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X-ray phase contrast imaging (XPCi) methods are sensitive to phase in addition to attenuation effects and, therefore, can achieve improved image contrast for weakly attenuating materials, such as often encountered in biomedical applications. Several XPCi methods exist, most of which have already been implemented in computed tomographic (CT) modality, thus allowing volumetric imaging. The Edge Illumination (EI) XPCi method had, until now, not been implemented as a CT modality. This article provides indications that quantitative 3D maps of an object’s phase and attenuation can be reconstructed from EI XPCi measurements. Moreover, a theory for the reconstruction of combined phase and attenuation maps is presented. Both reconstruction strategies find applications in tissue characterisation and the identification of faint, weakly attenuating details. Experimental results for wires of known materials and for a biological object validate the theory and confirm the superiority of the phase over conventional, attenuation-based image contrast.
© 2014 Optical Society of America
1. Introduction
For many decades after x-rays were discovered in 1895, radiography has suffered from two limitations: overlapping structures in projections, and poor contrast for low attenuating materials (e.g. soft tissues). While the former limitation was overcome by the development of computed tomography (CT) in the mid-70s, the latter to a good extent still applies today. One option to remove the second limitation is based on the observation that, when x-rays pass through an object, not only do they get attenuated, but they also undergo a phase shift. Both physical phenomena can be linked to the object’s complex refractive index:
where E is the x-ray energy. The parameters δ and β drive phase and attenuation effects, respectively. For lowly attenuating materials such as soft tissue, δ can be up to three orders of magnitude larger than β for energies used in biomedical imaging [1]. Unlike conventional radiographic techniques, which only exploit β, x-ray phase contrast imaging (XPCi) methods are sensitive to both parameters. Unsurprisingly, an improved image quality has been demonstrated for various soft tissue types [1]. Different XPCi methods exist [2–6], and most of them are already available also as CT modalities [7–10]. Among the existing techniques, the Edge Illumination (EI) XPCi method has high potential for widespread application, as it is based on a simple working principle [11], can be implemented both at synchrotrons and with conventional x-ray sources and scaled up to large fields of view [6], is relatively unaffected by environmental vibrations [12], has a high phase sensitivity [13, 14] and is dose efficient [15]. Despite these advantages, EI XPCi had, until now, not been implemented in CT mode. This article bridges this gap.In the following, the working principle of EI XPCi is described, followed by a theoretical discussion on the possibility to reconstruct 3D maps of the parameters δ and β, which finds application for example in multi-modal material identification [16]. Moreover, by adapting a strategy developed for an alternative XPCi method [17], we present a formula for a combined phase and attenuation reconstruction that does not require the separation of the two effects. Finally, experimental results are presented that confirm the theory, obtained for a custom-built wire phantom consisting of known materials and a biological object (a domestic wasp). These are the first quantitative tomographic EI XPCi images ever published. Entrance dose values recorded during the scans are also provided.
2. Theory
The working principle of EI XPCi is schematically shown in Fig. 1(a). A laminar beam traverses an object and, after a distance zod, impinges on a single detector row. An x-ray absorbing edge is placed in contact with the detector row such that a fraction of the sensitive area of each pixel is covered; consequently, a part of the laminar beam hits the pixels, while the other part hits the absorbing edge. As a consequence, a positive/negative refraction of the beam results in a higher/lower measured intensity and image contrast is, therefore, due to a combination of attenuation and refraction. Repositioning the edge such that it covers complementary parts of the pixels [Fig. 1(b)] has the effect of inverting the refraction contrast. In practice, it is convenient to use the bottom and top edges of a slit to achieve the two edge illumination conditions, as it makes it easy to switch between the two configurations. The intensity on the detector without an object is a function of the edge position. The corresponding curve (in the following referred to as C(y0), where y0 is the edge position), which can be measured by scanning the edge vertically through the beam while recording the intensity, is a monotonically increasing/decreasing function for the two edge/detector configurations in Figs. 1(a) and 1(b). The curves corresponding to the experimental EI XPCI setup described in Section 3 are shown in Fig. 2.
Fig. 1 The working principle of EI XPCi. A laminar beam falls partially on an x-ray absorbing edge and partially on a row of detector pixels. As a consequence, a sensitivity to refraction effects in the y-direction is achieved. The two complementary edge/detector row configurations in (a) and (b) are possible. The rotation of the object enables tomographic imaging.
Fig. 2 The curves, in the following denoted by C, show the intensity on the detector (here given as a fraction of the maximum achievable intensity) as a function of the edge position y0 for the edge/detector configurations displayed in Figs. 1(a) and 1(b). Positioning the edge at y0 = 0 corresponds to 50% of the maximum achievable intensity.
The refraction-induced displacement (D) and the transmission (T) of the beam can be expressed as [17]:
where the object is described in terms of the phase shift and attenuation it imposes on the beam: 𝒪 is the extent of the object and k is the wave number. Assuming a laminar beam of intensity I0 and the absorbing edge to be positioned at y0, in the case of negligible small angle scattering the intensity on the detector with an object can be described by [13]: When two projections are acquired with the two complementary edge/detector configurations [Figs. 1(a) and 1(b)] and processed according to a previously presented algorithm [13], μ and the first derivative of Φ can be extracted. Consequently, when the method is implemented in CT mode, i.e. the object is rotated and imaged at multiple angles (θ), the following sinograms can be obtained: where (ξ = xcosθ − zsinθ, η = y, ζ = xsinθ + zcosθ) is the frame of reference of the object. The line ℓ(x,y;θ;s) represents the path of an incoming x-ray. Equations (7) and (8) enable the reconstruction of maps of the phase and attenuation parameters δ and β via standard CT algorithms [18], with the former parameter requiring an additional integration step before or after CT reconstruction. The constant of integration can be fixed if a region exists above or below the object where δ is known (for instance, a homogeneous region where δ is constant).In some applications, a reconstruction without the prior separation of refraction and attenuation may be advantageous. Consider again Eq. (6), together with the definition of displacement and transmission [Eqs. (2) and (3)]. Taking the logarithm of each term yields:
The argument on the left describes a normalised projection, and the right term links it to the phase and attenuation parameters of the object. To be used in CT reconstruction, the right term of Eq. (9) must take the form of a line integral. The attenuation μ is a line integral by definition, and ln(C(y0)) is a constant. In order to write also the second summand as a line integral, we consider its linearisation around y0 through a first order Taylor expansion [17]: This approximation is valid if y0 corresponds to the approximately linear part of C (as here C″(y0) = 0), which is fulfilled in practice when y0 coincides with 50% of the maximum achievable illumination [Fig. (2)]. The validity is further restricted to small refraction angles, which is often the case in biomedical applications. The substitution of Eq. (10) into Eq. (9) and the acquisition of multiple projections in CT mode (rotation by θ) yields the sinogram: where the factor A = (zod/2) · (C′(y0)/C(y0)) is a constant that depends only on the imaging system. Equation (11) enables the reconstruction of: which is effectively a linear combination of (differential) phase and attenuation parameters.3. Experimental verification
The accuracy of the reconstructed phase and attenuation maps as obtained from Eqs. (7) and (8) and the validity of the mixed approach as predicted by Eq. (12) were tested experimentally at the SYRMEP beamline of the Elettra synchrotron facility in Trieste, Italy [19], using an EI XPCi setup as depicted in Figs. 1(a) and 1(b). Although it is of course possible, through the design of an appropriate set of masks, to exploit the configuration where the axis of rotation is orthogonal to the direction of phase sensitivity, we have decided for this first proof-of-concept experiment to investigate the case in which these two directions are parallel. This creates a situation in many ways analogue to that encountered in analyser-based imaging, an analogy which was recently demonstrated on a more formal basis [20]. This enabled us to investigate the reconstruction of both separate and mixed phase and attenuation maps in a single experiment. The situation where the axis of rotation is orthogonal to the direction of phase sensitivity will be investigated in future experiments.
The laminar beam was obtained by collimating the primary beam (120 (H) × 4 (V) mm2) using a slit (Huber GmbH, Rimsting, Germany) with an opening of 20 (H) × 0.02 (V) mm2. The slit was placed at approximately 22 m from the source, which has full width at half maximum dimensions of 0.28 (H) × 0.08 (V) mm2. The “PICASSO” single photon-counting Si strip detector, developed by the Instituto Nazionale di Fisica Nucleare (INFN, Italy) and based on the Mithen ASICS [21–23], with a 210 (H) × 0.3 (V) mm2 active surface and pixel size of 50 (H) × 300 (V) μm2 was located approximately 0.9 m downstream of the slit and positioned so as to match the orientation of the laminar beam. A tungsten edge, thick enough to absorb all x-rays at the used energies, was placed in front of the detector, leaving half of each pixel uncovered. The object stage, consisting of a rotator (PI GmbH, Karlsruhe, Germany), two goniometers (Kohzu Precision Co. Ltd., Kawasaki Kanagawa, Japan) and a vertical translation stage (Newport Corporation, Irvine, California, USA), was located 0.7 m upstream of the edge/detector row combination. A channel-cut Si (1,1,1) crystal was used to monochromatise the beam to a nominal photon energy of 25 keV with a fractional bandwidth of 0.2%.
The custom-built phantom consisted of five wires of the diameters and materials listed in Tab. 1, which were tilted with respect to the vertical direction to ensure that refraction occurred in the direction of sensitivity (y-axis in Fig. 1). For the sake of generality, the materials were chosen to range from weakly to highly attenuating. In order to account for potential material impurities and deviations from nominal density values, as a rough estimate, a ± 5% uncertainty was assigned to the density values for this proof-of-principle experiment, implying an approximate ± 5% uncertainty of the nominal values of the phase and attenuation parameters δ and β. These were determined according to [24].
Table 1. Specification of the materials used in the wire phantom, ordered from lowest to highest absorbing. A ± 10 – 20 % error on the diameter was declared by the supplier and a ± 5% deviation from the nominal density values was assumed to account for potential material impurites and incorrect density measurement, implying an approximate ± 5% uncertainty on the nominal phase and attenuation parameters, which were calculated according to [24].
Projections were acquired for both edge/detector row configurations [Figs. 1(a) and 1(b)] with an exposure time of 0.1 s. The sample was rotated over 180 degrees with a 0.5 degree angular step and scanned vertically with a 5 μm displacement. The object rotation stage was operated in continuous mode, i.e. it kept rotating also during the vertical displacement of the object. Sinograms corresponding to the two opposing edge/detector configurations [Figs. 1(a) and 1(b)] were therefore acquired with an angular offset. Moreover, sinograms for vertically adjacent slices were acquired with an angular offset. Generally, this does not pose a problem for the separation of phase and attenuation contributions, and for the CT reconstruction, as long as the angular views in the sinograms can be re-ordered such that they all start at the same angle. However, in our specific case, the angular offset was 11.06 degrees, which is not an exact multiple of the used angular step (0.5 degrees). As a consequence of this, it was impossible to re-order the views in the sinograms such that they all started at the same angle, and a slight offset always remained. As a consequence, small errors occurred during the extraction of phase and attenuation contributions via [13]. Moreover, since the reconstructed transverse slices were slightly misaligned, the integration, which is part of the retrieval of the δ-maps, generated strong stripe artefacts.
After the sinograms were aligned with the best possible accuracy, separate phase and attenuation sinograms [Eqs. (7) and (8)] were generated using a peviously presented algorithm [13]. Experimental errors on beam energy (0.2%) and distances (<1%) were considered negligible in comparison with the much larger uncertainties caused by the setup problems described above. Maps of δ and β were reconstructed with the standard filtered back-projection formula (FBP); in the case of the δ-map, an additional integration step was performed afterwards to convert differential into absolute phase. The constant of integration was fixed by assuming that δ = 0 for the air surrounding the wires in the phantom; this was considered reasonable since, at the used x-ray energy, the δ-value for air is typically three orders of magnitude smaller than those of the imaged materials. For the analysis of the quantitative accuracy of the maps, a region of interest was selected within each wire and the reconstructed δ- and β-values were averaged inside it. This was repeated for 100 slices of the reconstructed maps, covering 0.5 mm of the phantom vertically. The extracted phase and attenuation parameters were averaged over these slices and compared to the nominal ones listed in Tab. 1.
Afterwards, sinograms Smixed [Eq. (11)] were generated directly from the projections containing combined phase and attenuation contrast, acquired in only one of the edge/detector row configurations. A mixed CT reconstruction was obtained via the FBP. In order to validate the relationship predicted by Eq. (12), a linear combination of the separately reconstructed differential phase and attenuation maps available from the previous step was computed using the appropriate scaling factors k and A, and compared with the mixed reconstruction.
The imaging and reconstruction of separate phase and attenuation as well as mixed maps were repeated for a domestic wasp. In this case, a lower beam energy (20 keV) was used to ensure sufficient contrast also in the attenuation case, and the vertical scanning was performed with a larger (13 μm) displacement to be able to cover the entire insect (∼ 4 mm) within an acceptable acquisition time.
4. Results and discussion
The reconstructed maps of phase and attenuation parameters for the imaged wires are shown in Figs. 3(a) and 3(b). Although all wires were scanned at once, they are displayed in separate figures in order to avoid the lowly refracting/attenuating wires to be masked by the highly refracting/attenuating ones. As the integration in the reconstruction process introduces artefacts due to unidirectional error propagation in general [25] and because we faced the additional challenge of slightly misaligned sinograms with respect to the angular scanning range, the wires were segmented and the background subtracted in the phase maps and, for consistency, the same processing was also applied to the attenuation maps. Residues of these artefacts are still visible in form of non-uniform grey levels within some wires. The elliptical shapes of the wires’ cross-sections are a result of their inclined orientation.
Fig. 3 Reconstructed maps of the phase and attenuation parameters δ (a) and β (b) for a custom-built wire phantom. The maps correspond to the following wires: 1. nylon 6, 2. PEEK, 3. PBT, 4. sapphire, 5. titanium. The plots underneath show the retrieved parameters plotted against nominal ones for the wires’ materials at 25 keV. The results of a linear regression analysis on the retrieved values is also shown. Moreover, the plots contain δ- and β-values that were retrieved from simulated data.
Figures 3(c) and 3(d) show the retrieved δ- and β-values plotted against the nominal ones of the involved materials at 25 keV. Please note the ± 5% uncertainty on the nominal values. The error bars on the retrieved values show one standard deviation of all values that were averaged to obtain the results. All retrieved values follow a reasonable trend, although slight discrepancies from the ideal case can be observed (see linear trend lines in Figs. 3(c) and 3(d)). Moreover, the error bars on the retrieved values, in particular on the retrieved δ-values, are large. We believe that this uncertainty and the slight offset to the nominal values can be attributed almost entirely to the experimental problems described in Section 3. It can be noted that higher uncertainties on the retrieved δ-values are observed for stronger refractive properties of the material. This can also be attributed to the experimentally observed slice alignment problem described above: the angular misalignment of the CT slices results in an accumulation of slightly erroneous differential phase data during the integration, the magnitude of which is proportional to the δ-value of the material. To further support our hypothesis that the problems in the retrieved δ- and β-values are caused by the experimental problems, we replicated the experiment with an EI XPCI simulation software [27]. The δ- and β-values that were retrieved from the simulated data are shown in the plots in Figs. 3(c) and 3(d), and are indeed in a much better agreement with the ideal case than the experimental data. This, together with the trend of the experimental results, indicates that the reconstruction of quantitatively accurate maps of δ and β from EI XPCI data is in general feasible, although the data presented here can only be regarded a preliminary experimental verification due to the specific problems of the used experimental setup. Please also note the wide range of δ-values which the method allows to retrieve: as previously observed in a non-CT study, this is a consequence of the fact that the method does not require fine grating periods to be phase sensitive [26].
Figures 4(a) and 4(b) show coronal slices extracted from the phase and attenuation maps of a domestic wasp (only the thorax and head regions are shown). The phase map generally appears smoother than the attenuation one, which is due to the integration. Apart from the slight blurring, most features, and in particular the structure of the insect’s thorax, are much better visualised in the phase map than in the attenuation one. Some thin features in the attenuation map are only visible due to free space propagation phase effects in the direction parallel to the edge (manifesting as black and white fringes, for example the shell of the thorax), which are present due to the spatial coherence of the synchrotron beam. Profiles across the thorax are plotted below the maps [Figs. 4(c) and 4(d)]. The contrast-to-noise ratio (CNR) for the thorax region covered by these profiles was calculated according to:
where σ denotes the standard deviation. The CNR in the phase profile (CNR = 36.2) is clearly higher than in the attenuation one (CNR = 4.4), which demonstrates the superiority of phase over attenuation contrast for low attenuating materials such as biological tissues. It should be noted that the signal in EI is, to a good approximation and for most experimental setups, independent from the pixel size [28], which explains why a much superior CNR was achieved for phase images compared to attenuation ones despite the relatively large pixel size. Free space propagation effects (sharp peaks on the edges), which also cause the negative values in the plot, were not included in the calculation of the CNR for the attenuation image.Fig. 4 Coronal slices extracted from the reconstructed maps of the phase (a) and attenuation (b) parameters for a domestic wasp. The images show only the front half of the insect. Profiles across the thorax are shown below the respective map (c,d).
A transverse slice of the mixed reconstruction (fmixed) for the wire phantom is shown in Fig. 5(a). A sub-region of the phantom containing only three of the wires was selected. In Fig. 5(b), another map obtained through the linear combination of the separately reconstructed maps of ∂δ/∂η and β, scaled by A= −1.42·104 and the wave number according to Eq. (12), is shown for the same sub-region. Visually, the images look identical. Profiles across the lowest (nylon 6) and highest (titanium) attenuating wires, shown Figs. 5(c) and 5(d), show a quantitative match for both materials. This demonstrates the robustness of the method for a broad range of materials. The extremely high attenuation of titanium explains the streak artefacts around the titanium wire in the maps.
Fig. 5 Reconstructed mixed (a) and linearly combined (b) differential phase and absorption maps for a sub-region of the wire phantom. Profiles across the titanium and nylon 6 wires are extracted from both maps and plotted on the right (c,d).
Coronal slices extracted from the mixed and linearly combined maps for the domestic wasp (again only showing the thorax and head region) are shown in Figs. 6(a) and 6(b). Again, visually, the two maps are identical. The profiles across the insect’s thorax and head, shown in Figs. 6(c) and 6(d), quantitatively confirm the accuracy of the match. It should be noted that the profiles are in very good agreement even for the complex structure within the insect’s head. The slight discrepancies on the edges are due to unavoidable differences in the experimental conditions between the acquisitions in the two opposing edge/detector configurations (i.e. minimal variation of the working point y0). A similar effect was already observed in [17].
Fig. 6 Coronal slices extracted from the mixed (a) and linearly combined (b) differential phase and absorption maps for a domestic wasp. Profiles across the insect’s thorax and head were extracted from both maps and plotted underneath (c,d).
Figure 7 shows a volumetric rendering of the mixed map, generated with the open source software 3DSlicer (www.slicer.org). The contrast has been windowed so as to expose the outer (antennas, wings) as well as some of the internal structure (thorax, abdomen) of the insect. A high level of detail is visible.
Fig. 7 Volume rendering of the mixed map (phase and attenuation contrast) of a domestic wasp. The contrast has been windowed so as to expose also the internal structure of the insect.
The entrance dose per projection and vertical scanning step was 0.26 mGy for the 25 keV beam and 0.51 mGy for the 20 keV beam. It should be noted that, although the latter is higher due to the decreased x-ray energy, both values are considerably lower than what was previously reported for some XPCi techniques [29]. We find this offers promising opportunities for phase CT imaging of biological samples such as for example small animals, as the total entrance dose for all performed scans lies below or is at least comparable with the acceptability limits for this application [30]. Generally speaking, it should be noted that no dose optimisation strategy was adopted for this preliminary study. Moreover, the separate maps were reconstructed from twice as many projections as the mixed maps, which, as the statistics were kept the same, resulted in doubling the dose. It can be possible, however, to achieve the reconstruction of the separate maps at the same low dose as the mixed maps by reducing the exposure time by a factor of two, since eventually pairs of projections are processed together, which can compensate for the reduced statistics.
5. Conclusion
We have presented the first quantitative tomographic images reconstructed from EI XPCi data. Among these were maps of the phase and attenuation parameters of a custom-built phantom. Their quantitative accuracy was assessed by a comparison to nominal values for the involved materials, and a reasonable agreement was found within the limits of the peculiarities of the experimental setup. The presented images also confirm a model that enables the reconstruction of maps showing combined phase and attenuation contrast. The theory was tested for weakly and highly attenuating materials and for the complex structure of a biological object. A good match between theory and experiment was obtained in all cases.
The choice to reconstruct individual, separate maps of phase and attenuation parameters or combined ones depends on the specific application. The two approaches have very different advantages and, in some cases, one could be better suited than the other. Quantitative imaging, as required for the identification of different tissue types and other materials, must be based on the separate reconstruction of phase and attenuation maps. The attenuation term serves as the monochromatic analogue of the Hounsfield unit, while the phase parameter, characterised by enhanced soft tissue contrast, serves as its monochromatic phase analogue. However, this requires the acquisition of two full independent datasets, in opposite edge/detector configurations. On the other hand, the reconstruction of mixed maps requires a single dataset, with no movement or set-up modification required apart from sample rotation. This is thus intrinsically a faster modality, requiring a simplified acquisition procedure. While not fully quantitative, this approach still provides edge-enhanced attenuation images, which can helpful for identifying faint, low attenuating details when the extraction of strictly quantitative information is not needed.
The flexibility in the reconstruction and the relatively low entrance doses make tomographic EI XPCi particularly suitable for biological samples. Combined with its compatibility with laboratory-based x-ray sources, we envisage the method being a useful tool for biomedical imaging in the future, for example for the study of small animals.
Acknowledgments
This work was supported by the UK Engineering and Physical Sciences Research Council (Grant Nos. EP/G004250/1, EP/I021884/1 and EP/L001381/1). P.C.D. and M.E. are supported by Marie Curie Career Integration Grant Nos. PCIG12-GA-2012-333990 and PCIG12-GA-2012-334056 within the Seventh Framework Programme of the European Union.
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