A single-exposure quantitative method of x-ray phase contrast imaging, suitable for animal in vivo observations, is described and shown experimentally both for a known static sample and an ex vivo biological airway. The ability to acquire the desired information within a single exposure is important for dynamic samples, as is sufficient sensitivity to reveal small variations in the composition or thickness of such a sample. This approach satisfies both these needs by analyzing how a reference grid pattern is deformed by the presence of the sample, similar to a Shack-Hartmann sensor. By resolving the shift of the pattern into horizontal and vertical components, a quantitative phase depth map is recovered, sensitive to both sharp edges as well as low phase gradients.
©2011 Optical Society of America
X-rays are widely used for the non-invasive examination of biological form and function, from basic research to the clinic. While in the past contrast was primarily achieved through absorption only, phase contrast x-ray imaging (PCXI) techniques [1, 2] now enable soft tissue structures, such as the airways, to be visualized [3, 4]. PCXI of dynamic samples , as used in biomedical studies of live animals, places restrictions on the choice of imaging technique since any sample movement means that a single-exposure method is desirable. Therefore propagation-based methods of x-ray phase contrast are often preferred, which provide good edge visibility , but have a lower sensitivity to phase gradients than interferometric , analyzer-based [7–9] or Talbot effect grating methods [10–12]. These more sensitive methods are able to produce single-exposure differential phase contrast images, but typically require multiple exposures for quantitative phase retrieval, in order to separate phase information from absorption information. Single-exposure analyzer-based PCXI is possible given a uniform field of illumination and a homogeneous sample [13, 14] or by recording the transmitted and diffracted images simultaneously [15, 16], but these techniques present some experimental difficulties in beam stability and alignment.
Here we present a single-exposure imaging method that determines the sample phase depth by analysing how the sample deforms a high visibility reference x-ray pattern. This technique uses commercially available grids to produce this reference pattern and does not have strong requirements regarding beam uniformity or precise alignment of optics. It is also capable of resolving phase features smaller than the grid spacing and retrieving quantitative phase depth maps.
These capabilities are realised by extending to two phase gradient directions the single grating technique described in our earlier publication , and applying a Fourier method of integration to retrieve phase depth from the two resulting perpendicular phase derivative maps [18–20].
The mechanism used here to realize phase contrast is as seen in a Shack-Hartmann sensor [21, 22], traditionally used for wavefront reconstruction in visible light optics. Such a sensor may study the phase of the incident wave by illuminating an array of lenslets that are perpendicular to the optic axis and looking at how the resulting focal spots are transversely displaced relative to their position when illuminated by a normally-incident planar wave. Each transverse focal-spot displacement is a result of the oblique incident angle of the wavefront on that lenslet, therefore the phase across the incident wave may be calculated from the displacements of many lenslets. Applications of a Shack-Hartmann sensor have included wavefront diagnosis to correct telescopes for atmospheric turbulence in astronomical imaging  and the study of vortices in laser light . Analysis techniques have included centroid approaches for determining how lenslet focus points are displaced, the use of Zernike polynomials and Karhunen-Loeve functions as wavefront basis functions , as well as Fourier-based demodulation techniques . The Shack-Hartmann sensor has been translated to the field of x-rays optics by using an array of x-ray lenslets  or a regular array of holes .
A similar concept for phase visualization was meanwhile shown for visible light imaging, using a high-contrast grating or grid as a reference pattern rather than an array of focus points. Images have been reconstructed by simple reference subtraction  and by Takeda’s Fourier shifting method [29–32].
Takeda’s method of analysis has also been used to reconstruct images from x-ray exposures where a reference pattern is created either from a regular attenuation grid  or from regular fringes created using a Talbot interferometer . Both of these methods enable single-exposure image reconstruction, but place restrictions on sample size, without showing quantitative reconstruction. Liu et al.  analyzed a similar distorted reference pattern using a modified hybrid input-output algorithm.
Note that multiple-grating based phase contrast imaging [10–12,19] employs a different method of phase visualization, where the Talbot effect provides contrast, and the specially designed gratings are stepped relative to each other transversely with one exposure required at each step to extract phase information.
While the set-up proposed in our paper is similar to that used by Wen et al. , their samples were larger (hence on a different length scale relative to the grid period) and sit within the field of view, so they are able to employ Takeda’s method of analysis [29–32]. The sub-micron pixel size used in our imaging means that the grid frequency is not sufficiently separated from the sample feature size and therefore Takeda’s method of Fourier analysis  may not be used. Our interrogation window based method therefore enables imaging of small features within a sample without requiring a grid of even smaller spacing. Several advantages are obtained from using a local method rather than a frequency-based method. These include sensitivity to sharp edges, the ability to use an irregular or imperfect grid and the ability to image and reconstruct a section of the sample without boundary requirements.
The use of a single attenuating grid before the sample is similar to a micro-beam therapy set-up where 25 μm wide beams are used to irradiate tumors  (comparable to the 32 μm grating gaps used here). This suggests possibilities of simultaneous therapy and imaging, although the x-ray energy used in micro-beam therapy is typically higher than that used for imaging.
It is anticipated that our single-exposure method will be of most use in imaging small or otherwise weakly contrasting features within a dynamic sample. In particular this enables biomedical researchers to monitor the evolution of such features in living systems in real time, for example to study the effects of a treatment or some other intervention.
The experimental set-up is as seen in Fig. 1 , where an attenuating grid (overlaid Gilder 400 lppi 3.05mm diameter gold electron microscopy bar grids) is placed immediately upstream of the sample. The high contrast reference pattern from the grid is modified by phase gradients in the sample (note that placing the grid immediately downstream of the sample would also produce suitable images, but would result in an unnecessarily higher sample dose). With propagation from the sample to the detector, these phase gradients will distort the reference intensity pattern. The integrated phase shift φ at each point on the nominally planar sample exit surface can then be found by determining the transverse shift S of the reference grid pattern, resolved into horizontal shift Sx and vertical shift Sy.
From the geometry in Fig. 1
By comparing the grid-and-sample image with a reference grid-only image taken separately, the magnitude of the reference pattern shift S may be determined at each pixel (x,y) within the sample image. The pixel size need not be reduced during analysis, given the continuous nature of the reference pattern (unlike traditional Shack-Hartmann sensors which track the motion of a finite number of features, each much greater than the pixel size) and the real-space analysis (unlike Fourier analysis techniques). For each pixel (x,y) in the real-space image, the shift (S) required to achieve the position of closest correlation between the reference grid-only image and a sample of the grid-and-sample image, centered at that pixel (x,y), is found and resolved into the horizontal and vertical components (Sx and Sy). The method, as performed at each pixel (x,y), is illustrated schematically in Fig. 2 , which shows the top right corner of a grid hole, with the effects of Fresnel diffraction from the grid itself. The grid-only image is shown with a red color-map and the interrogation window from the grid-and-sample image is shown superimposed with a blue color-map.
In Fig. 2 a ) the grid-and-sample interrogation window is shown in place as observed, and it can be seen that the image of the grid pattern has been displaced (as indicated by the blue arrow) due to the presence of the sample. The blue grid-and-sample image interrogation window is shifted relative to the grid-only image (Fig. 2 b )) to locate the position which corresponds most closely with the grid-only image. This position is found by taking the maximum of the cross-correlation of the local grid-only image (shown in red color-map) and the grid-and-sample image interrogation window (shown in blue color-map). Each cross-correlation is implemented in the Fourier domain for computational speed, and then normalized as described by Lewis , with normalization required to account for the large variations in intensity across the interrogation window. As in Fig. 2 c ), the shift required for alignment, S, may then be described in terms of Sx and Sy, the reference pattern shift in each of the two perpendicular directions.
Both the reference and sample grid images may also be interpolated at a fraction of the pixel size before the interrogation window analysis in order to enable sub-pixel shifts (in the example shown in this paper the image was interpolated at half the pixel size).
Recording the local reference pattern shifts Sx and Sx for each pixel (x,y) within the grid-and-sample image provides two differential phase contrast images, as seen in the experimental example of Fig. 3 , where the edges of two 1.5 mm diameter Perspex spheres are imaged with 0.18 μm pixels (full experimental parameters given in the caption to Fig. 3). These shifts may then be converted into the angles θx and θy using Eq. (1).
For radiation with wavenumber k, the sample phase depth φ(x,y) will deflect the incident rays by angles θx and θy according to,
Combining Eqs. (1) and (2) and assuming a single material, so that φ = -kδT, where δ is the real refractive index decrement from 1 and T is the projected thickness , the thickness gradients may be retrieved in both transverse directions,
Equation (3) will therefore convert the two shift images (Fig. 3 a ) and b)) into thickness gradient images. These images may be integrated to give the thickness T(x,y), by making use of the Fourier derivative theorem,18–20]
This identity can be verified by substituting the Fourier derivative theorem (Eq. (4) into Eq. (6) itself. The constant of integration came by setting the Fourier origin (kx = 0, ky = 0) of the quantity in square brackets such that the recovered projected thickness T was equal to zero in the area outside the sample (thereby avoiding a singularity at the Fourier origin). This amounts to taking the Cauchy principle value of the integral transform that is represented in discrete form by Eq. (4).
3. Reconstructed thickness maps
Using the method described in Sec. 2, the projected thickness map of the 1.5mm diameter Perspex spheres of Fig. 3 (experimental parameters in the caption) was reconstructed, as shown in Fig. 4 a ). Note that Fig. 4 a ) appears much less noisy than Fig. 3, since integration is inherently insensitive to low-amplitude high frequency noise in a derivative image.
The reconstructed thickness is then compared to the theoretical thickness of two adjacent perfectly spherical Perspex spheres in Fig. 4 b ), showing very good correspondence with a normalized RMS error of 3.2%. Note that the sample has scratches on the surface of what are not perfect spheres, so differences in the retrieved and “theoretical” thickness do not necessarily correspond to inaccuracies in the retrieval. Figure 4 c ), a surface plot of the reconstructed thickness, demonstrates how well the sharp edges of the spheres are reconstructed, as well as the areas of lower phase gradient towards their centers.
The phase gradient that may be resolved using this method is obtained substituting the detectable transverse shift, a, into Eq. (3),
Therefore there is greater sensitivity to weak gradients at longer propagation distances, and with detectors of higher spatial resolution (small a). If shifts of down to the size of one pixel are detectable, evaluation of Eq. (7) gives that thickness gradients of down to 0.45 should be detected in the example shown in Figs. 3 and 4 (). This gradient would occur at the point as marked by the arrow in Fig. 3 a ). As seen in Fig. 3, even lower phase gradients may be detected using sub-pixel alignment, as the high contrast reference pattern permits transverse shifts of less than a pixel to be detected (a shift of 0.18 μm pixels ÷ 2 corresponds to a gradient of 0.225, also marked in Fig. 3). The maximum resolvable thickness gradient detectable (without using phase unwrapping) may be found by evaluating Eq. (7) with the grid period as a, yielding a gradient of 9170 (i.e. 0.9 cm/μm) in this example.
Given that tissue has a similar composition by mass and hence a similar refractive index to Perspex , similar thickness gradients should be detectable in biomedical imaging. In the same way that thickness variations may be detected, a refractive index decrement change of 45% (e.g. δ=4.0×10−10 to δ=5.8×10−10) would be observable. This is of relevance to our ongoing study imaging the airway surface [40–42]. We seek to observe micron-size changes in the airway surface liquid (ASL) depth, seen during disease states including cystic fibrosis, as a marker for assessing airway health and disease treatments. The ASL differs in refractive index from the surrounding tissue by a similar percentage .
To test the utility of our method for imaging the airway surface, a trachea (a long cylindrical airway), excised from a mouse, was maintained in a 0.7% saline bath with tubing connected at each end to maintain the trachea lumen as air-filled, and the surrounding saline warmed to body temperatures to approximate surrounding tissue (Fig. 5 ). This ex vivo arrangement removed complications from the overlying structures (skin, fur etc.) encountered in intact animal imaging. Although the trachea was excised, the airway surface physiology (ion balance of the airway liquid layer and underlying cilia) remains functioning for several hours when the temperature and hydration are appropriately maintained . A ventilator moved moist air through the airway during imaging, mimicking normal mouse breathing. 25 keV x-rays were used with 0.18 μm pixels, 1 meter propagation from sample to detector and the 64 μm period gold grating.
Figure 5 a ) shows a propagation-based phase contrast image of one side of the excised airway, where a strong interference fringe is observed between the air-filled lumen and the trachea tissue. The less-distinct interface between the tissue and the surrounding water is also visible, as is a speckle pattern from the cartilage rings protecting the airway. The Tranport of Intensity Equation (TIE)  was used to create a phase retrieved image , as shown in Fig. 5 b ). It is seen here that the quality of the TIE reconstruction is strongly dependent on a uniform field of illumination – despite the fact that the original image was divided by a flat field image in an attempt to remove variations in the incident illumination, low frequency artifacts persist in the reconstructed phase image.
Figure 5 c ) shows the same airway imaged and d) reconstructed using the method described in Section 2. Given that the phase gradients are predominantly in the horizontal direction, a 45° diagonal grating is sufficient for reconstruction .
It is now not only possible to see and compare different regions; air, tissue and liquid, but also to look at the projected thickness and curvature of the airway. The ability to differentiate between tissue and the saline bath is promising for our ongoing work differentiating the ASL from tissue at high resolution.
It can be seen that the dark propagation-based interference fringe along the inside of the airway disrupts the interrogation window analysis in this region (although this does not debilitate the thickness reconstruction). These artifacts could be reduced by introducing a spinning random phase screen close to the sample, which would decrease the absolute value of the complex degree of coherence , and hence suppress the propagation-based phase contrast .
A method of imaging phase gradients and recovering projected phase shift with x-rays has been demonstrated, using a simple set-up consisting of a source, a single grid and a CCD detector. Contrast is achieved through a Shack-Hartmann-like mechanism, by determining how a sample distorts a reference grid pattern.
Requiring only a single exposure to reconstruct the sample, this technique is well-suited to observing live samples, and the lack of need for any boundary conditions means that a small feature within a larger biological system may be easily observed in vivo over time. While propagation-based PCXI may be preferred for observing small high-contrast features (e.g. debris moving along the airway surface ), this grid technique may be preferred for visualizing weaker features on a larger scale (e.g. airway surface structure and radius of curvature). This would be of value to other airway studies, for example looking at how the airways constrict/expand with asthma.
The authors would like to thank the Japan Synchrotron Radiation Research Institute for the privilege of using the SPring-8 facilities for these experiments and scientists Yoshio Suzuki, Akihisa Takeuchi, Kentaro Uesugi and Naoto Yagi for their assistance. We acknowledge travel funding provided by the International Synchrotron Access Program (ISAP) managed by the Australian Synchrotron. The ISAP is an initiative of the Australian Government being conducted as part of the National Collaborative Research Infrastructure Strategy. K. Morgan acknowledges the support of a postgraduate publication award at Monash University, D. Paganin acknowledges the Australian Research Council and K. Siu the National Health and Medical Research Council. We thank Richard Carnibella for his advice on efficiently computing normalized cross correlations, and acknowledge his and Aidan Jamison’s assistance during the experiment, as well as Andreas Fouras for the use of his CCD camera. Finally, we thank David Parsons and Martin Donnelley for their vital roles in the airway imaging project.
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