Over the last few years, differential phase-contrast x-ray computed tomography (PC-CT) using a hard x-ray grating interferometer and polychromatic x-ray tube sources has been developed. The method allows for simultaneous determination of the attenuation coefficient and the refractive index decrement distribution inside an object in three dimensions. Here we report experimental results of our investigation on the quantita-tiveness and accuracy of this method. For this study, a phantom consisting of several tubes filled with chemically well-defined liquids was built and measured in PC-CT. We find, that the measured attenuation coefficients and refractive index decrements closely match calculated, theoretical values. Moreover, the study demonstrates, how substances with similar attenuation coefficient or refractive index decrement, can be uniquely distinguished by the simultaneous, quantitative measurement of both quantities.
© 2009 Optical Society of America
Phase-contrast x-ray imaging techniques are used to increase contrast in weakly absorbing biological samples [1, 2]. For a long time, phase-contrast computed tomography (CT) could only be performed using highly brilliant synchrotron radiation sources [1, 2, 3, 4, 5, 6]. Quantitative measurements were reported [7, 8, 9, 10, 11, 12, 13] and even verified in a few cases using sychrotron radiation in combination with diffraction enhanced imaging (DEI) , propagation-based phase-contrast imaging , and phase-contrast microscopy using zone plates [11, 12].
Recently, the use of laboratory x-ray sources for phase-contrast imaging became feasible and has been demonstrated to provide excellent results for both microscopic specimens using phase propagation techniques [13, 14, 15] and macroscopic specimens using the grating interferometer approach [16, 17, 18, 19, 20]. This work is particularly based on the grating interferometer approach that is compatible with low-brilliance x-ray sources. Here we present experimental results of a study on the quantitativeness and accuracy of the reconstructed three-dimensional distribution of the attenuation coefficients and refractive index decrements in a self-built phantom. The present work was largely motivated by the fact that modern medical CT applications increasingly rely on a quantitative interpretation of the tomographic gray-scale images, e.g., in the assessment of bone density in the context of osteoporosis.
2. Materials and methods
The refractive index of a three-dimensional object can be described by its complex refractive index distribution n(x,y, z)=1-δ(x,y, z)+iβ(x,y, z), where x, y, and z describe the coordinate system of the sample. In conventional absorption-contrast CT the imaginary part β is measured in form of the linear attenuation coefficient µ of the x-rays. A combination of the Radon transform of the object with Beer-Lambert’s law describes the transmission projection in one plane z=z 0 [21, 22]
where µ(x′,y′)=4π/λ· β(x′,y′), λ is the x-ray wave length, and x′ and y′ belong to a coordinate system which is rotated by the projection angle Θ around the z-axis with respect to x and y. Integration is carried out along the x-ray path over the extension of the sample of diameter d. Note that we have omitted the variable z to simplify the writing. The mass attenuation coefficient (µ/ρ) is defined by the ratio of the linear attenuation coefficient µ and density ρ. For mixtures that consist of I components, each with mass attenuation coefficient (µ/ρ)i, with i=1, …, I, the resulting mass attenuation coefficient is given by
where Wi is the weight fraction of the i-th component and the resulting linear attenuation coefficient simply is µsubst=(µ/ρ)subst·ρsubst, with the density of the substance ρsubst.
In differential phase-contrast imaging, the variations of the real part of the refractive index 1-δ of the object are detected by analyzing the slight refraction of x-rays caused by the object. In addition to the absorption-contrast projection in Eq. (1), we measure the differential phase-contrast projection of refraction angles 
where ΦΘ(x′)=2π/λ∫d 0 δ(x′,y′)dy′ is the spatially dependent, total relative phase shift of the x-ray wave front acquired during its propagation through the sample.
The refractive index decrement δ depends on the x-ray wavelength λ and on the density of the sample as 
where re=2.82·10−15m is the classical electron radius, Ni is the atomic density of type i atoms given as atoms per unit volume, and f 1 i is the real part of their atomic scattering factor in the forward direction. If the photon energy of the incident x-ray radiation lies considerably above the absorption edges, we may put f 1 i=Zi, the total number of electrons in the atom and the sum in Eq. (4) then describes the electron density inside the sample. To obtain δ from given elemental weight fractions Wi, we substitute Ni=(Wi·NA/Ai)ρsubst, with the atomic mass Ai of atom i and the Avogadro’s number NA, into Eq. (4).
For the reconstruction of the object’s original complex refractive index distribution, or more explicitly of µ(x,y) and δ(x,y), from the two sets of projection images T Θ(x′) and α Θ(x′), respectively, a filtered back-projection reconstruction algorithm is usually applied . For the transmission projections T Θ(x′) in Eq. (1) the corresponding filtered back-projection reconstruction can be written as [21, Chapter 3, Eq. (31)]
where 𝓕[ln T Θ(ω)] represents the Fourier transform of the logarithm of the normalized transmission projection, 𝓕−1 denotes the inverse Fourier transform operator, and 𝓕[k (ω)]≡|ω| is the Fourier representation of the filter function for line projection integrals.
where 𝓕[α Θ(ω)] represents the Fourier transform of the projections of the measured deflection angles and Θ[h(ω)]=i·sgn(ω)/(2π) is the Fourier transform of the filter function for gradient projections, where here sgn(ω) is the sign function. Please note that for a correct reconstruction of the phase signal the approximation of a weakly scattering sample was made . Since the presence of a strong scatterer will completely destroy the interference pattern of the grating interferometer, this approximation can be considered as fulfilled when a differential phase contrast can be measured. In a fan-beam setup the sensitivity of the grating interferometer is reduced by the factor r 1/l [18, Eq. (8)], where r 1 is the source-to-sample distance and l is the source-to-G1 distance. To correct for this effect, the recorded phase projections were renormalized by multiplication with (r 1/l)−1=[(1.4 m-0.078 m)/(1.4 m)]−1=1.059 prior to reconstruction.
We built a phantom consisting of thirteen small cylindrical low-density polyethylene (PELD) tubes with an outer diameter of 8 mm and a volume of 0.5 ml that we filled with fluids of well defined chemical composition (compare Table 1). To cover a wide range of attenuation coefficients and refractive index decrements, we mixed different pure liquids (ethanol, water, glycerol) and salts (NaI, NaCl).
Differential phase-contrast x-ray imaging and CT was carried out at the Laboratory for Micro- and Nanotechnology, Paul Scherrer Institut, Switzerland, using an x-ray grating interferometer as schematically shown in Fig. 1. The interferometer consists of a source grating G0, a phase grating G1, and an analyzer absorption grating G2. The source grating G0, which is placed close to the x-ray tube anode, allows for the use of x-ray sources with square-millimeter-sized focal points . The signal is formed within the two gratings G1 and G2 due to the Talbot self-imaging effect . The interferometer that was used here was designed for an x-ray energy of 28 keV and consisted of three gratings with respective periods of p 0=14.2 µm, p 1=3.5 µm, and p2=2.0 µm and respective structure heights of h 0=42 µm, h 1=36 µm, and h 2=26 µm. The distance between G0 and G1 was 1.40 m and the distance between G1 and G2 was 0.198 m, corresponding to the fifth fractional Talbot distance. The gratings were produced
on 4 inch silicon wafers in a fabrication process involving photolithography, deep etching into silicon, and (for the absorption gratings G0 and G2) electroplating of gold .
As x-ray source we used a Seifert ID 3000 x-ray generator operated at 40 kV and 25 mA with a tungsten (W) line focus tube (DX-W8×0.4-L). Due to the inclination of the target with respect to the optical axis of 6°, the effective source size was 0.8 (h)×0.4 (v)mm2. A PILATUS 100K pixel detector consisting of an array of 487×195 pixels with a pixel size of 0.172×0.172 mm2 was used to record images. Its detection efficiency determined by the probability of absorbing an x-ray in the 320-µm-thick Si sensor is ~10% at 28 keV.
The measurements were carried out at room temperature in a water tank (of PMMA, 50 mm thick) filled with demineralized water. The water tank, with the fluid phantom hanging into it from the top, was placed in front of the G1 grating, with the cylindrical tubes of the phantom parallel to the rotation axis z (compare Fig. 1). For the tomographic scan 361 projections were taken for projection angles from Θ=0° to 360°. For each projection 16 phase steps were recorded with an exposure time of 10 seconds per image.
To obtain the phase-contrast and the conventional projection, a set of raw images was recorded for different positions of the grating G1. From this so-called phase-stepping scan (described in more detail in ) both projections were computed. These projections can directly be used as input for the filtered back-projection reconstruction algorithms, i.e., numerical implementations of Eqs. (5) and (6). Since our measurement was carried out in a water tank, we denote the reconstructed x-ray attenuation coefficients as Δµ(x,y) and the measured refractive index decrements as Δδ(x,y) to indicate that these values give the difference to the respective values of water.
The calculation of theoretical values for µ and δ for each fluid was performed using their density at room temperature  and their elemental weight fractions, both listed in Table 1. In the following we denote calculated values by an index ‘c’ e.g. as µc and δc. For the dilution series of NaCl in water the density as a function of salt concentration was used in the calculation. Since we could not find tabulated data for the density of the dilution series of NaI in ethanol and for the ethanol-glycerol mixtures, consequently we did not calculate theoretical values for these mixtures. To calculate δc we used tabulated data of Kissel  and for the calculation of µc, we used the data of Plechaty et al. . Since our measurements were carried out in a water tank, i.e. relative to water, we give the theoretical values relative to those of water as Δµc=µc-µc(H2O) and Δδc=δc-δc(H2O) for comparison with the experimental results.
Both µ and δ strongly depend on the x-ray energy. For the comparison of measured and calculated values we thus have to select an effective photon energy, for which we calculate the theoretical data. The measurement of µ and δ is generally carried out with two different effective x-ray energies due to two main reasons: 1.) The image formation processes of the conventional and the phase-contrast data are intrinsically different, and 2.) the specific energy-dependent efficiency of the x-ray optical gratings leads to a different weighting of the initial energy spectrum (for more details, see [30, 31]). Thus, we use two different effective x-ray energies Eµ and Eδ, which we determine from the measured data of ethanol as described below since we used a photon-counting but not energy despersive detector for the measurements.
3. Results and discussion
Reconstructions of Δµ and Δδ from the same slice of the phantom are shown in Figs. 2(a) and 2(b), respectively. For the specific reconstructions shown in Fig. 2, 55 tomographic slices were averaged (along the z-axis) to increase the signal-to-noise ratio. From these reconstructions experimental values for Δµ and Δδ were determined from the mean value over a circular averaging region of 30 pixels diameter (containing 709 pixels) within each tube. Also the standard deviations σµ and σδ over all pixels were determined. The measured values together with the standard deviations are given in Table 2. Figure 3 shows a scatter plot of the measured Δµ and Δδ values.
The tube with demineralized water in the phantom serves as a control of the zero value. Water should be measured as zero, since the measurement was carried out in a water tank, i.e., relative to water. We find that the measured values for water Δµ=0.023±0.019·10−2mm−1 and Δδ=-0.015±0.049·10−8 fall approximately within one standard deviation from zero. A second very important result is the fact that the measured values for Δµ and Δδ change in the correct proportion depending on the concentration for all three fluid series H2O-NaCl, EtOH-NaI, and EtOH-Gly.
Effective energies for the µ and the δ measurements were determined by matching the measured and the calculated data for ethanol, which has comparitively large signals in both µ and δ. The minimum difference between measured and calculated values was obtained for Eµ=30.1 keV and Eδ=28.3 keV, both determined with 0.1 keV resolution. Note that both effective energies and especially Eδ are close to the interferometer design energy of 28 keV. Using these effective energies we observe that for most fluids the measured values closely match the theoretically calculated values. Only the fluids with high µ and δ values (H2O-NaCl5, H2O-NaCl10 and Gly) show a trend to have too low measured δ values. This might be attributed to beam-hardening effects or to a slightly wrong effective energy Eδ due to error in the ethanol measurement that was used for the determination of the effective energies.
From the reconstructions in Fig. 2 and the plot in Fig. 3 it is obvious, that fluids can be much better distinguished by using both complementary signals instead of only one. Here we are interested in the contrast between two fluids in the absorption and the phase-contrast signal. Contrast can be described by the contrast-to-noise ratio
where Sa and Sb represent the measured signals Δµ (or Δδ) of fluids a and b, respectively, and σS=(σ 2 a+σ 2 b)1/2 is the standard deviation of the signal difference |Sa-Sb|, with the standard deviation σa and σb of the respective signals. For example for the fluids H2O-NaCl1.25 and glycerol (Gly) we find CNRµ=(0.358-0.247)/(0.0222+0.0282)1/2=3.12 looking at the attenuation coefficients, but looking at the refractive index decrements instead, we find a much higher contrast-to-noise ratio of CNRδ=(5.831-0.122)/(0.0772+0.0532)1/2=61.07. Similarly, we find for the fluids H2O and EtOH-NaI1.25 contrast-to-noise ratios of CNRµ=1.44 and CNRδ=70.16. These substances can thus hardly be distinguished by their attenuation coefficients alone, but easily by looking at their refractive index decrements. Vice versa, the fluids EtOH and EtOH-NaI1.25 show less contrast for the refractive index decrement (CNRδ=4.79) than for the attenuation coefficient (CNRµ=22.04), but can be easily distinguished by looking at both complementary signals.
Our results clearly illustrate that PC-CT as a combined CT method, which simultaneously yields phase-contrast and attenuation-contrast images, provides significantly more and unique information than any of the techniques alone. We have particularly shown that our approach can yield quantitative volume information of the distribution of the refractive index decrement δ and the attenuation coefficient µ. By combining both signals, the absorption and the phase-contrast signal, it is possible to distinguish substances with weak contrast in either one of the signal channels and to generally improve the specificity of the measurement using the second signal provided in PC-CT. The presented experimental results obtained from a self-built fluid phantom agree well with the theoretical expectation. In particular because of the quantitativeness of the results, demonstrated with a conventional polychromatic x-ray tube source, we believe that this method is of great interest for a wide range of quantitative x-ray CT applications, including future medical diagnostics, industrial non-destructive testing, and other research areas.
We would like to thank Anastasia Savouchkina and Patrick Farquet for their help in preparing the chemical solutions.
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