Open Access
Add to BibSonomy
Abstract
X-ray dark-filed imaging is a powerful approach to quantify the dimension of micro-structures of the object. Often, a series of dark-filed signals have to be measured under various correlation lengths. For instance, this is often achieved by adjusting the sample positions by multiple times in Talbot-Lau interferometer. Moreover, such multiple measurements can also be collected via adjustments of the inter-space between the phase gratings in dual phase grating interferometer. In this study, the energy resolving capability of the dual phase grating interferometer is explored with the aim to accelerate the data acquisition speed of dark-filed imaging. To do so, both theoretical analyses and numerical simulations are investigated. Specifically, the responses of the dual phase grating interferometer at varied X-ray beam energies are studied. Compared with the mechanical position translation approach, the combination of such energy resolving capability helps to greatly shorten the total dark-field imaging time in dual phase grating interferometer.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
From a grating based X-ray interferometer system [1–4], three images containing distinct contrast can be easily obtained from the same group of phase stepping data. They are the absorption contrast, differential phase contrast (DPC), and dark-field (DF) contrast. As two complimentary contrast information, the DPC signal is able to provide superior soft tissue contrast [5–9], and the dark-field signal is particularly sensitive to certain fine structures on micrometer scale such as microcalcifications [10–12]. In addition, it is demonstrated that the X-ray dark-field signal might be useful for the diagnosis of emphysema [13], pulmonary fibrosis [14], and chronic obstructive pulmonary disease (COPD) [15]. Furthermore, dark-field imaging is also the foundation of the so-called tensor computed tomography [16], which is able to render the three-dimensional orientations of the inner micro-structures that are hard to be revealed by the conventional absorption imaging.
To estimate the dimensions of such micro-particles [17–19], usually, a set of dark-field signals correspond to different system correlation lengths need to be acquired. To do so, the imaging object is translated between the source grating and the phase grating (or alternatively between the phase grating and the analyzer grating) of the Talbot-Lau interferometer. As expected, the multiple mechanical translations would prolong the entire data acquisition time of dark-filed imaging. Moreover, the radiation dose efficiency of dark-field imaging with Talbot-Lau interferometer gets reduced as the analyzer grating stops more than half amount of the incident X-ray photons. Recently, several innovative X-ray interferometer systems having higher radiation dose efficiency have been experimentally demonstrated. Compared with the Talbot-Lau interferometer, such new X-ray interferometer systems use two or three phase gratings [20–22] to generate the diffraction fringes that have relatively large periods. For instance, Miao et. al. [20,21] showed for the first time that a three phase grating interferometer system may have superior sensitivity than the Talbot-Lau interferometer in resolving ultra-small DPC signals. In addition, Kagias et. al. [22] demonstrated for the first time that the particle size can be determined from dual phase grating interferometer based dark-field imaging without translating the sample. Instead, dark-field signals correspond to different system correlation lengths are acquired by slightly changing the inter-space between the two phase gratings. Nevertheless, the multiple mechanical translations of phase gratings would still prolong the entire data acquisition time of dark-filed imaging in dual phase grating interferometer.
The energy responses [23] of grating interferometer has been demonstrated important for improving the imaging performance of the interferometer system such as increasing the signal-to-noise ratio (SNR) of the differential phase contrast (DPC) image [24] and enhancing the material decomposition capability for spectral CT imaging [25]. For dual phase grating interferometer, the energy response is also essential [26]. In stead of studying the energy response with respect to the inter-space between the two phase gratings, the relationship between the energy response and the X-ray beam energy is investigated in this study. With it, the feasibility of performing energy resolving dark-filed imaging [27] is explored for dual phase grating interferometer with the purpose to shorten the entire data acquisition time. Essentially, the energy resolving dark-field imaging method allows one to acquire more than one single dark-field signals at different X-ray beam energies. When combined with the aforementioned mechanical position translation approach, as a consequence, the energy resolving capability would help to save a lot of dark-field imaging time in dual phase grating interferometer.
The rest of this paper is arranged as follows: the section II briefly introduces the energy dependent imaging model for dual phase grating interferometer, the section III presents the results of numerical simulations performed at two beam energies, the section IV and V present the imaging model and results of energy resolving dark-field imaging, discussions are presented in the section VI, and the conclusion is made in the section VII.
2. Energy dependent imaging model
In the following theoretical discussions, a dual phase grating interferometer system is assumed, see Fig. 1. By default, the X-ray wave emitted from a single slit source is assumed to be polychromatic. Moreover, the two one-dimensional (1D) phase gratings add $\pi$ phase shift for X-ray beam with wavelength of $\lambda _0$. With the standard Kirchhoff’s diffraction theory, the final X-ray intensity on the detector plane can be approximated [26] as follows:
Herein, $\sigma$ denotes the width of the slit X-ray source, $p_d$ represents the dimension of a single detector element. In addition, the terms f$_{1}$ and f$_{2}$ are written as [28]:
In Eq. (1), it is assumed that the final X-ray intensity $\text {I}_{d}(x_{d})$ only contains the lower diffraction orders such as $\pm 1$ and $\pm 2$. Specifically, terms A$_{1}$ corresponds to the diffraction order of $\pm 1$, and A$_{2}$ corresponds to the diffraction order of $\pm 2$. Based on Eq. (1), the fringe visibility is defined [26] as:
Interestingly, Eq. (1) indicates that the periods of the detected diffraction fringes contain two main frequency components: one is two times higher than the other one. Moreover, the change of X-ray beam energy would lead to variations to the visibility of the diffraction fringe. Namely, Eq. (1) demonstrates that the visibility of a dual phase grating interferometer system depends on the X-ray beam energy E. Indeed, such energy dependent responses provide an important opportunity to acquire multiple energy resolving dark-filed images at a fixed sample position in dual phase grating interferometer system.
3. Numerical study
Numerical simulations are performed to investigate the energy responses of a dual phase grating interferometer system. Specifically, it is assumed that the designed working energies of the $\pi$ phase grating are 30.0 keV and 60 keV, see Table 1 for more details. To compare, Talbot-Lau interferometer with similar system distance and identical working energies are also simulated, see Table 2 for more details.
Table 1. Key parameters of the dual phase grating interferometer systems.
Table 2. Key parameters of the Talbot-Lau interferometer systems.
The energy dependence of A$_1\left ( {\lambda } \right )$ and A$_2\left ({\lambda } \right )$ are plotted in Fig. 2(a) and Fig. 2(b), respectively. It is found that the amplitude of A$_1\left ( {\lambda } \right )$ slowly increases to its maximum around some certain X-ray wavelengths. As a contrary, A$_2\left ( {\lambda } \right )$ oscillates sinusoidally with the same amplitude at different X-ray wavelengths. Quantitatively, A$_1\left ( {\lambda } \right )$ is equal to 0 at $m\lambda _0$ ($m$ is an integer), but A$_2\left ( {\lambda } \right )$ reaches to 0 when the incident wavelength is equal to $2m\lambda _0$. For other wavelength values, the contribution of A$_1\left ( {\lambda } \right )$ and A$_2\left ( {\lambda } \right )$ differs with respect to $\lambda$. The combined plot of A$_1\left ( {\lambda } \right )$ and A$_2\left ( {\lambda } \right )$ is shown in Fig. 2(c). Since A$_1\left ( {\lambda } \right )$ and A$_2\left ( {\lambda } \right )$ have opposite behaviors at $(2t+1)\lambda _0$ ($t$ is an integer), therefore, the minimum values in A$_1\left ( {\lambda } \right )$ are compensated by A$_2\left ( {\lambda } \right )$ and become local maximum after combination.
Fig. 2. Plots of (a) term A$_1\left ( {\lambda } \right )$, (b) term A$_2\left ( {\lambda } \right )$ and (c) combination of A$_1\left ( {\lambda } \right )$ and A$_2\left ( {\lambda } \right )$ at different X-ray wavelengths.
The diffraction fringes and energy responses of dual phase grating interferometer under different monochromatic X-ray beams are shown in Fig. 3. The designed working energy of the $\pi$ phase gratings in Fig. 3(a) is 30 keV, and the designed working energy of the $\pi$ phase gratings in Fig. 3(b) is 60 keV. The $d_3$ varies from 0.5 m to 1.5 m with 1.0 cm sampling interval. Specifically, results in Fig. 3(a) are obtained at 20.0 keV, 30.0 keV and 45.0 keV X-ray beam, respectively. Moreover, results in Fig. 3(b) are obtained at 40.0 keV, 60.0 keV and 90.0 keV X-ray beam, respectively. It can be observed that the distribution of diffraction fringes after G$_2$ grating is spatially continuous. The corresponding line profiles are plotted at $d_3 = 100.81$ cm. As seen, the period of the fringes obtained at 20.0 keV and 40 keV ($1.5\pi$ phase grating) is two times larger than the ones obtained at 30 keV and 60 keV ($\pi$ phase grating). At 45 keV and 90 keV ($0.67\pi$ phase grating), the generated diffraction fringes look less uniform. Due to the appearance of small peaks, as a result, the fringe periods at 45 keV and 90 keV look similar to the ones obtained at 30 keV and 60 keV. Additionally, the fringe visibility varies as the beam energy changes, see Fig. 3(c)-(d). As expected, both systems reach to the maximum at their own designed working energy, i.e., 30 keV and 60 keV. As the beam energy goes lower, it is found that fringe visibility with high values can appear again at some certain energies. On the contrary, the fringe visibility decreases as the beam energy goes higher than the designed energy. Results plotted in Fig. 3(c) demonstrate that the dual phase grating interferometer and the Talbot-Lau interferometer have comparable energy responses at low working energies, e.g., 30 keV. However, the dual phase grating interferometer has superior energy responses than the Talbot-Lau interferometer at high working energies, e.g., 60 keV, see the results in Fig. 3(d). The reduction of energy responses, i.e., fringe visibility, in Talbot-Lau interferometer is mainly due to the decreased absorption ratio of gold on the analyzer grating as the beam energy increases [27].
Fig. 3. Diffraction fringes and energy responses of the dual phase grating interferometer under different monochromatic X-ray beams. The designed working energy of the $\pi$ phase gratings is 30 keV for the results obtained in the first row, and the designed working energy of the $\pi$ phase gratings is 60 keV for the results obtained in the second row. The entire energy responses of the designed 30 keV interferometer is plotted in (c), and the entire energy responses of the designed 30 keV interferometer is plotted in (d). They are compared with the energy responses of Talbot-Lau interferometer.
In addition, the impact of polychromatic X-ray beam is studied as well. Herein, an energy resolving photon counting detector (PCD) is assumed with distinct energy window settings. The X-ray beam spectra are estimated in SpekCalc, one with 45 kVp tube potential (0.2 mm aluminum (Al) and 0.1 mm copper (Cu) filtration) and the other with 90 kVp tube potential (0.2 mm Al and 0.6 mm Cu filtration), see the plots in Fig. 4(a)-(b). The selected energy windows for the low energy spectrum are: [18 keV, 23 keV], [28 keV, 33 keV] and [36 keV, 41 keV], and for the high energy spectrum are: [37 keV, 42 keV], [57 keV, 62 keV] and [67 keV, 72 keV]. The simulated diffraction results are depicted in Fig. 4(c)-(d). As seen, similar results are obtained as for the monochromatic ones depicted in Fig. 3. For instance, the distributions of diffraction fringes after the second phase grating are spatially continuous. The fringe periods obtained from the [18 keV, 23 keV] and [37 keV, 42 keV] energy windows are approximately two times larger than the ones obtained from the rest energy windows.
Fig. 4. Diffraction fringes and energy responses of the dual phase grating interferometer under two polychromatic X-ray spectra. The designed working energy of the $\pi$ phase gratings is 30 keV for the results obtained in the second row, and the designed working energy of the $\pi$ phase gratings is 60 keV for the results obtained in the third row.
4. Signal model for dark-field imaging
The quantitative dark-field imaging model for dual phase grating interferometer is primarily inherited from Strobl [18]. Assuming the radius of the hard microsphere sample is $r$, and the real-space autocorrelation function can be expressed as follows:
Herein, $\sigma _{DF}^2$ represents the scattering cross-section of the microsphere and is numerically equal to the maximum dark-field signal of the given particle. Essentially, multiple sets of dark-field signals can be simultaneously obtained by varying the mean beam energy $\bar {E}$ (corresponds to different correlation lengths) in dual phase grating system.
5. Experiments and results
Experiments are carried out on our dual phase grating interferometer system. The system includes a micro-focus X-ray tube (L9421-02, Hamamatsu, Japan), a CMOS X-ray detector (MX510XG, XIMEA, Germany) with $4.6$ $\mu$m pixel dimension. The periods of the two phase gratings are $2.86$ $\mu$m with 50% duty cycle. They are designed to induce $\pi$ phase shift at 17.0 keV. The optimal system geometry is determined [28] as follows: $d_1=d_3= 23.5$ cm, and $d_2 = 2.98$ cm. The $0.512 \pm 0.02 \mu m$ sized silicon dioxide microsphere (EPRUI Technology, Shanghai, China) is diluted in water with a volume fraction of $2.52\,{\%}$. The solution is filled in a 2.0 ml centrifuge tube with inner diameter of 9.4 mm. In total, 11 number of phase steps are utilized.
Voltage modulation is performed to generate two different beam energies during the experiments to mimic the energy resolving responses of a true PCD. Specifically, two distinct tube voltages are used: low energy 30 kVp and high energy 50 kVp (with additional $0.1$ mm copper (Cu) filter), corresponding to two different mean beam energies of 17.0 keV and 34.0 keV. With the low energy X-ray beam, the system works as a dual $\pi$ phase grating interferometer. At the high energy X-ray beam, as a contrary, the system works as a dual $0.5\pi$ phase grating interferometer. Meanwhile, the silicon dioxide solution sample is mechanically moved to four different locations, namely, $16$ cm, $18$ cm, $20$ cm and $22$ cm apart from the detector plane. As a result, eight dark-field projections can be collected only by moving the sample for four times, saving half of the total data acquisition time.
The diffraction fringes acquired at low energy and high energy are depicted in Fig. 5(a) along with their line profiles. Moreover, the extracted dark-field projection is shown in Fig. 5(b), and the vertically averaged profile is plotted below. Data fitting is used to minimize the signal noise and make a precise estimation of the dark-filed signal. Finally, regression analysis is performed with respect to the acquired eight dark-field signals to estimate the particle size $r$. Specifically, the solid fitting curve plotted in Fig. 5(c) is defined by Eq. (9). The fitting results show that the scanned silicon dioxide microsphere has a diameter of $0.537 \pm 0.06 \mu m$, which is in good agreement with the actual particle size.
Fig. 5. The experimental dark-field imaging results. (a) The diffraction fringes obtained for low energy and high energy X-ray beams. (b) The dark field image obtained at low energy when the sample is 16.0 cm apart from the detector. (c) The measured experimental data and the fitting curve with regard to Eq. (9).
6. Discussions
In this study, the feasibility of performing energy resolving dark-filed imaging is investigated for dual phase grating interferometer system. Numerical simulations are carried out under two conditions to study the period and the visibility of the diffraction fringes. Results show that the fringes generated from dual $1.5\pi$ phase grating interferometer outperform the ones generated from dual $0.5\pi$ phase grating system with higher fringe visibility and signal uniformity, see the results in Fig. 3(a)-(b). Physical experiments are carried out to demonstrate the capability of performing energy resolving dark-filed imaging with accelerated data acquisition speed for dual phase grating interferometer system. As expected, results demonstrate that the micro-bubble size can be accurately predicted from the energy resolved dark-filed projections that are acquired with only half number of mechanical movements. Thereby, the entire data acquisition period of dark-field imaging could be significantly reduced in dual phase grating interferometer.
The current study has the following limitations. First, the energy resolving dark-field imaging capability is not directly demonstrated with PCD due to the lack of such hardware in our laboratory. To compensate, two different beam energies are manually selected by modulating the tube voltage and beam filtration. Obviously, this limits the detection of multiple dark-field projections under different energies at the same time. When a PCD is utilized, more than two different dark-field projections can be simultaneously obtained from a single scan. Second, it is hard to make our present dual phase grating system work at lower beam energies such as 11.3 keV (corresponds to $1.5\pi$ phase shift) or 6.8 keV (corresponds to $2.5\pi$ phase shift). Therefore, we chose to let it work at 34.0 keV (corresponds to $0.5\pi$ phase shift). However, the fringe visibility obtained at 34.0 keV decreases significantly. To overcome such difficulty, it would be desired to design a dual $\pi$ phase grating interferometer system working at higher energy, e.g., 60 keV, to allow multiple measurements of the dark-field imaging with even less number of mechanical movements of the sample.
7. Conclusion
In conclusion, the energy resolving dark-field imaging approach is demonstrated to collect multiple sets of dark-field signals from a dual phase grating interferometer with half (or even less) number of mechanical movements of the sample. As a consequence, fast dark-field imaging would be achieved for dual phase grating interferometer in the future.
Funding
National Natural Science Foundation of China (12027812); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515111031); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021362).
Acknowledgments
The authors would like to thank Prof. Mei Hong and Dr. Tong Li at Peking University Shenzhen Graduate School for providing the silicon dioxide microsphere samples.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81(17), 3287–3289 (2002). [CrossRef]
2. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(2), L866–L868 (2003). [CrossRef]
3. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13(16), 6296–6304 (2005). [CrossRef]
4. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat. Phys. 2(4), 258–261 (2006). [CrossRef]
5. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase contrast x-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996). [CrossRef]
6. A. Momose, “Phase tomography by X-Ray Talbot interferometry,” Jpn. J. Appl. Phys. 44(9R), 6355 (2005). [CrossRef]
7. M. Bech, T. H. Jensen, R. Feidenhans, O. Bunk, C. David, and F. Pfeiffer, “Soft-tissue phase-contrast tomography with an x-ray tube source,” Phys. Med. Biol. 54(9), 2747–2753 (2009). [CrossRef]
8. T. H. Jensen, A. Böttiger, M. Bech, et al., “X-ray phase-contrast tomography of porcine fat and rind,” Meat Sci. 88(3), 379–383 (2011). [CrossRef]
9. K. Li, Y. Ge, J. Garrett, N. Bevins, J. Zambelli, and G.-H. Chen, “Grating-based phase contrast tomosynthesis imaging: proof-of-concept experimental studies,” Med. Phys. 41(1), 011903 (2013). [CrossRef]
10. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-x-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7(2), 134–137 (2008). [CrossRef]
11. G. Anton, F. Bayer, M. W. Beckmann, et al., “Grating-based darkfield imaging of human breast tissue,” Z. Med. Phys. 23(3), 228–235 (2013). [CrossRef]
12. Z. Wang, N. Hauser, G. Singer, M. Trippel, R. A. Kubik-Huch, C. W. Schneider, and M. Stampanoni, “Non-invasive classification of microcalcifications with phase-contrast x-ray mammography,” Nat. Commun. 5(1), 3797 (2014). [CrossRef]
13. K. Hellbach, A. Yaroshenko, F. G. Meinel, et al., “In vivo dark-field radiography for early diagnosis and staging of pulmonary emphysema,” Invest. Radiol. 50(7), 430–435 (2015). [CrossRef]
14. A. Yaroshenko, F. G. Meinel, M. Bech, et al., “Pulmonary emphysema diagnosis with a preclinical small-animal x-ray dark-field scatter-contrast scanner,” Radiology 269(2), 427–433 (2013). [CrossRef]
15. K. Willer, A. A. Fingerle, L. B. Gromann, et al., “X-ray dark-field imaging of the human lung–a feasibility study on a deceased body,” PLoS One 13(9), e0204565 (2018). [CrossRef]
16. A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” Europhys. Lett. 105(3), 38002 (2014). [CrossRef]
17. S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50(22), 4310–4319 (2011). [CrossRef]
18. M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4(1), 7243 (2014). [CrossRef]
19. R. K. Panta, M. F. Walsh, S. T. Bell, N. G. Anderson, A. P. Butler, and P. H. Butler, “Energy calibration of the pixels of spectral x-ray detectors,” IEEE Trans. Med. Imaging 34(3), 697–706 (2015). [CrossRef]
20. H. Miao, A. A. Gomella, K. J. Harmon, E. E. Bennett, N. Chedid, S. Znati, A. Panna, B. A. Foster, P. Bhandarkar, and H. Wen, “Enhancing tabletop x-ray phase contrast imaging with nano-fabrication,” Sci. Rep. 5(1), 13581 (2015). [CrossRef]
21. H. Miao, A. Panna, A. A. Gomella, E. E. Bennett, S. Znati, L. Chen, and H. Wen, “A universal moiré effect and application in x-ray phase-contrast imaging,” Nat. Phys. 12(9), 830–834 (2016). [CrossRef]
22. M. Kagias, Z. Wang, K. Jefimovs, and M. Stampanoni, “Dual phase grating interferometer for tunable dark-field sensitivity,” Appl. Phys. Lett. 110(1), 014105 (2017). [CrossRef]
23. M. Engelhardt, C. Kottler, O. Bunk, C. David, C. Schroer, J. Baumann, M. Schuster, and F. Pfeiffer, “The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources,” J. Microsc. 232, 145–157 (2008). [CrossRef]
24. Y. Ge, R. Zhang, K. Li, and G.-H. Chen, “Improving radiation dose efficiency of x-ray differential phase contrast imaging using an energy-resolving grating interferometer and a novel rank constraint,” Opt. Express 24(12), 12955–12968 (2016). [CrossRef]
25. X. Ji, R. Zhang, K. Li, and G.-H. Chen, “Dual energy differential phase contrast CT (DE-DPC-CT) imaging,” IEEE Trans. Med. Imaging 39(11), 3278–3289 (2020). [CrossRef]
26. A. Yan, X. Wu, and H. Liu, “Predicting fringe visibility in dual-phase grating interferometry with polychromatic x-ray sources,” J. X-Ray Sci. Technol. 28(6), 1055–1067 (2020). [CrossRef]
27. K. Taphorn, L. Kaster, T. Sellerer, A. Hötger, and J. Herzen, “Spectral X-ray dark-field signal characterization from dual-energy projection phase-stepping data with a Talbot-Lau interferometer,” Sci. Rep. 13(1), 767 (2023). [CrossRef]
28. Y. Ge, J. Chen, P. Zhu, et al., “Dual phase grating based x-ray differential phase contrast imaging with source grating: theory and validation,” Opt. Express 28(7), 9786–9801 (2020). [CrossRef]
