Coherent X-ray scatter imaging and its applications in biomedical science and industry

X-ray Fokker-Planck equation for paraxial imaging

The Fokker-Planck equation can be used in a partially-coherent imaging context to model the evolution of the intensity of a paraxial x-ray wave field with propagation. This forms a natural generalisation of the transport-of-intensity equation. The x-ray Fokker-Planck equation can simultaneously account for both propagation-based phase contrast, and the diffusive effects of sample-induced small-angle x-ray scattering, when forming an x-ray image of a thin sample. Two derivations are given for the Fokker-Planck equation associated with x-ray imaging, together with a Kramers-Moyal generalisation thereof. Both equations are underpinned by the concept of unresolved speckle due to unresolved sample micro-structure. These equations may be applied to the forward problem of modelling image formation in the presence of both coherent and diffusive energy transport. They may also be used to formulate associated inverse problems of retrieving the phase shifts due to a sample placed in an x-ray beam, together with the diffusive properties of the sample. The domain of applicability for the Fokker-Planck and Kramers-Moyal equations for paraxial imaging is at least as broad as that of the transport-of-intensity equation which they generalise, hence the technique is also expected to be useful for paraxial imaging using visible light, electrons and neutrons.

Interior x-ray diffraction tomography with low-resolution exterior information

X-ray diffraction tomography (XDT) resolves spatially-variant XRD profiles within macroscopic objects, and provides improved material contrast compared to the conventional transmission-based computed tomography (CT). However, due to the small diffraction cross-section, XDT suffers from long imaging acquisition time, which could take tens of hours for a full scan using a table-top x-ray tube. In medical and industrial imaging applications, oftentimes only the XRD measurement within a region-of-interest (ROI) is required, which, together with the demand to reduce imaging time and radiation dose to the sample, motivates the development of interior XDT systems that scan and reconstruct only an internal region within the sample. The interior problem does not have a unique solution, and a direct inversion on the truncated projection data often leads to large reconstruction errors in ROI. To reduce the truncation artifacts, conventional attenuation-based interior reconstruction problems rely on a known region or piecewise constant constraint within the ROI. Here we propose a quasi-interior XDT scheme that incorporates a small fraction of projection information from the exterior region to assist ROI reconstruction. In the phantom simulation, a small amount (17% of exterior region) of added exterior projection data improves the reconstruction quality by ~50%. The addition of exterior samplings in the experiment demonstrates improved spatial and XRD profile reconstructions compared to total-variation-based reconstruction or sinogram extrapolation. We expect our quasi-interior XDT to obviate the requirement on prior knowledge of the object or its support, and to allow the ROI reconstruction to be performed with the fast, widely-used filtered back-projection algorithm for easy integration into real-time XDT imaging modules.

Detection with polychromatic x-ray pencil beam illumination: information-theoretic bounds

Non-destructive testing (NDT) by x-ray imaging is commonly used for finding manufacturing defects, cargo inspection, or security screening. These tasks can be regarded as examples of a detection problem where a target is either present or not. Task-specific information (TSI) [J. Opt. Soc. Am. A24, B25 (2007)JOAOD60740-323210.1364/JOSAA.24.000B25; Appl. Opt.47, 4457 (2008)APOPAI0003-693510.1364/AO.47.004457] bounds, an information-theoretic based metric, are presented for a threat detection task. A system using polychromatic x-ray pencil beam object illumination and energy-resolving detectors for both absorption and diffraction measurements is employed for this task. Water and diesel are two liquids chosen as non-threat and threat materials, respectively, for this study. Three different threat class configurations are examined: a homogeneous object with fixed thickness, a homogeneous object with stochastic thickness, and a dual-material object (i.e., representing a target and clutter) with stochastic thickness, where the threat material has a fixed thickness. We find for the threat class composed of a dual-material object that a minimum threat thickness of 4.5 cm is needed to achieve a desired TSI≥0.7 using a joint absorption and diffraction measurement.

X-ray diffraction tomography with limited projection information

X-ray diffraction tomography (XDT) records the spatially-resolved X-ray diffraction profile of an extended object. Compared to conventional transmission-based tomography, XDT displays high intrinsic contrast among materials of similar electron density and improves the accuracy in material identification thanks to the molecular structural information carried by diffracted photons. However, due to the weak diffraction signal, a tomographic scan covering the entire object typically requires a synchrotron facility to make the acquisition time more manageable. Imaging applications in medical and industrial settings usually do not require the examination of the entire object. Therefore, a diffraction tomography modality covering only the region of interest (ROI) and subsequent image reconstruction techniques with truncated projections are highly desirable. Here we propose a table-top diffraction tomography system that can resolve the spatially-variant diffraction form factor from internal regions within extended samples. We demonstrate that the interior reconstruction maintains the material contrast while reducing the imaging time by 6 folds. The presented method could accelerate the acquisition of XDT and be applied in portable imaging applications with a reduced radiation dose.

Non-invasive classification of breast microcalcifications using x-ray coherent scatter computed tomography

We investigate the use of energy dispersive x-ray coherent scatter computed tomography (ED-CSCT) as a non-invasive diagnostic method to differentiate between type I and type II breast calcifications. This approach is sensitive to the differences of composition and internal crystal structure of different types of microcalcifications. The study is carried out by simulating a CSCT system with a scanning pencil beam, considering a polychromatic x-ray source and an energy-resolving photon counting detector. In a first step, the multidimensional angle and energy distributed CSCT data is reduced to the projection-space distributions of only a few components, corresponding to the expected target composition: adipose, glandular tissue, weddellite (calcium oxalate) for type I calcifications, and hydroxyapatite for type II calcifications. The maximum-likelihood estimation of scatter components algorithm used, operating in the projection space, takes into account the polychromatic source, the detector response function and the energy dependent attenuation. In the second step, component images are reconstructed from the corresponding estimated component projections using filtered backprojection. In a preliminary step the coherent scatter differential cross sections for hydroxyapatite and weddellite minerals were determined experimentally. The classification of type I or II calcifications is done using the relative contrasts of their components as the criterion. Simulation tests were carried out for different doses and energy resolutions for multiple realizations. The results were analyzed using relative/receiver operating characteristic methodology and show good discrimination ability at medium and higher doses. The noninvasive CSCT technique shows potential to further improve the breast diagnostic accuracy and reduce the number of breast biopsies.

Full-field energy-dispersive powder diffraction imaging using laboratory X-rays

A laboratory instrument with the ability to spatially resolve energy-dispersed X-ray powder diffraction patterns taken in a single snapshot has been developed. The experimental arrangement is based on a pinhole camera coupled with a pixelated spectral X-ray detector. Collimation of the diffracted beam is defined by the area of the footprint of a detector pixel and the diameter of the pinhole aperture. Each pixel in the image, therefore, contains an energy-dispersed powder diffraction pattern. This new X-ray imaging technique enables spatial mapping of crystallinity, crystalline texture or crystalline phases from within a sample. Validation of the method has been carried out with a back-to-back comparison with crystalline texture mapping local to a friction stir weld in an aluminium alloy taken using synchrotron radiation.

Snapshot molecular imaging using coded energy-sensitive detection

We demonstrate a technique for measuring the range-resolved coherent scatter form factors of different objects from a single snapshot. By illuminating the object with an x-ray pencil beam and placing a coded aperture in front of a linear array of energy-sensitive detector elements, we record the coherently scattered x-rays. This approach yields lateral, range, and momentum transfer resolutions of 1 mm, 5 mm, and 0.2 nm⁻¹, respectively, which is sufficient for the distinguishing a variety of solids and liquids. These results indicate a path toward real-time volumetric molecular imaging for non-destructive examination in a variety of applications, including medical diagnostics, quality inspection, and security detection.

Achieving a stable time response in polymeric radiation sensors under charge injection by X-rays

Existing inorganic materials for radiation sensors suffer from several drawbacks, including their inability to cover large curved areas, lack of tissue-equivalence, toxicity, and mechanical inflexibility. As an alternative to inorganics, poly(triarylamine) (PTAA) diodes have been evaluated for their suitability for detecting radiation via the direct creation of X-ray induced photocurrents. A single layer of PTAA is deposited on indium tin oxide (ITO) substrates, with top electrodes selected from Al, Au, Ni, and Pd. The choice of metal electrode has a pronounced effect on the performance of the device; there is a direct correlation between the diode rectification factor and the metal-PTAA barrier height. A diode with an Al contact shows the highest quality of rectifying junction, and it produces a high X-ray photocurrent (several nA) that is stable during continuous exposure to 50 kV Mo Kalpha X-radiation over long time scales, combined with a high signal-to-noise ratio with fast response times of less than 0.25 s. Diodes with a low band gap, 'Ohmic' contact, such as ITO/PTAA/Au, show a slow transient response. This result can be explained by the build-up of space charge at the metal-PTAA interface, caused by a high level of charge injection due to X-ray-induced carriers. These data provide new insights into the optimum selection of metals for Schottky contacts on organic materials, with wider applications in light sensors and photovoltaic devices.

X-ray colour imaging

A prototype X-ray colour imaging system has been assembled using the principle of tomographic energy-dispersive diffraction imaging (TEDDI). The new system has been tested using samples of nylon-6, aluminium powder and deer antler bone. Non-destructive three-dimensional images of the test objects have been reconstructed on a 300 microm scale with an associated diffraction pattern at each voxel. In addition, the lattice parameters of the polycrystalline material present in the sampled voxels have been determined using full pattern refinement methods. The use of multiple diffracted parallel colour X-ray beams has allowed simultaneous spatially resolved data collection across a plane of the sample. This has simplified the sample scan motion and has improved data collection times by a factor scaling with the number of detector pixels. The TEDDI method is currently limited to thin samples (approx. 1-2mm) with light atoms owing to the very low detection efficiency of the silicon detector at X-ray energies above 25 keV. We describe how these difficulties can be removed by using semiconductor detectors made from heavier atomic material.

Hard-X-ray dark-field imaging using a grating interferometer

Imaging with visible light today uses numerous contrast mechanisms, including bright- and dark-field contrast, phase-contrast schemes and confocal and fluorescence-based methods. X-ray imaging, on the other hand, has only recently seen the development of an analogous variety of contrast modalities. Although X-ray phase-contrast imaging could successfully be implemented at a relatively early stage with several techniques, dark-field imaging, or more generally scattering-based imaging, with hard X-rays and good signal-to-noise ratio, in practice still remains a challenging task even at highly brilliant synchrotron sources. In this letter, we report a new approach on the basis of a grating interferometer that can efficiently yield dark-field scatter images of high quality, even with conventional X-ray tube sources. Because the image contrast is formed through the mechanism of small-angle scattering, it provides complementary and otherwise inaccessible structural information about the specimen at the micrometre and submicrometre length scale. Our approach is fully compatible with conventional transmission radiography and a recently developed hard-X-ray phase-contrast imaging scheme. Applications to X-ray medical imaging, industrial non-destructive testing and security screening are discussed.

Energy dispersive X-ray diffraction as a means to identify illicit materials: A preliminary optimisation study

Energy dispersive X-ray diffraction is proposed as a suitable non-destructive method to rapidly identify illicit drugs in parcels. A preliminary data set of 7 illicit drug samples and a possible cutting agent has been collected with a range of count times using a tungsten target X-ray source, a high resolution HpGe detector and a variable geometry diffraction cell. These results have been used to calibrate and train multivariate analysis software to predict the drug content in previously unseen spectra.

Small angle scatter imaging from wide beam diffraction patterns

In this paper we report on the extension of the technique of mapping small angle x-ray scatter (SAXS) across a soft material specimen several millimetres square. In the conventional SAXS mapping technique a pencil beam of x-rays is raster scanned over the specimen with the scatter pattern recorded from each point in the raster. In our technique a wide, parallel beam is used, speeding up the data collection time considerably. An image processing algorithm is used to separate the scatter pattern features from individual points along the line of the beam. To test the efficacy of the technique a phantom was constructed using gelatin and rat tail tendon collagen. Collagen fibres in the phantom were arranged in quarters horizontally, diagonally and vertically leaving one quarter with just gelatin. The phantom was used to collect both raster scanned sets of SAXS patterns spaced at 0.25 mm horizontally and vertically and also a wide beam data set. The width of the beam in this case was approximately 7 mm. Using the third-order diffraction of rat tail tendon intensity data were gathered from each SAXS pattern and used to construct a map. Data from the raster scan image and that from the wide beam are compared. Finally using a phantom made from dehydrated rat tail tendon and paraffin wax a tomographic slice constructed using data from SAXS patterns is shown.

A semianalytic model to extract differential linear scattering coefficients of breast tissue from energy dispersive x-ray diffraction measurements

The goal of this work is to develop a technique to measure the x-ray diffraction signals of breast biopsy specimens. A biomedical x-ray diffraction technology capable of measuring such signals may prove to be of diagnostic use to the medical field. Energy dispersive x-ray diffraction measurements coupled with a semianalytical model were used to extract the differential linear scattering coefficients [mus(x)] of breast tissues on absolute scales. The coefficients describe the probabilities of scatter events occuring per unit length of tissue per unit solid angle of detection. They are a function of the momentum transfer argument, x=sin(theta/2)/X, where theta=scatter angle and lambda=incident wavelength. The technique was validated by using a 3 mm diameter 50 kV polychromatic x-ray beam incident on a 5 mm diameter 5 mm thick sample of water. Water was used because good x-ray diffraction data are available in the literature. The scatter profiles from 6 degrees to 15 degrees in increments of 1 degrees were measured with a 3 mm x 3 mm x 2 mm thick cadmium zinc telluride detector. A 2 mm diameter Pb aperture was placed on top of the detector. The target to detector distance was 29 cm and the duration of each measurement was 10 min. Ensemble averages of the results compare well with the gold standard data of A. H. Narten ["X-ray diffraction data on liquid water in the temperature range 4 degrees C-200 degrees C," ORNL Report No. 4578 (1970)]. An average 7.68% difference for which most of the discrepancies can be attributed to the background noise at low angles was obtained. The preliminary measurements of breast tissue are also encouraging.

Optimum momentum transfer arguments for x-ray forward scatter imaging

In our research program we have shown through modeling, related numerical calculations, and experimental measurements that there exists a potential use of scattered radiation for medical x-ray imaging. Each incident photon of wavelength lambda which scatters at a small angle theta with respect to its initial direction of travel has a change in momentum characterized by the photon momentum transfer argument x = lambda(-1) sin(theta/2). In this work, we show that in order to maximize the signal-to-noise ratio (SNR) obtained with scattered x rays, one must detect photons with specific x values. Using a photon counting detector to distinguish 2-cm-thick polymethyl methacrylate and nylon targets situated within a 15-cm-diam spherical water phantom with an 80 kV beam yields experimentally SNR/square root(K(air)c) = 12.8 +/- 0.2 (mJ/kg)(-1/2) when using the photons between x = 0.5 and 0.7 nm(-1). Here K(air)c is the air collision kerma and the average momentum transfer argument, x, is calculated by weighting x by the incident photon fluence distribution. The model predicts a value of SNR/square root(K(air)c) = 12.9 (mJ/kg)(-1/2). If we choose to form the signal with the range in x extended to be from 0.5 to 1.0 nm(-1) then, despite the detection of more scattered photons, experimentally SNR/square root(K(air)c) decreases by 38% to 7.9 +/- 0.3 (mJ/kg)(-1/2). The model predicts a value of 9.46 (mJ/kg)(-1/2). Results for energy integrating detectors are in general similar to those for photon counters, but there exist cases where a significant decrease in SNR can occur. For example, for measurements in air with the two plastics at theta = 3 degrees the SNR for an energy integrator was found to be 52% that of a photon counter. Numerical calculations predict that the effects of spectral blur can be significant when a narrow angular range is used for detection. Preliminary numerical predictions for breast tissues suggest a potential use of x-ray scatter in the field of mammography.

Material-specific analysis using coherent-scatter imaging

Coherent-scatter computed tomography (CSCT) is a novel imaging method we are developing to produce cross-sectional images based on the low-angle (<10 degrees) scatter properties of tissue. At diagnostic energies, this scatter is primarily coherent with properties dependent upon the molecular structure of the scatterer. This facilitates the production of material-specific maps of each component in a conglomerate. Our particular goal is to obtain quantitative maps of bone-mineral content. A diagnostic x-ray source and image intensifier are used to acquire scatter patterns under first-generation CT geometry. An accurate measurement of the scatter patterns is necessary to correctly identify and quantify tissue composition. This requires corrections for exposure fluctuations, temporal lag in the intensifier, and self-attenuation within the specimen. The effect of lag is corrected using an approximate convolution method. Self-attenuation causes a cupping artifact in the CSCT images and is corrected using measurements of the transmitted primary beam. An accurate correction is required for reliable density measurements from material-specific images. The correction is shown to introduce negligible noise to the images and a theoretical expression for CSCT image SNR is confirmed by experiment. With these corrections, the scatter intensity is proportional to the number of scattering centers interrogated and quantitative measurements of each material (in g/cm3) are obtained. Results are demonstrated using both a series of poly(methyl methacrylate) (PMMA) sheets of increasing thickness (2-12 mm) and a series of 5 acrylic rods containing varying amounts of hydroxyapatite (0-0.400 g/cm3), simulating the physiological range of bone-mineral density (BMD) found in trabecular bone. The excellent agreement between known and measured BMD demonstrates the viability of CSCT as a tool for densitometry.

Updating of form factor tabulations for coherent scattering of photons in tissues

An updating of photon transport modelling in tissues is carried out by including the effect of molecular interference in the coherent (Rayleigh) scattering. To this end, the present tabulations--which permit us to obtain the linear differential scattering coefficient of compounds from a simple weighted sum of the elemental components--are integrated by adding files for a limited set of molecular interference functions. This set originates from a four-component model which is found to be capable of reproducing human tissues in situations involving bony and soft tissues. The proposed procedure overcomes, in the computation, the hindrance that the dependence on molecular interference effects leads every tissue to have its own diffraction pattern, which is not easily obtained by means of measurements or calculations.