X-ray differential phase contrast imaging methods, including projection imaging and the corresponding computed tomography (CT), have been implemented using a Talbot interferometer and either a synchrotron beam line or a low brilliance x-ray source generated by a stationary-anode x-ray tube. From small-angle scattering events which occur as an x-ray propagates through a medium, a signal intensity loss can be recorded and analyzed for an understanding of the micro-structures in an image object. This has been demonstrated using a Talbot-Lau interferometer and a stationary-anode x-ray tube. In this paper, theoretical principles and an experimental implementation of the corresponding CT imaging method are presented. First, a line integral is derived from analyzing the cross section of the small-angle scattering events. This method is referred to as small-angle scattering computed tomography (SAS-CT). Next, a Talbot-Lau interferometer and a rotating-anode x-ray tube were used to implement SAS-CT. A physical phantom and human breast tissue sample were used to demonstrate the reconstructed SAS-CT image volumes.
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In recent years, important progress has been made in x-ray differential phase contrast CT imaging [1–11] using a Talbot interferometer. The method was implemented using either synchrotron beam lines [1,2,12–16] with high brilliance and high spatial and spectral coherence or a conventional x-ray tube [4–7,11,17,18] with low brilliance. When this method is compared with other phase contrast imaging methods, such as diffraction enhanced imaging [19–24] or in-line holography [25–28], the requirements of spatial coherence and spectral coherence are reduced. A conventional x-ray tube with a focal spot on the order of a millimeter does not provide sufficient spatial coherence length for Talbot interferometry. However, this technical hurdle can be easily overcome [3,4] by additional beam collimation using an absorption linear grating with slit opening of several microns. As a result, a conventional medical x-ray tube can be used to implement phase contrast imaging. The result is a reduction in the distance from the x-ray focal spot to detector compared with in-line holography. A micro-focus x-ray tube  can also be used to implement differential phase contrast CT with a compact system design.
When an x-ray beam interacts with a medium, x-ray photons may be absorbed by the medium via photoelectric effect. X-ray photons may also be scattered by electrons via inelastic Compton scattering. These scattered photons have a wide distribution across the entire image object. The above two processes are the physical foundation of traditional absorption x-ray CT imaging method and have been well studied [30,31]. Due to the wave nature of the x-ray beam, a phase shift is also induced when the beam propagates through the medium. As a result, the incident x-ray beam will deviate from its incident direction, leading to refraction of the beam. This phenomenon is the physical foundation of and has been explored in diffraction enhanced imaging and differential phase contrast imaging methods.
In differential phase contrast imaging, an implicit assumption is that the medium can be considered as many uniform sub-regions such that wave front distortion is the primary effect when the beam propagates through the medium. This condition is challenged by small-angle scattering (SAS) events in the above piece-wise uniform assumption. In fact, when the mean size of the “particulate scatter” [32,33] is much greater than the wavelength of the coherent x-ray beam, x-ray small-angle coherent scattering events will cause an intensity reduction in a small cone shape region around the incident beam direction . The scattering angle , for a given x-ray energy, and thus wave number, k, is inversely proportional to the size of the scatter, a [33,34].
In the hard x-ray energy regime, the x-ray wavelength is shorter than 0.1 nm. Thus, the scatterers with a mean size from tens of nanometers to tens or hundreds of micrometers will contribute to the SAS cross section. As a result, analysis of SAS may provide a powerful tool to understand the micro-structure of the medium. In fact, small-angle x-ray scattering techniques have been well-established for studying features of colloidal size . It has been widely used in determining the structure of biological samples such as proteins, lipoproteins and membranes, and nucleic acids.
In Talbot-Lau interferometer based differential phase contrast imaging experiments, the above SAS effect is encoded in the detected intensity profile [5,8]. Due to the nature of SAS, it is expected that the intensity reduction caused by SAS is manifested as a change in the amplitude of the intensity modulation profile:1–4]. It has been argued  that the ratio of the modulation visibility, , measured with and without an image object can be defined as the dark field imaging signal caused by the SAS events. Indeed, experimental results demonstrate that some micro-structures can be observed in the dark field signal defined above, but not in the absorption or differential phase contrast signals. These findings indicate the rich contrast information provided by a single data acquisition process with absorption contrast, phase contrast, and SAS-induced dark field contrast all present. The above three contrast mechanisms may potentially provide complimentary information about the image object.
Currently, two important issues remain to be understood in SAS-induced dark field imaging contrast: the first is how the above measured modulation visibility is related to the microscopic theory of the SAS physics, and the second deals with the possibility of a tomographic reconstruction of the SAS signal measurements to reconstruct the local distribution of the microscopic structure of small-angle scatterers. In order to address these issues, an explicit imaging model for SAS imaging is needed and one needs to know whether and how the measured SAS signal can be written as a line integral of some physical quantities or a combination of the physical quantities of the small-angle scatterers.
In this paper, we present a theoretical and experimental study to address the above two issues. A microscopic model is developed to demonstrate that the modulation visibility, , is indeed the intensity reduction factor caused by the small-angle scatterers. The logarithmic of the final intensity reduction factor is a line integral of a combination of physical quantities used to characterize the scatterer size and number density. As a result, a tomographic reconstruction algorithm for absorption CT can be used to directly reconstruct the physical parameters to understand the local distribution of the small-angle scatterers. Finally, we conducted experiments using both a physical phantom and a breast tissue sample to demonstrate the above imaging models.
2. Small-angle scattering imaging model
2.1 Notations and small-angle approximations
Assume that the incident x-ray beam is described by a wave vector k with amplitude , where λ is the wavelength. After the incident wave is scattered once, the exit wave vector is determined by the wave vector transfer q:
Since the SAS event under consideration is elastic, . The angle between k and is the scattering angle θ. Using a simple geometric analysis and small-angle approximation,
Under the same approximation, the solid angle can be approximated:
Namely, an integral over a solid angle also represents an integral of all possible wave vector transfers q under the small-angle approximation.
2.2 Recursive relation of intensity change in the small-angle scattering process
According to the scattering theory, after a scattering event, the probability of finding one photon in a solid angle in the direction is given by. Here is determined by the form factor of the scatterer, where r 0 is the classical electron radius.
When an x-ray wave propagates through a medium along the z-direction, it experiences many SAS events. Assume the x-ray wave experiences (n-1) SAS events by the time it reaches position z along the propagation direction (Fig. 1 ). The wave is then scattered one more time within line element dz such that the wave experiences n SAS events at the position.
The angular distribution of photons elastically scattered (n-1) times is , the probability of finding one photon in a unit solid angle with a wave vector transfer q 1. Suppose q 2 is the wave vector transfer due to the single SAS event which occurred in the distance dz. The probability distribution of finding one scattered photon in a unit solid angle is given by 
Although the integral is formally written as over all possible directions, as shown in Eq. (4), the above integral also implies that the integral is performed over the amplitude of the vector q 1 when small scattering angle approximation is used. Equation (5) therefore implies that the probability of scattering n times is a convolution between probability of scattering (n-1) times events and a consecutive single scattering event.
As is well known in scattering theory, the Fourier transform of scattering intensity distribution gives the corresponding normalized signal intensity measured at the detector plane. Therefore, if we define a Fourier transform,Eq. (5) can be rewritten into the following recursive relation,
Using Eq. (7), the following important relation can be derived:
Namely, the final effect of the SAS events on signal intensity is a simple product of the effect of all individual scattering events . Note that this is significantly different from intuitive arguments [36,37] where the effect of SAS is treated as a convolution between a broadening kernel and the signal intensity without SAS.
2.3 Gaussian approximation and signal intensity reduction factor
For a single scattering event, the resulting angular distribution can be approximated as a Gaussian function :
A straightforward calculation of the Fourier integral gives the corresponding spatial distribution of the signal intensity,
Therefore, when an individual small-angle scattering event occurs, based on the characteristic size of the scatterers, the final signal intensity is reduced by a Gaussian factor.
2.4 Fundamental imaging equation in SAS-CT
In this subsection, based on the above theoretical framework, the basic imaging equation for SAS-CT is derived. Note that the probability of one SAS event within a propagation distance is proportional toEq. (8) and the Gaussian approximation in Eq. (10), the final Gaussian reduction factor due to small-angle scattering events is
Equation (13) states that after all single SAS events, within the Gaussian approximation, the final signal intensity is reduced by an Gaussian factor compared to the signal intensity which has not been affected by SAS processes. The final Gaussian factor with the effective width, R eff, is given by
Based on Eq. (1), the modulation visibility with an object in place is defined by
Since the spatial variable r represents the transverse distance measured from the incident wave direction, Eq. (13) dictates that there is no signal reduction at r = 0. This corresponds to the DC component in signal intensity Eq. (1) for the differential phase contrast CT data acquisition. Therefore, the normalized modulation visibility, V SAS, is nothing but the signal intensity reduction factor given by Eq. (13):
From this, the fundamental imaging equation for SAS-CT is obtained:
Equation (19) gives the relation of a physically measurable quantity, lnV SAS, and the microscopic nature of the small-angle scatterers, in terms of both their characteristic size and local density.
2.5 Image reconstruction algorithm in SAS-CT
The fundamental imaging equation of SAS-CT, Eq. (19), is of the same form as that for absorption CT. Therefore, the well-known filtered backprojection (FBP) algorithm with a ramp filter can be used to reconstruct SAS-CT images. By extension, an absorption cone-beam CT image reconstruction algorithm can be directly applied to SAS-CT without modifications. In this paper, the FDK reconstruction algorithm  is used to reconstruct images in the experimental studies described below.
3. Experimental methods
All data for the results presented in this paper were collected from a Talbot-Lau interferometer experimental setup constructed at the University of Wisconsin-Madison, as shown in Fig. 2 . The data acquisition system is similar to those previously reported , except for the use of a rotating-anode x-ray tube, compared with a stationary-anode tube used by other groups.
The system subcomponents used in the experimental setup include x-ray gratings, a rotating-anode x-ray tube, a flat-panel x-ray detector, and a rotating motion stage to enable tomographic acquisitions. The grating portion of the setup is comprised of three linear gratings fabricated using techniques described in literature . The first grating, labeled G0, is an absorption grating, which divides the x-ray beam exiting from the tube into an array of spatially coherent line sources, allowing the use of large focal spot (~1 mm) x-ray sources. The second grating, G1, is a phase grating, fabricated using a wet-etch procedure in Si. G1 is designed to introduce a π-phase shift at the mean energy of the x-ray beam for half of the incident x-rays. The final grating, G2, is an absorption grating, fabricated the same way as G1, but with the additional step of electroplating gold into the grating slits. This grating acts as an analyzer so that the replicated fringe pattern from G1 is converted to an intensity distribution at the detector plane. The fringe pattern occurs with a spatial period of 4.5 μm, which is too small to be resolved directly. In order to measure intensity modulation at the detector plane, a phase stepping approach is used [1–4]. In this work, 8 phase steps were used, sampled over the 4.5 μm period.
The x-ray tube used is a Varian G1592 with a 0.3 mm nominal focal spot size, which is connected to a CPI Indico 100 generator. Note that the demonstration of the use of a conventional rotating-anode x-ray tube is important if the technique is to prove useful outside the laboratory. As this is an interferometer-based design, it is sensitive to displacements of the system elements, with relative position changes as small as 10 nm between some components being detectable. This results in stability requirements that are orders of magnitude more demanding than conventional absorption-based imaging systems. The rotating-anode tube allows for much higher x-ray outputs than would be possible with stationary-anode tubes, thus allowing faster scans.
Two different image objects were studied: a calcification phantom and diseased human breast tissue. An illustration of the calcification phantom is shown in Fig. 3 . The calcification phantom is constructed from a PMMA cylinder with an inner diameter of 22.2 mm and wall thickness of 1.65 mm. The cylinder is filled with 15 g/100 mL-H2O of beef-hide gelatin and contains discrete layers of calcifications of different sizes. The calcifications are composed of calcium hydroxylapatite, and their arrangement in each layer is also shown in the illustration. The human breast tissue was acquired from a full mastectomy which was fixed in a formalin solution. A sample of the tissue containing known-diseased portions was removed by a pathologist to be scanned with the grating interferometer system.
For the CT acquisition, 360 projections were taken at 1° increments. Each projection had an exposure time of 40 seconds. The detector is a Rad-icon Shad-o-Box 2048, with 48 × 48 μm2 pixels across a 1024 × 2048 array. The tube potential was 40 kVp, with a continuous tube current of 20 mA. Once the intensity modulation is recorded, the data is processed to extract the SAS-CT and absorption projections.
In general, objects which have large amounts of small-angle scattering, regardless of x-ray attenuation, will have a substantial SAS-CT component. Using the relationship in Eq. (16), Eq. (1) can be rewritten:
so that the contributions of x-ray attenuation, small-angle scatter, and phase shift can all be separated. In addition to the known absorption contrast CT, SAS-CT image slices can be reconstructed using Eq. (19) as the fundamental imaging equation.
In order to reconstruct the projection data, the pixels are binned 2 × 2 for an effective pixel size of 96 × 96 μm2. As a result of the cone beam geometry of the imaging system, the absorption and SAS-CT images are reconstructed by the FDK reconstruction algorithm commonly used in cone-beam CT . The reconstructed image matrix is 500 × 500 × 500, with a voxel size of (80 μm)3. The resulting images show two different quantities, with the absorption image showing the usual linear attenuation, and the SAS-CT image showing a map of the local small-angle scattering characteristics.
4. Experimental results
The SAS-CT and attenuation contrast reconstructions of the calcification phantom are shown in Fig. 4 . The reconstructed layer shown in the figure is the 150-212 μm layer. Both the attenuation contrast and SAS-CT image show good visibility of the calcifications, though the contrast is higher in the SAS-CT image due to the removal of the background gelatin material. The attenuation image shows contributions from all parts of the phantom, including the gelatin background. This demonstrates a possible advantage of the SAS-CT contrast mechanism, the removal of relatively homogeneous material from an image object while leaving high contrast scattering objects.
SAS-CT and attenuation contrast reconstructions of human breast tissue are shown in Fig. 5 . An oil cyst is present in both of the reconstructions. This type of cyst is categorized by its spherical shape and hollow center. A large portion of the cyst is composed of calcium, which contributes significantly to the attenuation image. The structure of the cyst along with attenuation contributes to the SAS-CT contrast seen in the left hand image. As was the case with the calcification phantom, both contrast mechanisms allow for easy visualization of the object, though again the attenuation image has signal contribution from the background material which is not present in the SAS-CT image. This reconstruction shows the potential for SAS-CT imaging in breast imaging and the ability of the SAS-CT contrast mechanism to isolate signal presence from small-angle scatters like calcifications.
5. Discussion and conclusions
The physics of SAS events must be considered in order to develop an imaging equation for SAS-CT. A derivation of the imaging equation allows for the reconstruction process to reconstruct a physically-meaningful quantity using standard CT reconstruction methods. The reconstructed quantity in the case of SAS-CT is related to the small-angle scattering cross section, the small-angle scatter number density, and the characteristic scatterer size.
Using the derived imaging equation, SAS-CT images were reconstructed for comparison with corresponding phase contrast and absorption contrast CT images. The SAS-CT reconstructions provided unique and complementary information, enhancing the overall evaluation of the image object. Objects which contain small-angle scattering components do not necessarily provide enough attenuation contrast for easy detection in absorption-based imaging. SAS-CT imaging may be able to resolve small-scale, low-attenuation scattering structures in an otherwise low-contrast, relatively-uniform material. The potential for this was shown with both physical phantom and breast tissue results, where the SAS-CT contrast mechanism removed signal contributions from a relatively homogeneous background.
The technique used in this study allows for the simultaneous acquisition of SAS-CT, differential phase, and absorption contrast CT images, due to the fact that reconstructions of each type can be completed from a single data set. This would be critical in future clinical applications, where repeated scans for better disease detection would not be feasible.
In summary, in this paper, a microscopic derivation was developed for small-angle scattering computed tomography for a Talbot-Lau interferometer data acquisition method. It was clearly demonstrated that the logarithm of the modulation visibility in differential phase contrast imaging is related to the line integral of microscopic nature of the small-angle scatterers. This relation provides a foundation for extending the projection imaging method to tomographic reconstruction of physical quantities of small-angle scattering events. Both physical phantoms and breast tissue samples were scanned to demonstrate the SAS-CT principles developed in this paper.
Further work is planned to quantify the SAS-CT contrast mechanism in order to determine the physical aspects of the image object and to further investigate the potential for SAS-CT reconstructions in medical and industrial applications.
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