form of edge enhancement at the edges of object pixels. High and equal sampling is necessary to avoid this. In the simulation framework described in Section 6.1 the projection approximation is used, meaning that the contribution from attenuation and phase shift are added at once in the exit plane. The object model is in other words an (2D) image. The most memory efficient option would then be to use an object model with coarse sampling and upsample the 2D image prior to propagation. As is evident from Fig. 6.5c this fails to fully remove the artifacts seen in Fig. 6.5b. More smoothing can be applied in X-RAY IMAGING SIMULATIONS | 61

(a) (b)

Figure 6.6. Simulated chest X-ray with PBI. (a) Overview. (b) Example of section (8 × 8 cm2) from reader study. the upsampling procedure, but this will add too much blurring before the artifacts are removed. This leaves upsampling of the full 3D model. This is not trivial as boundaries between features of many different sizes must be preserved as the coarse voxelation is removed. A straight-forward algorithm for this is shown in Paper B. By processing small subvolumes and materials separately, the coarse voxelation can be removed while limiting memory requirements. Parallelization by processing on subvolumes speeds up the process significantly, but the computational cost is nevertheless very high. The important advantage of this approach compared to MC or ray tracing is that this preprocessing is done once for a model. The remaining WP is fast and most parameters, such as 훿, 휇, energy, propagation distance, source and detector characteristics, etc. can be adjusted in this stage.

6.3 Virtual clinical trials

A clinical trial is an established process to estimate the efficacy and safety of new devices, procedures and drugs for clinical use. It is one step in a long development. Trials are important, but time consuming and expensive and therefore a bottleneck in new development. Clinical trials for a new drug can cost billions of USD and take years. Conducting virtual clinical trials (VCT), i.e., substituting real patients for a computer model is therefore an appealing prospect. Both time and cost could be drastically reduced and patients would not be exposed to risk. The challenge in realizing this is the complexity of the human body. It is therefore not possible for virtual clinical trials to completely substitute conventional clinical trials in the near future. They could, however, 62 | X-RAY IMAGING SIMULATIONS

speed up the development prior to a final clinical trial and improve the chances of success [173]. Clinical imaging is one area where VCT could be realized soon as mainly structure is of interest. No advanced intracellular or intercellular chemical processes are necessary. A sufficiently detailed, varied and flexible human disease model is still challenging.

In Paper C, a small VCT, or virtual clinical study, was conducted to investigate if the edge enhancement in PBI could be useful for radiologists in finding pulmonary nodules. The 4D Extended Cardiac-Torso (XCAT) was used as a human model [171, 174]. This NURBS-based model was converted to a voxel-based phantom with 100 µm sampling. Using the upsampling and simulation process described in Paper B, 15 chest X-ray images were created, each in three configurations of energy and propagation distance, yielding 45 images in total. The three configurations were phase contrast (60 keV monochromatic radiation and 푧 = 12 m), conventional radiography (120 kVp tungsten spectrum and 푧 = 0 m), and monochromatic radiography (60 keV and 푧 = 0 m). Two different absorption images (푧 = 0 m) were simulated to decouple the impact of the spectrum. 240 sections, three of each configuration, were extracted and used in a reader study. Two active radiologists reviewed the 240 sections and marked potential pulmonary nodules.

The sensitivity for the three configurations were 0.83 for conventional chest X-ray, 0.85 for monochromatic chest X-ray and 0.84 for phase-contrast chest X-ray. The result was consistent for both radiologists and indicates that the three configurations had the same benefit to the radiologist for this particular task. One hypothesis prior to the study was that the overlapping edge enhancement could result in more false-positives, but this was not observed. The radiologists were also asked to rate the image quality of all section and phase contrast received a lower value on average. This can probably be par- tially attributed to that the radiologists were not accustomed to see edge enhancement in images. This also relates to a general problem in these types of studies, that the readers are used to looking at images created with the conventional technique, but not with new one. They can in other words be biased against the new technique. SMALL-ANIMAL IMAGING ON COMPACT SYSTEMS | 63

Chapter 7

Small-animal imaging on compact systems

Phase-contrast X-ray imaging was first developed at synchrotrons. The superior flux and coherence enabled new types of imaging not possible in the small laboratory. For a broader use, translation to compact sources is essential as access to synchrotrons is limited. This chapter describes two applications in small-animal imaging where compact high-brightness sources can enable pre-clinical imaging and provide a stepping stone on the path towards clinical imaging.

7.1 Visualizing vascular canals in bone

Microstructure in bone plays an important role in bone remodeling (see Fig. 7.1a). In general, bone imaging can be done well with attenuation-based imaging, but phase contrast can improve contrast at high resolution and at low attenuating regions close to the bone. Phase contrast can also be important in dose critical in-vivo bone imaging [175]. In Paper D, microstructures in a rat femur were imaged at high resolution, which has been done previously at synchrotrons [176]. The vascular canals, in humans called the Haversian system, and lacunae, the microscopic cavities where single osteocytes live, were observed (see Fig. 7.1b), and the vascular network could be segmented.

High-resolution bone imaging is a difficult task as the strong absorption in bone reduces the flux considerably, especially for low energies that give the strongest phase signal. Using a MetalJet D2 (Excillum AB, Stockholm, Sweden) source and a detector with thin scintillator, similar results as synchrotron experiments could be achieved, but at the cost of about 30 times 64 | SMALL-ANIMAL IMAGING ON COMPACT SYSTEMS

(a) (b)

Figure 7.1. High-resolution bone imaging. (a) Tomographic section of a rat femur injected with Tricalcium phosphate (TCP), acting as a synthetic bone graft to aid regrowth of bone in a fracture. Scale bar 500 µm. (b) Vascular canals with lacunae spread around. Scale bar 100 µm. longer exposure time [176]. This shows nevertheless that compact sources can offer a more accessible alternative for this application.

7.2 In-vivo lung tomography

In-vivo tomography of rodents is done routinely on laboratory sources, but resolution and overall quality is worse than in-situ imaging. This is mainly due to three factors: dose limitations, source limitations, and cardio-respiratory motion. The dose determines roughly what SNR is possible at a given resolution. The higher the resolution, the lower the SNR. The source power must be high to keep exposure times short. For PBI, the spot size is crucial, which results in a need for high-brightness sources. Cardio-respiratory motion is unavoidable and it is dealt with by gating – acquiring images at certain points (phases) in the cardio-respiratory cycle. It can be done with two different schemes (see Fig. 7.2): prospective and retrospective gating. The latter means that one acquires plenty of images and then picks out which ones belong to the desired respiratory phase. This can be done based on a signal measured during the acquisition (extrinsic) or by analyzing movement in the images (intrinsic). A problem with retrospective gating is that it can be inefficient in terms of dose since some data might be discarded. This is not the case in prospective gating as images only are acquired on a trigger during the desired phase. Prospective gating can also be done in two modes: free breathing and mechanical ventilation. Free breathing means that the SMALL-ANIMAL IMAGING ON COMPACT SYSTEMS | 65

trigger delay acquisition interval lung pressure lung pressure

time time

(a) (b)

Figure 7.2. Respiratory gating. The blue curve is the lung pressure. A drop in pressure indicates inspiration and the following peak indicates expiration. (a) Prospective gating. The triggers for image acquisition (dashed lines) are synced with the respiratory cycle either by monitoring the natural breathing or by forcing a steady pattern with ventilation. Here the former is shown with a fixed delay between inspiration (dip in curve) and trigger. (b) Retrospective gating, the image acquisition occurs with a regular interval, not synced with the respiration. Images are sorted on position in respiratory cycle by using the respiratory signal (extrinsic) or analyzing the images (intrinsic). tomographic system waits for the mouse to reach the right phase in the cycle before quickly acquiring an image. In mechanical ventilation (or forced breathing) the mouse is intubated and connected to a respirator. The respirator is then synced with the imaging system. The advantage of mechanical ventilation is the predictable cycle and that the end inspiration (lungs fully inflated) can be kept still a short time to obtain an image. Free breathing must be done on end-expiration (lungs deflated) as this is when the lungs are naturally still. The predictable behavior and good quality has made mechanical ventilation a common choice in in-vivo imaging, but there are drawbacks. Mechanical ventilation can cause ventilator-induced lung injury (VILI). This includes adverse effects such as inflammation and pulmonary edema [177]. This can affect the result if longitudinal studies are conducted, i.e., repeated measurement on the same mouse. Clearly, a free-breathing mouse is preferable as long as the cardio-respiratory motion does not impair the image quality.

In Paper E, high-resolution tomography of free-breathing mice was demonstrated with a MetalJet D2 source. A pneumatic sensor placed on the mouse provided a clear respiratory signal which could be transferred to the detector, rotation stage, and X-ray shutter to synchronize exposure and rotation in a full tomography. The source power was 200 W and the exposure time 150 ms. The total X-ray dose was 450 mGy which is low enough to allow for longitudinal studies. To evaluate the resolution in 66 | SMALL-ANIMAL IMAGING ON COMPACT SYSTEMS

(a) (b)

~100 µm

~75 µm

Figure 7.3. High-resolution lung images acquired on a free breathing mouse. (a) Axial slice in the tomogram. Scale bar 5 mm (b) Magnified section. Scale bar 500 µm. (c) 3D segmentation of rib cage (gray) and lungs (blue), (d) 3D segmentation of tracheobronchial tree. Small scale bar is 100 µm. The segmentation (c-d) is reproduced from Paper E. SMALL-ANIMAL IMAGING ON COMPACT SYSTEMS | 67

the lungs, Fourier shell correlation (FSC) (see Appendix A) was used as a complement to studying the size of small visible features. The FSC-estimated resolution was about 34 µm. This agreed well with visual observation of edges with a thickness around 33 µm (between 10% and 90% of intensity shift). A segmentation of the tracheobronchial tree (see Fig. 7.3) could be performed with standard segmentation software (Amira 6.3, Thermo Fisher Scientific, US) down to airways smaller than 100 µm.

CONCLUSIONS AND OUTLOOK | 69

Chapter 8

Conclusions and outlook

Developments to include complex objects in propagation-based phase-contrast X-ray imaging have been presented in this Thesis. Three problems on this theme have been addressed, multi-material phase retrieval, simulations for clinical imaging, and translation of small-animal imaging to compact laboratory systems.

The first area, multi-material phase retrieval, was considered in PaperA. The quantitative nature of a method was shown and its reduction of artifacts compared to a more involved method. These methods did, however, assume monochromatic radiation for quantitative imaging, which rules out laboratory sources. Creating a quantitative method for polychromatic radiation that is stable at low SNR and requires little manual adjustment will be an important step towards a versatile and reliable PBI. One possible path is using spectral information, which will be aided by recent and ongoing development of energy resolving detectors. Creating stable solutions for low SNR imaging can, however, be a challenge as SNR is even lower for individual energy bins. Iterative methods will most likely be more common, which can reduce manual and time-consuming adjustment and artifacts.

The second area, simulations for clinical imaging, was considered in Papers B and C. The presented two-step strategy enables fast simulation with existing voxel-based models and virtual clinical trials to evaluate use of PBI. The strategy was shown in a small VCT testing the benefit of edge enhancement in lung nodule detection. Despite the advantages of the proposed strategy, processing still requires substantial computer power. Increasing computational power and ongoing model development makes the use of VCTs more likely in the future. Although this work has focused 70 | CONCLUSIONS AND OUTLOOK

on voxel-based models as this is a good option today, surface models are more flexible and efficient in storage. Both types are likely to beusedin the coming years, but with better methods for ray tracing the advantage will probably tilt towards surface models.

The third area, translation to compact sources, was considered in Papers D and E. Examples were shown of how imaging can be moved from synchrotron to compact laboratory setups, even for difficult problems such as tomography of free-breathing mice. Mechanical ventilation is not likely to disappear due to challenges associated with imaging free-breathing mice. The latter should, however, serve as an important alternative to eliminate the impact of ventilation on experimental results. Continued translation and simplification of compact laboratory PBI is reliant on source and detector technology. Good progress has been made with direct detection and photon counting, although it is difficult to achieve small pixels with sufficient stopping power and small PSF. Concerning sources, the LMJ technology has enabled fast compact imaging at low energies, but more development is necessary to reach higher energies and acquisition times better compatible with pre-clinical imaging and clinical imaging.

The field of phase-contrast X-ray imaging has reached an exciting point where clinical trials using both synchrotrons and laboratory sources have been completed. Nevertheless, more progress in phase retrieval, simulations, detector technology, high-brightness sources, and experimental realization is necessary to establish PBI as a standard technique in clinical imaging. FOURIER TRANSFORM | 71

Appendix A

Fourier Transform

The Fourier Transform is a mathematical transformation founded in the concept that many continuous and discontinuous functions can be described by an infinite sum of sine waves [178]. The transformation can decompose functions of space to functions of spatial frequency, and is in two variables defined as

∞ 퐹 (푢, 푣) = ∬ 푓(푥, 푦) e−2휋푖(푥푢+푦푣)d푥d푦. (A.1) −∞ The function 푓 of the spatial coordinates 푥 and 푦, becomes 퐹 which is a function of the spatial frequencies 푢 and 푣. Note that the Fourier transform of a function often is denoted with a capital letter. The function 퐹 exists in the so-called frequency space or Fourier space. To return to real space an inverse transform is defined as

∞ 푓(푥, 푦) = ∬ 퐹 (푢, 푣) e2휋푖(푢푥+푦푣)d푢d푣. (A.2) −∞ For convenience, the Fourier transform will be denoted 픉 and its inverse 픉−1, i.e.

퐹 (푢, 푣) = 픉 {푓(푥, 푦)} 푓(푥, 푦) = 픉−1 {퐹 (푢, 푣)} . (A.3)

Convolution theorem The are numerous uses for the Fourier transform. In this work the convolution theorem has mainly been used. It states that the Fourier transform of a convolution between two functions is the same as the product of the Fourier transform of each of the two functions, 픉{푓 ∗ 푔} = 픉{푓} ⋅ 픉{푔}, (A.4) 72 | FOURIER TRANSFORM

where ∗ denotes a convolution. This means that an alternative to performing a convolution is to multiply the functions in Fourier space and inverse transform,

푓 ∗ 푔 = 픉−1 {픉{푓} ⋅ 픉{푔}} . (A.5) This is highly relevant for image processing as a convolution is a common, but slow operation. Multiplications and efficient algorithms for discrete Fourier transform, so-called fast Fourier transform (FFT), are on the other the other hand, very fast.

Fourier ring/shell correlation Fourier ring correlation is a method to estimate image resolution. It was developed for electron microscopy in the 1980s [179,180]. Two images of the same object are Fourier transformed and the normalized cross-correlation (NCC) for concentric rings around the origin for the two images are calculated. Each NNC value, from a pair of rings, is a point on a curve which is a function of frequency (see Fig. A.1). Large and clear features will have a strong correlation between the images so the NCC is high for low frequencies. With increasing frequency the curve will drop and eventually reach a threshold, below which it is mostly noise. FRC can be summarized with the formula

∗ ∑ 퐹1(푟)퐹2(푟) (푟 ) = 푟∈푟푖 , FRC 푖 √ (A.6) √ 2 2 (∑ |퐹1(푟)| ) (∑ |퐹2(푟)| ) ⎷ 푟∈푟푖 푟∈푟푖 ∗ where 퐹1(푟) and 퐹2(푟) are the Fourier transformed images, denotes the complex conjugate, 푟 is a position (frequency) in Fourier space and 푟푖 positions on a ring. To consider image volumes, the FRC is easily extended to the Fourier shell correlation (FSC). A critique against the FRC is that the threshold can be somewhat arbitrary. 1/2-bit threshold curve is considered a ”general purpose indicator of interpretable resolution” [181]. FOURIER TRANSFORM | 73

(a) (b) (c)

1 FRC 0.8 1/2 bit threshold Resolution point 0.6

0.4

FRC 0.2

-0.2

-0.4 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Spatial frequency [pixels-1]

(d)

Figure A.1. FRC shown for simple line patterns. (a) Line patterns with 6, 8, and 10 pixels half-period lines. (b) Line patterns with added blurring (PSF = 13 pixels) and noise. (c) Fourier transform of (b) shown as logarithm of absolute value. (d) Fourier ring correlation curve for (b) and an identical image with same noise level. The threshold curve is intersected at about 0.0767 which corresponds to the PSF (1/0.0767 ≈ 13). Note that the 8 pixels half-period lines still can be seen with a 13 pixel PSF.

SUMMARY OF PAPERS | 75

Summary of papers

This Thesis is based on the five papers listed below. The results include theoretical work on phase retrieval methods, simulations of clinical imaging and laboratory imaging using high-brightness liquid-metal-jet sources. The author was the main responsible for papers A to D, which include design of studies, derivations and theoretical work, simulations, experiments, analysis, and writing. In Paper E the author contributed to the design and execution of the experiments, analysis of experimental results, and writing.

Paper A: Comparison of quantitative multi-material phase-retrieval algorithms in propagation-based phase-contrast X-ray tomography This paper compares two phase retrieval methods, the linear and the parallel. The mathematical derivation of the linear method is shown and that the method can give quantitative information. A qualitative and quantitative comparison shows also that the linear method preserves image quality better.

Paper B: In Silico Phase-Contrast X-Ray Imaging of Anthropomorphic Voxel-Based Phantoms This paper proposes a processing strategy to enable use of large detailed phantoms in phase-contrast simulations. Sampling is identified as a key problem in these wave-propagation simulations. To use an existing voxel- based phantom, an upsampling procedure is described, which creates high- resolution thickness maps of all materials as input for the simulations. The strategy is shown on a mammography phantom.

Paper C: Propagation-based phase-contrast CXR: a virtual clinical study This paper investigates the benefit of propagation-based phase-contrast for radiologists in identifying pulmonary nodules, i.e., small potentially malig- nant growths in the lungs. Chest X-rays are simulated using the XCAT 76 | SUMMARY OF PAPERS

phantom and the strategy presented in paper B. A reader study with radiologists is conducted to test sensitivity of phase contrast compared to conventional chest X-ray.

Paper D: Cellular-Resolution Imaging of Microstructures in Rat Bone using Laboratory Propagation-Based Phase-Contrast X-ray Tomography This paper shows that microstructure in the bones of small animals, such as rats, can be investigated with a compact imaging system. Lacunae, tiny cavities of single osteocyte cells and the vascular network can be imaged. Previously this type of imaging has been done at synchrotrons, but with sufficient stability and brightness it can also be done with compact sources.

Paper E: Resolving the terminal bronchioles in free-breathing mice: propagation-based phase-contrast CT This paper shows that high-resolution in-vivo tomography on mice can be performed without mechanical ventilation on laboratory sources. Lung tomograms of free-breathing mice with a resolution around 30 µm are acquired with a 450 mGy dose. The resolution is further analyzed with Fourier shell correlation and 3D segmentation of the tracheobronchial tree down to sub-100 µm airways is shown. ACKNOWLEDGEMENTS | 77

Acknowledgements

With this short note I would like to acknowledge a few people who have given me advice and help throughout my research work, for which I am sincerely grateful.

Prof. Hans Hertz, my main supervisor, who through his vision and experience helped guide me, while at the same time allowing me to explore my own ideas.

Assoc. Prof. Anna Burvall, my co-supervisor, whose sharp mind and eye for detail has lead to many illuminating discussions on physics, mathematics, and scientific writing.

Past and present members of the Hard X-ray group: Jakob Larsson, William Twengström, Thomas Schromm, Jakob Reichmann, and especially Jenny Romell and Kian Shaker, with whom I have shared many long hours in the lab.

Collaborators around the world: Dr. Susanne Lewin (Uppsala University), Assist. Prof. Ehsan Abadi (Duke University, U.S.), and Docent Sven Nyrén and Dr. Bariq Al-Amiry (Karolinska Institute).

Dr. Kentaro Uesugi and Dr. Masato Hoshino (SPring-8, Japan), who kindly let me visit SPring-8 and learn about synchrotron experiments.

Prof. Kjell Carlsson for pointing out parts in this Thesis that could be improved.

All the members of the BioX group, who make it a warm and cheerful place to take one’s first steps in the world of science.

REFERENCES | 79

References

[1] W. C. Röntgen, “Über eine neue Art von Strahlen”, Sitzungsberichten der Würzburger Physik.-medic. Gesellschaft 9, 132–141 (1895). [2] “Eine sensationelle Entdeckung”, Die Presse, January 5 (1896).

[3] W. C. Röntgen, “On a new kind of rays”, Nature 53(1369), 274–277 (1896).

[4] E. B. Frost, “Experiments on the x-rays”, Science 3(59), 235–236 (1896).

[5] “X rays find a needle in a foot”, The New York Times, February 15 (1896).

[6] W. König, 14 Photographien mit Röntgen-Strahlen. Leipzig: Verlag von Johan Ambrosius Barth (1896).

[7] G. Alvaro, “I vantaggi pratici della scoperta di röntgen in chirurgia”, Giornale Medico del Regio Esercito 44(5), 385–394 (1896). [8] “Epokgörande upptäckt”, Dagens Nyheter, January 9 (1896).

[9] “Röntgenfotografier i Upsala”, Upsala Nya Tidning, February 10 (1896).

[10] “Röntgen-strålar i Sverige”, Svenska Dagbladet, February 14 (1896).

[11] T. Stenbeck, “Om den Röntgenska upptäckten af x-strålarne och dess historiska utveckling”, Hygiea - Medicinsk och Farmacevtisk Må- nadsskrift 58(2) (1896). [12] F. Henschen, Min långa väg till Salamanca: en läkares liv. Stockholm: Bonnier (1957).

[13] “Vid k. vetenskapssocietetens sammanträde”, Svenska Dagbladet, November 3 (1896). 80 | REFERENCES

[14] “Röntgens upptäckt tillämpad i hjärnkirurgien”, Hallandsposten, February 6 (1897).

[15] S. E. Henschen, K. G. Lennander, and T. Stenbeck, “Om Röntgens strålar i hjärnkirurgiens tjänst.”, Nordiskt Medicinskt Arkiv, Fest- band 30 (1897).

[16] S. E. Henschen, “Die Röntgen-Strahlen im Dienste der Hirnchirurgie”, XII Congrès International De Médecine (Moscou 1897) 4 (1899).

[17] T. Stenbeck, Röntgenstrålarne i medicinens tjenst. Stockholm: Wahlström & Widstrand (1900).

[18] K. Ångström, “Om röntgenstrålarna: deras framställning och förhis- toria”, in Föreningen Heimdals Folkskrifter Nr. 51, Stockholm: F. & G. Beijers Bokförlagsaktiebolag, (1898).

[19] J. Daniel, “The x-rays”, Science 3(67), 562–563 (1896).

[20] E. A. Codman, “The cause of burns from x-rays”, The Boston Medical and Surgical Journal 135(24), 610–611 (1896).

[21] G. Frei, “Deleterious effects of x-rays on the human body”, Electrical Review, New York 29, 95 (1896).

[22] W. Rollins, “Notes on X-light: Vacuum Tube Burns”, Boston Medical and Surgical Journal 146 (1902).

[23] O. E. Langland and R. P. Langlais, “Early pioneers of oral and max- illofacial radiology”, Oral Surg. Oral Med. Oral Pathol. Oral Ra- diol. 80(5), 496–511 (1995).

[24] W. Rollins, “Notes on X-light: X-light can kill animals”, Boston Med- ical and Surgical Journal 144 (1901).

[25] W. Rollins, Notes on X-light. Boston, MA: The University Press (1904).

[26] A. A. Michelson and E. W. Morley, “On the Relative Motion of the Earth and the Luminiferous Ether”, Am. J. Sci. 34(203), 333–345 (1887).

[27] J. J. Thomson, “Longitudinal electric waves, and röntgen’s x rays”, Proceedings of the Cambridge Philosophical Society 9, 49–61 (1896).

[28] G. G. Stokes, “On the nature of the röntgen rays”, Proceedings of the Cambridge Philosophical Society 9, 215–216 (1896). REFERENCES | 81

[29] C. G. Barkla, “Polarisation in Röntgen Rays”, Nature 69(1794), 463 (1904).

[30] C. G. Barkla, “Secondary röntgen radiation”, Nature 71(1845), 440 (1905).

[31] C. G. Barkla, “Secondary rontgen rays and atomic weight”, Na- ture 73(1894), 365–365 (1906).

[32] H. Haga and C. H. Wind, “Die Beugung der Röntgenstrahlen”, An- nalen der Physik 304(8), 884–895 (1899).

[33] H. Haga and C. H. Wind, “Die Beugung der Röntgenstrahlen”, An- nalen der Physik 315(2), 305–312 (1903).

[34] M. Eckert, “From X-rays to the h-hypothesis: Sommerfeld and the early quantum theory 1909–1913”, The European Physical Journal Special Topics 224(10), 2057–2073 (2015).

[35] A. Sommerfeld to W. Wien. May 13th (1905). Deutschen Museum Archive NL 56, 010.

[36] B. Walter and R. Pohl, “Zur Frage der Beugung der Röntgenstrahlen”, Annalen der Physik 330(4), 715–724 (1908).

[37] B. Walter and R. Pohl, “Weitere versuche über die beugung der rönt- genstrahlen”, Annalen der Physik 334(7), 331–354 (1909).

[38] P. P. Koch, “Über die Messung der Schwärzung photographis- cher Platten in sehr schmalen Bereichen. Mit Anwendung auf die Messung der Schwärzungsverteilung in einigen mit Röntgenstrahlen aufgenommenen Spaltphotogrammen von Walter und Pohl”, Annalen der Physik 343(8), 507–522 (1912).

[39] A. Sommerfeld, “Über die Beugung der Röntgenstrahlen”, Annalen der Physik 343(8), 473–506 (1912).

[40] W. Wien, “Über eine Berechnung der Wellenlänge der Röntgen- strahlen aus dem Planckschen Energie- Element”, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch- Physikalische Klasse 5, 598–601 (1907).

[41] A. Bavais, “Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l’espace”, J. Ecole Polytech 19, 1–128 (1907). 82 | REFERENCES

[42] P. P. Ewald, Fifty Years of X-ray Diffraction. Utrecht: International Union of Crystallography, N.V.A. Oosthoek’s Uitgeversmaatschappij (1962).

[43] W. Friedrich, P. Knipping, and M. v. Laue, “Interferenz-erscheinungen bei röntgenstrahlen”, Sitzungsberichte der Bayerischen Akademie der Wissenschaften 1912(14), 303–322 (1912).

[44] M. v. Laue, “Eine quantitative Prüfung der Theorie für die Interferenz-Erscheinungen bei Röntgenstrahlen”, Sitzungsberichte der Bayerischen Akademie der Wissenschaften 1912(16), 363–373 (1912).

[45] Nobelstiftelsen, Les Prix Nobel en 1914–1918. Stockholm: P.A. Norstedt och Söner (1920).

[46] W. Stenström, Experimentelle Untersuchungen der Röntgenspectra. PhD thesis, Lund University, Lund, (1919).

[47] J. G. Brown, “Refraction and reflection of x-rays”, in X-Rays and Their Applications, 147–156, Boston, MA: Springer US, (1966).

[48] A. Larsson, M. Siegbahn, and I. Waller, “Der experimentelle nach- weis der brechung von röntgenstrahlen”, Naturwissenschaften 12(52), 1212–1213 (1924).

[49] A. Larsson, M. Siegbahn, and I. Waller, “The refraction of x-rays”, Phys. Rev. 25(2), 235 (1925).

[50] Nobelstiftelsen, Les Prix Nobel en 1924–1925. Stockholm: P.A. Norstedt och Söner (1926).

[51] I. Waller, Theoretische Studien zur Interferenz- und Dispersionsthe- orie der Röntgenstrahlen. PhD thesis, Uppsala University, Uppsala, (1925).

[52] M. Born, “Theoretical investigations on the relation between crystal dynamics and x-ray scattering”, Reports on Progress in Physics 9(1), 294–333 (1942).

[53] G. Kellström, “Experimentelle untersuchungen über interferenz und beugungserscheinungen bei langwelligen röntgenstrahlen”, Nova Acta Soc. Sci. Upsal. ser. IV, 8 (1932).

[54] I. McNulty, “The future of x-ray holography”, Nucl. Instrum. Methods Phys. Res. A 347(1), 170–176 (1994). REFERENCES | 83

[55] D. Gabor, “A New Microscopic Principle”, Nature 161(4098), 777–778 (1948). [56] J. Ambrose and G. N. Hounsfield, “Computerized transverse axial tomography”, Brit. J. Radiol. 46(542), 148–149 (1973). [57] G. N. Hounsfield, “Computerized transverse axial scanning (tomography): Part 1. Description of system”, Brit. J. Radiol. 46(552), 1016–1022 (1973). [58] R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in x-ray computerised tomography”, Phys. Med. Biol. 21(5), 733–744 (1976). [59] J. Rienitz, “Schlieren experiment 300 years ago”, Nature 254(5498), 293–295 (1975). [60] T. Birch, The history of the Royal society of London for improving of natural knowledge, from its first rise. In which the most considerable of those papers communicated to the society, which have hitherto not been published, are inserted in their proper order, as a supplement to the Philosophical transactions. VOL. III. London: A. Millar (1757). [61] T. Young, “I. The Bakerian Lecture. Experiments and Calculations relative to physical Optics”, Philos. Trans. R. Soc. 94, 1–16 (1804). [62] L. Zehnder, “Ein neuer Interferenzrefraktor”, Zeitschrift für Instru- mentenkunde 11, 275–285 (1891). [63] L. Mach, “Ueber einen Interferenzrefraktor”, Zeitschrift für Instru- mentenkunde 12, 89–93 (1892). [64] P. Krehl and S. Engemann, “August toepler — the first who visualized shock waves”, Shock Waves 5(1), 1–18 (1995). [65] A. Toepler, Beobachtungen nach einer neuen optischen Methode — Ein Beitrag zur Experimental-Physik. Bonn: Max Cohen & Sohn (1864). [66] F. Zernike, “How I Discovered Phase Contrast”, Science 121(3141), 345–349 (1955). [67] Nobelstiftelsen, Les Prix Nobel en 1953. Stockholm: Almqvist & Wiksell International (1954). [68] U. Bonse and M. Hart, “An x‐ray interferometer”, Appl. Phys. Lett. 6(8), 155–156 (1965). 84 | REFERENCES

[69] M. Hart and U. Bonse, “Interferometry with x rays”, Physics To- day 23(8), 26–31 (1970).

[70] A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation”, Rev. Sci. Instrum. 66(12), 5486–5492 (1995).

[71] S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Steven- son, “Phase-contrast imaging using polychromatic hard X-rays”, Na- ture 384(6607), 335–338 (1996).

[72] R. Fitzgerald, “Phase-sensitive X-ray imaging”, Physics Today 53, 23–26 (2000).

[73] A. Einstein and L. Infeld, The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta. United Kingdom: Cambridge University Press (1938).

[74] Bureau International des Poids et Mesures, SI Brochure: The Inter- national System of Units (SI). 9 ed. (2019).

[75] M. Berger, J. Hubbell, S. Seltzer, J. Chang, J. Coursey, R. Sukumar, D. Zucker, and K. Olsen, “XCOM: Photon Cross Sections Database (version 1.5)”, [Online] Available: http://physics.nist.gov/xcom [2021, February 14]. National Institute of Standards and Technology, Gaithersburg, MD.

[76] L. Meitner, “Über die Entstehung der 훽-Strahl-Spektren radioaktiver Substanzen”, Zeitschrift für Physik 9, 131–144 (1922).

[77] P. V. Auger, “Sur les rayons 훽 secondaires produits dans un gaz par des rayons X”, Comptes Rendus Acad. Sci. 177(3), 169–171 (1923).

[78] M. Siegbahn, “Relations between the k and l series of the high- frequency spectra”, Nature 96(2416), 676–676 (1916).

[79] J. Als-Nielsen and D. McMorrow, Elements of Modern X-ray Physics. Chichester: John Wiley & Sons, Inc., 2 ed. (2011).

[80] A. H. Compton, “A quantum theory of the scattering of x-rays by light elements”, Phys. Rev. 21(5), 483–502 (1923).

[81] J. T. Bushberg, J. A. Seibert, E. M. Leidholdt, and J. M. Boone, The Essential Physics of Medical Imaging. Philadelphia, PA, USA: Lippincott Williams & Wilkins, 3 ed. (2012). REFERENCES | 85

[82] United States Army, Multiservice Tactics, Techniques, and Procedures for Treatment of Nuclear and Radiological Casualties (2014). [83] United Nations Scientific Committee on the Effects of Atomic Radi- ation, Sources and effects of ionizing radiation. New York: United Nations (2008). [84] M. Tubiana, L. E. Feinendegen, C. Yang, and J. M. Kaminski, “The linear no-threshold relationship is inconsistent with radiation biologic and experimental data”, Radiology 251(1), 13–22 (2009). [85] D. Attwood and A. Sakdinawat, X-Rays and Extreme Ultraviolet Ra- diation. United Kingdom: Cambridge University Press, 2 ed. (2016). [86] W. Duane and F. L. Hunt, “On x-ray wave-lengths”, Phys. Rev. 6(2), 166–172 (1915). [87] H. A. Kramers, “XCIII. On the theory of X-ray absorption and of the continuous X-ray spectrum”, Philos. Mag. 46(275), 836–871 (1923). [88] D. E. Grider, A. Wright, and P. K. Ausburn, “Electron beam melting in microfocus x-ray tubes”, J. Phys. D: Appl. Phys. 19(12), 2281–2292 (1986). [89] Excillum AB, Stockholm, Sweden, http://www.excillum.com. [90] O. Hemberg, M. Otendal, and H. M. Hertz, “Liquid-metal-jet anode electron-impact x-ray source”, Appl. Phys. Lett. 83(7), 1483–1485 (2003). [91] D. H. Larsson, P. A. C. Takman, U. Lundström, A. Burvall, and H. M. Hertz, “A 24 kev liquid-metal-jet x-ray source for biomedical applications”, Rev. Sci. Instrum. 82(12), 123701 (2011). [92] A. Thompson, D. Attwood, E. Gullikson, M. Howells, K.-J. Kim, J. Kirz, J. Kortright, I. Lindau, Y. Liu, P. Pianetta, A. Robinson, J. Scofield, J. Underwood, G. Williams, and H. Winick, X-Ray Data Booklet. Center for X-Ray Optics and Advanced Light Source, 3 ed. (2009). [93] Z. Huang and R. D. Ruth, “Laser-electron storage ring”, Phys. Rev. Lett. 80(5), 976–979 (1998). [94] R. J. Loewen, A Compact Light Source: Design and Technical Fea- sibility Study of a Laser-Electron Storage Ring X-Ray Source. PhD thesis, Stanford Linear Accelerator Center, Stanford University, Stan- ford, CA, (2003). 86 | REFERENCES

[95] E. Eggl, M. Dierolf, K. Achterhold, C. Jud, B. Günther, E. Braig, B. Gleich, and F. Pfeiffer, “The munich compact light source: initial performance measures”, J. Synchrotron Radiat. 23(5), 1137–1142 (2016). [96] B. Günther, R. Gradl, C. Jud, E. Eggl, J. Huang, S. Kulpe, K. Achter- hold, B. Gleich, M. Dierolf, and F. Pfeiffer, “The versatile X-ray beam- line of the Munich Compact Light Source: design, instrumentation and applications”, J. Synchrotron Radiat. 27(5), 1395–1414 (2020). [97] P. Favier, L. Amoudry, K. Cassou, K. Dupraz, A. Martens, H. Monard, and F. Zomer, “The compact x-ray source ThomX”, Proc. SPIE 10387, 1038708 (2017). [98] L. Faillace, R. G. Agostino, A. Bacci, R. Barberi, A. Bosotti, F. Broggi, P. Cardarelli, S. Cialdi, I. Drebot, V. Formoso, M. Gam- baccini, M. Ghedini, D. Giannotti, D. Giove, F. Martire, G. Met- tivier, P. Michelato, L. Monaco, R. Paparella, G. Paternó, V. Petrillo, F. Prelz, E. Puppin, M. R. Conti, A. R. Rossi, P. Russo, A. Sarno, D. Sertore, A. Taibi, and L. Serafini, “Status of compact inverse Compton sources in Italy: BriXS and STAR”, Proc. SPIE 11110, 1111005 (2019). [99] F. Albert and A. G. R. Thomas, “Applications of laser wakefield accelerator-based light sources”, Plasma Phys. Control. Fu- sion 58(10), 103001 (2016). [100] S. Kneip, C. McGuffey, F. Dollar, M. S. Bloom, V. Chvykov, G. Kalintchenko, K. Krushelnick, A. Maksimchuk, S. P. D. Mangles, T. Matsuoka, Z. Najmudin, C. A. J. Palmer, J. Schreiber, W. Schu- maker, A. G. R. Thomas, and V. Yanovsky, “X-ray phase contrast imaging of biological specimens with femtosecond pulses of betatron radiation from a compact laser plasma wakefield accelerator”, Appl. Phys. Lett. 99(9), 093701 (2011). [101] J. Wenz, S. Schleede, K. Khrennikov, M. Bech, P. Thibault, M. Heigoldt, F. Pfeiffer, and S. Karsch, “Quantitative x-ray phase- contrast microtomography from a compact laser-driven betatron source”, Nat. Commun. 6(1), 7568 (2015). [102] J. M. Cole, D. R. Symes, N. C. Lopes, J. C. Wood, K. Poder, S. Alatabi, S. W. Botchway, P. S. Foster, S. Gratton, S. Johnson, C. Kamperidis, O. Kononenko, M. D. Lazzari, C. A. J. Palmer, D. Rusby, J. Sanderson, M. Sandholzer, G. Sarri, Z. Szoke-Kovacs, L. Teboul, J. M. Thompson, J. R. Warwick, H. Westerberg, M. A. Hill, REFERENCES | 87

D. P. Norris, S. P. D. Mangles, and Z. Najmudin, “High-resolution μCT of a mouse embryo using a compact laser-driven x-ray betatron source”, Proc. Natl. Acad. Sci. U.S.A. 115(25), 6335–6340 (2018).

[103] F. Albert, A. G. R. Thomas, S. P. D. Mangles, S. Banerjee, S. Corde, A. Flacco, M. Litos, D. Neely, J. Vieira, Z. Najmudin, R. Bingham, C. Joshi, and T. Katsouleas, “Laser wakefield accelerator based light sources: potential applications and requirements”, Plasma Phys. Con- trol. Fusion 56(8), 084015 (2014).

[104] D. H. Larsson, W. Vågberg, A. Yaroshenko, A. Önder Yildirim, and H. M. Hertz, “High-resolution short-exposure small-animal laboratory x-ray phase-contrast tomography”, Sci. Rep. 6(1), 39074 (2016).

[105] C. W. E. van Eijk, “Inorganic scintillators in medical imaging”, Phys. Med. Biol. 47(8), R85–R106 (2002).

[106] M. J. Kitchen, G. A. Buckley, T. E. Gureyev, M. J. Wallace, N. Andres-Thio, K. Uesugi, N. Yagi, and S. B. Hooper, “CT dose reduction factors in the thousands using x-ray phase contrast”, Sci. Rep. 7(1), 15953 (2017).

[107] T. E. Gureyev, Y. I. Nesterets, A. Kozlov, D. M. Paganin, and H. M. Quiney, “On the ”unreasonable” effectiveness of transport of intensity imaging and optical deconvolution”, J. Opt. Soc. Am. A 34(12), 2251–2260 (2017).

[108] D. M. Paganin, Coherent X-Ray Optics. Oxford University Press (2006).

[109] M. R. Teague, “Deterministic phase retrieval: a Green’s function solution”, J. Opt. Soc. Am. 73(11), 1434–1441 (1983).

[110] A. Momose, “Recent Advances in X-ray Phase Imaging”, Jpn. J. Appl. Phys. 44(9R), 6355–67 (2005).

[111] A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics”, Phys. Med. Biol. 58(1), R1–R35 (2012).

[112] S. W. Wilkins, Y. I. Nesterets, T. E. Gureyev, S. C. Mayo, A. Pogany, and A. W. Stevenson, “On the evolution and relative merits of hard x-ray phase-contrast imaging methods”, Philos. Trans. R. Soc. A 372(2010), 20130021 (2014). 88 | REFERENCES

[113] T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Compari- son of two x-ray phase-contrast imaging methods with a microfocus source”, Opt. Express 21(25), 30183–30195 (2013).

[114] A. Momose, “Demonstration of phase-contrast X-ray computed tomography using an X-ray interferometer”, Nucl. Instr. Meth. Phys. Res. A 352(3), 622–628 (1995).

[115] K. Goetz, M. P. Kalashnikov, Y. A. Mikhaĭlov, G. V. Sklizkov, S. I. Fe- dotov, E. Foerster, and P. Zaumseil, “Measurements of the parameters of shell targets for laser thermonuclear fusion using an x-ray schlieren method”, Soviet Journal of Quantum Electronics 9, 607–610 (1979).

[116] E. Förster, K. Goetz, and P. Zaumseil, “Double crystal diffractome- try for the characterization of targets for laser fusion experiments”, Kristall und Technik 15(8), 937–945 (1980).

[117] T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays”, Nature 373(6515), 595–598 (1995).

[118] V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object”, J. Phys. D: Appl. Phys. 28(11), 2314–2317 (1995).

[119] D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Say- ers, “Diffraction enhanced x-ray imaging”, Phys. Med. Biol. 42(11), 2015–2025 (1997).

[120] H. Talbot, “LXXVI.Facts relating to optical science. No. IV”, Philos. Mag. Series 3. 9(56), 401–407 (1836).

[121] A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray talbot interferometry”, Jpn. J. Appl. Phys. 42, L866–L868 (2003).

[122] T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer”, Opt. Express 13(16), 6296–6304 (2005).

[123] F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources”, Nat. Phys. 2(4), 258–261 (2006). REFERENCES | 89

[124] F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönni- mann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer”, Nat. Mater. 7(2), 134–137 (2008).

[125] H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based x-ray phase-contrast imaging using fourier transform phase retrieval”, Opt. Express 19(4), 3339–3346 (2011).

[126] G. Sato, T. Kondoh, H. Itoh, S. Handa, K. Yamaguchi, T. Naka- mura, K. Nagai, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional gratings-based phase-contrast imaging using a conventional x-ray tube”, Opt. Lett. 36(18), 3551–3553 (2011).

[127] F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard x-ray phase tomography with low-brilliance sources”, Phys. Rev. Lett. 98(10), 108105 (2007).

[128] M. Bech, O. Bunk, T. Donath, R. Feidenhans'l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography”, Phys. Med. Biol. 55(18), 5529–5539 (2010).

[129] K. Willer, A. Fingerle, W. Noichl, F. De Marco, M. Frank, T. Ur- ban, R. Schick, A. Gustschin, B. Gleich, J. Herzen, T. Koehler, A. Yaroshenko, T. Pralow, G. Zimmermann, B. Renger, A. Sauter, D. Pfeiffer, M. Makowski, E. Rummeny, P. Grenier, and F. Pfeiffer, “X-ray dark-field chest imaging can detect and quantify emphy-sema in copd patients”, medRxiv (2021).

[130] K. S. Morgan, D. M. Paganin, and K. K. W. Siu, “X-ray phase imaging with a paper analyzer”, Appl. Phys. Lett. 100(12), 124102 (2012).

[131] I. Zanette, T. Zhou, A. Burvall, U. Lundström, D. H. Larsson, M. Zdora, P. Thibault, F. Pfeiffer, and H. M. Hertz, “Speckle-based x- ray phase-contrast and dark-field imaging with a laboratory source”, Phys. Rev. Lett. 112(25), 253903 (2014).

[132] A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field”, Med. Phys. 28(8), 1610–1619 (2001).

[133] M. Endrizzi, P. C. Diemoz, T. P. Millard, J. L. Jones, R. D. Speller, I. K. Robinson, and A. Olivo, “Hard x-ray dark-field imaging with 90 | REFERENCES

incoherent sample illumination”, Appl. Phys. Lett. 104(2), 024106 (2014).

[134] A. Olivo and R. Speller, “A coded-aperture technique allowing x- ray phase contrast imaging with conventional sources”, Appl. Phys. Lett. 91(7), 074106 (2007).

[135] P. R. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, “Phase and absorption retrieval using incoherent x-ray sources”, Proc. Natl. Acad. Sci. U.S.A. 109(35), 13922–13927 (2012).

[136] G. K. Kallon, M. Wesolowski, F. A. Vittoria, M. Endrizzi, D. Basta, T. P. Millard, P. C. Diemoz, and A. Olivo, “A laboratory based edge- illumination x-ray phase-contrast imaging setup with two-directional sensitivity”, Appl. Phys. Lett. 107(20), 204105 (2015).

[137] F. Krejci, J. Jakubek, and M. Kroupa, “Hard x-ray phase contrast imaging using single absorption grating and hybrid semiconductor pixel detector”, Rev. Sci. Instrum. 81(11), 113702 (2010).

[138] F. Krejci, J. Jakubek, and M. Kroupa, “Single grating method for low dose 1-d and 2-d phase contrast x-ray imaging”, J. Instrum. 6(01), C01073 (2011).

[139] J. W. Goodman, Introduction to Fourier optics. Roberts & Company Publishers, 3rd ed. (2005).

[140] R. Grella, “Fresnel propagation and diffraction and paraxial wave equation”, J. Opt. 13(6), 367–374 (1982).

[141] P. Cloetens, R. Barrett, J. Baruchel, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging”, J. Phys. D: Appl. Phys. 29(1), 133–146 (1996).

[142] A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source”, Rev. Sci. Instrum. 68(7), 2774–2782 (1997).

[143] S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, “X-ray phase-contrast microscopy and microtomography”, Opt. Express 11(19), 2289–2302 (2003).

[144] Y. I. Nesterets, S. W. Wilkins, T. E. Gureyev, A. Pogany, and A. W. Stevenson, “On the optimization of experimental parameters for x- ray in-line phase-contrast imaging”, Rev. Sci. Instrum. 76(9), 093706 (2005). REFERENCES | 91

[145] I. Häggmark, “3-material phase retrieval in phase-contrast x-ray tomography”, Master’s thesis, KTH Royal Institute of Technology, Stockholm (2016).

[146] J. Radon, “Über die Bestimmung von Funktionen durch ihre Inte- gralwerte längs gewisser Mannigfaltigkeiten”, Berichte über die Ver- handlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig 69, 262–277 (1917).

[147] J. Radon, “On the determination of functions from their integral values along certain manifolds”, IEEE T. Med. Imaging 5, 170–176 (1986).

[148] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging. New York: IEEE Press (1988).

[149] M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in x-ray ct”, Physica Medica 28(2), 94–108 (2012).

[150] L. A. Feldkamp, L. C. Davis, and J. W. Kress, “Practical cone-beam algorithm”, J. Opt. Soc. Am. A 1(6), 612–619 (1984).

[151] J. F. Barrett and N. Keat, “Artifacts in CT: Recognition and Avoid- ance”, RadioGraphics 24(6), 1679–1691 (2004).

[152] H. Nyquist, “Certain topics in telegraph transmission theory”, Trans- actions of the American Institute of Electrical Engineers 47(2), 617–644 (1928).

[153] C. E. Shannon, “Communication in the presence of noise”, Proceedings of the IRE 37(1), 10–21 (1949).

[154] R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy”, Proc. Roy. Soc. Lond. A. 317(1530), 319–340 (1970).

[155] R. Hegerl and W. Hoppe, “Influence of Electron Noise on Three- dimensional Image Reconstruction”, Zeitschrift für Naturforschung A 31(12), 1717–1721 (1976).

[156] W. L. Wagner, F. Wuennemann, S. Pacilé, J. Albers, F. Arfelli, D. Dreossi, J. Biederer, P. Konietzke, W. Stiller, M. O. Wielpütz, A. Accardo, M. Confalonieri, M. Cova, J. Lotz, F. Alves, H.-U. Kauc- zor, G. Tromba, and C. Dullin, “Towards synchrotron phase-contrast lung imaging in patients – a proof-of-concept study on porcine lungs 92 | REFERENCES

in a human-scale chest phantom”, J. Synchrotron Radiat. 25(6), 1827–1832 (2018).

[157] A. Burvall, U. Lundström, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography”, Opt. Express 19(11), 10359–10376 (2011).

[158] P. Cloetens, W. Ludwig, J. Baruchel, D. Van Dyck, J. Van Lan- duyt, J. P. Guigay, and M. Schlenker, “Holotomography: Quantitative phase tomography with micrometer resolution using hard synchrotron radiation x rays”, Appl. Phys. Lett. 75(19), 2912–2914 (1999).

[159] M. Langer, P. Cloetens, A. Pacureanu, and F. Peyrin, “X-ray in- line phase tomography of multimaterial objects”, Opt. Lett. 37(11), 2151–2153 (2012).

[160] D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object”, J. Microsc. 206(1), 33–40 (2002).

[161] M. Beltran, D. Paganin, K. Uesugi, and M. Kitchen, “2D and 3D X-ray phase retrieval of multi-material objects using a single defocus distance”, Opt. Express 18(7), 6423–6436 (2010).

[162] M. Ullherr and S. Zabler, “Correcting multi material artifacts from single material phase retrieved holo-tomograms with a simple 3D Fourier method”, Opt. Express 23(25), 32718–32727 (2015).

[163] A. Ruhlandt and T. Salditt, “Three-dimensional propagation in near- field tomographic X-ray phase retrieval”, Acta Crystallogr. A 72(2), 215–221 (2016).

[164] A. Badal and A. Badano, “Accelerating monte carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit”, Med. Phys. 36(11), 4878–4880 (2009).

[165] Y. Sung, W. P. Segars, A. Pan, M. Ando, C. J. R. Sheppard, and R. Gupta, “Realistic wave-optics simulation of x-ray phase-contrast imaging at a human scale”, Sci. Rep. 5(1), 12011 (2015).

[166] U. Lundström, D. H. Larsson, A. Burvall, P. A. C. Takman, L. Scott, H. Bismar, and H. M. Hertz, “X-ray phase contrast for CO2 microan- giography”, Phys. Med. Biol. 57(9), 2603–2617 (2012). REFERENCES | 93

[167] K. S. Morgan, K. K. W. Siu, and D. M. Paganin, “The projection approximation and edge contrast for x-ray propagation-based phase contrast imaging of a cylindrical edge”, Opt. Express 18(10), 9865–9878 (2010).

[168] K. S. Morgan, K. K. Siu, and D. M. Paganin, “The projection approximation versus an exact solution for x-ray phase contrast imaging, with a plane wave scattered by a dielectric cylinder”, Opt. Com- mun. 283(23), 4601–4608 (2010).

[169] B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digi- mouse: a 3d whole body mouse atlas from CT and cryosection data”, Phys. Med. Biol. 52, 577–587 (2007).

[170] C. Lee, D. Lodwick, J. Hurtado, D. Pafundi, J. L. Williams, and W. E. Bolch, “The UF family of reference hybrid phantoms for computational radiation dosimetry”, Phys. Med. Biol. 55(2), 339–363 (2009).

[171] W. P. Segars, G. Sturgeon, S. Mendonca, J. Grimes, and B. M. W. Tsui, “4d xcat phantom for multimodality imaging research”, Med. Phys. 37(9), 4902–4915 (2010).

[172] M.-C. Gosselin, E. Neufeld, H. Moser, E. Huber, S. Farcito, L. Ger- ber, M. Jedensjö, I. Hilber, F. D. Gennaro, B. Lloyd, E. Cherubini, D. Szczerba, W. Kainz, and N. Kuster, “Development of a new generation of high-resolution anatomical models for medical device evalua- tion: the virtual population 3.0”, Phys. Med. Biol. 59(18), 5287–5303 (2014).

[173] E. Abadi, W. P. Segars, B. M. W. Tsui, P. E. Kinahan, N. Bottenus, A. F. Frangi, A. Maidment, J. Lo, and E. Samei, “Virtual clinical trials in medical imaging: a review”, J. Med. Imaging 7(4), 1–40 (2020).

[174] E. Abadi, W. P. Segars, G. M. Sturgeon, J. E. Roos, C. E. Ravin, and E. Samei, “Modeling lung architecture in the XCAT series of phantoms: Physiologically based airways, arteries and veins”, IEEE T. Med. Imaging 37(3), 693–702 (2018).

[175] K. D. Harrison and D. M. L. Cooper, “Modalities for visualization of cortical bone remodeling: The past, present, and future”, Front. Endocrinol. 6, 122 (2015).

[176] H. M. Britz, Y. Carter, J. Jokihaara, O. V. Leppänen, T. L. Järvinen, G. Belev, and D. M. Cooper, “Prolonged unloading in growing rats 94 | REFERENCES

reduces cortical osteocyte lacunar density and volume in the distal tibia”, Bone 51(5), 913–919 (2012). [177] A. S. Slutsky and V. M. Ranieri, “Ventilator-induced lung injury”, N. Engl. J. Med. 369(22), 2126–2136 (2013). [178] J. Fourier, Théorie analytique de la chaleur. Paris: F. Didot (1822). [179] W. O. Saxton and W. Baumeister, “The correlation averaging of a reg- ularly arranged bacterial cell envelope protein”, J. Microsc. 127(2), 127–138 (1982). [180] M. van Heel, W. Keegstra, W. Schutter, and van Bruggen E.F.J., “Arthropod hemocyanin studies by image analysis”, in Life Chemistry Reports Suppl. 1, Structure and Function of Invertebrate Respiratory Proteins, EMBO Workshop, Leeds (E. Wood, ed.), 69–73, (1982). [181] M. van Heel and M. Schatz, “Fourier shell correlation threshold cri- teria”, J. Struct. Biol. 151(3), 250–262 (2005).