Grating-based X-ray phase contrast imaging using polychromatic laboratory sources

Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 342–345

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Grating-based X-ray phase contrast imaging using polychromatic laboratory sources Zhili Wang a , Kun Gao b , Peiping Zhu a , Qingxi Yuan a , Wanxia Huang a , Kai Zhang a , Youli Hong a , Xin Ge b , Ziyu Wu a,b,∗ a b

Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China National Synchrotron Radiation Laboratory, University of Science and Technology of China, Hefei 230026, China

a r t i c l e

i n f o

Article history: Available online 21 December 2010 Keywords: X-ray imaging Phase contrast Talbot–Lau interferometer Polychromatic

a b s t r a c t X-ray phase contrast imaging has been demonstrated to have an improved contrast over conventional absorption imaging for those weakly absorbing objects. However, most of the hard X-ray phase-sensitive imaging has so far been impractical with laboratory available X-ray sources. Grating-based phase imaging approach has the prominent advantage that polychromatic laboratory X-ray generators can be efficiently used in a Talbot–Lau configuration. Through numerical simulations, we demonstrate here the efficient use of polychromatic X-ray laboratory sources for differential phase contrast imaging. The presented results explain why in recently reported experiments, polychromatic X-ray tubes could be efficiently used in a Talbot–Lau interferometer. Furthermore, the results indicate that the fractional Talbot distance is not limited by the polychromaticity of the X-ray source. Since the sensitivity of phase measurements is proportional to the fractional Talbot distance, the image quality for phase measurements can be further improved. Finally, the potential optimizations of the imaging system are discussed from an application perspective, taking into consideration both available X-ray flux and compactness of the system. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Phase contrast X-ray imaging, which uses the phase shift as imaging signal, provides a method to perform detailed visualizations of weakly absorbing objects with an increased sensitivity [1–3]. Various phase contrast X-ray imaging methods have been developed since the 1990s. They can be categorized into interferometric imaging [4,5], propagation-based imaging (PBI) [6,7], analyzer-based imaging (ABI) [8–11] and grating-based imaging (GBI) [12–21]. Among them, GBI has been the most promising phase imaging method since it has been demonstrated that it may be efficiently combined with polychromatic laboratory Xray sources [15,17,18,20–22]. With grating-based approach, it is capable of providing absorption, phase-contrast and dark-field information simultaneously [13,15,17,21], and enabling threedimensional reconstructions of the linear absorption coefficient, the refractive index and linear diffusion coefficient of the object with a single tomographic scan [22].

∗ Corresponding author at: Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Yuquan 19B, Shijingshan District, Beijing 100049, China. E-mail addresses: [email protected], [email protected] (Z. Wu). 0368-2048/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.elspec.2010.12.009

A breakthrough in X-ray phase contrast imaging has been made by using the Talbot–Lau interferometer approach, which mainly consists of three transmission gratings. It has been demonstrated to provide excellent experimental results using polychromatic laboratory X-ray sources [15,17,18,20–22]. A recent work has shown the ability of the Talbot–Lau interferometer approach to provide quantitative and accurate imaging of a liquid contrast phantom [20]. Based on numerical simulations, Engelhardt et al. have demonstrated that the Talbot interferometer can be well combined with polychromatic laboratory X-ray sources [23]. Here we present a study on the efficiency of the Talbot–Lau interferometer using a polychromatic X-ray tube generator. The present work is mainly motivated by the fact that only polychromatic laboratory X-ray tube generators are available in modern medical applications. 2. Methods As shown in Fig. 1, the setup of an X-ray Talbot–Lau interferometer consists of a source grating G0, a phase grating G1 and an analyzer absorption grating G2 [15,17]. The source grating G0, placed close to the X-ray generator, allows the use of laboratory X-ray sources with square-millimeter-sized focal spots. Spatially coherent X-rays from each subsource created by G0 irradiate the Talbot interferometer (constructed by G1 and G2) downstream. A

Z. Wang et al. / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 342–345

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modulation, are, for an incident spherical wave [27], zN =

L · Np21 /22 

(4)

L − Np21 /22 

where the Talbot order N is an odd integer for phase gratings. The factor , dependent on the type of the phase grating, satisfies  = 2 for a ␲ phase grating used here. To ensure a constructive overlap of the Moiré patterns derived from each subsource, the geometry of the set-up should satisfy the condition [15], zN L = p0 p2

(5)

And the observed Moiré fringe pattern has the following form: I(x, y, zN ; ) =

 cn () n

Fig. 1. Set-up of an X-ray Talbot–Lau interferometer, consisting of a source grating G0 with a period of p0 , a phase grating G1 with a period of p1 and an analyzer absorption grating G2 with a period of p2 .

Moiré fringe pattern is generated by the superposition of the selfimage of G1 [24] and the absorption grating G2. The period of G0 is chosen so that each subsource contributes constructively to the image-formation process. With an object typically placed in front of G1, the Moiré fringe pattern is deformed owing to the refraction at the object and recorded by the detector just behind G2. Absorption, phase-contrast and scattering information of the object can be extracted from the same projection images taken at each view angle during a tomographic scan. Assuming a quasi-monochromatic illumination with unit intensity, a Moiré fringe pattern can be observed behind G2 [25] I(x, y, zN ; ) =



an ()

 z n  N p2

n

 exp 2i



n (xg + y) p2

(1)

where  is the X-ray wavelength, an () is the Fourier coefficient determined by the gratings G1 and G2 and the spatial coherence of the X-rays,  and xg are the relative inclination and displacement, respectively, of G2 against G1. In Eq. (1),  is the reduced spectral degree of coherence of the incident X-rays, and can be calculated by means of van Cittert-Zernike theorem [26]. In the case of a Talbot–Lau interferometer, a uniform intensity distribution is a good approximation since the laboratory X-ray source has a focal spot much larger than the source grating period p0 . Under this approximation, the reduced spectral degree of coherence is given by 

 z n  N p2

 =

T0 (; ) exp(−i(2/L)(zN n/p2 ))d



 =

c () m m



T0 (; )d





exp i2((m/p0 ) − (zN /L)(n/p2 ) d



T0 (; )d (2)

where cm () is the Fourier coefficient of the transmittance function T0 (x ; ) of G0, T0 (x; ) =

 m



cm () exp 2i

mx p0



(3)

The fractional Talbot distances zN , i.e., the positions along the optical axis at which the self-image pattern exhibits a maximum

c0 ()



an () exp 2i



n (xg + y) p2

(6)

Under polychromatic illumination, the observed Moiré fringe is a weighted integral of the Moiré patterns over all contributing energies within the effective energy spectrum,



I(x, y, zN ) =

dI(x, y, zN ; )weff ()

(7)

where weff () is the normalized effective energy spectrum of the X-rays, which is calculated by weff () = w() · () · ()

(8)

where w() is the photon energy spectrum of an X-ray tube, () represents the X-ray transmission of objects in the beam path, and () describes the spectral efficiency of the detector. Dealing with polychromaticity effects, the following issues should be taken into consideration [13,14]: 1, the complex transmission function of an object is wavelength dependent. Assuming the whole spectrum of the X-rays is far from absorption edges of the materials constituting the object, a phase shift induced by the object varies linearly with the wavelength and a linear absorption coefficient varies as the third power of the wavelength; 2, the phase induced by the phase grating varies linearly with the X-ray wavelength, in the case of the whole spectrum of the X-rays being far from the absorption edges of the materials of the gratings. The transmittance of the lines in the absorption gratings are wavelength dependent, since the linear absorption coefficient varies as the third power of the wavelength; 3, the fractional Talbot distances are approximately inversely proportional to the X-ray wavelength, as shown in Eq. (4). The distance between G1 and G2 can only be equal to one of the fractional Talbot distances of the central wavelength. 3. Numerical results and discussion The following numerical simulations are carried out to provide a quantitative insight into the problem of the efficiency of a Talbot–Lau interferometer using a polychromatic laboratory X-ray generator. As a figure of merit, the visibility of the Moiré fringe is analyzed, which is defined as V = (Imax − Imin )/(Imax + Imin ), where Imax and Imin are correspondingly the maximum and minimum intensity values in the Moiré patterns. The set-up assumed for the simulations resembles that applied for the measurements presented by Pfeiffer et al. [17]. The relative inclination is  = 1.3◦ and the relative displacement is xg = p2 /4. The distances L and zN are varied to demonstrate the change of the visibility with increasing Talbot order. Our simulation used the energy spectrum for a tungsten (W) target at 40 kV tube voltage [28]. For the calculation of the effective spectrum weff () according to Eq. (8), the spectral efficiency of the PILATUS 100K pixel detector is considered, and approximated by

344

Z. Wang et al. / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 342–345 Table 1 Characteristics of the Moiré fringe patterns shown in Fig. 3.

Monochromatic 28 keV The first fractional Talbot distance (L = 1.57 m, zN = 43 mm) The third fractional Talbot distance (L = 4.7 m, zN = 129 mm) The fifth fractional Talbot distance (L = 7.85 m, zN = 215 mm)

Fig. 2. Normalized effective energy spectrum of a tungsten target at 40 kV tube voltage. It comprises the anode self-absorption, a 1-mm-thick beryllium window in the beam path, the spectral efficiency of the 320-␮m-thick Si sensor and the beam hardening due to the 810-␮m-thick Si wafers of the three gratings and a 1-␮m-thick gold layer on the analyzer grating [31]. An additional 3-mm-thick Al filtering was used to match the designed energy of 28 keV. Note that the effective spectral density has a low energy cutoff at 17 keV due to the detector threshold setting [29].

the photon absorption probability in the 320-␮m-thick Si sensor [17,29]. To calculate the spectral efficiency (), we used tabulated data of the mass absorption coefficient from Hubbell and Seltzer [30]. Fig. 2 shows the normalized effective energy spectrum weff (), which attains its maximum value at approximately 28 keV and its half-maximum values at 21 keV and 34 keV, respectively. We need to underline that due to the detector threshold setting, the effective spectral density has a low energy cutoff at 17 keV [29]. Fig. 3 shows the simulated Moiré patterns assuming monochromatic incident radiation of 28 keV (Fig. 3(a)) as well as using the generated spectrum shown in Fig. 2 at different fractional Talbot distances (Fig. 3(b)–(d)). The characteristics of the images

Imax

Imin

V

0.8119 0.7903

0.3896 0.4185

35.15% 30.76%

0.7518

0.4566

24.33%

0.7221

0.4755

20.59%

(including maximum and minimum intensities and the visibility) are summarized in Table 1. Analysis of Fig. 3 and the data in Table 1 shows that for the generated spectrum shown in Fig. 2, the Moiré fringe patterns are fairly insensitive to the X-rays polychromaticity. According to Table 1, the visibility of the Moiré fringe corresponding to the first fractional Talbot distance is just slightly reduced compared to that of monochromatic radiation of 28 keV. Even in the worst case, the visibility at the fifth 5th Talbot distance remains more than 20%, a value sufficient for experimental applications [19–22]. This result confirms reports of recent experiments pointing out that a Talbot–Lau interferometer well matches commercially available X-ray tube generators [15,17,20–22]. As a consequence, a widespread application of the method can be expected, such as biomedical imaging, material science, homeland security etc. Furthermore, the sensitivity of phase measurements is proportional to the fractional Talbot distance, and the phase image quality can be further improved by increasing this distance. Our result demonstrates that when using polychromatic laboratory X-ray sources, high sensitivity for phase measurements can be obtained with an acceptable image contrast. This issue is innovative and, because of low absorption contrast and relatively low variations of the refractive index for biological samples, extremely attractive for foreseen biomedical imaging applications. Despite the great achievement by the Talbot–Lau interferometer approach, the layout of the imaging system can be further optimized from the point of view of applications. We consider the following equations derived from classical optical theories: the fractional Talbot distances for a plane wave, Eq. (9), and the geometrical magnification factor, Eq. (10). dN = N M=

p21

(9)

22 

p2 L + zN = L p1 /

(10)

By combining Eqs. (4) and (5), we can derive the following expressions relating the pitch p0 of the source grating G0 and p1 of the phase grating G1,



L + p0 = Np2 p1 =

Np2

p0 p2 (p0 + p2 )/

L + zN =

Fig. 3. Simulated Moiré fringe patterns assuming monochromatic incident X-rays of 28 keV (a) as well as using the generated spectrum at the first (b), the third (c) and the fifth fractional Talbot distance. All the intensity images are displayed on the same linear scale.

 L 2

p0 + p2 L p0

+

2L N

(11) (12) (13)

where the pitch of the grating G2, p2 , is limited by the fabrication technologies. As shown in Eq. (13), for a given distance L, the total distance L + zN can only be decreased by decreasing p0 , the period of the source grating G0. From Eq. (11), we can see that the period of G0 can be decreased by either decreasing the X-ray wavelength  or increasing the Talbot order N. However, decreasing the period p0 results in reduction of available X-ray flux and, as a result, in increase of exposure time. Notwithstanding the decrease

Z. Wang et al. / Journal of Electron Spectroscopy and Related Phenomena 184 (2011) 342–345

of the total distance with the X-ray wavelength, the decrease of the wavelength is negative for the contrast, since the differential contrast is proportional to . However, note that the total absorbed dose reduces significantly with decrease of  (the linear absorption coefficient is approximately proportional to 3 ). Thus there are two definite tradeoffs to be made, the first is between the exposure time and the compactness of the system, and the second one is between the image contrast and the absorbed dose. 4. Conclusions

[3] [4] [5] [6] [7] [8] [9] [10] [11]

The functional capability of a Talbot–Lau interferometer using polychromatic laboratory X-ray generators has been analyzed based on numerical simulations. The results show that the visibility of the Moiré fringe is fairly insensitive to the polychromaticity of the X-ray source. Even at a high-order fractional Talbot distance, the Moiré fringe visibility remains to be sufficient for performing experiments. And thus a high sensitivity of phase measurements can be achieved. Together with the results obtained in the Talbot interferometer [22], we may conclude that grating-based phase contrast imaging can be well combined with standard X-ray tube generators. This method can find widespread applications in areas where phase contrast imaging is demanded, but is currently unavailable, such as biological and biomedical imaging, homeland security, etc.

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Acknowledgements

[23]

This work was financially supported by the National Outstanding Youth Fund (Project No. 10125523 to Z.W.), the Knowledge Innovation Program of the Chinese Academy of Sciences (KJCX2YW-N42), the Key Important Project of the National Natural Science Foundation of China (10734070), the National Natural Science Foundation of China (NSFC 10774144 and 10979055) and the National Basic Research Program of China (2009CB930804).

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