Theory and preliminary experimental verification of quantitative edge illumination x-ray phase contrast tomography C. K. Hagen,1,∗ P. C. Diemoz,1 M. Endrizzi,1 L. Rigon,2,3 D. Dreossi,4 F. Arfelli,2,3 F. C. M. Lopez,2,3 R. Longo,2,3 and A. Olivo1 1 Department

of Medical Physics and Bioengineering, University College London, Malet Place, Gower Street, London WC1E 6BT, UK 2 Department of Physics, University of Trieste, Via Valerio 2, 34127 Trieste, Italy 3 Instituto Nazionale di Fisica Nucleare, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy 4 Sincrotrone Trieste SCpA, S.S. 14 km 163.5, 34012 Basovizza (TS), Italy ∗ [email protected]

Abstract: 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 OCIS codes: (340.7440) X-ray imaging; (110.6960) Tomography; (340.6720) Synchrotron radiation.

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Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7989

7. A. Momose, T. Takeda, Y. Itai and K. Hirano, “Phase-contrast x-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996). 8. P. Cloetens, R. Barrett, J. Baruchel, J. Guigay and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996). 9. F. Dilmanian, Z. Zhong, B. Ren, X. Wu, L. Chapman, I. Orion and W. Thomlinson, “Computed tomography of x-ray index of refraction using the diffraction enhanced imaging method,” Phys. Med. Biol. 45, 933–946 (2000). 10. 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, 6296–6304 (2005). 11. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. Menk, S. Pani, M. Prest, P. Poporat, L. Rigon, G. Tromba, E. Vallazza and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001). 12. T.P. Millard, M. Endrizzi, K. Ignatyev, C.K. Hagen, P.R.T. Munro, R. Speller and A. Olivo, “Method for the automatization of the alignment of a laboratory based x-ray phase contrast edge illumination system,” Rev. Sci. Instrum. 84, 083702 (2013). 13. P.C. Diemoz, M. Endrizzi, C.E. Zapata, Z. Pesic, C. Rau, A. Bravin, I. Robinson and A. Olivo, “X-ray phase contrast imaging with nanoradian angular resolution,” Phys. Rev. Lett. 110, 138105 (2013). 14. P.C. Diemoz, C.K. Hagen, M. Endrizzi and A. Olivo, “Sensitivity of laboratory based implementations of edge illumination x-ray phase-contrast imaging,” Appl. Phys. Lett. 103, 244104 (2013). 15. A. Olivo, S. Gkoumas, M. Endrizzi, C.K. Hagen, M.B. Szafraniec, P.C. Diemoz, P.R.T. Munro, K. Ignatyev, B. Johnson, J.A. Horrocks, S.J. Vinnicombe, J.L. Jones and R.D. Speller, “Low-dose phase contrast mammography with conventional sources,” Med. Phys. 40(9), 090701 (2013). 16. J. Herzen, T. Donath, F. Pfeiffer, O. Bunk, C. Padeste, F. Beckmann, A. Schreyer and C. David, “Quantitative phase-contrast tomography of a liquid phantom using a conventional x-ray tube source,” Opt. Express 17(2), 10010 (2009). 17. P.C. Diemoz, P. Coan, I. Zanette, A. Bravin, S. Lang, C. Glaser and T. Weitkamp, “A simplified approach for computed tomography with an x-ray grating interferometer,” Opt. Express 19(3), 1691–1698 (2011). 18. A.C. Kak and M. Slaney, “Principles of computerized tomographic imaging,” IEEE Press (1988). 19. G. Tromba, R. Longo, A. Abrami, F. Arfelli, A. Astolfo, P. Bregant, F. Brun, K. Casarin, V. Chenda, D. Dreossi, M. Hola, J. Kaiser, L. Manchini, R. Menk, E. Quai, E. Quaia, L. Rigon, T. Rokvic, N. Sodini, D. Sanabor, E. Schultke, M. Tonutti, A. Vascotto, F. Zanconati, M. Cova and E. Castelli, “The SYRMEP beamline of Elettra: clinical mammography and biomedical applications,” AIP Conference Proceedings 1266, 18–23 (2010). 20. P.R.T. Munro, C.K. Hagen, M.B. Szafraniec and A. Olivo, “A simplified approach to quantitative coded aperture x-ray phase imaging,” Opt. Express 21(9), 11187–11201 (2013). 21. L. Rigon, F. Arfelli, A. Astolfo, A. Bergamaschi, D. Dreossi, R. Longo, R.H. Menk, B. Schmitt, E. Valazza and E. Castelli, “A single-photon counting “edge-on” silicon detector for synchrotron radiation mammography,” Nuclear Instruments and Methods in Physics Research Section A 608(1), S62–S65 (2009). 22. F.C. Lopez, L. Rigon, R. Longo, F. Arfelli, A. Bergamaschi, R.C. Chen, D. Dreossi, B. Schmitt, E. Valazza and E. Castelli, “Development of a fast read-out system of a single photon counting detector for mammography with synchrotron radiation,” JINST 6, C12031 (2011). 23. A. Mozzanica, A. Bergamaschi, R. Dinapoli, F. Gozzo, B. Henrich, P. Kraft, B. Patterson and B. Schmitt, “MythenII: a 128 channel single photon counting readout chip,” Nuclear Instruments and Methods in Physics Research Section A 607(1), 250–252 (2009). 24. B.L. Henke, E.M. Gullikson and J.C. Davis, “X-ray interactions: photoabsorption, scattering, transmission and reflection at E = 50-30000 eV, Z = 1-92,” Atomic Data and Nuclear Data Tables 54, 181–342 (1993). 25. T. Thuering, P. Modregger, B.R. Pinzer, Z. Wang and M. Stampanoni, “Non-linear regularized phase retrieval for unidirectional x-ray differential phase contrast radiography,” Opt. Express 19(25), 25545–25558 (2011). 26. P.R.T. Munro, L. Rigon, K. Ignatyev, F.C.M. Lopez, D. Dreossi, R.D. Speller and A. Olivo, “A quantitative, non-interferometric x-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013) 27. F.A. Vittoria, P.C. Diemoz, M. Endrizzi, L. Rigon, F.C. Lopez, D. Dreossi, P.R.T. Munro and A. Olivo, “Strategies for efficient and fast wave optics simulation of coded-aperture and other x-ray phase-contrast imaging methods,” Appl. Opt. 52(28), 6940–6947 (2013). 28. Diemoz et al., manuscript submitted. 29. T. Saam, J. Herzen, H. Hetterich, S. Fill, M. Willner, M. Stockmar, K. Achterhold, I. Zanette, T. Weitkamp, U. Schueller, S. Auweter, S. Adam-Neumair, K. Nikolaou, M. Reiser, F. Pfeiffer and F. Bamberg, “Translation of atherosclerotic plaque phase-contrast CT imaging from synchrotron radiation to a conventional lab-based x-ray source,” PLOS ONE 8(9), e73513 (2013). 30. R.E. Guldberg, A.S.P. Lin, R. Coleman, G. Robertson and C. Duvall, “Microcomputed tomography imaging of skeleteral development and growth,” Birth Defects Research C 72, 250–259 (2004).

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Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7990

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: n(E) = 1 − δ (E) + iβ (E),

(1)

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. #204867 - $15.00 USD (C) 2014 OSA

Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7991

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.

The refraction-induced displacement (D) and the transmission (T ) of the beam can be expressed as [17]: D(x, y) =

zod ∂ Φ(x, y) · , k ∂y

(2)

T (x, y) = e−μ (x,y) ,

(3)

where the object is described in terms of the phase shift and attenuation it imposes on the beam: Φ(x, y) = k ·



μ (x, y) = 2k ·

O



δ (x, y, z) dz,

O

(4)

β (x, y, z) dz.

(5)

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]: I(x, y) = I0 · T (x, y) ·C(y0 + D(x, y)),

(6)

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: S phase (x, y; θ ) =

∂Φ ∂ (x, y; θ ) = k · ∂y ∂y

Sabs (x, y; θ ) = μ (x, y; θ ) = 2k ·



 (x,y;θ ;s)

(x,y;θ ;s)

δ (ξ , η , ζ ) ds

β (ξ , η , ζ ) ds,

(7) (8)

where (ξ = x cos θ − z sin θ , η = y, ζ = x sin θ + z cos θ ) is the frame of reference of the object. The line (x, y; θ ; s) represents the path of an incoming x-ray. Equations (7) and (8) enable #204867 - $15.00 USD (C) 2014 OSA

Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7992

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 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:      I(x, y) zod ∂ Φ(x, y) = −μ (x, y) + ln C y0 + · − ln(C(y0 )). (9) ln I0 ·C(y0 ) k ∂y 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]:    zod ∂ Φ(x, y) C (y0 ) zod ∂ Φ(x, y) · ≈ ln (C(y0 )) + · · . (10) ln C y0 + k ∂y C(y0 ) k ∂y 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:      I(x, y; θ ) ∂ 1 = δ (ξ , η , ζ ) ds, (11) Smixed (x, y; θ ) = − ln k · β (ξ , η , ζ ) − A · 2 I0 ·C(y0 ) ∂η (x,y;θ ;s) 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: fmixed (ξ , η , ζ ) = k · β (ξ , η , ζ ) − A ·

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∂ δ (ξ , η , ζ ), ∂η

(12)

Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7993

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) x 4 (V) mm2 ) using a slit (Huber GmbH, Rimsting, Germany) with an opening of 20 (H) x 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) x 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) x 0.3 (V) mm2 active surface and pixel size of 50 (H) x 300 (V) μ m2 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

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].

material nylon 6 PEEK PBT sapphire (99.9%) titanium (>99.6%)

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diam. (μ m) 150 ± 20% 450 ± 20% 180 ± 20% 250 ± 20% 250 ± 10%

chemical formula C6 H11 NO C19 H14 O3 C12 H12 O4 Al2 O3 Ti

density (g/cm3 ) 1.13 ± 5% 1.29 ± 5% 1.31 ± 5% 3.99 ± 5% 4.51 ± 5%

δ (10−7 ) 4.11 ± 5% 4.49 ± 5% 4.59 ± 5% 12.95± 5% 13.87± 5%

β (10−10 ) 1.35± 5% 1.50± 5% 1.64± 5% 17.03± 5% 143.27± 5%

Received 16 Jan 2014; revised 10 Mar 2014; accepted 17 Mar 2014; published 28 Mar 2014 7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007989 | OPTICS EXPRESS 7994

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]. 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 μ m 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 (

Theory and preliminary experimental verification of quantitative edge illumination x-ray phase contrast tomography.

X-ray phase contrast imaging (XPCi) methods are sensitive to phase in addition to attenuation effects and, therefore, can achieve improved image contr...
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