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Tomographic images of breast tissues obtained by Compton scattering: An analytical computational study
Radiation Physics and Chemistry 116 (2015) 273–277
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Tomographic images of breast tissues obtained by Compton scattering: An analytical computational study M. Antoniassi a,b,n, M.E. Poletti a, A. Brunetti c a
Departamento de Física, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Ribeirão Preto, São Paulo, Brazily Departamento Acadêmico de Física, Universidade Tecnológica Federal do Paraná, Curitiba, Brazil c Dipartimento di Scienze Politiche, Scienze della Comunicazione e Ingegneria dell' Informazione, Università di Sassari, Italy b
H I G H L I G H T S
An analytical computational simulation of Compton scattering tomography was studied. Results of simulations were compared with those obtained by Monte Carlo simulation. A reconstruction algorithm was implemented for reconstructing the scattering images. Electron density images of breast tissues (simulated breast phantom) were obtained. Influence of parameters (sample size, beam energy and glandularity) was studied.
art ic l e i nf o
a b s t r a c t
Article history: Received 1 October 2014 Received in revised form 27 January 2015 Accepted 30 January 2015 Available online 31 January 2015
In this work, we studied by analytical simulation the potential of a Compton scatter technique for breast imaging application. A Compton scattering tomography system was computationally simulated in order to provide the projection data (scattering signal) for the image reconstructions. The simulated projections generated by the analytical proposed method were validated through comparison with those obtained by Monte Carlo simulation. Electron density images were obtained from the scattering signal using a reconstruction algorithm implemented for the system geometry. Finally, the quality of the reconstructed images was evaluated for different sample sizes, beam energies, and tissue compositions (glandularities). & 2015 Elsevier Ltd. All rights reserved.
Keywords: Compton scattering Electron density Breast cancer Mammography Tomography
1. Introduction Compton scattering imaging is a technique that allows obtaining the spatial distribution of electron density (ρe) of the material examined (Golosio et al., 2004). It can be used as an alternative to the conventional transmission techniques (Carlsson, 1999) when the latter is unable to provide a discrimination in terms of linear attenuation coefficient (m) between the different constituents of the sample. Different methods have been proposed using Compton scattering imaging technique (Cesareo et al., 1992; Harding, 1997), but, particularly, few of them were applied for breast imaging (Aviles et al., 2011; VanUytven et al., 2008).
n
Corresponding author. E-mail address: [email protected] (M. Antoniassi).
http://dx.doi.org/10.1016/j.radphyschem.2015.01.038 0969-806X/& 2015 Elsevier Ltd. All rights reserved.
The main problems encountered in the computational study of this kind of imaging technique are related to the time required for the simulations (usually performed by Monte Carlo techniques) and to the reconstruction of the images from the scatter signal (Golosio et al., 2004) that is generally much more difficult than the reconstruction in transmission techniques, mainly due to self-attenuation effects in the sample (Brunetti and Golosio, 2001). In the present work, we studied using a fast analytical simulator the potential of a Compton scatter technique for breast imaging application. A Compton scattering tomography system was computationally simulated in order to provide the projection data (scattering signal) for the image reconstructions. Electron density images were obtained from the scattering signal using a reconstruction algorithm implemented for the system geometry. Finally, the quality of the reconstructed images was evaluated for different sample sizes, beam energies, and tissue compositions (glandularities).
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M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
2. Materials and method
2.2. Image reconstructions
2.1. Simulation of tomography system
2.2.1. Transmission tomography The transmission tomography is necessary to determine the attenuation properties, i.e., the map of the linear attenuation coefficient m(x,y) of the sample, which is used to correct the selfattenuation effects in the Compton scatter tomography. The transmission projection Pt(s,θ) can be defined as a function of the intensities of incident (I0) and transmitted (It) photons detected by the transmission detector at each angle of rotation (θ) as (Golosio et al., 2003)
The simulation of the tomography system was implemented using Matlab software. Fig. 1 represents schematically the simulated Compton scatter tomography experiment. The x and y axes are fixed in the reference frame of the sample. The s and u axes are parallel to the translation direction and to the beam respectively, and they are related to x and y by a rotation of an angle θ. The sample was rotated at uniform angular steps (1.25°), totalizing 288 angular views. At each angle, the sample was translated (1 mm steps) and the transmitted and scattered signal, at each translational step, were acquired respectively by the transmission and scatter detectors. The simulated X-ray source produced a monoenergetic beam of 1 1 mm2 square cross section. Three different monoenergetic beams at incident energies (E0) of 40, 60 and 80 keV were investigated in this work. The two scatter detectors (4 20 cm2) were simulated such that the total scatter signal at each translational step was the sum of the signal obtained by each one of the detectors. The parallel distance between the beam and the scatter detectors was always 18 cm. For simulation of the samples (phantoms), it was used cylinders of 12, 14 and 16 cm diameter and 9 cm height. Three sample compositions, simulating breast of different glandularities (g) were studied: (i) g ¼25% (ρe ¼ 3.185 1023 e /cm3), (ii) g ¼50% or 50–50 breast (ρe ¼3.267 1023 e /cm3) and (iii) g ¼75% (ρe ¼3.353 1023 e /cm3), where g indicates the percentage of glandular tissue in a mixture of glandular and adipose tissues. Cylindrical insets (see Fig. 1): A (adipose, ρe ¼3.108 1023 e /cm3), B (glandular, ρe ¼3.444 1023 e /cm3), C (carcinoma, ρe ¼3.452 1023 e /cm3) and D (micro-calcification content, MCC, ρe ¼3.562 1023 e /cm3) of 2.5 cm diameter and 1 cm height were placed in the central height region of the sample. The elemental composition and physical density of adipose and glandular breast tissues were taken from Hammerstein et al. (1979) and of carcinoma, from VanUytven et al. (2008). The MCC was simulated as a tissue region composed of a mixture of 95% glandular tissue and 5% calcium hydroxyapatite.
⎛ I0 ⎞ Pt (s , θ) = ln ⎜ ⎟ ⎝ It (s , u) ⎠
∫ μ (s, u) du = ∫ μ (x, y) δ (s − x cos θ − y sin θ) dx dy , =
(1)
which corresponds to the Radon transform of the map m(x,y) (Golosio et al., 2003). There are different algorithms which can be used to obtain m(x, y) from the acquired projections (Hendee and Ritenour, 2002). In this work, the transmission images were reconstructed using the filtered back-projection algorithm (Shepp and Logan, 1974). 2.2.2. Compton scatter tomography The contribution of a small path du along the beam to the detected scatter signal is given by (Brunetti and Golosio, 2001)
dIs = I0 f (s , u, θ) p (s , u) g (s , u, θ),
(2)
where: (i) f(s,u,θ) is the probability that a photon coming from the source (us) reaches the point u. It depends on the linear attenuation coefficient m0 of the sample at the beam energy (E0) and can be written as
⎛ f (s , u, θ) = exp ⎜ − ⎝
∫u
u s
⎞ μ 0 (s , u‵) du‵⎟, ⎠
(3)
(ii) p(s,u) is the probability of an incident photon being inelastically (Compton) scattered along du. For high momentum transfer values (i.e., for high incident energy and/or high scattering angle), the effects of electron binding are small, and the Compton scattering probability p(s,u) can be written as a function of the electron density (ρe) and the Klein–Nishina cross-section (sKN):
p (s , u) = ρe (s , u) σ KN du,
(4)
(iii) g(s,u,θ) is the probability that a photon Compton scattered from (s,u) reaches the detector (Brunetti et al., 2002):
g (s , u , θ ) =
Fig. 1. Compton scatter tomography arrangement.
1 4π
∫Ω
dΩ det
⎛ 1 ⎛ dσ KN ⎞ ⎜ ⎟ exp ⎜ − σ KN ⎝ dΩ ⎠ ⎝
det
⎞
∫(s, u) μ s (l) dl⎟⎠,
(5)
where Ωdet is the solid angle subtended by the scatter detectors from the interaction point and ms is the linear attenuation coefficient at the Compton scattered energy (Ec). The integral along dl, in the exponential function, is calculated from (s,u) to the detector surface element enclosed by the solid angle dΩ. Defining the attenuation correction K at the point (s,u) as K (s, u, θ) = f (s, u, θ) g (s, u, θ), the total scattered radiation reaching the detector (projection), normalized by the factor I0sKN, is obtained summing the contributions from each element du of the sample along the incidence line:
M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
∫ ρe (s, u) K (s, u, θ) du = ∫ ρe (x, y) K (x, y , θ) δ (s − x cos θ − y sin θ) dx dy .
2.3. Validation and study of the images
Ps (s , θ) =
(6)
Eq. (6) is similar to (1), except by the presence, inside the integral, of the attenuation correction K depending on the rotation angle, θ. In fact, it is often referred to as the generalized Radon transform of ρe(x,y) (Golosio et al., 2003). Then, analogously to the transmission tomography, it is possible to use the filtered backprojection algorithm to reconstruct the image from the scatter projections. The result of the reconstruction gives, instead of the original map of electron density, ρe(x,y), an approximated image, ~ ~ I (x, y) = ρe (x, y) K (x, y), distorted by the effects of the attenuation inside the sample. The tilde above I represents the operation of ~ filtered back-projection and K (x, y) is an attenuation related function, independent of angle θ, obtained as a result of the filtered back-projection process. ~ This function K (x, y), although not known a priori, can be approximated by the average over rotational angle of the attenuation correction K(x,y,θ) (Brunetti and Golosio, 2001; Hogan et al., 1991):
1 ~ K (x , y ) ≅ 2π
∫0
2π
K (x , y , θ ) dθ .
(7)
In this way, the map of electron density can be approximately obtained from I˜ applying the attenuation correction K˜ as
~ I (x , y ) ρ~e (x, y) = ~ . K (x , y )
275
The analytical simulation model proposed in this work was validated comparing the scattering projections, Ps(s, θ ¼0°), of a homogenous sample (50–50 breast) obtained by it and those obtained by Monte Carlo simulation using the XRMC code (Golosio et al., 2014). The study of the Compton scatter tomographic images was done by assessing different parameters of the simulation: (i) sample size, (ii) incident energy and (iii) sample glandularity (composition).
3. Results and discussion 3.1. Validation of the simulation method Fig. 2 compares the scattering projections for θ ¼0° and E0 ¼60 keV of a homogenous sample (50–50 breast) obtained by the simulation model proposed in this work and those obtained by Monte Carlo simulation, where the relative count represents the ratio between the number of detected and incident photons. As can be observed, the results obtained by the present method are in good agreement with those provided by Monte Carlo simulation with mean differences about 0.4%. 3.2. Reconstructed images
(8)
This approximation works well when the self-attenuation is low, i.e., for small samples and/or small values of m (Brunetti et al., 2004; Hogan et al., 1991). When the self-attenuation effects are considerable, a secondary correction is needed to compensate them. The second correction implemented in this work is an adaptation for the present Compton scatter technique of the iterative correction method applied for radionuclide computed tomography (Chang, 1978). It is an iterative procedure that consists in reprojecting the primary corrected image ρ~e (x, y), using Eq. (6), to form a new set of projections. After this, a set of error projections is obtained by subtracting each new projection from its corresponding original projection. An error image is produced by applying the filtered back-projection technique in this set of error projections and after correcting the resulting image by the same attenuation correction (Eq. (8)) used in the primary correction. This error image is after added to the primary corrected image to form a final electron density image. Since this secondary correction process is an iterative procedure, the reconstructed image can be reprojected again and the process repeated. In this work the final images were obtained after two iterations. In the present study, both the analytical simulation of the projections and the reconstruction algorithm were implemented using the software Matlab. In the analytical simulations, the maps m0(x,y) and ms(x,y) were calculated from the elemental composition data of the tissues using the XCOM program (Berger et al., 1998). For the reconstruction procedures, ms and m0 cannot be known a priori. In this way, the map m0(x,y) was obtained from the simulated transmission tomography (at the incident energy) and ms(x, y), for each Compton scattering energy, was estimated from m0(x,y) using the ratio of the linear attenuation coefficients of a reference material at the Compton and incident energies: ms(Ec)¼m0(E0)[m ref(Ec)/mref(E0)]. In this work the 50–50 breast was chosen as the reference material.
The reconstructed electron density images obtained for 80 keV and (a) 12 cm, (b) 14 cm, and (c) 16 cm sample diameters are shown in Fig. 3. The contrasts in the insert regions (adipose, glandular, carcinoma and MCC) with respect to the background (50–50 breast) are about 4.9%, 5.4%, 5.7% and 9.0% respectively and the inaccuracy in the electron density values in all this insert regions is less than 0.8% for all investigated sample diameters. Fig. 4 presents the intensity profiles along the central (a) x and (b) y axes of the 40, 60 and 80 keV images obtained for 14 cm sample diameter. It can be seen from Fig. 4 that the reconstructed pixel values are in good agreement with the original (theoretical) ones (electron density values). The inaccuracy of the electron density values calculated in the region of the inserts for 40, 60 and 80 keV are less than 1.5%, 1.1% and 0.7% respectively. The better accuracy for higher energies is due to lower attenuation effects in these conditions. As the energy increases, the linear attenuation coefficients of the tissues
Fig. 2. Scattering projections for θ ¼0° of a homogenous sample (50–50 breast) obtained by this work and by Monte Carlo simulation.
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Fig. 3. Electron density images obtained for 80 keV incident energy and (a) 12 cm, (b) 14 cm, and (c) 16 cm sample diameters.
Fig. 4. Intensity profiles of the 40, 60 and 80 keV electron density images: (a) x-axis profile and (b) y-axis profile.
M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
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References
Fig. 5. Intensity profiles along the central y-axis in the region of the carcinoma tissue obtained for 80 keV, 14 cm sample diameter and three different sample glandularities: 25%, 50% and 75%.
decrease and therefore the inherent errors due to approximations in the reconstruction model (e.g. estimation of ms from m0), or in the attenuation correction procedures are smaller. Fig. 5 compares the intensity profiles along the central y-axis in the region of the carcinoma tissue obtained for 80 keV, 14 cm sample diameter and three different sample glandularities: 25%, 50% and 75%. Comparing the contrast values (8.4%, 5.7% and 3%) of the carcinoma for different glandularities (25%, 50% and 75%), it is observed that if the glandularity increases, the image contrast decreases. This reduction of contrast, which in practical terms implies the diminution of the detectability of the carcinoma in the breast, is due to similarities between the electron densities of glandular and carcinoma breast tissues (Antoniassi et al., 2010; Ryan et al., 2005). It is important to mention that 95% of women have breast glandularity less than 45% (Yaffe et al., 2009), condition in which the results present contrast values higher than 5.7%.
4. Conclusion In this work, we studied by analytical simulation the potential of a Compton scatter technique for breast imaging application. Comparisons using Monte Carlo technique showed that the analytical simulation method is adequate to obtain single scattering projections for the proposed Compton scatter tomographic system. The reconstruction algorithm showed to be able to produce electron density images of breast tissues with good accuracy (errorso1.5%) for all sample size and incident energy investigated in this work. Further studies should be done in order to evaluate the use polyenergetic beams and verify the influence of multiple scattering and statistical noise on the quality of the reconstructed images.
Acknowledgements The authors would like to acknowledge the support of the Brazilian agency Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (Processos: 2012/05587-7 and 2013/05079-4).
Antoniassi, M., Conceição, A.L.C., Poletti, M.E., 2010. Characterization of breast tissues using Compton scattering. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 619, 375–378. Aviles, J.E.A., Pistorius, S., Elbakri, I.A., Gordon, R., Ahmad, B., 2011. A 1st generation scatter CT algorithm for electron density breast imaging which accounts for bound incoherent, coherent and multiple scatter: a Monte Carlo study. J. X-Ray Sci. Technol. 19, 477–499. Berger, M.J., Hubbell, J.H., Seltzer, S.M., Chang, J., Coursey, J.S., Sukumar, R., Zucker, D.S., Olsen, K., 1998. XCOM: photon cross sections database. In: NIoSaT, (NIST) (Ed.), XCOM: Photon Cross Sections Database. National Institute of Standards and Technology (NIST). Brunetti, A., Cesareo, R., Golosio, B., Luciano, P., Ruggero, A., 2002. Cork quality estimation by using Compton tomography. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. Atoms 196, 161–168. Brunetti, A., Cesareo, R., Rao, D.V., Golosio, B., 2004. Geometrical factor influence on Compton profile measurement for biological samples. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 526, 584–592. Brunetti, A., Golosio, B., 2001. Software for X-ray fluorescence and scattering to mographic reconstruction. Comput. Phys. Commun. 141, 412–425. Carlsson, C.A., 1999. Imaging modalities in x-ray computerized tomography and in selected volume tomography. Phys. Med. Biol. 44, R23. Cesareo, R., Hanson, A.L., Gigante, G.E., Pedraza, L.J., Mathaboally, S.Q.G., 1992. Interaction of keV photons with matter and new applications. Phys. Rep. 213, 117–178. Chang, L.-T., 1978. A method for attenuation correction in radionuclide computed tomography. IEEE Trans. Nucl. Sci. 25, 638–643. Golosio, B., Brunetti, A., Cesareo, R., 2004. Algorithmic techniques for quantitative Compton tomography. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. Atoms 213, 108–111. Golosio, B., Schoonjans, T., Brunetti, A., Oliva, P., Masala, G.L., 2014. Monte Carlo simulation of X-ray imaging and spectroscopy experiments using quadric geometry and variance reduction techniques. Comput. Phys. Commun. 185, 1044–1052. Golosio, B., Simionovici, A., Somogyi, A., Lemelle, L., Chukalina, M., Brunetti, A., 2003. Internal elemental microanalysis combining x-ray fluorescence, Compton and transmission tomography. J. Appl. Phys. 94, 145–156. Hammerstein, G.R., Miller, D.W., White, D.R., Masterson, M.E., Woodard, H.Q., Laughlin, J.S., 1979. Absorbed radiation dose in mammography. Radiology 130, 485–491. Harding, G., 1997. Inelastic photon scattering: effects and applications in biomedical science and industry. Radiat. Phys. Chem. 50, 91–111. Hendee, W.R., Ritenour, E.R., 2002. Computed Tomography, Medical Imaging Physics, 4th ed. Wiley-Liss, New York, pp. 251–263. Hogan, J.P., Gonsalves, R.A., Krieger, A.S., 1991. Fluorescent computer tomography: a model for correction of X-ray absorption. IEEE Trans. Nucl. Sci. 38, 1721–1727. Ryan, E.A., Farquharson, M.J., Flinton, D.M., 2005. The use of Compton scattering to differentiate between classifications of normal and diseased breast tissue. Phys. Med. Biol. 50, 3337–3348. Shepp, L.A., Logan Jr., B.F., 1974. The Fourier reconstruction of a head section. Trans. Nucl. Sci. 21, 21–43. VanUytven, E., Pistorius, S., Gordon, R., 2008. A method for 3D electron density imaging using single scattered x-rays with application to mammographic screening. Phys. Med. Biol. 53, 5445–5459. Yaffe, M.J., Boone, J.M., Packard, N., Alonzo-Proulx, O., Huang, S.-Y., Peressotti, C.L., Al-Mayah, A., Brock, K., 2009. The myth of the 50–50 breast. Med. Phys. 36, 5437–5443.
Contents lists available at ScienceDirect
Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem
Tomographic images of breast tissues obtained by Compton scattering: An analytical computational study M. Antoniassi a,b,n, M.E. Poletti a, A. Brunetti c a
Departamento de Física, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Ribeirão Preto, São Paulo, Brazily Departamento Acadêmico de Física, Universidade Tecnológica Federal do Paraná, Curitiba, Brazil c Dipartimento di Scienze Politiche, Scienze della Comunicazione e Ingegneria dell' Informazione, Università di Sassari, Italy b
H I G H L I G H T S
An analytical computational simulation of Compton scattering tomography was studied. Results of simulations were compared with those obtained by Monte Carlo simulation. A reconstruction algorithm was implemented for reconstructing the scattering images. Electron density images of breast tissues (simulated breast phantom) were obtained. Influence of parameters (sample size, beam energy and glandularity) was studied.
art ic l e i nf o
a b s t r a c t
Article history: Received 1 October 2014 Received in revised form 27 January 2015 Accepted 30 January 2015 Available online 31 January 2015
In this work, we studied by analytical simulation the potential of a Compton scatter technique for breast imaging application. A Compton scattering tomography system was computationally simulated in order to provide the projection data (scattering signal) for the image reconstructions. The simulated projections generated by the analytical proposed method were validated through comparison with those obtained by Monte Carlo simulation. Electron density images were obtained from the scattering signal using a reconstruction algorithm implemented for the system geometry. Finally, the quality of the reconstructed images was evaluated for different sample sizes, beam energies, and tissue compositions (glandularities). & 2015 Elsevier Ltd. All rights reserved.
Keywords: Compton scattering Electron density Breast cancer Mammography Tomography
1. Introduction Compton scattering imaging is a technique that allows obtaining the spatial distribution of electron density (ρe) of the material examined (Golosio et al., 2004). It can be used as an alternative to the conventional transmission techniques (Carlsson, 1999) when the latter is unable to provide a discrimination in terms of linear attenuation coefficient (m) between the different constituents of the sample. Different methods have been proposed using Compton scattering imaging technique (Cesareo et al., 1992; Harding, 1997), but, particularly, few of them were applied for breast imaging (Aviles et al., 2011; VanUytven et al., 2008).
n
Corresponding author. E-mail address: [email protected] (M. Antoniassi).
http://dx.doi.org/10.1016/j.radphyschem.2015.01.038 0969-806X/& 2015 Elsevier Ltd. All rights reserved.
The main problems encountered in the computational study of this kind of imaging technique are related to the time required for the simulations (usually performed by Monte Carlo techniques) and to the reconstruction of the images from the scatter signal (Golosio et al., 2004) that is generally much more difficult than the reconstruction in transmission techniques, mainly due to self-attenuation effects in the sample (Brunetti and Golosio, 2001). In the present work, we studied using a fast analytical simulator the potential of a Compton scatter technique for breast imaging application. A Compton scattering tomography system was computationally simulated in order to provide the projection data (scattering signal) for the image reconstructions. Electron density images were obtained from the scattering signal using a reconstruction algorithm implemented for the system geometry. Finally, the quality of the reconstructed images was evaluated for different sample sizes, beam energies, and tissue compositions (glandularities).
274
M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
2. Materials and method
2.2. Image reconstructions
2.1. Simulation of tomography system
2.2.1. Transmission tomography The transmission tomography is necessary to determine the attenuation properties, i.e., the map of the linear attenuation coefficient m(x,y) of the sample, which is used to correct the selfattenuation effects in the Compton scatter tomography. The transmission projection Pt(s,θ) can be defined as a function of the intensities of incident (I0) and transmitted (It) photons detected by the transmission detector at each angle of rotation (θ) as (Golosio et al., 2003)
The simulation of the tomography system was implemented using Matlab software. Fig. 1 represents schematically the simulated Compton scatter tomography experiment. The x and y axes are fixed in the reference frame of the sample. The s and u axes are parallel to the translation direction and to the beam respectively, and they are related to x and y by a rotation of an angle θ. The sample was rotated at uniform angular steps (1.25°), totalizing 288 angular views. At each angle, the sample was translated (1 mm steps) and the transmitted and scattered signal, at each translational step, were acquired respectively by the transmission and scatter detectors. The simulated X-ray source produced a monoenergetic beam of 1 1 mm2 square cross section. Three different monoenergetic beams at incident energies (E0) of 40, 60 and 80 keV were investigated in this work. The two scatter detectors (4 20 cm2) were simulated such that the total scatter signal at each translational step was the sum of the signal obtained by each one of the detectors. The parallel distance between the beam and the scatter detectors was always 18 cm. For simulation of the samples (phantoms), it was used cylinders of 12, 14 and 16 cm diameter and 9 cm height. Three sample compositions, simulating breast of different glandularities (g) were studied: (i) g ¼25% (ρe ¼ 3.185 1023 e /cm3), (ii) g ¼50% or 50–50 breast (ρe ¼3.267 1023 e /cm3) and (iii) g ¼75% (ρe ¼3.353 1023 e /cm3), where g indicates the percentage of glandular tissue in a mixture of glandular and adipose tissues. Cylindrical insets (see Fig. 1): A (adipose, ρe ¼3.108 1023 e /cm3), B (glandular, ρe ¼3.444 1023 e /cm3), C (carcinoma, ρe ¼3.452 1023 e /cm3) and D (micro-calcification content, MCC, ρe ¼3.562 1023 e /cm3) of 2.5 cm diameter and 1 cm height were placed in the central height region of the sample. The elemental composition and physical density of adipose and glandular breast tissues were taken from Hammerstein et al. (1979) and of carcinoma, from VanUytven et al. (2008). The MCC was simulated as a tissue region composed of a mixture of 95% glandular tissue and 5% calcium hydroxyapatite.
⎛ I0 ⎞ Pt (s , θ) = ln ⎜ ⎟ ⎝ It (s , u) ⎠
∫ μ (s, u) du = ∫ μ (x, y) δ (s − x cos θ − y sin θ) dx dy , =
(1)
which corresponds to the Radon transform of the map m(x,y) (Golosio et al., 2003). There are different algorithms which can be used to obtain m(x, y) from the acquired projections (Hendee and Ritenour, 2002). In this work, the transmission images were reconstructed using the filtered back-projection algorithm (Shepp and Logan, 1974). 2.2.2. Compton scatter tomography The contribution of a small path du along the beam to the detected scatter signal is given by (Brunetti and Golosio, 2001)
dIs = I0 f (s , u, θ) p (s , u) g (s , u, θ),
(2)
where: (i) f(s,u,θ) is the probability that a photon coming from the source (us) reaches the point u. It depends on the linear attenuation coefficient m0 of the sample at the beam energy (E0) and can be written as
⎛ f (s , u, θ) = exp ⎜ − ⎝
∫u
u s
⎞ μ 0 (s , u‵) du‵⎟, ⎠
(3)
(ii) p(s,u) is the probability of an incident photon being inelastically (Compton) scattered along du. For high momentum transfer values (i.e., for high incident energy and/or high scattering angle), the effects of electron binding are small, and the Compton scattering probability p(s,u) can be written as a function of the electron density (ρe) and the Klein–Nishina cross-section (sKN):
p (s , u) = ρe (s , u) σ KN du,
(4)
(iii) g(s,u,θ) is the probability that a photon Compton scattered from (s,u) reaches the detector (Brunetti et al., 2002):
g (s , u , θ ) =
Fig. 1. Compton scatter tomography arrangement.
1 4π
∫Ω
dΩ det
⎛ 1 ⎛ dσ KN ⎞ ⎜ ⎟ exp ⎜ − σ KN ⎝ dΩ ⎠ ⎝
det
⎞
∫(s, u) μ s (l) dl⎟⎠,
(5)
where Ωdet is the solid angle subtended by the scatter detectors from the interaction point and ms is the linear attenuation coefficient at the Compton scattered energy (Ec). The integral along dl, in the exponential function, is calculated from (s,u) to the detector surface element enclosed by the solid angle dΩ. Defining the attenuation correction K at the point (s,u) as K (s, u, θ) = f (s, u, θ) g (s, u, θ), the total scattered radiation reaching the detector (projection), normalized by the factor I0sKN, is obtained summing the contributions from each element du of the sample along the incidence line:
M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
∫ ρe (s, u) K (s, u, θ) du = ∫ ρe (x, y) K (x, y , θ) δ (s − x cos θ − y sin θ) dx dy .
2.3. Validation and study of the images
Ps (s , θ) =
(6)
Eq. (6) is similar to (1), except by the presence, inside the integral, of the attenuation correction K depending on the rotation angle, θ. In fact, it is often referred to as the generalized Radon transform of ρe(x,y) (Golosio et al., 2003). Then, analogously to the transmission tomography, it is possible to use the filtered backprojection algorithm to reconstruct the image from the scatter projections. The result of the reconstruction gives, instead of the original map of electron density, ρe(x,y), an approximated image, ~ ~ I (x, y) = ρe (x, y) K (x, y), distorted by the effects of the attenuation inside the sample. The tilde above I represents the operation of ~ filtered back-projection and K (x, y) is an attenuation related function, independent of angle θ, obtained as a result of the filtered back-projection process. ~ This function K (x, y), although not known a priori, can be approximated by the average over rotational angle of the attenuation correction K(x,y,θ) (Brunetti and Golosio, 2001; Hogan et al., 1991):
1 ~ K (x , y ) ≅ 2π
∫0
2π
K (x , y , θ ) dθ .
(7)
In this way, the map of electron density can be approximately obtained from I˜ applying the attenuation correction K˜ as
~ I (x , y ) ρ~e (x, y) = ~ . K (x , y )
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The analytical simulation model proposed in this work was validated comparing the scattering projections, Ps(s, θ ¼0°), of a homogenous sample (50–50 breast) obtained by it and those obtained by Monte Carlo simulation using the XRMC code (Golosio et al., 2014). The study of the Compton scatter tomographic images was done by assessing different parameters of the simulation: (i) sample size, (ii) incident energy and (iii) sample glandularity (composition).
3. Results and discussion 3.1. Validation of the simulation method Fig. 2 compares the scattering projections for θ ¼0° and E0 ¼60 keV of a homogenous sample (50–50 breast) obtained by the simulation model proposed in this work and those obtained by Monte Carlo simulation, where the relative count represents the ratio between the number of detected and incident photons. As can be observed, the results obtained by the present method are in good agreement with those provided by Monte Carlo simulation with mean differences about 0.4%. 3.2. Reconstructed images
(8)
This approximation works well when the self-attenuation is low, i.e., for small samples and/or small values of m (Brunetti et al., 2004; Hogan et al., 1991). When the self-attenuation effects are considerable, a secondary correction is needed to compensate them. The second correction implemented in this work is an adaptation for the present Compton scatter technique of the iterative correction method applied for radionuclide computed tomography (Chang, 1978). It is an iterative procedure that consists in reprojecting the primary corrected image ρ~e (x, y), using Eq. (6), to form a new set of projections. After this, a set of error projections is obtained by subtracting each new projection from its corresponding original projection. An error image is produced by applying the filtered back-projection technique in this set of error projections and after correcting the resulting image by the same attenuation correction (Eq. (8)) used in the primary correction. This error image is after added to the primary corrected image to form a final electron density image. Since this secondary correction process is an iterative procedure, the reconstructed image can be reprojected again and the process repeated. In this work the final images were obtained after two iterations. In the present study, both the analytical simulation of the projections and the reconstruction algorithm were implemented using the software Matlab. In the analytical simulations, the maps m0(x,y) and ms(x,y) were calculated from the elemental composition data of the tissues using the XCOM program (Berger et al., 1998). For the reconstruction procedures, ms and m0 cannot be known a priori. In this way, the map m0(x,y) was obtained from the simulated transmission tomography (at the incident energy) and ms(x, y), for each Compton scattering energy, was estimated from m0(x,y) using the ratio of the linear attenuation coefficients of a reference material at the Compton and incident energies: ms(Ec)¼m0(E0)[m ref(Ec)/mref(E0)]. In this work the 50–50 breast was chosen as the reference material.
The reconstructed electron density images obtained for 80 keV and (a) 12 cm, (b) 14 cm, and (c) 16 cm sample diameters are shown in Fig. 3. The contrasts in the insert regions (adipose, glandular, carcinoma and MCC) with respect to the background (50–50 breast) are about 4.9%, 5.4%, 5.7% and 9.0% respectively and the inaccuracy in the electron density values in all this insert regions is less than 0.8% for all investigated sample diameters. Fig. 4 presents the intensity profiles along the central (a) x and (b) y axes of the 40, 60 and 80 keV images obtained for 14 cm sample diameter. It can be seen from Fig. 4 that the reconstructed pixel values are in good agreement with the original (theoretical) ones (electron density values). The inaccuracy of the electron density values calculated in the region of the inserts for 40, 60 and 80 keV are less than 1.5%, 1.1% and 0.7% respectively. The better accuracy for higher energies is due to lower attenuation effects in these conditions. As the energy increases, the linear attenuation coefficients of the tissues
Fig. 2. Scattering projections for θ ¼0° of a homogenous sample (50–50 breast) obtained by this work and by Monte Carlo simulation.
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Fig. 3. Electron density images obtained for 80 keV incident energy and (a) 12 cm, (b) 14 cm, and (c) 16 cm sample diameters.
Fig. 4. Intensity profiles of the 40, 60 and 80 keV electron density images: (a) x-axis profile and (b) y-axis profile.
M. Antoniassi et al. / Radiation Physics and Chemistry 116 (2015) 273–277
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References
Fig. 5. Intensity profiles along the central y-axis in the region of the carcinoma tissue obtained for 80 keV, 14 cm sample diameter and three different sample glandularities: 25%, 50% and 75%.
decrease and therefore the inherent errors due to approximations in the reconstruction model (e.g. estimation of ms from m0), or in the attenuation correction procedures are smaller. Fig. 5 compares the intensity profiles along the central y-axis in the region of the carcinoma tissue obtained for 80 keV, 14 cm sample diameter and three different sample glandularities: 25%, 50% and 75%. Comparing the contrast values (8.4%, 5.7% and 3%) of the carcinoma for different glandularities (25%, 50% and 75%), it is observed that if the glandularity increases, the image contrast decreases. This reduction of contrast, which in practical terms implies the diminution of the detectability of the carcinoma in the breast, is due to similarities between the electron densities of glandular and carcinoma breast tissues (Antoniassi et al., 2010; Ryan et al., 2005). It is important to mention that 95% of women have breast glandularity less than 45% (Yaffe et al., 2009), condition in which the results present contrast values higher than 5.7%.
4. Conclusion In this work, we studied by analytical simulation the potential of a Compton scatter technique for breast imaging application. Comparisons using Monte Carlo technique showed that the analytical simulation method is adequate to obtain single scattering projections for the proposed Compton scatter tomographic system. The reconstruction algorithm showed to be able to produce electron density images of breast tissues with good accuracy (errorso1.5%) for all sample size and incident energy investigated in this work. Further studies should be done in order to evaluate the use polyenergetic beams and verify the influence of multiple scattering and statistical noise on the quality of the reconstructed images.
Acknowledgements The authors would like to acknowledge the support of the Brazilian agency Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (Processos: 2012/05587-7 and 2013/05079-4).
Antoniassi, M., Conceição, A.L.C., Poletti, M.E., 2010. Characterization of breast tissues using Compton scattering. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 619, 375–378. Aviles, J.E.A., Pistorius, S., Elbakri, I.A., Gordon, R., Ahmad, B., 2011. A 1st generation scatter CT algorithm for electron density breast imaging which accounts for bound incoherent, coherent and multiple scatter: a Monte Carlo study. J. X-Ray Sci. Technol. 19, 477–499. Berger, M.J., Hubbell, J.H., Seltzer, S.M., Chang, J., Coursey, J.S., Sukumar, R., Zucker, D.S., Olsen, K., 1998. XCOM: photon cross sections database. In: NIoSaT, (NIST) (Ed.), XCOM: Photon Cross Sections Database. National Institute of Standards and Technology (NIST). Brunetti, A., Cesareo, R., Golosio, B., Luciano, P., Ruggero, A., 2002. Cork quality estimation by using Compton tomography. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. Atoms 196, 161–168. Brunetti, A., Cesareo, R., Rao, D.V., Golosio, B., 2004. Geometrical factor influence on Compton profile measurement for biological samples. Nucl. Instrum. Methods Phys. Res. Sect. A: Accel. Spectrom. Detect. Assoc. Equip. 526, 584–592. Brunetti, A., Golosio, B., 2001. Software for X-ray fluorescence and scattering to mographic reconstruction. Comput. Phys. Commun. 141, 412–425. Carlsson, C.A., 1999. Imaging modalities in x-ray computerized tomography and in selected volume tomography. Phys. Med. Biol. 44, R23. Cesareo, R., Hanson, A.L., Gigante, G.E., Pedraza, L.J., Mathaboally, S.Q.G., 1992. Interaction of keV photons with matter and new applications. Phys. Rep. 213, 117–178. Chang, L.-T., 1978. A method for attenuation correction in radionuclide computed tomography. IEEE Trans. Nucl. Sci. 25, 638–643. Golosio, B., Brunetti, A., Cesareo, R., 2004. Algorithmic techniques for quantitative Compton tomography. Nucl. Instrum. Methods Phys. Res. Sect. B: Beam Interact. Mater. Atoms 213, 108–111. Golosio, B., Schoonjans, T., Brunetti, A., Oliva, P., Masala, G.L., 2014. Monte Carlo simulation of X-ray imaging and spectroscopy experiments using quadric geometry and variance reduction techniques. Comput. Phys. Commun. 185, 1044–1052. Golosio, B., Simionovici, A., Somogyi, A., Lemelle, L., Chukalina, M., Brunetti, A., 2003. Internal elemental microanalysis combining x-ray fluorescence, Compton and transmission tomography. J. Appl. Phys. 94, 145–156. Hammerstein, G.R., Miller, D.W., White, D.R., Masterson, M.E., Woodard, H.Q., Laughlin, J.S., 1979. Absorbed radiation dose in mammography. Radiology 130, 485–491. Harding, G., 1997. Inelastic photon scattering: effects and applications in biomedical science and industry. Radiat. Phys. Chem. 50, 91–111. Hendee, W.R., Ritenour, E.R., 2002. Computed Tomography, Medical Imaging Physics, 4th ed. Wiley-Liss, New York, pp. 251–263. Hogan, J.P., Gonsalves, R.A., Krieger, A.S., 1991. Fluorescent computer tomography: a model for correction of X-ray absorption. IEEE Trans. Nucl. Sci. 38, 1721–1727. Ryan, E.A., Farquharson, M.J., Flinton, D.M., 2005. The use of Compton scattering to differentiate between classifications of normal and diseased breast tissue. Phys. Med. Biol. 50, 3337–3348. Shepp, L.A., Logan Jr., B.F., 1974. The Fourier reconstruction of a head section. Trans. Nucl. Sci. 21, 21–43. VanUytven, E., Pistorius, S., Gordon, R., 2008. A method for 3D electron density imaging using single scattered x-rays with application to mammographic screening. Phys. Med. Biol. 53, 5445–5459. Yaffe, M.J., Boone, J.M., Packard, N., Alonzo-Proulx, O., Huang, S.-Y., Peressotti, C.L., Al-Mayah, A., Brock, K., 2009. The myth of the 50–50 breast. Med. Phys. 36, 5437–5443.