Abstract

The demand for quantitative medical imaging is increasing in the ongoing digitalization. Conventional computed tomography (CT) is energy-dependent and therefore of limited comparability. In contrast, dual-energy CT (DECT) allows for the determination of absolute image contrast quantities, namely the electron density and the effective atomic number, and is already established in clinical radiology and radiation therapy. Grating-based phase-contrast computed tomography (GBPC-CT) is an experimental X-ray technique that also allows for the measurement of the electron density and the effective atomic number. However, the determination of both quantities is challenging when dealing with polychromatic GBPC-CT setups. In this paper, we present how to calculate the effective atomic numbers with a polychromatic, laboratory GBPC-CT setup operating between 35 and 50\,kVp. First, we investigated the accuracy of the measurement of the attenuation coefficients and electron densities. For this, we performed a calibration using the concept of effective energy. With the reliable experimental quantitative values, we were able to evaluate the effective atomic numbers of the investigated materials using a method previously shown with monochromatic X-ray radiation. In detail, we first calculated the ratio of the electron density and attenuation coefficient, which were experimentally determined with our polychromatic GBPC-CT setup. Second, we compared this ratio with tabulated total attenuation cross sections from literature values to determine the effective atomic numbers. Thus, we were able to calculate two physical absolute quantities -- the electron density and effective atomic number -- that are in general independent of the specific experimental conditions like the X-ray beam spectrum or the setup design.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

F. Sardanelli, “Trends in radiology and experimental research,” Eur. Radiol. Exp. 1, 1 (2017).
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T. Hyodo, M. Hori, P. Lamb, K. Sasaki, T. Wakayama, Y. Chiba, T. Mochizuki, and T. Murakami, “Multimaterial decomposition algorithm for the quantification of liver fat content by using fast-kilovolt-peak switching dual-energy CT: Experimental validation,” Radiology 282(2), 381–389 (2017).
[Crossref]

2016 (4)

S. Leng, A. Huang, J. M. Cardona, X. Duan, J. C. Williams, and C. H. McCollough, “Dual-energy CT for quantification of urinary stone composition in mixed stones: A phantom study,” Am. J. Roentgenol. 207(2), 321–329 (2016).
[Crossref]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realisation of high-sensitivity laboratory X-ray grating-based phase-contrast computed tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

2015 (4)

M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
[Crossref] [PubMed]

D. Hahn, P. Thibault, A. Fehringer, M. Bech, T. Koehler, F. Pfeiffer, and P. B. Noël, “Statistical iterative reconstruction algorithm for X-ray phase-contrast CT,” Sci. Rep. 5, 10452 (2015).
[Crossref] [PubMed]

J. K. van Abbema, M.-J. van Goethem, M. J. W. Greuter, A. van der Schaaf, S. Brandenburg, and E. R. van der Graaf, “Relative electron density determination using a physics based parameterization of photon interactions in medical DECT,” Phys. Med. Biol. 60(9), 3825–3846 (2015).
[Crossref] [PubMed]

A. Sarapata, M. Willner, M. Walter, T. Duttenhofer, K. Kaiser, P. Meyer, C. Braun, A. Fingerle, P. B. Noël, F. Pfeiffer, and J. Herzen, “Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum,” Opt. Express 23(1), 523–535 (2015).
[Crossref] [PubMed]

2014 (4)

B. Y. Hur, J. M. Lee, W. Hyunsik, K. B. Lee, I. Joo, J. K. Han, and B. I. Choi, “Quantification of the fat fraction in the liver using dual-energy computed tomography and multimaterial decomposition,” J. Comput. Assist. Tomogr. 38(6), 845–852 (2014).
[Crossref] [PubMed]

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
[Crossref] [PubMed]

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
[Crossref] [PubMed]

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
[Crossref]

2013 (2)

2012 (2)

M. L. Taylor, R. L. Smith, and R. D. Franich, “Robust calculation of effective atomic numbers: The Auto-Zeff software,” Med. Phys. 39(4), 2473–4209 (2012).
[Crossref]

A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
[Crossref] [PubMed]

2011 (2)

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
[Crossref] [PubMed]

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray grating interferometer and synchrotron radiation,” Phys. Status Solidi A 208(11), 2526–2532 (2011).
[Crossref]

2010 (2)

G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
[Crossref] [PubMed]

Z. Qi, J. Zambelli, N. Bevins, and G.-H. Chen, “Quantitative imaging of electron density and effective atomic number using phase contrast CT,” Phys. Med. Biol. 55(9), 2669–2677 (2010).
[Crossref] [PubMed]

2009 (2)

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(12), 10010–10018 (2009).
[Crossref] [PubMed]

D. T. Boll, N. A. Patil, E. K. Paulson, E. M. Merkle, W. N. Simmons, S. A. Pierre, and G. M. Preminger, “Renal stone assessment with dual-energy multidetector CT and advanced postprocessing techniques: Improved characterization of renal stone composition—pilot study,” Radiology 250(3), 813–820 (2009).
[Crossref] [PubMed]

2008 (2)

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
[Crossref] [PubMed]

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, 134–137 (2008).
[Crossref] [PubMed]

2007 (1)

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

2006 (1)

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, 258–261 (2006).
[Crossref]

2005 (2)

2003 (2)

A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11(19), 2303–2314 (2003).
[Crossref] [PubMed]

M. Torikoshi, T. Tsunoo, M. Sasaki, M. Endo, Y. Noda, Y. Ohno, T. Kohno, K. Hyodo, K. Uesugi, and N. Yagi, “Electron density measurement with dual-energy X-ray CT using synchrotron radiation,” Phys. Med. Biol. 48(5), 673–685 (2003).
[Crossref] [PubMed]

2002 (1)

C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
[Crossref]

2000 (1)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
[Crossref]

1986 (1)

H. Q. Woodard and D. R. White, “The composition of body tissues,” Br. J. Radiol. 59(708), 1209–1218 (1986).
[Crossref] [PubMed]

1979 (1)

G. T. Herman, “Correction for beam hardening in computed tomography,” Phys. Med. Biol. 24(1), 81–106 (1979).
[Crossref] [PubMed]

1976 (1)

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in X-ray computerised tomography,” Phys. Med. Biol. 21(5), 733–744 (1976).
[Crossref] [PubMed]

1975 (1)

E. C. McCullough, “Photon attenuation in computed tomography,” Med. Phys. 2, (6)307–320 (1975).
[Crossref] [PubMed]

1965 (1)

R. C. Murty, “Effective atomic numbers of heterogeneous materials,” Nature 207, 398–399 (1965).
[Crossref]

1946 (1)

W. Spiers, “Effective atomic number and energy absorption in tissues,” Br. J. Radiol. 19, 52–63 (1946).
[Crossref] [PubMed]

Achterhold, K.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
[Crossref] [PubMed]

Akkara Veetil, B. M.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
[Crossref] [PubMed]

Alvarez, R. E.

R. E. Alvarez and A. Macovski, “Energy-selective reconstructions in X-ray computerised tomography,” Phys. Med. Biol. 21(5), 733–744 (1976).
[Crossref] [PubMed]

Auweter, S.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
[Crossref] [PubMed]

Baruchel, J.

I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray grating interferometer and synchrotron radiation,” Phys. Status Solidi A 208(11), 2526–2532 (2011).
[Crossref]

Bäumer, C.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
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Bech, M.

D. Hahn, P. Thibault, A. Fehringer, M. Bech, T. Koehler, F. Pfeiffer, and P. B. Noël, “Statistical iterative reconstruction algorithm for X-ray phase-contrast CT,” Sci. Rep. 5, 10452 (2015).
[Crossref] [PubMed]

M. Willner, M. Bech, J. Herzen, I. Zanette, D. Hahn, J. Kenntner, J. Mohr, A. Rack, T. Weitkamp, and F. Pfeiffer, “Quantitative X-ray phase-contrast computed tomography at 82 keV,” Opt. Express 21(4), 4155–4166 (2013).
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A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
[Crossref] [PubMed]

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, 134–137 (2008).
[Crossref] [PubMed]

Beckmann, F.

Bequé, D.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
[Crossref]

Berger, M. J.

M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker, and K. Olsen, “XCOM: Photon cross section database,” NBSIR 87–3597 (2010).

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, and G.-H. Chen, “Quantitative imaging of electron density and effective atomic number using phase contrast CT,” Phys. Med. Biol. 55(9), 2669–2677 (2010).
[Crossref] [PubMed]

Birnbacher, L.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental realisation of high-sensitivity laboratory X-ray grating-based phase-contrast computed tomography,” Sci. Rep. 6, 24022 (2016).
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M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
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Boll, D. T.

D. T. Boll, N. A. Patil, E. K. Paulson, E. M. Merkle, W. N. Simmons, S. A. Pierre, and G. M. Preminger, “Renal stone assessment with dual-energy multidetector CT and advanced postprocessing techniques: Improved characterization of renal stone composition—pilot study,” Radiology 250(3), 813–820 (2009).
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Bongartz, T.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
[Crossref] [PubMed]

Brandenburg, S.

J. K. van Abbema, M.-J. van Goethem, M. J. W. Greuter, A. van der Schaaf, S. Brandenburg, and E. R. van der Graaf, “Relative electron density determination using a physics based parameterization of photon interactions in medical DECT,” Phys. Med. Biol. 60(9), 3825–3846 (2015).
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Braun, C.

M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
[Crossref] [PubMed]

A. Sarapata, M. Willner, M. Walter, T. Duttenhofer, K. Kaiser, P. Meyer, C. Braun, A. Fingerle, P. B. Noël, F. Pfeiffer, and J. Herzen, “Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum,” Opt. Express 23(1), 523–535 (2015).
[Crossref] [PubMed]

Brönnimann, C.

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, 134–137 (2008).
[Crossref] [PubMed]

Bruyndonckx, P.

A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
[Crossref] [PubMed]

Bunk, O.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
[Crossref] [PubMed]

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(12), 10010–10018 (2009).
[Crossref] [PubMed]

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, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

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, 258–261 (2006).
[Crossref]

Burvall, A.

Cardona, J. M.

S. Leng, A. Huang, J. M. Cardona, X. Duan, J. C. Williams, and C. H. McCollough, “Dual-energy CT for quantification of urinary stone composition in mixed stones: A phantom study,” Am. J. Roentgenol. 207(2), 321–329 (2016).
[Crossref]

Chabior, M.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
[Crossref] [PubMed]

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
[Crossref]

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
[Crossref] [PubMed]

Chang, J.

M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker, and K. Olsen, “XCOM: Photon cross section database,” NBSIR 87–3597 (2010).

Chen, G.-H.

Z. Qi, J. Zambelli, N. Bevins, and G.-H. Chen, “Quantitative imaging of electron density and effective atomic number using phase contrast CT,” Phys. Med. Biol. 55(9), 2669–2677 (2010).
[Crossref] [PubMed]

Chiba, Y.

T. Hyodo, M. Hori, P. Lamb, K. Sasaki, T. Wakayama, Y. Chiba, T. Mochizuki, and T. Murakami, “Multimaterial decomposition algorithm for the quantification of liver fat content by using fast-kilovolt-peak switching dual-energy CT: Experimental validation,” Radiology 282(2), 381–389 (2017).
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Choi, B. I.

B. Y. Hur, J. M. Lee, W. Hyunsik, K. B. Lee, I. Joo, J. K. Han, and B. I. Choi, “Quantification of the fat fraction in the liver using dual-energy computed tomography and multimaterial decomposition,” J. Comput. Assist. Tomogr. 38(6), 845–852 (2014).
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Cloetens, P.

Coman, J.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
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Coulon, P.

G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
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Coursey, J. S.

M. J. Berger, J. H. Hubbell, S. M. Seltzer, J. Chang, J. S. Coursey, R. Sukumar, D. S. Zucker, and K. Olsen, “XCOM: Photon cross section database,” NBSIR 87–3597 (2010).

Cozzini, C.

A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
[Crossref]

Crowson, C. S.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
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David, C.

T. Zhou, U. Lundström, T. Thüring, S. Rutishauser, D. H. Larsson, M. Stampanoni, C. David, H. M. Hertz, and A. Burvall, “Comparison of two X-ray phase-contrast imaging methods with a microfocus source,” Opt. Express 21(25), 30183–30195 (2013).
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M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
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I. Zanette, T. Weitkamp, S. Lang, M. Langer, J. Mohr, C. David, and J. Baruchel, “Quantitative phase and absorption tomography with an X-ray grating interferometer and synchrotron radiation,” Phys. Status Solidi A 208(11), 2526–2532 (2011).
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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(12), 10010–10018 (2009).
[Crossref] [PubMed]

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, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, C. Kottler, O. Bunk, and C. David, “Hard X-Ray phase tomography with low-brilliance sources,” Phys. Rev. Lett. 98, 108105 (2007).
[Crossref] [PubMed]

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, 258–261 (2006).
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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).
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C. David, B. Nöhammer, H. Solak, and E. Ziegler, “Differential X-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett. 81, 3287–3289 (2002).
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Davis, J. M.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
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Diaz, A.

Dill, S.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
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Donath, T.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
[Crossref] [PubMed]

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(12), 10010–10018 (2009).
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Dorscheid, R.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
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Duan, X.

S. Leng, A. Huang, J. M. Cardona, X. Duan, J. C. Williams, and C. H. McCollough, “Dual-energy CT for quantification of urinary stone composition in mixed stones: A phantom study,” Am. J. Roentgenol. 207(2), 321–329 (2016).
[Crossref]

Duttenhofer, T.

Duvdevani, M.

G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
[Crossref] [PubMed]

Eikenberry, E. F.

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, 134–137 (2008).
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Eliahou, R.

G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
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Endo, M.

M. Torikoshi, T. Tsunoo, M. Sasaki, M. Endo, Y. Noda, Y. Ohno, T. Kohno, K. Hyodo, K. Uesugi, and N. Yagi, “Electron density measurement with dual-energy X-ray CT using synchrotron radiation,” Phys. Med. Biol. 48(5), 673–685 (2003).
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Fehringer, A.

D. Hahn, P. Thibault, A. Fehringer, M. Bech, T. Koehler, F. Pfeiffer, and P. B. Noël, “Statistical iterative reconstruction algorithm for X-ray phase-contrast CT,” Sci. Rep. 5, 10452 (2015).
[Crossref] [PubMed]

Fingerle, A.

M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
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A. Sarapata, M. Willner, M. Walter, T. Duttenhofer, K. Kaiser, P. Meyer, C. Braun, A. Fingerle, P. B. Noël, F. Pfeiffer, and J. Herzen, “Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum,” Opt. Express 23(1), 523–535 (2015).
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Fior, G.

M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
[Crossref] [PubMed]

Fishman, E.

J. Henes, M. Fuld, E. Fishman, M. Horger, and J. Fritz, “Dual-energy computed tomography of the knee, ankle, and foot: Noninvasive diagnosis of gout and quantification of monosodium urate in tendons and ligaments,” Semin. Musculoskelet. Radiol. 20(01), 130–136 (2016).
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Fitzgerald, R.

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2000).
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Franich, R. D.

M. L. Taylor, R. L. Smith, and R. D. Franich, “Robust calculation of effective atomic numbers: The Auto-Zeff software,” Med. Phys. 39(4), 2473–4209 (2012).
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Franz, W. B.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
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Fritz, J.

J. Henes, M. Fuld, E. Fishman, M. Horger, and J. Fritz, “Dual-energy computed tomography of the knee, ankle, and foot: Noninvasive diagnosis of gout and quantification of monosodium urate in tendons and ligaments,” Semin. Musculoskelet. Radiol. 20(01), 130–136 (2016).
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Fuld, M.

J. Henes, M. Fuld, E. Fishman, M. Horger, and J. Fritz, “Dual-energy computed tomography of the knee, ankle, and foot: Noninvasive diagnosis of gout and quantification of monosodium urate in tendons and ligaments,” Semin. Musculoskelet. Radiol. 20(01), 130–136 (2016).
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Glazebrook, K. N.

T. Bongartz, K. N. Glazebrook, S. J. Kavros, N. S. Murthy, S. P. Merry, W. B. Franz, C. J. Michet, B. M. Akkara Veetil, J. M. Davis, T. G. Mason, K. J. Warrington, S. R. Ytterberg, E. L. Matteson, C. S. Crowson, S. Leng, and C. H. McCollough, “Dual-energy CT for the diagnosis of gout: an accuracy and diagnostic yield study,” Ann. Rheum. Dis. 74(6), 1072–1077 (2014).
[Crossref] [PubMed]

Gofrit, O. N.

G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
[Crossref] [PubMed]

Grandl, S.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
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Graser, A.

A. Spek, F. Strittmatter, A. Graser, P. Kufer, C. Stief, and M. Staehler, “Dual energy can accurately differentiate uric acid-containing urinary calculi from calcium stones,” World J. Urol. 34(9), 1297–1302 (2016).
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Greuter, M. J. W.

J. K. van Abbema, M.-J. van Goethem, M. J. W. Greuter, A. van der Schaaf, S. Brandenburg, and E. R. van der Graaf, “Relative electron density determination using a physics based parameterization of photon interactions in medical DECT,” Phys. Med. Biol. 60(9), 3825–3846 (2015).
[Crossref] [PubMed]

Grünzweig, C.

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, 134–137 (2008).
[Crossref] [PubMed]

Hahn, D.

D. Hahn, P. Thibault, A. Fehringer, M. Bech, T. Koehler, F. Pfeiffer, and P. B. Noël, “Statistical iterative reconstruction algorithm for X-ray phase-contrast CT,” Sci. Rep. 5, 10452 (2015).
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M. Willner, M. Bech, J. Herzen, I. Zanette, D. Hahn, J. Kenntner, J. Mohr, A. Rack, T. Weitkamp, and F. Pfeiffer, “Quantitative X-ray phase-contrast computed tomography at 82 keV,” Opt. Express 21(4), 4155–4166 (2013).
[Crossref]

Han, J. K.

B. Y. Hur, J. M. Lee, W. Hyunsik, K. B. Lee, I. Joo, J. K. Han, and B. I. Choi, “Quantification of the fat fraction in the liver using dual-energy computed tomography and multimaterial decomposition,” J. Comput. Assist. Tomogr. 38(6), 845–852 (2014).
[Crossref] [PubMed]

Hellerhoff, K.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
[Crossref] [PubMed]

Henes, J.

J. Henes, M. Fuld, E. Fishman, M. Horger, and J. Fritz, “Dual-energy computed tomography of the knee, ankle, and foot: Noninvasive diagnosis of gout and quantification of monosodium urate in tendons and ligaments,” Semin. Musculoskelet. Radiol. 20(01), 130–136 (2016).
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G. T. Herman, “Correction for beam hardening in computed tomography,” Phys. Med. Biol. 24(1), 81–106 (1979).
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Herrmann, C.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
[Crossref] [PubMed]

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D. Hahn, P. Thibault, A. Fehringer, M. Bech, T. Koehler, F. Pfeiffer, and P. B. Noël, “Statistical iterative reconstruction algorithm for X-ray phase-contrast CT,” Sci. Rep. 5, 10452 (2015).
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A. Sarapata, M. Willner, M. Walter, T. Duttenhofer, K. Kaiser, P. Meyer, C. Braun, A. Fingerle, P. B. Noël, F. Pfeiffer, and J. Herzen, “Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum,” Opt. Express 23(1), 523–535 (2015).
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A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
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M. Willner, M. Bech, J. Herzen, I. Zanette, D. Hahn, J. Kenntner, J. Mohr, A. Rack, T. Weitkamp, and F. Pfeiffer, “Quantitative X-ray phase-contrast computed tomography at 82 keV,” Opt. Express 21(4), 4155–4166 (2013).
[Crossref]

A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
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M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
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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(12), 10010–10018 (2009).
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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, 258–261 (2006).
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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).
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G. Hidas, R. Eliahou, M. Duvdevani, P. Coulon, L. Lemaitre, O. N. Gofrit, D. Pode, and J. Sosna, “Determination of renal stone composition with dual-energy CT: In vivo analysis and comparison with X-ray diffraction,” Radiology 257(2), 394–401 (2010).
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D. T. Boll, N. A. Patil, E. K. Paulson, E. M. Merkle, W. N. Simmons, S. A. Pierre, and G. M. Preminger, “Renal stone assessment with dual-energy multidetector CT and advanced postprocessing techniques: Improved characterization of renal stone composition—pilot study,” Radiology 250(3), 813–820 (2009).
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J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
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Reiser, M.

M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
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J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Bäumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Med. Biol. 53(15), 4031–4047 (2008).
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A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
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M. Willner, G. Fior, M. Marschner, L. Birnbacher, J. Schock, C. Braun, A. Fingerle, P. B. Noël, E. J. Rummeny, F. Pfeiffer, and J. Herzen, “Phase-contrast hounsfield units of fixated and non-fixated soft-tissue samples,” PLoS ONE 10(8), e0137016 (2015).
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Sarapata, A.

A. Sarapata, M. Willner, M. Walter, T. Duttenhofer, K. Kaiser, P. Meyer, C. Braun, A. Fingerle, P. B. Noël, F. Pfeiffer, and J. Herzen, “Quantitative imaging using high-energy X-ray phase-contrast CT with a 70 kVp polychromatic X-ray spectrum,” Opt. Express 23(1), 523–535 (2015).
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A. Sarapata, M. Chabior, C. Cozzini, J. I. Sperl, D. Bequé, O. Langner, J. Coman, I. Zanette, M. Ruiz-Yaniz, and F. Pfeiffer, “Quantitative electron density characterization of soft tissue substitute plastic materials using grating-based X-ray phase-contrast imaging,” Rev. Sci. Instr. 85(10), 103708 (2014).
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M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
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M. Torikoshi, T. Tsunoo, M. Sasaki, M. Endo, Y. Noda, Y. Ohno, T. Kohno, K. Hyodo, K. Uesugi, and N. Yagi, “Electron density measurement with dual-energy X-ray CT using synchrotron radiation,” Phys. Med. Biol. 48(5), 673–685 (2003).
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Schroer, C.

M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
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A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
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M. Chabior, T. Donath, C. David, O. Bunk, M. Schuster, C. Schroer, and F. Pfeiffer, “Beam hardening effects in grating-based X-ray phase-contrast imaging,” Med. Phys. 38(3), 1189–1195 (2011).
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D. T. Boll, N. A. Patil, E. K. Paulson, E. M. Merkle, W. N. Simmons, S. A. Pierre, and G. M. Preminger, “Renal stone assessment with dual-energy multidetector CT and advanced postprocessing techniques: Improved characterization of renal stone composition—pilot study,” Radiology 250(3), 813–820 (2009).
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M. Willner, J. Herzen, S. Grandl, S. Auweter, D. Mayr, A. Hipp, M. Chabior, A. Sarapata, K. Achterhold, I. Zanette, T. Weitkamp, A. Sztrókay, K. Hellerhoff, M. Reiser, and F. Pfeiffer, “Quantitative breast tissue characterization using grating-based X-ray phase-contrast imaging,” Phys. Med. Biol. 59(7), 1557–1571 (2014).
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A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. 109(39), 15691–15696 (2012).
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M. L. Taylor, R. L. Smith, and R. D. Franich, “Robust calculation of effective atomic numbers: The Auto-Zeff software,” Med. Phys. 39(4), 2473–4209 (2012).
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Figures (5)

Fig. 1
Fig. 1 Schematic laboratory GBPC-CT setup. The setup consists of the X-ray source, three X-ray gratings, namely the source grating, the phase grating, and the analyzer grating, followed by the Pilatus II detector. The sample, which is immersed in a water container, is put directly in front of the phase grating. This figure is licensed under the Creative Commons Attribution (CC BY) and first published in Willner et al. [29].
Fig. 2
Fig. 2 Results of the phantom measurement at different source spectra. Axial slices of two scans at 35 kVp and at 50 kVp. The phantom consists of five different materials (1: LDPE, 2: POM, 3: PEEK, 4: Nylon, and 5: PMMA) embedded in a water surrounding. The absorption coefficients are displayed on the left and the electron density on the right. The quantitative GBPC-CT values can be found in Table 1 and the mass densities and chemical compositions of the materials can be found in Table 2. The attenuation coefficient µ is displayed linearly in an interval of [0.2, 0.6] in 1/cm and the electron density ρe in an interval of [300, 450] in e/nm3. The experimental values presented here and in the other tables were determined as the mean of a certain volume and the respective error as the total standard deviation of the same data.
Fig. 3
Fig. 3 Scatter plots of the total linear attenuation coefficients versus the electron density of the two different setup voltages shown in Table 1. The experimental linear attenuation coefficients µ are plotted against the corresponding electron densities ρe obtained from GBPC-CT measurements at 35 kVp (a) and 50 kVp (b). Standard deviations are indicated by error bars and the red solid line represents the linear attenuation coefficient for Compton scattering µincoh.
Fig. 4
Fig. 4 Determination of the effective atomic number for two different energies. The ratio represents both sides of Eq. (15). The right side of Eq. (15) is the theoretical ratio of the atomic number over the total cross section Z/σtot and shown by the continuous lines, which are based on the spline interpolated tabulated data. The left side of Eq. (15) is the ratio of the experimentally determined electron density over the linear attenuation coefficient ρe/µ of single elements, which is marked by the crosses (‘x’) and uses the experimental results from the phantom measurements shown in Fig. 2.
Fig. 5
Fig. 5 Phantom experiments with different surrounding liquids. Quantitative GBPC-CT results of phantom measurements with different surrounding liquids visualize the effect of the polychromatic X-ray spectrum on the measured values of the electron density and the effective atomic number due to beam hardening. The top row shows the electron density ρe, the bottom row the effective atomic number Zeff. Subfigures (a) and (d) show an axial slice of the phantom immersed in water, subfigures (b) and (e) show the same phantom measured in oil, and subfigures (c) and (f) represent the experiment in the 5% NaCl solution as surrounding liquid. The components and results of the phantom are listed in Tables 3 and 4. The electron density ρe is linearly displayed in an interval of [300, 400] in e/nm3 and the effective atomic number Zeff in an interval of [5, 9]. The tube voltage was 40 kVp.

Tables (4)

Tables Icon

Table 1 Electron density values and linear attenuation coefficients of different polymers and water. The data was evaluated from phantom measurements at tube voltages of 35 kVp and 50 kVp (see Fig. 2). The theoretical reference values can be found in [37, 38]. The mass densities and chemical compositions of the materials can be found in Table 2.

Tables Icon

Table 2 Comparison of the determined effective atomic number Zeff with reference values. The theoretical effective atomic numbers of Ref. 1 were calculated from the elemental compositions of the materials according to Eq. (16). Ref. 2 uses a factor of 3.50 in Eq. (16) [41], the mass density ρ is based on the data provided by the manufacturer [45]. Nylon is of type 6,6.

Tables Icon

Table 3 Relative comparison of the electron density of the liquids phantom in dependence of different surrounding liquids at 40 kVp. Results of three phantom measurements with different surrounding liquids are shown. With exception of PMMA and water, no theoretical reference values are given (see Table 1).

Tables Icon

Table 4 Relative comparison of the effective atomic number of the liquids phantom in dependence of different surrounding liquids at 40 kVp. Results of three phantom measurements with different surrounding liquids are shown. With exception of PMMA and water, no theoretical reference values are given (see Table 2).

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

p = p 0 + a p 0 3 ,
n = 1 δ + i β ,
δ = 2 π r 0 2 c 2 E 2 ρ e ,
β = μ 2 k ,
μ ads = μ rel + μ H 2 O ( E eff , μ ) ,
1 k Φ x = p 2 2 π d ϕ ,
Φ = k δ rel d z ,
δ abs = δ rel + δ H 2 O ( E eff , δ ) .
ρ e = E eff , δ 2 2 π r 0 2 c 2 δ abs .
ρ e = ρ N A w i Z i w i A i ,
ρ e = ρ N A Z A .
μ = ρ N A A σ tot ( E , Z ) ,
σ tot ( E , Z ) = σ ph ( E , Z ) + σ coh ( E , Z ) + σ incoh ( E , Z ) .
μ = ρ e σ tot ( E , Z ) Z .
ρ e μ = Z eff σ tot ( E , Z ) ,
Z eff = i ( w i Z i ) 2.94 2.94 ,

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