- Email: [email protected]
Compositional images from the Diffraction Enhanced Imaging technique
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957 www.elsevier.com/locate/nima
Compositional images from the Diffraction Enhanced Imaging technique M.O. Hasnaha,, Z. Zhongb, C. Parhamc, H. Zhangd, D. Chapmand,e a
Department of Math and Physics, P.O. Box 2713, Qatar University, Doha, Qatar National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 19763, USA c Biomedical Engineering, University of North Carolina, Chapel Hill, NC 27599, USA d Division of Biomedical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A9 e Anatomy and Cell Biology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E5 b
Received 29 August 2006; received in revised form 14 November 2006; accepted 22 November 2006 Available online 2 January 2007
Abstract Diffraction Enhanced Imaging (DEI) derives X-ray contrast from absorption, refraction, and extinction. While the refraction angle image of DEI represents the gradient of the projected mass density of the object, the absorption image measures the projected attenuation (mt) of an object. Using a simple integral method it has been shown that a mass density image (rt) can be obtained from the refraction angle image. It then is a simple matter to develop a combinational image by dividing these two images to create a m/r image. The m/r is a fundamental property of a material and is therefore useful for identifying the composition of an object. In projection X-ray imaging the m/r image identifies the integrated composition of the elements along the beam path. When applied to DEI computed tomography (CT), the image identifies the composition in each voxel. This method presents a new type of spectroscopy based in radiography. We present the method of obtaining the compositional image, the results of experiments in which we verify the method with known standards and an application of the method to breast cancer imaging. r 2006 Elsevier B.V. All rights reserved. PACS: 41.50.+h; 42.30.Va; 61.10.Nz; 87.59.Bh Keywords: Composition of material; Spectroscopy; Computed tomography; Diffraction Enhanced Imaging; X-ray
1. Introduction
be expressed as
The X-ray absorption property of an element is defined by the attenuation cross-section of the element, commonly referred to as the mass attenuation coefficient, m/r. This quantity is used to calculate the attenuation of X-rays by materials (either elemental or a composition of elements). It requires knowledge of the density of the material and, in the case of a composite, the additional knowledge of the mass fractions of the elements in the material [1]. The projected absorption and projected density images can
mt ¼
Corresponding author. Tel.: +1 974 582 2891.
E-mail address: [email protected] (M.O. Hasnah). 0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.11.066
N X ðm=ri Þri ti i¼1
rt ¼
N X
ri ti .
ð1Þ
i¼1
for the case of N compositionally different materials. The sums would be replaced by integrals if the material were composed of a continuous distribution of materials. For simplicity, a discrete distribution of materials will be assumed. Expressed as a ratio: PN ðm=ri Þri ti mt ¼ i¼1 . (2) PN rt i¼1 ri ti
ARTICLE IN PRESS 954
M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
This property is determined in a pixel of cross-sectional area A, then, PN PN N ðm=ri Þmi X mt m i¼1 ðm=ri Þri ti A ¼ ¼ (3) ¼ i¼1 PN ri f i rt M r t A i i¼1 i¼1 i where M is the total projected mass in the pixel crosssectional area through the object, mi is the mass of each component of the material and fi is the mass fraction of the ith component. The last result is the well-recognized equation for the determination of the mass attenuation coefficient for a multi-component material. Diffraction Enhanced Imaging (DEI) is a radiographic technique that derives contrast from an object’s X-ray absorption, refraction gradient, and ultra small angle scatter-rejection (extinction) properties [2–4]. Compared with the absorption contrast of conventional radiography, the two additional contrast mechanisms of refraction and extinction allow the visualization of more features in objects [5,6]. The refraction gradient image is a result of the spatial gradients of the projected electronic density (and hence, mass density) of the object. A simple integral method was developed to prepare a projected density image [7,8] of the object: a rt image. The DEI apparent absorption image derives contrast from both absorption and some scattering properties of the object. For simplicity, we will temporarily ignore the effects of scattering and assume that this image is directly related to the projected absorption: a mt image. It is well known that DEI does not deliver a true refraction angle image or a true absorption image in the presence of an object that has ultra-small angle scattering. However, there are methods that recover this information by relying on multiple images acquired at a variety of analyzer settings [9,10]. Typically these methods determine the refraction angle by finding the centroid of the rocking curve peak and the absorption by integrating the images over the rocking angle. Traditional X-ray radiography measures and depicts the combined effects of mass attenuation and density. For example, an object will look exactly the same on X-ray radiograph as a different object with twice the mass attenuation and half the density. The ability to determine the projected absorption and density using the DEI method or multiple image methods presents a unique opportunity to explore combinations of the density and absorption images, specifically to determine a m/r image. What we present here is a novel method to extract some information regarding the composition of an object relying on the unique properties that DEI can deliver. 2. Theory 2.1. Absorption images The absorption image is a projection of the attenuating properties of all the materials along the paths of the X-rays. The averaged linear attenuation coefficient of a
spatially varying material through a fixed thickness t of material within a pixel xi, zj can be found by Z t mðxi ; zj Þt ¼ mðxi ; zj ; yÞ dy, (4) 0
where mðxi ; zj ; yÞ is the linear attenuation coefficient at the projected pixel location xi, zj, as a function of X-ray beam trajectory, y. The integral is carried out over the thickness of the object, t. The image is formed in the xz plane and is chosen to be consistent with the DEI setup shown in Fig. 1. The value of mðxi ; zj Þt can be obtained from either conventional radiography or DEI as mðxi ; zj Þt ¼ ln
Iðxi ; zj Þ I0
(5)
where Iðxi ; zj Þ is the intensity measured in pixel xi, zj and I0 is the incident intensity on the first surface of the object (assumed to be uniform and characterized by a single value). 2.2. Density images The refraction angle image from DEI is a measure of the gradient of the projected density of the object and can be expressed as [3,6]: Z q t q Dyðxi ; zj Þ ¼ K e r ðxi ; zj ; yÞ dy K e re ðxi ; zj Þt (6) qz 0 e qz where K e ¼ re l2 =2p, re is the classical electron radius (2.82 1015 m), l is the X-ray wavelength and t is the thickness (fixed) of the object. The quantity re ðxi ; zj Þt is the average electron density over the thickness t. When Eq. (6) is integrated over the direction z we obtain an electronic density image as [7], re ðxi ; zj Þt ¼ re ðxi ; z0 Þt þ
j X zpix l¼0
¼ re ðxi ; z0 Þt þ
Ke
j X zpix l¼0
Dyðxi ; zj Þ
q Ke qz Ke
Z
t
re ðxi ; zj ; yÞ dy ð7Þ 0
where zpix is the linear dimension of the detector pixel in the z-direction and is a result of the summation (integration) process. This image represents the electron density averaged
Detector Object
Analyzer y
z
Double Crystal Monochromator x
Synchrotron Beam
Fig. 1. Schematic representation of the experimental setup. The coordinate system used is indicated near the object. The incident X-ray beam travels along the y direction, the object is scanned along the z-direction, the image of the object is formed in the xz plane and the diffraction plane of the monochromator-analyzer crystal is the yz plane.
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
over the thickness of the object t. The second term is the change in electronic density from the beginning integration point (l ¼ 0 index). The first term is the required constant of integration. Thus the method requires knowledge of the electron density at the starting point of integration. Since this image is the result of a summation from the edge there are some restrictions on its application. First, it is best applied when the edge of the image field region extends beyond the object so that a known region can be used to determine the integration constant. Second, the sums are performed along all z columns of the image. This summation will suffer from noise due to statistical fluctuations in the refraction angle image and this point will be addressed in the following section. 3. Method In comparing Eqs. (4) and (7) it is clear that the ratio results in an image that is related to the absorption per unit electronic density (similar to a m/r), mðxi ; zj Þt re ðxi ; zj Þt ¼
re ðxi ; z0 Þt þ
Rt 0
mðxi ; zj ; yÞ dy
Rt l¼0 ðzpix =K e ÞK e ðq=qzÞ 0
Pj
re ðxi ; zj ; yÞ dy
. ð8Þ
The ratio represents the image of attenuation crosssection scaled to the electronic density and hence is the average attenuation cross-section per electron. Though the representation for the absorption in Eq. (8) is more natural, it is more common to use the m/r representation where r is the mass density of the material. The m/r has been well-studied and tabulated values of for m/r elements and composites at different X-ray energies are routinely available. Eq. (8) can be modified to mðxi ; zj Þt mðxi ; zj Þt ¼ 1 ¯ rðxi ; zj Þt ðZ=AÞ ure ðxi ; zj Þt Rt 0 mðxi ; zj ; yÞ dy h i ¼ Rt Pj 1 ¯ ðZ=AÞ u re ðxi ; z0 Þt þ l¼0 ðzpix =K e ÞK e ðq=qzÞ 0 re ðxi ; zj ; yÞ dy
ð9Þ ¯ where ðZ=AÞ is the average Z to A ratio for the material which can be approximated by 1/2 for mid-Z and higher Z elements and u is the nucleon mass (1.67 1027 kg). Eq. (9) can be used to convert the attenuation and refraction images into the projected compositional image from DEI. Specifically, mðxi ; zj Þt rðxi ; zj Þt ¼
lnðIðxi ; zj Þ=I 0 Þ . P 1 ¯ ðZ=AÞ u½re ðxi ; z0 Þt þ jl¼0 ðzpix =K e ÞDyðxi ; zj Þ ð10Þ
955
These equations will be used to interpret and analyze the data. 4. Results and discussion Experiments were performed at the National Synchrotron Light Source X15A beamline. The images of the test object were acquired at 30 keV using the DEI system as schematically shown in Fig. 1. The monochromator and analyzer used the Si (3,3,3) reflection which had a rocking curve width of approximately 2.0 mrad. The incident beam was approximately 1 mm high and 100 mm wide. The images were acquired in line scan mode with the diffracted beam from the analyzer being detected by an image plate (Fuji, HR V). The images plates were digitized in a Fuji BAS2500 reader at a pixel size of 50 mm. The test object was a 6.1 mm diameter glass rod immersed in a CsCl solution in a Lucite box with 1 mm thick walls. The Lucite box had internal dimension of 6.35 mm along the X-ray beam path. A sketch of the test object is shown in Fig. 2a. The concentration of CsCl in the solution was 0.24 g of CsCl per cm3 of water. This concentration is chosen so that the attenuation coefficient of the CsCl solution is the same as that of the glass at the imaging energy. As such, a radiograph of the phantom would yield no contrast for the glass rod. Images of the object were then obtained with low and high angle analyzer settings, DEI attenuation and refraction images were computed. Those images are shown in Figs. 2b (attenuation or mt image) and 2c (refraction angle or Dy image). As expected, the mt image yields the same value for the glass rod and the CsCl medium (Fig. 2b), except for the regions near the edges of the rod. The edge artifact arises from the strong refraction that occurs at the edges of objects whose refractive index is significantly different than the matrix material. This strong refraction affects the DEI algorithm and results in an inaccurate local value of mt and Dy [9]. The refraction angle image, Dy, has been integrated, statistically correct with values that do not streak the image, and scaled to give a mass density image shown in Fig. 2d. This image along with the projected attenuation image are used to give the projected compositional image, mt=rt. In all cases, the gray scale units are shown on the left side of the images. The boxed region shown in the projected compositional image is used to plot the measured mt=rt across the rod. This is shown in Fig. 3 as the solid line. Tabulated m/r values were used for theoretical calculations of mt=rt which is shown as the dashed line. There is a reasonable agreement between measured and theoretical values. Part of the disagreement between the two values arises from the strong refraction or ‘‘the edge effect’’ that occurs at the edges of the glass rod [9]. Again, an artifact due to edge effects appears at the rod edges in the line plot. An estimate of the error in this analysis is shown as the single error bar on the left of the experimental values. This error bar is
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
956
10 mm mm
µt gray scale
6m 6mm
2.8
CsCl solutions
glass rod
2.6 2.4
2.0
1
ρt gray scale
∆θ θ gray scale
2.2
0 -1
1.9 1.8 1.7 1.6 1.5
µ/ρ ρ gray scale (cm2/g)
-2
1.8 1.6 1.4 1.2
Fig. 2. Compositional image test object: (a) schematic representation of the object comprised of a glass rod and CsCl solution, (e) mt=rt image derived from the mt image of (b) and the rt image of (d), (c) refraction angle image which when integrated along vertical lines gives the rt image of (d).
2.0
− − µt / ρt (cm2/g)
1.5
1.0 measured calculated
0.5
0 0
2
4
6
8
10
12
Position (mm) Fig. 3. Comparison of measured and calculated mt=rt values from a line across the test object shown in Fig. 2. The error bar show on the left is indicative of the estimated errors in the data.
based on the measured noise in both the projected absorption and projected mass density images and represents a ‘‘two sigma’’ error. To show an application of the method, images were acquired of a mastectomy specimen at 40 keV. This
specimen was approximately 2 cm thick and was determined to have invasive cancer. Images of this specimen are shown in Fig. 4. The refraction angle image was obtained by the MIR method which is known to be less susceptible to artifacts than DEI is shown in Fig. 4b. The rt image was obtained using a constrained least squares method [8] to minimize streaks created by a simple integral method [7]. The mt=rt image of this specimen is given in Fig. 4d. An interesting feature of this image is the increased mt=rt at the location of the thick fibril. Most likely this fibril is partially calcified which is common in this type of cancer. This ability to determine relative composition is not possible by inspecting either the mt or rt images separately. A line cut through this feature is shown in Fig. 5 along with an estimate of the analysis error. The relative increase in mt=rt can be accounted for by a 5% hydroxyapatite concentration in the fibril. This increase is determined by assuming the fibril has the same thickness along the projection direction as it appears laterally in the image. 5. Conclusion We have shown that the DEI method can be used to develop a new compositional image to radiography. This image is directly related to a fundamental property of a material(s); the absorption per electron of a material. The compositional image is a property of the object at a specific energy. Additional information about the composition
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
0.2 ∆θ gray scale
0.85
µ tgray scale
957
0.80
0.75
a
0.70
0.1 0.0 -0.1 -0.2 -0.3
5mm
b
0.280
µ/ρ gray scale (cm
ρ t gray scale
3.15 3.10 3.05 3.00
0.260 0.250 0.240
c
2.95
0.270
d
Fig. 4. Measured mt=rt of a mastectomy specimen with invasive cancer: (a) mt image, (b) refraction angle image, Dy, (c) computed rt image and (d) mt=rt image. The raw data was taken at an imaging energy of 40 keV.
Acknowledgements
0.30
The authors would like to acknowledge the support from the Canada Research Chairs program (D.C. & H.Z.), Shaikha Bint Jabor Al Thani, Vice President for Academic Affairs & Chief Academic Officer, Qatar University (M.H.) and the US Department of Energy grant DE-AC02-DEAC02-98CH10886 (Z.Z.).
− − µt / ρt (cm2/g)
0.28
0.26
References
0.24
0.22
0.20
0
5
10 Position (mm)
15
Fig. 5. Measured mt=rt values from a line across the specimen in Fig. 4. The error bar show on the left is indicative of the estimated errors in the data.
may be obtained by applying the method described here at multiple X-ray energies to form a directly measured spectroscopic representation m=rðEÞ. The analysis should be easily applied to computed tomography. In this instance, the average composition could be used to identify material or tissue types in a voxel which could prove quite powerful in image segmentation.
[1] B.D. Cullity, Elements of X-ray Diffraction, first ed., AddisonWesley Publishing Co., Reading, MA, 1956. [2] D. Chapman, W. Thomlinson, R.E. Johnston, D. Washburn, E. Pisano, N. Gmur, Z. Zhong, R. Menk, F. Arfelli, D. Sayers, Phys. Med. Biol. 42 (11) (1997) 2015. [3] Z. Zhong, W. Thomlinson, D. Chapman, D. Sayers, Nucl. Instr. and Meth. A 450 (2–3) (2000) 556. [4] Ya.I. Nesterets, T.E. Gureyev, K.M. Pavlov, D.M. Paganin, S.W. Wilkins, Opt. Commun. 259 (1) (2006) 19. [5] R.A. Lewis, C.J. Hall, A.P. Hufton, S. Evans, R.H. Menk, F. Arfelli, L. Rigon, G. Tromba, D.R. Dance, I.O. Ellis, A. Evans, E. Jacobs, S.E. Pinder, K.D. Rogers, Br. J. Radiol. 76 (2003) 301. [6] E. Pagot, P. Cloetens, S. Fiedler, A. Bravin, P. Coan, J. Baruchel, J. Hartwig, W. Thomlinson, Appl. Phys. Lett. 82 (20) (2003) 3421. [7] M.O. Hasnah, C. Parham, E.D. Pisano, Z. Zhong, O. Oltulu, D. Chapman, Med. Phys. 32 (2) (2005) 549. [8] M.N. Wernick, Y. Yang, I. Mondal, D. Chapman, M. Hasnah, C. Parham, E. Pisano, Z. Zhong, Phys. Med. Biol. 51 (7) (2006) 1769. [9] O. Oltulu, Z. Zhong, M. Hasnah, M.N. Wernick, D. Chapman, J. Phys. D (Appl. Phys.) 36 (17) (2003) 2152. [10] M.N. Wernick, O. Wirjadi, D. Chapman, Z. Zhong, N.P. Galatsanos, Y. Yang, J.G. Brankov, O. Oltulu, M.A. Anastasio, C. Muehleman, Phys. Med. Biol. 48 (23) (2003) 3875.
Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957 www.elsevier.com/locate/nima
Compositional images from the Diffraction Enhanced Imaging technique M.O. Hasnaha,, Z. Zhongb, C. Parhamc, H. Zhangd, D. Chapmand,e a
Department of Math and Physics, P.O. Box 2713, Qatar University, Doha, Qatar National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 19763, USA c Biomedical Engineering, University of North Carolina, Chapel Hill, NC 27599, USA d Division of Biomedical Engineering, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5A9 e Anatomy and Cell Biology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E5 b
Received 29 August 2006; received in revised form 14 November 2006; accepted 22 November 2006 Available online 2 January 2007
Abstract Diffraction Enhanced Imaging (DEI) derives X-ray contrast from absorption, refraction, and extinction. While the refraction angle image of DEI represents the gradient of the projected mass density of the object, the absorption image measures the projected attenuation (mt) of an object. Using a simple integral method it has been shown that a mass density image (rt) can be obtained from the refraction angle image. It then is a simple matter to develop a combinational image by dividing these two images to create a m/r image. The m/r is a fundamental property of a material and is therefore useful for identifying the composition of an object. In projection X-ray imaging the m/r image identifies the integrated composition of the elements along the beam path. When applied to DEI computed tomography (CT), the image identifies the composition in each voxel. This method presents a new type of spectroscopy based in radiography. We present the method of obtaining the compositional image, the results of experiments in which we verify the method with known standards and an application of the method to breast cancer imaging. r 2006 Elsevier B.V. All rights reserved. PACS: 41.50.+h; 42.30.Va; 61.10.Nz; 87.59.Bh Keywords: Composition of material; Spectroscopy; Computed tomography; Diffraction Enhanced Imaging; X-ray
1. Introduction
be expressed as
The X-ray absorption property of an element is defined by the attenuation cross-section of the element, commonly referred to as the mass attenuation coefficient, m/r. This quantity is used to calculate the attenuation of X-rays by materials (either elemental or a composition of elements). It requires knowledge of the density of the material and, in the case of a composite, the additional knowledge of the mass fractions of the elements in the material [1]. The projected absorption and projected density images can
mt ¼
Corresponding author. Tel.: +1 974 582 2891.
E-mail address: [email protected] (M.O. Hasnah). 0168-9002/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2006.11.066
N X ðm=ri Þri ti i¼1
rt ¼
N X
ri ti .
ð1Þ
i¼1
for the case of N compositionally different materials. The sums would be replaced by integrals if the material were composed of a continuous distribution of materials. For simplicity, a discrete distribution of materials will be assumed. Expressed as a ratio: PN ðm=ri Þri ti mt ¼ i¼1 . (2) PN rt i¼1 ri ti
ARTICLE IN PRESS 954
M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
This property is determined in a pixel of cross-sectional area A, then, PN PN N ðm=ri Þmi X mt m i¼1 ðm=ri Þri ti A ¼ ¼ (3) ¼ i¼1 PN ri f i rt M r t A i i¼1 i¼1 i where M is the total projected mass in the pixel crosssectional area through the object, mi is the mass of each component of the material and fi is the mass fraction of the ith component. The last result is the well-recognized equation for the determination of the mass attenuation coefficient for a multi-component material. Diffraction Enhanced Imaging (DEI) is a radiographic technique that derives contrast from an object’s X-ray absorption, refraction gradient, and ultra small angle scatter-rejection (extinction) properties [2–4]. Compared with the absorption contrast of conventional radiography, the two additional contrast mechanisms of refraction and extinction allow the visualization of more features in objects [5,6]. The refraction gradient image is a result of the spatial gradients of the projected electronic density (and hence, mass density) of the object. A simple integral method was developed to prepare a projected density image [7,8] of the object: a rt image. The DEI apparent absorption image derives contrast from both absorption and some scattering properties of the object. For simplicity, we will temporarily ignore the effects of scattering and assume that this image is directly related to the projected absorption: a mt image. It is well known that DEI does not deliver a true refraction angle image or a true absorption image in the presence of an object that has ultra-small angle scattering. However, there are methods that recover this information by relying on multiple images acquired at a variety of analyzer settings [9,10]. Typically these methods determine the refraction angle by finding the centroid of the rocking curve peak and the absorption by integrating the images over the rocking angle. Traditional X-ray radiography measures and depicts the combined effects of mass attenuation and density. For example, an object will look exactly the same on X-ray radiograph as a different object with twice the mass attenuation and half the density. The ability to determine the projected absorption and density using the DEI method or multiple image methods presents a unique opportunity to explore combinations of the density and absorption images, specifically to determine a m/r image. What we present here is a novel method to extract some information regarding the composition of an object relying on the unique properties that DEI can deliver. 2. Theory 2.1. Absorption images The absorption image is a projection of the attenuating properties of all the materials along the paths of the X-rays. The averaged linear attenuation coefficient of a
spatially varying material through a fixed thickness t of material within a pixel xi, zj can be found by Z t mðxi ; zj Þt ¼ mðxi ; zj ; yÞ dy, (4) 0
where mðxi ; zj ; yÞ is the linear attenuation coefficient at the projected pixel location xi, zj, as a function of X-ray beam trajectory, y. The integral is carried out over the thickness of the object, t. The image is formed in the xz plane and is chosen to be consistent with the DEI setup shown in Fig. 1. The value of mðxi ; zj Þt can be obtained from either conventional radiography or DEI as mðxi ; zj Þt ¼ ln
Iðxi ; zj Þ I0
(5)
where Iðxi ; zj Þ is the intensity measured in pixel xi, zj and I0 is the incident intensity on the first surface of the object (assumed to be uniform and characterized by a single value). 2.2. Density images The refraction angle image from DEI is a measure of the gradient of the projected density of the object and can be expressed as [3,6]: Z q t q Dyðxi ; zj Þ ¼ K e r ðxi ; zj ; yÞ dy K e re ðxi ; zj Þt (6) qz 0 e qz where K e ¼ re l2 =2p, re is the classical electron radius (2.82 1015 m), l is the X-ray wavelength and t is the thickness (fixed) of the object. The quantity re ðxi ; zj Þt is the average electron density over the thickness t. When Eq. (6) is integrated over the direction z we obtain an electronic density image as [7], re ðxi ; zj Þt ¼ re ðxi ; z0 Þt þ
j X zpix l¼0
¼ re ðxi ; z0 Þt þ
Ke
j X zpix l¼0
Dyðxi ; zj Þ
q Ke qz Ke
Z
t
re ðxi ; zj ; yÞ dy ð7Þ 0
where zpix is the linear dimension of the detector pixel in the z-direction and is a result of the summation (integration) process. This image represents the electron density averaged
Detector Object
Analyzer y
z
Double Crystal Monochromator x
Synchrotron Beam
Fig. 1. Schematic representation of the experimental setup. The coordinate system used is indicated near the object. The incident X-ray beam travels along the y direction, the object is scanned along the z-direction, the image of the object is formed in the xz plane and the diffraction plane of the monochromator-analyzer crystal is the yz plane.
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
over the thickness of the object t. The second term is the change in electronic density from the beginning integration point (l ¼ 0 index). The first term is the required constant of integration. Thus the method requires knowledge of the electron density at the starting point of integration. Since this image is the result of a summation from the edge there are some restrictions on its application. First, it is best applied when the edge of the image field region extends beyond the object so that a known region can be used to determine the integration constant. Second, the sums are performed along all z columns of the image. This summation will suffer from noise due to statistical fluctuations in the refraction angle image and this point will be addressed in the following section. 3. Method In comparing Eqs. (4) and (7) it is clear that the ratio results in an image that is related to the absorption per unit electronic density (similar to a m/r), mðxi ; zj Þt re ðxi ; zj Þt ¼
re ðxi ; z0 Þt þ
Rt 0
mðxi ; zj ; yÞ dy
Rt l¼0 ðzpix =K e ÞK e ðq=qzÞ 0
Pj
re ðxi ; zj ; yÞ dy
. ð8Þ
The ratio represents the image of attenuation crosssection scaled to the electronic density and hence is the average attenuation cross-section per electron. Though the representation for the absorption in Eq. (8) is more natural, it is more common to use the m/r representation where r is the mass density of the material. The m/r has been well-studied and tabulated values of for m/r elements and composites at different X-ray energies are routinely available. Eq. (8) can be modified to mðxi ; zj Þt mðxi ; zj Þt ¼ 1 ¯ rðxi ; zj Þt ðZ=AÞ ure ðxi ; zj Þt Rt 0 mðxi ; zj ; yÞ dy h i ¼ Rt Pj 1 ¯ ðZ=AÞ u re ðxi ; z0 Þt þ l¼0 ðzpix =K e ÞK e ðq=qzÞ 0 re ðxi ; zj ; yÞ dy
ð9Þ ¯ where ðZ=AÞ is the average Z to A ratio for the material which can be approximated by 1/2 for mid-Z and higher Z elements and u is the nucleon mass (1.67 1027 kg). Eq. (9) can be used to convert the attenuation and refraction images into the projected compositional image from DEI. Specifically, mðxi ; zj Þt rðxi ; zj Þt ¼
lnðIðxi ; zj Þ=I 0 Þ . P 1 ¯ ðZ=AÞ u½re ðxi ; z0 Þt þ jl¼0 ðzpix =K e ÞDyðxi ; zj Þ ð10Þ
955
These equations will be used to interpret and analyze the data. 4. Results and discussion Experiments were performed at the National Synchrotron Light Source X15A beamline. The images of the test object were acquired at 30 keV using the DEI system as schematically shown in Fig. 1. The monochromator and analyzer used the Si (3,3,3) reflection which had a rocking curve width of approximately 2.0 mrad. The incident beam was approximately 1 mm high and 100 mm wide. The images were acquired in line scan mode with the diffracted beam from the analyzer being detected by an image plate (Fuji, HR V). The images plates were digitized in a Fuji BAS2500 reader at a pixel size of 50 mm. The test object was a 6.1 mm diameter glass rod immersed in a CsCl solution in a Lucite box with 1 mm thick walls. The Lucite box had internal dimension of 6.35 mm along the X-ray beam path. A sketch of the test object is shown in Fig. 2a. The concentration of CsCl in the solution was 0.24 g of CsCl per cm3 of water. This concentration is chosen so that the attenuation coefficient of the CsCl solution is the same as that of the glass at the imaging energy. As such, a radiograph of the phantom would yield no contrast for the glass rod. Images of the object were then obtained with low and high angle analyzer settings, DEI attenuation and refraction images were computed. Those images are shown in Figs. 2b (attenuation or mt image) and 2c (refraction angle or Dy image). As expected, the mt image yields the same value for the glass rod and the CsCl medium (Fig. 2b), except for the regions near the edges of the rod. The edge artifact arises from the strong refraction that occurs at the edges of objects whose refractive index is significantly different than the matrix material. This strong refraction affects the DEI algorithm and results in an inaccurate local value of mt and Dy [9]. The refraction angle image, Dy, has been integrated, statistically correct with values that do not streak the image, and scaled to give a mass density image shown in Fig. 2d. This image along with the projected attenuation image are used to give the projected compositional image, mt=rt. In all cases, the gray scale units are shown on the left side of the images. The boxed region shown in the projected compositional image is used to plot the measured mt=rt across the rod. This is shown in Fig. 3 as the solid line. Tabulated m/r values were used for theoretical calculations of mt=rt which is shown as the dashed line. There is a reasonable agreement between measured and theoretical values. Part of the disagreement between the two values arises from the strong refraction or ‘‘the edge effect’’ that occurs at the edges of the glass rod [9]. Again, an artifact due to edge effects appears at the rod edges in the line plot. An estimate of the error in this analysis is shown as the single error bar on the left of the experimental values. This error bar is
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
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Fig. 2. Compositional image test object: (a) schematic representation of the object comprised of a glass rod and CsCl solution, (e) mt=rt image derived from the mt image of (b) and the rt image of (d), (c) refraction angle image which when integrated along vertical lines gives the rt image of (d).
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based on the measured noise in both the projected absorption and projected mass density images and represents a ‘‘two sigma’’ error. To show an application of the method, images were acquired of a mastectomy specimen at 40 keV. This
specimen was approximately 2 cm thick and was determined to have invasive cancer. Images of this specimen are shown in Fig. 4. The refraction angle image was obtained by the MIR method which is known to be less susceptible to artifacts than DEI is shown in Fig. 4b. The rt image was obtained using a constrained least squares method [8] to minimize streaks created by a simple integral method [7]. The mt=rt image of this specimen is given in Fig. 4d. An interesting feature of this image is the increased mt=rt at the location of the thick fibril. Most likely this fibril is partially calcified which is common in this type of cancer. This ability to determine relative composition is not possible by inspecting either the mt or rt images separately. A line cut through this feature is shown in Fig. 5 along with an estimate of the analysis error. The relative increase in mt=rt can be accounted for by a 5% hydroxyapatite concentration in the fibril. This increase is determined by assuming the fibril has the same thickness along the projection direction as it appears laterally in the image. 5. Conclusion We have shown that the DEI method can be used to develop a new compositional image to radiography. This image is directly related to a fundamental property of a material(s); the absorption per electron of a material. The compositional image is a property of the object at a specific energy. Additional information about the composition
ARTICLE IN PRESS M.O. Hasnah et al. / Nuclear Instruments and Methods in Physics Research A 572 (2007) 953–957
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Fig. 4. Measured mt=rt of a mastectomy specimen with invasive cancer: (a) mt image, (b) refraction angle image, Dy, (c) computed rt image and (d) mt=rt image. The raw data was taken at an imaging energy of 40 keV.
Acknowledgements
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The authors would like to acknowledge the support from the Canada Research Chairs program (D.C. & H.Z.), Shaikha Bint Jabor Al Thani, Vice President for Academic Affairs & Chief Academic Officer, Qatar University (M.H.) and the US Department of Energy grant DE-AC02-DEAC02-98CH10886 (Z.Z.).
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References
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Fig. 5. Measured mt=rt values from a line across the specimen in Fig. 4. The error bar show on the left is indicative of the estimated errors in the data.
may be obtained by applying the method described here at multiple X-ray energies to form a directly measured spectroscopic representation m=rðEÞ. The analysis should be easily applied to computed tomography. In this instance, the average composition could be used to identify material or tissue types in a voxel which could prove quite powerful in image segmentation.
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