Augmenting Soft Tissue Contrast Using Phase-Contrast Techniques in Micro-Computed Tomography Imaging

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AUGMENTING SOFT TISSUE CONTRAST USING

PHASE-CONTRAST TECHNIQUES IN MICRO-COMPUTED TOMOGRAPHY IMAGING

Aino Reunamo Master’s Thesis University of Eastern Finland Department of Applied Physics January 24, 2019

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UNIVERSITY OF EASTERN FINLAND, Faculty of Science and Forestry Department of Applied Physics, Medical Physics

Aino Reunamo: Augmenting Soft Tissue Contrast using Phase-Contrast Techniques in Micro- Computed Tomography

Master’s Thesis of Natural Sciences, 62 pages

Supervisors: Professor Jari Hyttinen and Professor Ville Kolehmainen January 2019

Key words: X-ray imaging, micro-computed tomography, phase-contrast imaging Abstract

X-ray imaging and computed tomography is a widely used tool in clinics as well as in research applications. Micro-computed tomography (µCT) is used in research for visualization of cross-sectional slices of a sample as well as for reconstruction of data for 3D visualization.

Conventional X-ray imaging is based on the attenuation of X-rays in the sample and the technique is referred to as absorption imaging. This technique provides useful information when the differences in attenuation coefficients is sufficient resulting in good contrast. On the other hand, low density samples such as soft tissues consist of various structures with similar densities, resulting in poor contrast in absorption images. This necessitates the use of a stain or phase imaging to enhance contrast in the sample during imaging. Staining has some limitation as not all samples can be stained, stains can be toxic, and staining can be time consuming. Therefore, new imaging techniques are required to image low density samples without the use of a stain.

A promising imaging possibility on a µCT is the use of phase information of an X-ray beam to generate an image of the sample. This technique is known as phase-contrast imaging and it is based on the refraction of X-rays due to differences in refractive indices. The Zeiss Xradia MicroXCT-400 provides the possibility of using propagation-based phase-imaging. In this Master’s Thesis, the phase-contrast imaging protocol on the Zeiss Xradia MicroXCT- 400 was optimized using thin polylactic acid fibers in order to enhance the visibility of low density samples. The optimization consisted of source and detector distance, power, and voltage variation measurements. The projection images were analyzed for fiber visibility and contrast-to-noise ratios (CNR) were calculated. The results of the optimization were applied to two types of collagen samples embedded in air, ethanol, and water.

The results of the analysis showed enhanced contrast for phase-contrast images compared to absorption images. Not all of the results were in alignment with theory, which is most likely due to the specifications and non-ideal operation of the X-ray source. The most important results indicated that the source does not need to placed at the negative limit to obtain useful phase information and the visibility increases with increasing sample-to-detector distance.

For a magnification of 4x, the optimal detector distances, with sufficient CNRs and reasonable exposure time, for source distances of 160mm, 230mm, and 300mm are 80mm, 90mm, and 90mm, respectively. Finally, significantly enhanced contrast was obtained for the collagen sample embedded in water using phase-imaging techniques compared to absorption imaging.

Even though enhanced contrast was obtained using phase-imaging techniques, the technique is limited due to the focal spot size and voltage of the X-ray source. As phase and absorption data cannot be separated, the final images are a combination of both. As a result, the optimal imaging settings are one which provide absorption information without a significant increase in noise as well as one which provides the edge enhancement effect typical for propagation-based phase-imaging. Further improvement of contrast and the use of phase information would require the application of phase retrieval on the raw projection data or

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Tiivistelmä

Röntgenkuvantaminen ja tietokonetomografia on laajalti käytössä oleva kuvantamis- menetelmä sekä sairaalassa että tutkimusympäristössä. Mikrotietokonetomografiaa (µCT) käytetään erityisesti tutkimuksessa visualisoimaan näytteiden poikkileikkauksia sekä rekon- struoimaan 3D malleja. Perinteinen röntgenkuvaus perustuu säteiden vaimenemiseen näyt- teessä ja menetelmää kutsutaan absorptiokuvantamiseksi. Menetelmä antaa hyödyllistä tietoa näytteiden rakenteesta, kun vaimenemiskertoimien erot vierekkäisten rakenteiden välillä on tarpeeksi suuria. Toisaalta, esimerkiksi pehmytkudosnäytteet koostuvat rakenteista, joiden tiehyserot ovat pieniä, minkä vuoksi absorptiokuvien kontrasti on huono. Tämän vuoksi on usein tarpeen käyttää leimausta parantamaan röntgenkuvien kontrastia. Useissa sovelluksissa leimaus antaa hyödyllistä tietoa näytteestä, mutta leimauksella on myös negati- ivisia puolia. Kaikkia näytteitä ei ole mahdollista leimata, osa leimoista on toksisia ja vär- jäys on usein aikaa vievää. Tämän vuoksi matalatiheyksisten näytteiden kuvantamiseen on tarpeellista kehittää uusia kuvantamismenetelmiä, jotka eivät vaadi näytteen leimausta.

Vaiheinformaation hyödyntäminen röntgenkuvan muodostamiseen on lupaava uusi ku- vantamismenetelmä, jota voidaan hyödyntää µCT:ssä. Menetelmää kutsutaan faasikon- trastikuvantamiseksi ja se perustuu röntgensäteiden siroamiseen, mikä johtuu taitekertoimien eroavaisuuksista rajapinnoilla. Zeiss Xradia MicroXCT-400µCT:llä on mahdollisuus hyödyn- tää vaiheen etenemiseen perustuvaa faasikuvausta röntgenkuvien muodostamiseen. Tässä Pro Gradu -tutkielmassa optimoitiin faasikontrastiin perustuva kuvantamisprotokolla Zeiss Xradia MicroXCT-400 laitteella kontrastin parantamiseksi. Optimointiin käytettiin ohuita polylaktidi lankoja ja optimointi koostui lähteen ja detektorin etäisyyksien, tehon ja jännit- teen arvojen vaihtelusta. Eri arvoilla otettiin projektiokuva ja kuvista analysoitiin lankojen näkyvyys sekä osasta laskettiin kontrastin suhde kohinaan. Optimoinnin tuloksia sovellet- tiin kahdenlaiseen kollageeni näytteeseen, jotka kuvattiin tomografiamenetelmällä. Ensim- mäisessä tomografiassa näytteet olivat ilmassa, sen jälkeen ne upotettiin etanoliin ja vi- imeiseksi veteen.

Analyysin tulokset osoittivat, että faasikuvantamisella saatiin parannettua lankojen kontrastia verrattuna absorptiokuviin. Toisaalta, kaikki tulokset eivät olleet teorian mukaisia, mikä johtuu todennäköisesti röntgenlähteen teknisistä ominaisuuksista. Olennaisimmat tulokset osoittivat, että röntgenlähdettä ei tarvitse sijoittaa negatiiviseen maksimiin, jotta saadaan hyödyllistä faasi-informaatiota. Lisäksi, lankojen näkyvyys paranee, kun detek- toria siirretään kauemmaksi näytteestä. Kun käytettiin suurennoksena 4x, optimaalinen detektorin etäisyys lähteen etäisyyksille 160mm, 230mm ja 300mm olivat 80mm, 90mm ja 90mm vastaavasti. Näillä asetuksilla saatiin kohtuullinen kontrastin suhde kohinaan sekä valotusaika. Lisäksi, kollageeninäytteen näkyvyys parantui huomattavasti faasikuvantamis- menetelmällä verrattuna absorptiokuvaukseen, kun näyte oli upotettuna veteen.

Faasikontrastikuvantamisella saatiin parannettua röntgenkuvien kontrastia huomattavasti, mutta tekniikka on rajoittunut johtuen röntgenlähteen alimmasta mahdollisesta jännitteestä sekä röntgenlähteen polttopisteen koosta. Kyseinen µCT laite ei pysty erot- telemaan faasi- ja absorptioinformaatiota toisistaan, minkä vuoksi lopullinen röntgenkuva sisältää molemmat informaatiot. Tästä johtuen optimaaliset kuvantamisparametrit ovat sel- laiset, jotka tuottavat absorptioinformaation ilman kohinan määrän liiallista nousua, mutta samalla tuottavat reunoja tehostavan faasi-informaation, joka on tyypillinen vaiheen etenemiseen perustuvaan faasikuvantamismenetelmään. Faasikuvantamista kyseisellä µCT:llä voisi kehittää esimerkiksi yrittämällä "phase retrieval" -tekniikka, jossa haetaan aallon vaihe näytteen sisällä raa’asta projektiodatasta tai käyttämällä absorptio- ja faasiristikoita kuvan- tamislaitteistossa faasi-informaation keräämistä varten.

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Abbreviations

CNR Contrast-to-noise ratio

CT Computed Tomography

DPC Differential phase-contrast FBP Filtered backprojection

FOV Field of view

PLA Polylactic acid

ROI Region of interest

SNR Signal-to-noise ratio

TIE Transport of intensity equation

µCT micro-Computed Tomography

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Variables

A Atomic weight

β Imaginary refractive index decrement

d Distance between phase and amplitude gratings dm Fractional Talbot distance

d_T Talbot distance

δ Real refractive index decrement

E X-ray energy

f(x, y;z) Diffracted X-ray

I Intensity

Iin Incident intensity

I_R Incident intensity on analyzer crystal

i Imaginary unit

L Radius of spherical wave

l Path traveled by X-ray

l_coh Coherence length

λ X-ray wavelength

µ Linear attenuation coefficient

N Number of transmitted photons

N0 Number of incident photons

N_A Avogadro’s constant

n Index of refraction

p grating period

p0 Source grating period

p₁ Phase grating period

p₂ Absorption grating period

Φ Phase profile of wavefront

ϕ Phase shift

φ Oscillations phase

R1 Distance from X-ray source to object R₂ Distance from object to detector r₀ Classical electron radius

ρ Sample density

ρ_e Electron density

s Size of the focal spot of the X-ray source

σbg Standard deviation of background region of interest σ_c Compton scattering cross section

σ_pe Photoelectric effect cross section

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σ_pe Photoelectric effect cross section

Θ Refraction angle

T(x, y) Transmission function T(x, y) Projected thickness

ω Angular frequency

x¯_bg Average gray scale inside background region of interest x_i Gray scale value at pixeli

x¯s Average gray scale inside subject region of interest

Z Atomic number

Z_eff Effective atomic number

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Preface

This thesis has been carried out in the Computational Biophysics and Imaging Group (CBIG) led by professor Jari Hyttinen, at the Institute of Biosciences and Medical Technology, BioMediTech, which combines University of Tampere and Tampere University of Technology.

The thesis has been funded by the Center of Excellence (CoE) of the Body-on-Chip Research.

I wish to thank my supervisors professor Jari Hyttinen who took a chance on me and gave me the opportunity to work on this interesting topic in his research group and professor Ville Kolehmainen who kindly contacted Jari and inquired about the chance to work on my Master’s Thesis in Tampere. I also wish to thank all members of CBIG who have assisted me with this thesis. A special thank you to Markus Hannula, M.Sc. (Tech.), for always offering your assistance, as well as Ilmari Tamminen, M.Sc., for all the interesting conversations.

I would also like to thank all my lecturers from the Department of Applied Physics at the University of Eastern Finland along with all the friends I made during the past five years.

Without your support I would most likely not be writing this thesis. Thank you for all the motivation and encouragement you have given me when the workload has seemed impossible.

Furthermore, a sincere thank you to my parents for providing me with your unwavering love and support and for always believing in me. Thank you to my siblings for taking your time and visiting us in Kuopio, making us focus on something else than our studies.

The biggest thank you I owe to my beloved husband, Jouni. Thank you for staying calm when I was not, for providing me with your constant love and encouragement, and most importantly for taking care of our lovely daughter when I had to work long hours at the university. I could not have done this without you.

Roosa, thank you for being ours, you light up our world.

Aino Reunamo

Tampere, January 2019

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1 Introduction

X-ray imaging as well as computed tomography is a widely used imaging technique in medicine and research. X-ray techniques in clinical use are based on the attenuation of X-rays in the sample and is referred to as absorption imaging [11]. This technique is also widely used in micro-computed tomography in laboratory devices as well as synchrotrons [36]. Absorption imaging provides high quality images with good contrast when the attenuation between different regions differs sufficiently. On the other hand, when low density samples such as soft tissues and some biomaterials are imaged using absorption settings, the resulting contrast is poor. Various stains can be used to stain a sample or specific parts of it in order to enhance the contrast of poorly absorbing samples [34, 47]. This can reveal important structures for visualizations but on the other hand, not everything can be stained, some stains can destroy samples, and staining can be time-consuming [34]. Due to this, imaging methods for the visualization of soft tissues and poorly absorbing samples are of high importance. This could also be potentially useful in clinical use, in applications such as mammography [20].

Phase-contrast techniques have been studied as they have shown potential in providing enhanced contrast for low density samples [3, 28]. There are several different techniques for generating phase contrast using X-ray techniques, most of which require highly powerful X- ray sources and additional optical instruments [10, 33]. These techniques vary in complexity and feasibility. One such technique, propagation-based phase-contrast imaging, is a phase- imaging technique, which sets the least amount of requirements for the imaging setup and phase information can be generated using a polychromatic X-ray source [14]. This allows the technique to be investigated in many conventional laboratories without the need to image with highly powerful X-ray sources or synchrotrons. Therefore, the purpose of this Master’s Thesis was to optimize the phase-imaging protocol on the Xradia MicroXCT-400 (Zeiss, Pleasanton, CA, USA) and to apply and test the results on low density samples that could benefit from phase-contrast imaging. The main aspect of the optimization is to increase the spatial coherence length of the X-ray beam by varying the source distance, power output and voltage of the device. Detector distance variation is also included in the optimization. As phase-imaging studies have been primarily investigated with synchrotrons and X-ray sources with better specifications, little information was available on the capabilities of generating phase-information on similar X-ray sources. Even though the technique is not ideally suited for an X-ray source in the Zeiss Xradia MicroXCT-400, it could have the potential of providing enhanced contrast for low density samples.

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2 Radiography and Computed Tomography

2.1 Radiography

Radiography, or X-ray imaging as it is more commonly known as, is used in various applications, including medical imaging and research. The use of X-rays for imaging purposes was discovered by Wilhelm Röntgen in 1895 and has been since utilized in various fields of study. A typical imaging setup consists of an X-ray source, the sample to be imaged, and an X-ray detector. The image formed on the detector represents the distribution of X-rays in the sample. [11]

2.1.1 Interaction of Radiation with Matter

When an X-ray beam comes in contact with and penetrates matter, a fraction of the X- rays are attenuated from the beam due to interactions in the sample, such as absorption and scatter. The attenuation of X-rays depends on the effective atomic number of the sample, the thickness of the sample, and the sample density. The interactions resulting in attenuation occur by different mechanisms depending on the photon energy. These interactions are photoelectric effect, Compton and Rayleigh scattering, and pair production. [11, 49]

The photoelectric effect occurs when a photon with sufficient energy releases an electron from an electron shell of an atom. This occurs if the incident photon has energy equal to or more than the binding energy of the electron to be released. When the photon has ejected the electron, a vacancy in the electron shell is created, which is filled by an electron from a higher electron shell. This vacancy is further filled by another electron and the chain reaction continues until the vacancies are filled. When an electron fills the vacancy of an electron from a lower electron shell, a characteristic X-ray or an Auger electron is released as a result of the differences in electron binding energies. The higher the atomic number of the atom from where the electron is released, the higher is the energy of the characteristic radiation. As higher energy photons can easily penetrate the rest of the matter, better contrast is achieved for absorber materials with high atomic number. [11, 12, 49]

As the photoelectric effect is an absorption phenomenon, image quality is not degraded by scattered X-rays. The photoelectric effect is proportional to 1/E³ where E is the photon energy, which partly explains why image contrast decreases with increasing X-ray energy. [11]

The exception to this is at the absorption edges of a particular element. The probability of photoelectric absorption increases when the photon energy is just above the absorption edge and decreases when the energy is just below the absorption edge. Therefore, the probability of photoelectric absorption increases with energy at certain intervals when the photon energy is just above the absorption edge of the element. The electron binding energy at an electron shell corresponds to the absorption edge of that shell. [11, 49]

Another interaction mechanism between photons and matter is Compton scattering, which is the main interaction mechanism between soft tissue and photons in the diagnostic energy range [11]. The interaction occurs between the incoming photon and the outer electrons of the target material. An incident photon releases an outer electron and the photon is absorbed. As a result, the atom is ionized and the leftover energy is emitted as a new photon. This new photon has energy less than the incident photon and the its path is random, which is why the new photon is referred to as a scattered photon. The scattered photons can further interact through photoelectric absorption and scattering or it can travel through the matter without interaction. In this case it could be detected by the detector,

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resulting in misplaced information and noise in the final image. For Compton scattering to occur, the incident photon needs to have a significantly higher energy than the binding energy of the electron. As a result, the probability of Compton scattering relative to photoelectric absorption increases with increasing energy of the incident photon. [11, 12]

The remaining interactions mechanisms, Rayleigh scattering and pair production, are less common at diagnostic X-ray energies. Rayleigh scattering occurs when an incident photon excites an entire atom. As a result, a scattered photon of the same energy as the incident photon is emitted. The interactions primarily occurs at very low X-ray energies and the only diagnostic application, which uses low enough X-ray energies is mammography. [4, 11] On the other hand, imaging of biological samples often is done at low energies to enhance contrast. In these applications, the photon energies can be lower than 30 keV, which increases the amount of Rayleigh scattering. As mentioned before, the detection of the scattered photons degrades image quality. Pair production only occurs when photon energies are above 1.02 MeV, which is above diagnostic X-ray energies. [11]

2.1.2 Attenuation

The interactions between radiation an matter described previously result in the attenuation of photons in matter. The linear attenuation coefficient µ is used to describe the fraction of removed photons per unit thickness. An exponential equation is used to describe the relationship between the number of incident photons (N₀) and the amount of transmitted photons (N) passing through a thickness ofx [4, 11]

N =N0e^−µx. (1)

In addition to the thickness of the material, the density of the matter affects the attenuation of photons [4, 11]. If the density of a material is high, also the number of atoms is high and therefore the probability of interaction of photons with matter increases [11]. In order to incorporate density information into the attenuation coefficient, the linear attenuation coefficient can be normalized to unit density, after which it is called the mass attenuation coefficientµ/ρ[4, 11]. Equation (1) can be written using the mass attenuation coefficient as

N =N₀e⁻ (_µ

ρ

)ρx

. (2)

2.2 Computed Tomography

Even though traditional X-ray imaging offers valuable information of the imaged sample, the technique has some serious limitations. Firstly, an X-ray image is a 2D representation of a 3D object and is therefore a combination of structures found at different depths of the patient. Another limitation of X-ray imaging is the fact that attenuation is dependent on the attenuation coefficient of matter and on the thickness of the matter and the contribution of either one is not known. [49] Computed tomography (CT) was introduced in 1971 [4] and provides a solution to many of the limitations of conventional radiography. CT scanners are made for specific applications; a clinical CT scanner is used to image parts of the human body while smaller samples, such as tissue samples or biomaterials, are imaged using a smaller micro-CT scanner.

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2.2.1 CT Scanner and Image Acquisition

A typical clinical computed tomography imaging system consists of a rotating gantry, a housing surrounding the gantry, as well as the patient table which moves through the gantry.

The position of the patient table can be controlled from the computer, which controls the entire image acquisition procedure. A drawing of a CT scanner is depicted in Figure 1.

[11, 49] The gantry contains all the required systems for producing and detecting X-rays, as well as what is required for the conversion of analog data to digital signal [11].

The X-ray beam in a typical clinical CT scanner is a fan or cone shaped beam and collimators are used to define the size and shape of the beam [11]. After the X-ray beam has traversed through the subject, it is detected by an array of detector elements arranged in an arc on the opposite side of the X-ray tube [4, 11]. The detector arrangement defines the slice thickness, which is the width of the imaged slice in the patient [11]. In modern scanners, the X-ray tube and detector are attached to the rotating gantry so that during imaging they rotate synchronized [4, 11]. This enables the use of grids to remove scattered radiation [11].

A tomographic image is formed by passing an X-ray beam through the subject at a number of different angles. This is done by rotating the gantry 360^◦ around the sample and collecting the information on the other side. There are various ways of combining rotation and acquisition of the projections, the typical acquisition mode for many clinical applications is a helical one, in which the acquisitions is continuous and the patient table moves continuously.

The advantage of this method is the fast acquisitions and the patient radiation dose can be significantly reduced by adjusting the pitch of the patient table. The pitch describes the speed of the patient table relative to the gantry rotation. The number of projections required depends on the number of pixels for which the attenuation coefficients are calculated and is typically thousands of projections. A computer is used to reconstruct a 2D tomographic image from the 1D projections. [4, 11, 49] The 2D images can be thought of as attenuation maps of the different slices from the imaged sample. The slices can then be combined to form a 3D image. [49]

Figure 1: A drawing of the side view of a clinical CT scanner. The gantry houses the X-ray tube and detector elements. The patient table moves horizontally and vertically, so that it can be adjusted into the center of the bore and moved during image acquisition. The shaded red are depicts the location of the X-ray beam. Modified from [11].

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2.2.2 Micro-Computed Tomography

Various research fields use CT techniques to obtain information of the internal structures of for example small animals, materials, and tissue samples. The imaging is typically performed using micro-computed tomography (µCT), which can provide a resolution ranging from under one micrometer to tens of micrometers. The µCT setup consists of the X-ray source and detector as well as the sample placed in between them. In contrast to clinical CT devices, there are two alternate setups for µCT systems; either the X-ray source and detector or the sample is rotated to obtain the required projections. If the sample bed is rotated, the sample must be well fixed on the bed in order to avoid any movement of the sample, which deteriorates the final image quality. The different setups are depicted in Figure 2. [31]

X-ray sources used in µCT imaging are divided into two categories: laboratory X-ray sources and synchrotron radiation sources. A synchrotron provides a monochromatic, highly collimated radiation beam, which allows for high resolution images, but these are typically not practical for many laboratories. On the other hand, typical X-ray sources used in laboratories do not produce monochromatic light and therefore contain a wide variety of X-ray wavelengths. [31, 36] Briefly, laboratory X-ray sources produce X-rays by applying an elec- tric potential between a cathode and an anode. Electrons from the cathode travel to the anode and are accelerated by the voltage between the cathode and anode. When the electrons reach the anode, a small fraction of the kinetic energy of the electrons is converted to X-ray photons. This occurs when the electron comes in close contact with an atomic nucleus and the positive charge of the nucleus causes the electron to decelerate. This results in the electron losing some of its kinetic energy, which is emitted as an X-ray photon. The amount of X-ray radiation produced depends on the atomic number of the target anode as well as the kinetic energy of the electrons. [11] The basic principle of the X-ray source and detector are the same in clinical andµCT imaging but some differences in the setup are possible. For example, in aµCT scanner, the use of grids to remove scattered radiation could be optional as radiation regulations are not as strict and the device can also hold additional optical lenses for magnification [56].

Figure 2: Setups used inµCT scanners: (a) rotational sample stage setup and (b) rotational source and detector setup. Modified from [31].

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2.2.3 Tomographic reconstruction

Before the projection data is reconstructed, several procedures are typically conducted as preprocessing steps to the data set. Dead detector elements are identified and interpolation is used to replace the dead pixel data, scatter correction is applied to reduce the amount of noise in the image, and algorithms that detect areas with low signal can also be applied.

These areas typically contain a relatively high amount of noise and smoothening can be applied to these areas for noise reduction. Finally, the logarithmic attenuation of X-rays is corrected for after which the attenuation coefficients can be used in a linear fashion. [11]

The reconstruction of the projection data to obtain 2D images can be done using several algorithms. Some reconstruction methods include simple backprojection, filtered backprojection (FBP), Fourier-based reconstruction, and iterative reconstruction. The FBP method is the most common reconstruction method used in CT imaging, but iterative techniques are starting to replace FBP methods as algorithms become faster and computation power increases. [11, 49]

Backprojection refers to the calculation of the attenuation coefficients in all of the matrix pixels from the measured projections. In the simple backprojection, the retrieved value from the projection is inserted into each of the pixels in the trajectory. By taking several projections, the accuracy of the reconstruction improves but a radial blur remains in the image.

The FBP algorithm was introduced to correct for the blur resulting from the backprojection reconstruction. [4, 11, 49]

The FBP algorithm corrects for the blur using convolution. The process of undoing the blur using convolution is called deconvolution. [4, 11] Convolution is defined as

p^′(x) =

∫ ∞ x^′=−∞

p(x)h(x−x^′)dx^′ :=p(x)⊗h(x), (3) wherep(x) is the measured projection,h(x) is the deconvolution kernel, andp^′(x) is the new projection value. The deconvolution kernel, h(x), is chosen so that it corrects for the radial blurring. [11, 49] When the deconvolution has been conducted for all measured projections, a reasonable image can be reconstructed. Because convolution is a mathematical filter, the method is called filtered backprojection. Convolution can be performed faster using the Fourier transform, which is the basis of the Fourier-based reconstruction method. [4, 11, 49]

Iterative reconstruction is a rigorous algorithm for the reconstruction of a tomographic image. The algorithm begins from an initial guess, which could be a constant image or an image from a FBP reconstruction. The algorithm then updates the initial guess and successive iterations to form an accurate tomographic image. Forward projection values are generated from the iterated image and the values are compared to the measured projection values. [4, 11, 49] An error matrix is generated by calculating the difference between the generated projection values and the actual projection values. The error matrix is then used to update the next iteration and the goal is to minimize the error matrix as the iteration progresses. The iterative reconstruction has many benefits as it can use the obtained data better than other reconstruction algorithms. This means that it can produce images with the same quality with a lowered dose or more accurate images with the same dose compared to other reconstruction methods. [11]

2.3 Image Quality

In radiographic imaging, image quality is adjusted so, that the wanted information is clearly

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ulations need to be considered, especially in clinical settings. The obtainable image quality depends on various different factors including the X-ray source, sample, and the detector.

2.3.1 Spatial Resolution

Spatial resolution of an image refers to the amount of detail that can be resolved [11, 12].

The spatial resolution limit is the smallest detail, which can be resolved using a particular imaging system [11]. The resolution depends on various properties of the imaging system and the detector determines the maximum obtainable spatial resolution [49]. The pixel size of the detector sets a theoretical size limit on the smallest resolvable object in an image but because various other factors affect spatial resolution, the pixel size is typically not the limiting factor [11].

Resolution properties can be measured using different measures such as the point spread, line spread, and edge spread functions. The functions describe the imaging system’s response to a point source, line source, and a sharp edge, respectively, and all of the functions can be used to assess the spatial resolution properties of the image. Especially in clinical settings, the spatial resolution limit is regularly assessed but the assessment is typically performed as line pair measurements where the smallest resolvable set of line pairs is determined. This is the limiting spatial resolution of the system. [11]

In CT imaging where the final image is the result of mathematical reconstruction algorithm, the spatial resolution is dependent on the resolution properties of the imaging system, imaging procedure, and noise but also on the reconstruction algorithms and filters used. In addition, factors such as gantry motion compensation and patient motion contribute to the final spatial resolution of the image. [11]

2.3.2 Contrast

The amount of contrast present in a radiographic image depends on the differences in the intensity values of the image. The contrast in X-ray transmission imaging is produced by the differences in attenuation of the X-ray beam. Contrast resolution refers to the ability to detect small changes in intensity values that are the result of attenuation of X-rays in structures with relatively similar compositions. [11, 12]

The contrast in the final image, displayed contrast, is the result of subject and detector contrast. Subject contrast is the information in the X-ray signal after it has traversed through and interacted in the subject but before it reaches the detector. This is theoretically the best possible contrast obtainable but cannot be measured. The subject contrast depends on internal and external factors. Internal factors are the structures that give rise to contrast due to differences in attenuation, while the external factors are the settings of the imaging protocol, which can be set so that subject contrast is optimized. External factors are, for example, the voltage and settings of the X-ray source and subsequently the characteristics of the radiation produced. Additionally, the possible use of a contrast agent can be used to enhance X-ray absorption. When the X-rays reach the detector,the subject contrast is adjusted depending on the detector’s response to the X-rays. A look-up table is used to convert the larger gray scale obtained in the imaging to a lower gray scale of the monitor, which is used to view the radiographic images. This is the final displayed contrast. [11] A look-up table is not used inµCT imaging.

As mentioned previously, samples with low effective atomic numberZ_eff attenuate X-rays poorly. Especially in many research fields, but also in diagnostics, soft tissues are imaged to visualize their internal structures but due to the poor attenuation, in some cases only the

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outlines of the sample can be detected. Any information contained inside the tissue is lost due to the poor attenuation contrast. Contrast agents have been introduced to solve this problem and they are used to label the entire sample or a specific portion of a biological tissue with a high Z element to produce contrast. [36] In diagnostics, barium and iodine are mainly used as contrast agents while in research purposes elements such as gold, silver and osmium have been used to stain biological tissues [11, 36]. Although useful in many applications, contrast agents also have some downsides. The penetration of the stain into the tissue might be slow and it might not diffuse into the entire depth of the tissue [34]. This makes the imaging procedure slow as the contrast agent needs to be applied first and after it has diffused, the imaging can be conducted. In addition, some stains are toxic causing added risk to the person conducting the staining, while other stains dissolve certain tissues causing information to be lost [34, 47]. Toxicity is typically not an issue during imaging as the sample is contained in the device, although some exposure can occur during the handling of the sample.

To quantify image quality and its properties, a few parameters can be calculated. A measure of the level or amplitude of signal relative to the surrounding noise is the contrast- to-noise ratio (CNR). The ratio can be calculated using two regions of interests (ROIs), one chosen from the background containing only noise signal and one from the imaged subject.

CNR can be calculated using

CNR = (¯xs−x¯bg)

σ_bg , (4)

where ¯xsis the average gray scale inside the subject ROI, ¯xbg is the average gray scale inside the background ROI, and σ_bg is the standard deviation of the background ROI. The CNR is independent of the size of the object and because the average gray scales are used in the calculation, the measure is most applicable when the average signal level represents the entire imaged subject. [11]

A similar measure to CNR is the signal-to-noise ratio (SNR) but the size and shape of the imaged object are taken into account. Additionally, the object does not have to produce a homogeneous signal, only the background signal needs to be homogeneous. The SNR can be calculated using the equation

SNR =

∑

i(x_i−x¯_bg)

σ_bg , (5)

where (x_i−x¯_bg) represents the signal at pixeliif the average background signal is ¯x_bg. [11]

In CT imaging, contrast resolution of the final image is affected by factors such as exposure time, tube voltage, slice thickness, as well as reconstruction method and filter. If thicker images are combined, the noise level will be lower than for thin slices because of the larger amount of detected X-rays. In regards to reconstruction algorithms, an iterative reconstruction algorithm can reduce the noise levels compared to FBP reconstruction. [11]

The relationship between contrast resolution and spatial resolution can be visualized from a contrast-detail diagrams displayed in Figure 3. The circles decrease in size from right to left and the contrast of the circles decreases from top to bottom. As the noise level in the diagrams increases, the contrast as well as resolution decreases. The smallest circle with the lowest contrast becomes unresolvable first. [11]

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Figure 3: Contrast-detail diagrams where A is a noiseless diagram and the noise level is increased inB and further in C. The yellow line separates the circles which can be resolved (upper right corner) from the circles which cannot be resolved (bottom left corner). [11]

2.3.3 Noise

Noise is present in all radiographic images and there are several sources of noise including the detector system as well as the subject of interest. Image noise degrades the image quality but cannot be completely removed. Noise resulting from the scattering of X-rays as described in Section 2.1.1, is present in all radiographic images but can be significantly reduced with the use of grids placed in front of the detector [11, 12, 49]. Another common source of noise is electronic noise, which is present in all electronic systems and is typically additive [11, 12].

If the real or interesting signal level is low and electronic noise level is high, the resulting signal consists mainly of noisy signal [11]. This degrades image quality and details in the area where signal level is low will most likely not be resolved.

Another type of electronic noise is structured noise, which is due to the electronic systems of the detector. As the detector pixels are read by their own amplifier circuits, and they cannot be tuned to match the other amplifiers, it will cause in the different detectors to have different settings resulting in structured noise. Due to the fact that the noise is constant over time, structured noise can be corrected for. Additionally, the sample to be imaged gives rise to anatomical noise, which is the structure seen on the anatomic image but are not relevant to diagnosis or are outside the area of interest. Anatomical noise does not add any useful information to the final image but can partially be removed using subtraction imaging.

CT imaging significantly reduces the presence of anatomical noise as overlapping anatomic structures can be separated. [11]

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3 Phase-Contrast Imaging

A method for providing enhanced contrast to radiographic images of soft tissues and other low density samples has been the realization and use of various phase-contrast imaging techniques.

In contrast to conventional radiography, where the attenuation of X-rays is used to produce contrast, the phase information of the wave is used in phase-contrast imaging to provide contrast. The phase-contrast techniques are generally divided into three categories. The techniques vary in their experimental setup, feasibility, and complexity, but are based on the use of phase information to generate contrast in a radiographic image. [20]

3.1 Physics and Basic Principle

Phase-contrast imaging methods use the phase information of the wave to generate an image of the sample. Phase contrast arises from phase changes, which are the result of refraction of the X-rays due to the properties of the sample. As refraction takes place for all wavelengths of electromagnetic radiation, the relationships that are familiar for visible light also apply for X-ray radiation. [40]

The refraction of X-rays, far from absorption edges, in a sample can be described using a complex index of refraction

n= 1−δ−iβ, (6)

where n is the refractive index, δ is the real part of the refractive index decrement and is related to the phase shift of the wave,iis the imaginary unit, and β is the complex part of the refractive index decrement related to the absorption properties of the sample [40, 50].

The refractive index can also be expressed as

n= 1−δ+iβ (7)

depending on how the electromagnetic wave propagating in the z-direction is expressed as [14, 54]. The complex part of the refractive index β is related to the linear absorption coefficientµby

β = λ

4πµ, (8)

where λ is the X-ray wavelength [40, 50]. On the other hand, δ is related to the X-ray wavelength by

δ = r₀ρN_AZ

2πA λ² (9)

where r0 is the classical electron radius, ρ is the density of the sample material, N_A is Avogadro’s constant, A is the atomic weight of the material, and Z is the atomic number [14, 32, 40]. Additionally,β can be expressed in terms of electron densityρ_e as

β = ρeλ

4πZ(σpe+σc) (10)

whereσ_pe is the photoelectric cross section and σ_c is the Compton scattering cross sections [14]. Equation (9) can also be written in terms of electron densityρe as [14]

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δ = ρer0λ²

2π . (11)

As the imaginary part of the refractive index decrement decreases faster thanE⁻⁴due to the photoelectric effect and Compton scattering, and as is seen from Equation (11) the real part of the refractive index decrement decreases only withE⁻² (E∝1/λ), significantly enhanced contrast could be achieved for soft tissues even with higher energies [7, 14]. Figure 4 depicts the ratio between the imaginary and real part of the refractive index for carbon, oxygen and hydrogen, which constitute most of soft tissue [4]. It can be clearly seen that at diagnostic energies, the ratio is still over one thousand.

Figure 4: Energy dependence of the ratioδ/β for carbon (red), oxygen (blue), and hydrogen (green).

The phase shift that a homogeneous sample introduces in an incident ray is dependent on δand the thickness of the sample. An inhomogeneous sample can be treated to be composed of many infinitesimally thick homogeneous mediums and the amount of X-ray shift introduced in the incident wave is given by

ϕ= 2π λ

∫

δ(l)dl=r₀λ

∫

ρe(l)dl, (12)

where the integral is over the path, which the ray has traveled through. [14, 46, 50, 54]

It is clearly seen from Equation (11) that the phase shift introduced in the wave is only dependent on the distribution of electrons in the sample. In contrast, from Equation (10) it is obvious that the imaginary part of the refractive index is dependent on both electron density and atomic number and as a result, the attenuation coefficient is also dependent on both variables. 3D imaging can also be applied for phase-imaging by measuring the phase shifts at various different angles around the sample and therefore, the resulting image is a depiction of the electron density distribution in the sample [14].

Measuring the phase shift is very difficult in practice and therefore other measurable quantities directly related to the phase shift are often used to determine the amount of phase

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shift introduced. If the incident wave travels along the z-axis, the refraction can be observed in the xy-plane. The refraction angle Θ is directly related to the phase shift, and it can be calculated by

Θ(x, y) = λ 2π

∂

∂xϕ(x, y) (13)

This has been used in various optical applications to study properties of tissues and materials.

Due to the refraction angle being in the range of microradians, it is extremely difficult to apply the technique for X-ray applications. [14] As the measurement of phase information of X-rays is difficult in practice, various setup have been developed for the purpose, all with their own requirements. The setups differ in the requirements for stability, the components of the system and X-ray source, as well as the feasibility of the system and even the quantity that is measured.

3.2 Propagation-Based Phase-Contrast Methods

Of all the developed phase-contrast methods, propagation-based phase-contrast imaging or in-line phase-contrast imaging, sets the least amount of requirements for the imaging setup as it does not require the use of any additional optical devices. The method was introduced by A. Snigirev in 1995 on a synchrotron source and since then, the method has been shown to be applicable also on conventional polychromatic X-ray sources. [14, 45, 54] The method is based on the Fresnel diffraction phenomenon, which results in the formation of an holographic image. As the X-ray beam interacts with the sample, some of the incident photons are refracted. The refracted photons interfere at some distance from the sample, which results in the formation of the hologram. [14, 54]

The setup required for in-line phase-contrast imaging is the most simple of all the available phase-contrast imaging methods. The setup, depicted in Figure 5, consists of the X-ray source, the object to be imaged and the detector. For phase-contrast to be visible, the distance from the object to detector needs to be sufficient, greater than what is required for conventional absorption imaging [57]. In-line phase-imaging only requires spatial coherence from the X-ray beam, with chromatic coherence not being an essential requirement [44]. The spatial coherence lengthlcoh is defined as

l_coh= λR₁

s , (14)

whereR1 is the distance from the source to the object andsis the focal spot size of the X-ray source [7, 14, 57]. In order for phase-contrast to be visible, the lateral coherence length needs to be larger than the detail imaged [7]. As it can be seen from Equation (14), the coherence length decreases with increasing energy for a particular setup. Therefore increasing the distanceR₁ or decreasing X-ray energy will result in better phase contrast.

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Figure 5: Imaging setup for in-line phase-contrast imaging, consisting of the X-ray source, object and the detector. W₁ and W₂ are the wavefronts between the source and object and between the object and the detector, respectively. R₁ and R₂ are the distances between the source and object and between object and detector, respectively. The interference of the X-rays results in the intensity profileI at the detector. Modified from [57]

In order to reconstruct a tomographic image, the phase information in the hologram needs to be understood. A basic laboratory µCT typically reconstructs the tomographic image using only the intensity profile of the hologram and a conventional reconstruction algorithm.

As phase changes occur mainly at boundaries, phase images without phase retrieval enhance the boundaries of the sample [7]. For a higher quality reconstruction, the phase information of the hologram can be used to reconstruct a map of the refractive index in a process called phase retrieval. The theoretical background has been formulated by Pogany, Gao and Wilkins and they used Fourier optics in their formulation. [44] They proposed that the attenuation and phase shift of an X-ray wave can be described by a 2D transmission functionT(x, y)

T(x, y) =e^iϕ(x,y)−^µ(x,y)² , (15) whereϕ(x, y) and µ(x, y) are the phase and linear attenuation coefficient of the object for a wave propagating in the z-direction [14, 44, 54]. When a spherical plane wave travels through the sample and arrives at the detector from distancezfrom the source, the diffracted X-ray f(x, y;z), under the paraxial Fresnel diffraction theory, is given by

f(x, y;z) = i λze^ikz

∫ ∫

T(x₀, y₀)eⁱ^2z^k^[(x−x⁰⁾²^+(y−y⁰⁾²^], (16) where k = 2π/λ is the wave number and (x₀, y₀) are the coordinates on the object plane [14, 54]. The calculation of the integral is very difficult and in the first formulation of the theory, Pogany et al were only able to solve for samples that are weakly attenuating [µ(x, y) ≪ 1] and weak phase objects [|ϕ(x, y)| ≪1] [44]. Using these approximations, the transmission function can be approximated by [14, 44, 54]

T(x, y)≈1 +iϕ(x, y)−µ(x, y)

2 . (17)

Using this, the intensity at the image plane is obtained as I(x, y, z)≈1 +λz

2π∇²ϕ(x, y), (18)

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where∇² is the Laplacian operator defined as ∇² =∇ · ∇ and ∇= (_∂x^∂

1, . . . ,_∂x^∂

n) [14, 44].

As a result, the contrast in the hologram is formed by the intensity changes due to the distortions in the wave front. Additionally, as the intensity is related to the Laplace of the phase shift, propagation-based phase contrast imaging is typically used for edge enhancement as the second derivative measures, in this case, how the rate of rate of change of the refractive index changes.

Since these approximations are derived from the assumptions that µ(x, y) ≪ 1 and

|ϕ(x, y)| ≪ 1, the equations above are not valid in many cases and cannot be used to ac- curately reconstruct a tomographic image [14, 54]. As a result, a new formalism has been introduced as a basis to deal with attenuation and the phase-shift by Gureyev et al. [23].

The new formalism is based on the transport of intensity equation (TIE), which is derived from the paraxial wave equation approximation. The TIE links the phase to the derivative of the intensity along the direction of the wave propagation. The TIE is given by

∂I(x, y, z)

∂z =− λ 2π∇ ·[

I(x, y, z)∇ϕ(x, y, z)]

(19) where the ∇operator is with respect to x and y. [23, 48, 54] For the TIE to be valid, the paraxial approximation needs to hold but in addition, Gureyev et al. had to make a near field approximation

⏐

I(x, y, z) I(x, y,0)−1

⏐

≪1 (20)

whereI(x, y,0) is the intensity at the object plane. Using these assumption it was determined that

λz 2π [

− ∇²ϕ(x, y)− ∇ϕ(x, y)∇lnI(x, y,0) ]

= I(x, y, z)

I(x, y,0)−1. (21) Equation (21) is based on the TIE and it is valid when the near-field criterion is met. As with Equation (18), the assumption made do not hold in all cases and as a result, additional formalisms have been proposed for phase retrieval and reconstruction of the tomographic image. [23, 54] Many of these methods rely on the TIE as a basis but other methods aim to better model the Fresnel diffraction of the X-ray wavefronts.

Paganin et al. used the TIE as a basis for the derivation of a new method to obtain the intensity and phase information generated by a homogeneous object of interest. The intensity and phase at the object exit surface (z= 0) are related to the projected thickness, T(x, y), of the object on the plane in which the image is taken by

I(x, y, z = 0) =Iⁱⁿe^−µT(x,y) (22) ϕ(x, y, z= 0) =−2π

λ δT(x, y), (23)

where Iⁱⁿ is the incident intensity. These two equations are then substituted into the TIE, Equation (19), which is then simplified and estimated. The intensities at z=0 and z = R₂ are then represented as Fourier integrals and can be used to obtain the projected thickness T(x, y) after a few steps. From Equations (22)-(23), the intensity and phase can be obtained at the exit surface of the object. [39] This formalism was originally developed for X-ray microscopy used to image in the soft X-ray region but has been recently implemented in

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various attempts at phase retrieval using the TIE [24, 29, 58]. Gureyev etal. combined the use of Fourier optics and the TIE for phase retrieval. This combination still results in a linear technique, beneficial especially for computation purposes. They were also able to reconstruct images with higher quality using the combined formalism compared to reconstructions using either Fourier optics or TIE. [25]

Wu etal. developed a new formalism for in-line phase contrast imaging using the Fresnel diffraction theory. The formalism was derived for spherical waves, both in the real space and frequency space. A brief description of the formalism follows. First, the spherical X-ray wavefront arriving at a location on the detector plane is written as a Fresnel-Kirchoff integral, which is then related to the intensity at the detector plane. Next, the Fourier transform is taken of the intensity after which a general formula for in-line phase-contrast imaging is obtained. During the derivation of the formalism, it is assumed that the phase varies only moderately in a small distance, which is reasonable especially in clinical X-ray imaging, as the X-ray wavelengths are short. The general formula can also be simplified depending on the application and in addition, the results of Pogany etal. were obtained as a special case of the new formalism. The intensity in real space can be found by using the convolution theorem of the Fourier space and the retrieval of attenuation information is also possible. [54] In another study Wu etal. derived a phase retrieval algorithm based on the phase-attenuation duality. The duality is a result of both the phase and attenuation being determined by the electron density of the sample assuming that the X-ray energy is between 60-150 keV. Wu etal. tested their phase retrieval using computer simulations and obtained enhanced image contrast. [55]

Chen et al. proposed a new formalism to improve contrast in phase-contrast images, in which the projections have been obtained using a source with a finite size and non-uniform distribution. The theory is based on the Huygens-Fresnel principle and the Fresnel-Kirchhoff integral. Chen etal. were able to validate the method experimentally. [13]

A completely different approach to the reconstruction of the phase image was presented by A. Bronnikov. The method does not require a phase retrieval process and the reconstruction of the refractive index distribution is done directly from the intensity distribution measured from a single plane in the near-field region. The theory behind the method is based on the relationship between the 3D Radon transform of the object function and the 2D Radon transform of the intensity distributions. [8] The original theory has been extended and implemented by others [9, 18, 21, 22, 29].

Arhatari et al. obtained similar expressions as A. Bronnikov with more general con- siderations [1, 8, 9]. The phase retrieval and reconstruction algorithm was developed for weakly absorbing, homogeneous objects and it is based on the use of the TIE. The benefit of the algorithm is that only one tomographic data set in the Fresnel region is required for reconstruction. This is because some Fourier space operations, done both in phase retrieval and FBP reconstruction, can be combined. The process has added benefits as it limits the radiation dose and the amount of data acquired is smaller, increasing computation efficiency.

[1] In another study, Arhatari etal. developed a phase retrieval method using a laboratory X-ray source with a large bandwidth. This method is also based on the TIE. [2]

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3.3 Interferometric Phase-Contrast Methods 3.3.1 Crystal Interferometry

The entire concept of phase-contrast imaging using X-rays was first introduced by Bonse &

Hart in 1965 using a crystal interferometer setup [10]. The setup is constructed of a single, highly perfect piece of crystal. Typically the crystal is a silicon crystal and it is cut into three crystal plates. The geometry of the setup is referred to as the Laue-Laue-Laue (LLL) interferometer setup. The first crystal plate is the beam splitter S, the second plate is the transmission element M, and the last plate is the analyzer crystal A [10, 57]. A. Momose introduced the use of the LLL interferometer setup in CT imaging to obtain tomographic phase-contrast images [37].

The working principle of the setup is as follows. When the incoming monochromatic X- ray beam satisfies the Bragg diffraction conditions, the splitter crystal divides the beam into two divergent X-ray beams. When the separate beams arrive at the transmission crystal, they are again both divided into two separate beams. From these four beams, the two convergent beams overlap at the analyzer crystal and there the beams are again divided into two and interference occurs. [10] An object in the beam path, between the transmission element and the analyzer crystal, causes a phase shift in the incoming beam. The amount of phase shift can be determined with the use of a phase shifter placed in the reference beam.

The beam incident on the first interferometer needs to be monochromatic and therefore a monochromator is placed before the splitter crystal. A typical LLL interferometer setup is depicted in Figure 6. [14]

Due to the short wavelength of X-rays, the interferometer setup imposes strict requirements on the mechanical stability of the optical elements. Good stability is achieved by using a monolithic crystal but this in turn limits the field of view (FOV) to sizes under what is needed for clinical applications. [57] In addition, as a monochromatic beam is required for imaging the feasibility of the setup is low and the demands of the setup are further increased [14]. Due to these limitations of the crystal interferometer setup, grating-based differential phase-contrast (DPC) imaging has been introduced as an alternative interferometric method.

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Figure 6: A typical LLL interferometer setup consisting of three plate crystals; the splitter, transmission element or mirror, and analyzer crystal. The monochromator is placed before the splitter crystal and the detector panel is placed after the analyzer crystal. The object to be images is placed in the beam path between the mirror and analyzer crystals causing a phase shift in the beam. The phase shift can be determined by a phase shifter placed in the reference beam. [14]

3.3.2 Grating Interferometry

DPC imaging is based on the use of gratings as optical elements to obtain phase contrast from the refraction of X-rays using a setup called the Talbot interferometer [14]. The setup is named after the Talbot effect discovered by Henry Fox Talbot in 1836. He discovered that when a coherent wave propagated through a periodic structure, such as a grating, a self-image, also called the Moiré fringe pattern, of the structure is reconstructed behind the grating at distances

d_T= 2mp²

λ , (24)

where m = 1,2,3, . . ., p is the grating period, and dT is called the Talbot distance. In addition, the fractional Talbot effect was realized; a self-image of the phase grating is formed at distances

dm = (m−1/2)p²

4λ . (25)

Equation (25) can be expressed for any grating as d_m = mp²

2η²λ. (26)

The contrast of the resulting self-image is at maximum for a phase grating when m is odd and for an amplitude or absorption grating, the maximum is achieved whenm is even. The coefficientη=1 for both amplitude and phase gratings that give a phase shift ofπ/2 andη=2 if the phase shift isπ. [32]

A grating-based interferometry setup using a coherent X-ray source is depicted in Figure 7 [33]. The setup consists of an X-ray source, the sample to be imaged, a phase grating G₁, an

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absorption grating G₂, and a CCD detector. The two gratings are separated by a distanced, wheredmust be a fractional Talbot distance so that maximum of the interference pattern is observed at G₂. [14, 32, 33, 57] In theory, grating G₁ can be an absorption or a phase grating but greater efficiency is achieved using a pure phase grating. Therefore, a phase grating with a π phase shift is a good choice. [14, 32] When the incident wavefront comes across the first grating G₁, it is diffracted into the first two diffraction orders at each grating slit. The diffracted beams from neighboring slits interfere and generate a periodic intensity pattern in the plane of the absorption grating G₂, which contains the refraction angle information. If an object is place between the X-ray source and the phase grating, it will cause refraction of the X-rays and as a result the interference pattern is modified compared to the intensity pattern created only by the grating. By comparing these interference patterns, the phase information of the sample can be obtained. [14, 32, 33, 52, 57]

Figure 7: A grating-based interferometer setup with an coherent X-ray source. A sample placed in front of the X-ray source causes distortions in the incident X-ray wavefront. The phase grating G₁ causes diffraction in the wavefront, resulting in interference and therefore intensity modulations in the wavefront. The modulated wavefront is then analyzed by the absorption grating G₂. The distance between the gratings G₁ and G₂ is d. [33]

Pfeiffer et al. introduced the use of an incoherent X-ray source in DPC imaging by proposing the use of a source grating G₀ to create individually coherent but mutually incoherent sources. A typical setup, called the Talbot-Laue interferometer, using a source grating is depicted in Figure 8. The source grating can contain a large number of apertures, all of which create sources of required coherence, and therefore conventional X-ray sources with source size of 1mm² or more can be used. In order for the line sources created by the source grating to be beneficial for image-formation, the setup geometry should satisfy

p₀ =p₂· l

d, (27)

where p₀ is the source grating period, p₂ is the absorption grating period, l is the distance from the source grating to the phase grating, andd is the distance from the phase grating to the absorption grating. Final image resolution is given wd/l, which is determined by the source size, w. [42] Wang et al. studied the effect of a polychromatic X-ray source on the efficiency of the Talbot-Laue interferometer setup. The evaluation of the efficiency is interesting as the phase of the intensity oscillations is dependent on the wavelength of

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used to assess the efficiency in the simulations. The results of the research showed that the visibility of the fringe pattern is not highly sensitive to the polychromaticity of the X-ray source even at higher fractional Talbot distances. [51]

Figure 8: A grating-based interferometer setup using an incoherent X-ray source. The setup contains the X-ray source, a source grating G₀, a phase grating G₁, an analyzer grating G₂, the sample to be imaged, which is placed between gratings G₀and G₁, as well as the detector placed behind grating G₂.[42] Image modified from [42].

Phase-stepping is performed to separate the phase information of the signal from absorption information and other contributions, such as imperfections of the grating [52]. The total phase shift could also be obtained straight from the intensity profile by integration but phase-stepping provides a more precise way of extracting only the phase information [42].

Phase stepping is performed by moving one of the gratings, typically G₂, in the direction of its period (x in Figure 8) over one period of the grating. For every point of the phase- stepping scan an image is taken and from these, the average intensity of the period for each pixel can be found. From these averages, the oscillations phase φcan be determined for all pixels. The phasesφof the intensity oscillations in each pixel are related to the phase profile Φ of the wavefront, the wavelength λ, the period of the absorption grating period p₂, and the decrement of the real part of the refractive indexδ by

φ= λd p₂

∂Φ

∂x = 2πd p₂

∫ +∞

−∞

∂δ

∂x∂z. (28)

Asφ contains only phase information, the phase of the object can be obtained from Equa- tion (28) by integration. This phase-stepping procedure is done both with and without the sample in place to obtain the phase information of the sample. [33, 52] A total of three steps are required for the phase-stepping scan to obtain φ for a sinusoidal intensity oscillation, although typically more steps are acquired [28, 33, 52]. This is because the intensity curve has three different components: amplitude, phase and a constant [32]. Some drawbacks of the phase-stepping process include its time-consuming nature and the fact that multiple exposures have to be done to obtain the phase-stepping curve [59]. These limit the imple- mentation of the technique especially into clinical use but the technique has been widely used in research [3, 17, 27, 28, 43, 46].

The 3D distribution of the X-ray refractive index can be obtained by tomographic re-

Augmenting Soft Tissue Contrast Using Phase-Contrast Techniques in Micro-Computed Tomography Imaging