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Phase-contrast X-ray imaging (PCI) or Phase-sensitive X-ray imaging is a general term for different technical methods using information about changes in the phase of an X-ray beam that passes through an object to create images of this object. While standard X-ray imaging techniques like radiography or Computed Tomography (CT) rely on the loss of intensity of the X-ray beam when traversing the sample (attenuation) measured directly with an X-ray detector, in PCI the phase-shift of the beam caused by the sample (refraction) is not measured directly but transformed into intensity variations that can be revealed by a detector.^[1]

In addition to projection images (2D superimposition images of the sample), PCI like conventional transmission allows to perform tomography and reconstruct slices and three-dimensional images. Here the real part of the sample refractive index is reconstructed, which is causing the phase shift. For samples consisting of atoms with low atomic number Z, PCI is more sensitive to density variations in the sample than conventional transmission-based X-ray imaging. This results in images with improved soft-tissue contrast.^[2]

A variety of phase-contrast imaging techniques with X-rays has been developed in the last years, all based on the observation of the interference pattern between diffracted and undiffracted waves.^[3] The most common techniques are Crystal interferometry, propagation-based imaging, analyzer-based imaging and grating-based imaging (see below).

X-rays were firstly discovered by Wilhelm Conrad Roentgen in 1895 and are sometimes still referred to as "Roentgen rays". He found out that the "new kind of rays" had the ability to penetrate material which was opaque for visible light and recorded the first X-ray image, displaying the hand of his wife.^[4] He was awarded the first Nobel Prize in Physics in 1901 "in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him".^[5] Since that X-rays were used as an invaluable tool to non-destructively determine the inner-structre of different objects, but for a long time the information was gained only by measuring the transmitted intensity of the waves and the phase information was neglected.

The principle of Phase-contrast imaging in general was discovered by Frits Zernike during his work with diffraction gratings and visible light.^[6][7] The application of his knowledge to microscopy brought him the Nobel Prize in Physics in 1953. Ever since phase-contrast microscopy is an important field of Optical microscopy.

The transfer from visible light to X-rays took a long time due to the slow progress in improving the quality of the X-ray beam and the non-availability of X-ray optics (lenses). “In the 70s it was realized that the synchrotron radiation emitted from charged particles circulating in storage rings constructed for high energy nuclear physics experiments was potentially a much more intense and versatile source of X-rays.“ ^[8] The construction of synchrotrons exclusively for the production of X-rays and the progress in the development of optical elements for X-rays built the fundament for further advancement in X-ray Physics.

The pioneer work for the implementation of the phase contrast method to X-ray physics was presented in 1965 by Ulrich Bonse and Michael Hart from the Department of Materials Science and Engineering of Cornell University, New York^[9] in the form of a crystal interferometer, made from a large and highly perfect single crystal. (Insert disadvantages of this kind of interferometer here. E.g. small field of view, extemely high stability of every setup component at the atomic scale, ...) About 30 years later the japanese scientists Atsushi Momose, Tohoru Takeda and co-workers picked up this idea and refined it for application in biological imaging. ^{[10] [11]}

At the same time two other approaches to phase contrast imaging emerged. The propagation-based imaging technique was primarily introduced by the group of Anatoly Snigirev at the ESRF (European Synchrotron Radiation Facility) in Grenoble, France. ^[12] and by Wilkins and coworkers at CSIRO (Commonwealth Scientific and Industrial Research Organisation) Division of Material Science and Technology in Clayton, Australia. ^[13] The experimental setup, which consisted of an inline-configuration of an X-ray source, a sample and a detector and did not require any optical elements, was conceptually identical to the setup of Dennis Gabor's revolutionary work on Holography in 1948. ^[14]

An alternative approach called analyzer-based imaging was first explored in 1995 by Viktor Ingal and Elena Beliaevskaya at the X-ray laboratory in St. Petersburg, Russia ^[15] and by Tim Davis and colleagues at the CSIRO ^[16] This method uses a Bragg crystal as an angular filter to reflect only a small part of the beam fulfilling the Bragg condition onto a detector. Important contributions for the progress of this method have been made by a US collaboration of the research teams of Dean Chapman, Zhong Zhong and William Thomlinson, for example the extracting of an additional signal caused by ultrasmall angle scattering ^[17] and the first CT image made with analyzer-based imaging. ^[18]

The fourth approach the Grating-based interferometer makes use of the Talbot effect, discovered by Henry Fox Talbot in 1836.^[19] The so called Talbot-Lau interferometer was initially used in atom interferometry for instance by John F. Clauser and Shinfang Li in 1994.(^[20]) The first X-ray Grating Interferometers using synchrotron sources were developed by Christian David and colleagues from the Paul Scherrer Institute (PSI) in Villingen, Switzerland ^[21] and the group of Atsushi Momose from the University of Tokyo. ^[22] The next milestone in Grating-based imaging was the incorporation of Computed Tomography into grating interferometry in 2005 by the group of the PSI including Timm Weitkamp (Mention Christian David?). ^[23] In 2006 another great advancement was the transfer of the grating-based technique to laboratory X-ray tubes by Franz Pfeiffer and co-workers^[24] which fairly enlarged the technique's potential for clinical use. As in analyzer-based imaging a supplementary signal caused by scattering due to the porous microstructure of the sample termed "dark-field signal" was extracted and found to provide "complementary and otherwise inaccessible structural information about the specimen at the micrometre and submicrometre length scale."^[25] Recently a lot of research was done to improve the grating-based technique: Han Wen and his team analyzed animal bones and found out that the intensity of the dark-field signal depends on the orientation of the grid and this is due to the anisotropy of the bone structure. They also provided a new approach for signal extraction/acquisiton?? named "single-shot fourier analyis". ^[26] The grating-based phase-contrast CT field was extended by tomographic images of the dark-field signal ^[27] and time-resolved phase-contrast CT. ^[28] Also the first pre-clinical studies using grating-based phase-contrast X-ray imaging were published. Marco Stampanoni and his group examined native breast tissue with "Differential Phase-contrast Mammography" ^[29] and a team led by Dan Stutman investigated the use of Grating-based imaging to image the small joints of the hand. ^[30]

In conventional X-ray imaging only the drop in intensity by attenuation caused by an object in the X-ray beam is considered and the radiation is treated as rays like in geometrical optics. But when X-rays pass through an object not only their amplitude but their phase is altered as well. Instead of simple rays X-rays can also be treated as electromagnetic waves. An object then can be described by its complex refractive index (cf. ^[8]):

${\displaystyle n=1-\delta +i\beta }$.

The term δ is the decrement of the real part of the refractive index and the imaginary part β describes the absorption index or extinction coefficient. Note that in contrast to optical light the real part of the refractive index is less than unity, this is "due to the fact that the X-ray spectrum generally lies to the high-frequency side of various resonances associated with the binding of electrons"^[8]. The phase velocity inside the object is larger than the velocity of light c. This leads to a different behavior of X-rays in a medium compared to visible light (e.g. refractive angles have negative values) but does not contradict the law of relativity, "which requires that only signals carrying information do not travel faster than c. Such signals move with the group velocity, not the phase velocity, and it can be shown that the group velocity is in fact less than c"^[8].

The impact of the refraction index on the behavior of the wave can be demonstrated with a wave propagating in an arbitrary medium with refractive index n . For simplicity, a monochromatic plane wave with no polarization is assumed here. The wave propagates in direction normal to the surface of the medium, named z in this expample. The scalar wave function in vacuum is

${\displaystyle \Psi (z)=E_{0}e^{ikz}}$.

Within the medium the wavevector changes from k to nk. Now the wave can be described as:

${\displaystyle \Psi (z)=E_{0}e^{inkz}=E_{0}e^{i(1-\delta )kz}e^{-\beta kz}}$

where δkz is the phase shift and e^{-β kz} is an exponential decay factor decreasing the amplitude E₀ of the wave.^[8]

More generally the total phase shift of the beam progating a distance z can be calculated with the integral:

${\displaystyle \Phi (z)={\frac {2\pi }{\lambda }}\int \!\delta (z)\,\mathrm {d} z}$

Where λ is the wavelength of the incident X-ray beam. This formula means that the phase shift is the projection of the decrement of the real part of the refractive index in imaging direction. This fulfills the requirement of the Computed tomography/tomographic principle which states, that "the input data to the reconstruction algorithm should be a projection of a quantity f that conveys structural information inside a sample. Then, one can obtain a tomogram which maps the value f."(Momose et al 1998 ^[31] In other words in Phase contrast imaging a map of the real part of the refraction index δ(x,y,z) can be reconstructed with standard techniques like Filtered Back Projection which is similar/analog to conventional X-ray computed tomography where a map of the imaginary part of the refraction index can be retrieved.

To get information about the compounding/configuration of a sample, basically the density distribution of the sample, one has to relate the measured values for the refractive index to intrinsic parameters of the sample, such a relation is given by the following formulas:

${\displaystyle \beta ={\frac {\rho _{a}\sigma _{a}}{2k}}}$

where ρ_a is the atomic number density, σ_a the absorption cross section, k the length of the wave vector and

${\displaystyle \delta ={\frac {\rho _{a}p}{k}}}$

where ρ_a is the atomic number density, p the phase shift cross section, k the length of the wave vector.

Far from the absorption edges (caused by the photoelectric effect/ energies near the resonance frequency of the medium??), dispersion effects can be neglected; this is the case for light elements (atomic numer Z<40) that are the components of human tissue and X-ray energies above 20 keV, which are typically used in medical imaging. Assuming these conditions the absorption cross section is approximately given by

${\displaystyle \sigma _{a}=0.02[barn]\left({\frac {k_{0}}{k}}\right)^{3}Z^{4}}$

where 0.02 is a constant given in Barn, the typical unit of particle interaction cross section area, k the length of the wave vector, k₀ the length of a wave vector with wavelength of 1 Angstrom and Z the atomic number. ^[32] The valid formula under these conditions for the phase shift cross section is:

${\displaystyle p={\frac {2\pi Zr_{0}}{k}}}$

where Z is the atomic number, k the length of the wave vector, and r₀ the classical electron radius.

This results in the following expressions for the two parts of the complex refraction index:

${\displaystyle \delta ={\frac {\rho _{a}p}{k}}={\frac {2\pi \rho _{a}Zr_{0}}{k^{2}}}}$

${\displaystyle \beta ={\frac {\rho _{a}\sigma _{a}}{2k}}=0.01[barn]\rho _{a}k_{0}^{3}\left({\frac {Z}{k}}\right)^{4}}$

Inserting typical values of human tissue in the formulas given above shows that δ is generally three orders of magnitude larger than β within the diagnostic X-ray range. This implicates that the phase-shift of an X-ray beam propagating through tissue may be much larger than the loss in intensity thus making PCI more sensitive to density variations in the tissue than absorption imaging.ma^[33] This is illustrated in figure3(Lewis2006 figure 1).

Due to the proportionalities

${\displaystyle \beta \propto k^{-4}}$, ${\displaystyle \delta \propto k^{-2}}$

the advantage of phase contrast over conventional absorption contrast even grows with increasing energy, making it possible to reduce the exposure dose while imaging soft tissue.^[33]

While for visible light the real part of n can deviate strongly from unity (n of glass in visible light ranges from 1.5 to 1.8) the difference from unity for X-rays in different media, is generally of the order 10^-5. Thus the refraction angles caused at the boundary between two isotropic media calculated with Snell's formula are also very small. The consequence of that is that refraction angels of X-rays passing through a tissue sample cannot be detected directly and are usually determined indirectly by "observation of the interference pattern between diffracted and undiffracted waves produced by spatial variations of the real part of the refractive index."(Weon 2006,^[3])

Crystal Interferometry or sometimes also called X-ray interferometry is the oldest but also the most complex method for experimental realization. It consists of three beam splitters in Laue geometry aligned parallel to each other. (see figure3: picture from bech or zanetti maybe modified with phase stepping wedge)The incident beam, which usually is collimated and filtered by a monochromator ( Bragg crystal) before, is split at the first crystal by Laue diffraction into two coherent beams, a reference beam which remains undisturbed and a beam passing through the sample. The second crystal acts as a transmission mirror and causes the beams to converge towards one another. The two beams meet at the plane of the third crystal, which is sometimes called, the analyzer crystal, and create an interference pattern the form of which depends on the optical path difference between the two beams caused by the sample. This interference pattern is detected with an X-ray detector behind the analyzer crystal. ^[9],^[34]

By putting the sample on a rotation stage and recording projections from different angles the 3D-distribution of the refractive index and thus tomographic images of the sample can be retrieved. ^[31] In contrast to the methods below, with the crystal interferometer the phase itself is measured and not a spatial change/derivative of it. To retrieve the phase shift out of the interference patterns a technique called phase-stepping or fringe scanning is used: a phase shifter (with the shape of a wedge) is introduced in the reference beam. The phase shifter creates straight interference fringes with regular intervals so called carrier fringes. When the sample is placed in the other beam, the carrier fringes are displaced. The phase shift caused by the sample corresponds to the displacement of the carrier fringes. Several interference patterns are recorded for different shifts of the reference beam and by analyzing them the phase information modulo 2π can be extracted.^{[34] [31]}. This ambiguity of the phase is called the Phase wrapping effect and can be removed by so called Phase-unwrapping techniques ^[35] These techniques can be used when the signal-to-noise ratio of the image is sufficiently high and phase variation is not too abrupt. ^[36]

Alternatively to the fringe scanning method, the Fourier-transform method can be used to extract the phase shift information with only one interferogram, thus shortening the exposure time, but this has the disadvantage of limiting the spatial resolution by the spacing of the carrier fringes.^[37]

X-ray interferometry is considered to be the most sensitive to the phase shift, of the 4 methods, consequently providing the highest density resolution in range of mg/cm³. ^[36]. But due to its high sensitivity, the fringes created by a strongly phase-shifting sample may become unresolvable; to overcome this problem a new approach called "coherence-contrast X-ray imaging" has been developed recently, where instead of the phase shift the change of the degree of coherence caused by the sample is relevant for the contrast of the image ^[38].

A general limitation for the spatial resolution of this method is given by the blurring in the analyzer crystal which arises from dynamical refraction, i.e. the angular deviation of the beam due to the refraction in the sample is amplified about ten thousand times in the crystal, because the beam path within the crystal depends strongly on its incident angle. This effect can be reduced by thinning down the analyzer crystal, e.g. with an analyzer thickness of 40 μm a resolution of about 6 μm was calculated. Alternatively the Laue crystals can be replaced by Bragg crystals, so the beam doesn´t pass through the crystal but is reflected on the surface.^[39]

Another constraint of the method is the requirement of a very high stability of the setup; the alignment of the crystals must be very precise and the path length difference between the beams should be smaller than the wavelength of the X-rays; to achieve this the interferometer is usually made out of a highly perfect single block of silicon by cutting out two grooves. By the monolithic production the very important spatial lattice coherence between all three crystals can be maintained relatively well but it limits the field of view to a small size,(e.g. 5cm x 5cm for a 6-inch ingot) and because the sample is normally placed in one of the beam paths the size of the sample itself is also constrained by the size of the silicon block. ^[9][40]. Recently developed configurations, using two crystals instead of one, enlarge the field of view considerably, but are even more sensitive to mechanical instabilities.^[41][42]

An additonal difficulty of the crystal interferometer is that the Laue crystals filter out most of the incoming radiation, thus requiring a high beam intensity or very long exposure times. ^[43] That limits the use of the method to highly brilliant X-ray sources like synchrotrons.

According to the constraints on the setup the crystal interferometer works best for high-resolution imaging of small samples which cause small or smooth phase gradients.

Analyzer-based imaging (ABI) is also known as diffraction-enhanced imaging (DEI) and multiple-image radiography (MIR)^[44](Momose refers to this technique as "Refraction-based technique"^[36] which can be easily confused with "Refraction-enhanced imaging" which is another term for Probagation-based imaging, see ^[45] Its setup consists of a monochromator(usually a single or double crystal) in front of the sample and a analyzer crystal positioned in Bragg geometry between the sample and the detector.(see figure...)This analyzer crystal acts as an angular filter for the radiation coming from the sample. When these X-rays hit the analyzer crystal the condition of Bragg diffraction is satisfied only for a very narrow range of incident angles. Refracted X-rays within this range will be reflected depending on the incident angle. The dependency of the reflected intensity on the incident angle is called a rocking curve and is an intrinsic feature of the crystal.(see figure) The typical angular acceptance is from a few microradians to tens of microradians and is related to the full width at half maximum (FWHM) of the rocking curve of the crystal.

When the scattered or refracted X-rays have incident angles outside this range they will not be reflected at all and don´t contribute to the signal. 

If the analyzer is perfectly aligned with the monochomator and thus positioned to the peak of the rocking curve, a standard X-ray radiograph with enhanced contrast is obtained because there is no blurring by scattered photons. If, otherwise, the analyzer is oriented at a small angle with respect to the monochromator (typically about half of the FWHM of the rocking curve) then X-rays refracted in the sample by a smaller angle will be reflected less, and X-rays refracted by a larger angle will be reflected more. Thus the contrast is based on different refraction angles in the sample. For small phase gradients the refraction angle can be expressed as

${\displaystyle \Delta \alpha ={\frac {1}{k}}{\frac {\partial \phi (x)}{\partial x}}}$ and therefore, instead of the phase itself, the first derivative of the phase front is measured. Usually sensitive to only one component of the phase gradient leading to possible ambiguities in phase estimation. Contrary to the former methods ABI usually provides phase information only in the diffraction direction, but is not sensitive to angular deviations on the plane perpendicular to the diffraction plane. This sensitivity to only one component of the phase gradient can lead to ambiguities in phase estimation.^[46]

So the

Propagation-based imaging (PBI) is the most common name for this technique but it is also called In-line Holography, Refraction- enhanced Imaging (cf.^[45] , sometimes Analyzer-based imaging is referred to as Refraction-based imaging as well ^[36]) or Phase-contrast radiography. The latter denomination derives from the fact that the experimental setup of this method is basically the same as in conventional radiography. It consists of an in-line arrangement of an X-ray source, the sample and an X-ray detector and no other optical elements are required. The only difference is that the detector is not placed immediately behind the sample but further away, so the radiation refracted by the sample can interfere with the unchanged beam.^[12] This simple setup and the low stability requirements provides a big advantage of this method over other methods discussed here.

Under spatially coherent illumination and an intermediate distance between sample and detector an interference pattern with "Fresnel fringes" is created; i.e. the fringes arise in the free space propagation in the Fresnel regime, which means that for the distance between detector and sample the approximation of Kirchhoff's diffraction formula for the near field, the Fresnel diffraction equation is valid. In contrast to crystal inteferometry the recorded interference fringes in PBI are not proportional to the phase itself but to the second derivative ( the Laplace operator) of the phase of the wavefront. Therefore the method is most sensitive to abrupt changes in the decrement of the refractive index. This leads to stronger contrast outlining the surfaces and structural boundaries of the sample (edge enhancement) compared with a conventional radiogram.^{[13] [47]}

PBI can be used to enhance the contrast of an absorption image, in this case the phase information in the image plane is lost but contributes to the image intensity. However it is also possible to separate the phase and the attenuation contrast, i.e. to reconstruct the distribution of the real and imaginary part of the refractive index separately. The unambiguous determination of the phase of the wave front, the so called Phase retrieval involves measuring one or more images of the object after varying a parameter of the imaging system such as the sample-detector distance or the x-ray energy. A common realization of Phase retrieval is the recording of several images at different detector-sample distances and using an algorithm based on the Transport of intensity equation to reconstruct the phase distribution, but this approach suffers from amplified noise for low spatial frequencies and thus slowly varying components may not be accurately recovered. There are several more approaches for phase retrieval and a good overview of them is given in ^[48], ^[49]

Tomographic reconstructions of the 3D distribution of the refraction index or "Holotomography" is implemented by rotating the sample and recording for each projection angle a series of images at different distances.^[50]

A high resolution detector is required to resolve the interference fringes, which practically limits the field of view of this technique. The achieved spatial resolution is relatively high in comparison to the other methods and is mainly limited by the blurring caused by the presence of the fringes.^[51]

As mentioned before, for the formation of the fresnel fringes, the constraint on the spatial coherence of the used radiation is very strict, which limits the method to small or very distant sources, but in contrast to Crystal interferometry and Analyzer-based imaging the constraint on the temporal coherence, i.e the polychromaticity is quite relaxed.(spectral width/wavelength=1)^[46]. Consequently the method cannot solely be applied to synchrotron sources but also to polycromatic laboratory X-ray sources providing sufficient spatial coherence, like Microfocus X-ray tubes.^[13]

In general the image contrast provided by this method is lower than in the other methods discussed here, especially if the density variations in the sample are small. Due to its strength in enhancing the contrast at boundaries, it´s well suited for imaging fiber or foam samples but not for biological samples containing soft tissue. ^[52]