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Geometrical factor influence on Compton profile measurement for biological samples
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
Geometrical factor influence on Compton profile measurement for biological samples A. Brunettia,b,*, R. Cesareoa,b, D.V. Raoa, B. Golosioa,b a
Dipartimento di Matematica e Fisica, Universita" di Sassari, Via Vienna 2, 07100 Sassari, Italy b Sezione I.N.F.N., Cagliari, Italy
Received 1 December 2003; received in revised form 9 February 2004; accepted 19 February 2004
Abstract The Compton profile is a correction of the Klein–Nishina cross-section for the motion of the electrons. This correction modifies the shape of the Compton peak in the spectrum of the scattered photons and it depends on the atomic wave functions of the electrons inside the sample. Thus, the Compton profile can be used as a probe for the electronic structure of atoms or molecules. However, the shape of the Compton peak is also influenced from the geometrical factors or apertures of collimators used in the experimental setup. Since the energy of the Compton scattered photons depends on the scattering angle, in principle, the best choice is to collimate the detector as much as possible, but, as a drawback, this means also a drastic reduction of the photon flux at the detector. This paper deals with a study of the influence of the geometrical factor on the discrimination of different biological elements. The results can be extended to other materials. Some results from reference biological samples are reported and discussed. r 2004 Elsevier B.V. All rights reserved. PACS: 07.05.t; 29.30.Kv; 78.70.Ck Keywords: Scattering; Material discrimination; X-ray
1. Introduction X- and g-rays, in the energy range of interest for medical and biological applications, interact with matter according to three fundamental ways: photoelectric, coherent (or Rayleigh) and incoherent (or Compton) scattering. Usually, the measured signal corresponds to transmitted radiation, i.e. the fraction of initial radiation not interacting *Corresponding author. Tel.: +39-79-229481; fax: +39-79229482. E-mail address: [email protected] (A. Brunetti).
with the sample. This is the case, for example, of mammographic, radiographic and tomographic measurements. In all these cases, the photons interacting with the sample and reaching the detector contribute to background. Among these interactions the Compton scattering is usually the most disturbing one. However, the scattered radiation also contains useful information and these positive properties have been studied in the last 30 years [1–16]. In spite of this, only a few applications have been reported [12–16] using the information due to the ‘‘Compton profile’’.
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.02.029
ARTICLE IN PRESS A. Brunetti et al. / Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592 6 Compton Profile of Al 5
J(pz)
4 3 2 1
-20
-15
-10
-5
(a)
0 pz
5
10
15
20
3.50E-036 Al, E = 10 keV, = 180˚
3.00E-036 2.50E-036 DDCS (barns)
Compton scattering is produced by the interaction of a photon with a electron usually supposed to be initially at rest and free. The probability to have a Compton interaction is described by the Klein–Nishina formula [17]. However, in real sample the electrons are moving and they are bounded to the atom. These effects must be considered in the cross-section and they constitute an important source of information about the sample composition [18,19]. The electron movement produces an energy spread of the Compton scattered photons, while the binding effect produces edges in the energy spectrum. Thus, even in an experimental condition of very small collimation aperture, the shape of the Compton peak will be different from a delta. An example of a Compton profile and Compton peak shape are reported in Fig. 1. The shape in Fig. 1a is obtained by using the Impulse Approximation (IA), the simplest of the models available. The exact profile exhibits an asymmetry called Compton defect [20,21]. In all our calculations we used the IA framework that can be considered valid for large momentum transfer. This approximation is normally used in the papers and Monte-Carlo codes reported in literature [22–24]. The shape of the Compton peak close to the maximum is essentially due to the interaction of the photon with the more external electrons and this part of the peak will be influenced by the chemical bound of the elements inside the sample. On the contrary, the two tiles (high and low energies) of the peak will be due to the interactions with the most internal electrons. In the tiles could be observed also the effect of the binding. Thus, the two tiles contain the information about the Z number of the scatterer. The incoming photon can also be affected by more than one interaction. The percentage of photons having more than one interaction with respect to those having just one is varying with the energy of the radiation and the nature and thickness of the sample, but usually it is ranging from few percents up to several 10th percent [25,26]. The influence depends also on the part of the Compton peak we are analyzing, because the tile at low energies (left tile) with respect to the center of the peak will be influenced more than the
585
2.00E-036 1.50E-036 1.00E-036 5.00E-037 0.00E+000 9000
(b)
9500
10000
Energy (eV)
Fig. 1. Example of Compton profile: aluminium. (a) Simulated Compton profile obtained by using the approach described in Ref. [30]; (b) Double differential cross-section by using a monochromatic line at 10 keV and a scattering angle of 180 . It is observable the edge due to the electron binding energy from most internal shell.
tile at energies higher than the center energy [14]. However, if the sample is small sized such as found in synchrotron measurements, the multiple scattering can be neglected. In our paper we intend to explore the Compton profile from the point of view of the optimization of the experimental parameters, with particular attention to the optimization of geometrical effects for the discrimination of different biological samples. The geometrical parameters have a great importance in the measurement because they are tightly bounded to the counting rate and therefore to the measurement time and the dose absorbed by the sample. This subject has been studied using different approaches by Paatero et al. [27] and
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Hanson and Gigante [28]. The first paper discusses the possibility to perform a deconvolution of the spectrum based on a priori information and the second paper is just a geometrical study of the energy spreading due to the source and detector aperture. Our approach is different: we start from the double differential cross-section that depends on the scattering angle and on the energy of the scattered photon. Then the aperture of the detector is included by integration of the double differential cross-section. We consider both the case of monochromatic and that of polychromatic incident X-ray excitation. Our approach provides a good approximation of the true shape of the detected spectrum. Moreover, the procedure is faster from a computational point of view than the approach reported in Ref. [28]. Several simulations are reported by using reference biological samples under solid angle apertures and X-ray excitation.
Compton profile, following the approach due to Ribberfors and Berggren [30]. The cross-section described in Eq. (1) does not consider the binding energy of the electrons. The binding effects modify Eq. (1) in the following manner d2 s r20 E 0 E E0 2 1 þ cos y þ ¼ ð1 cos yÞ dO dE 0 2_K E mc2 ! X 0 ni Ji ðpz ÞYðE E Ub Þ ð2Þ i
where ni is the number of electrons in the ith shell, Jðpz Þ has been expressed as a sum of contributions from each shell, Ub is the corresponding binding energy and Y is the Heaviside function (step function). In order to obtain the differential crosssection (DCS) the double differential cross-section can be integrated over the solid angle formed by the detector and the interaction point Z
2. Methods The determination of the influence of the geometrical factors on the Compton peak requires the introduction of the double differential Compton cross-section s; which can be expressed as d2 s r20 E E E0 2 ¼ 1 þ cos y þ dO dE 0 2_K E 0 mc2 ð1 cos yÞ Jðpz Þ ð1Þ where O is the solid angle, E the incident photon energy, E 0 the scattered photon energy, r0 the electron radius, K the modulus of the X-ray scattering vector, y the scattering angle, m the mass of electron, c the light speed, pz the projection of the initial momentum p of the electron on the scattering vector, Jðpz Þ is the Compton profile value at p ¼ pz : The Compton profile Jðpz Þ has been tabulated by Biggs et al. [29] according to the Hartree–Fock model. However, it is possible to simulate with a good approximation the Compton profile values by using just the Compton profile value at pz ¼ 0 and a linear interpolation of the integrated
Odetec
d2 s dO dO dE 0
ð3Þ
where Odetec is the solid angle from the interaction point to the detector. The integral can be also expressed as Z
ymax
2p ymin
d2 s sin y dy: dO dE 0
ð4Þ
The angle y is influenced by the finite aperture of the detector collimator. All these equations have been coded by using the C code described in [31] and they have been modified in this paper in order to introduce the possibility to use a polychromatic X-ray beam. This code is analytical and it allows to simulate every kind of Compton interaction as well as to separate the contribution from each atomic shell. It is also possible to select the energy step both in the X-ray spectra and multi-channel analyzer (MCA). In this way the energy resolution of the detector may be considered. Only first-order interactions may be simulated, because higher-order analytical simulations are difficult and have high computational cost.
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1.00E-037
3. Results
30˚
9.00E-038 8.00E-038
The equations above described have been applied to several biological reference samples:
7.00E-038
H 2O (70%) K 2HPO460˚ (30%) 60˚
20˚
10˚
4.00E-038
5˚
2.00E-038
1˚
1.00E-038 0.00E+000 50000
55000
(a)
60000
Energy (eV)
4.50E-038 4.00E-038 3.50E-038
30˚
H 2O (70%) K 2HPO460˚ (30%) 90˚
20˚ 10˚
3.00E-038
DCS (barn)
For the simulated spectra shown in Figs. 2–10 the monochromatic line shown at 59.54 keV from Am241 has been used, while for Figs. 11 and 12 a source of 20 keV was used. However, a lower energy such as the 20 keV (near to the molybdenum K-line) allows to take an advantage from the information about the absorption edges. All the reported data need to be convoluted for the detector response. Figs. 2 report the results of simulations with K2HPO4. In Fig. 2a, the detector is supposed to be at 60 . Fig. 2a shows the effect of the different aperture of the detector on the shape of the peak. While up to 5 the peak appears to be a symmetric Gaussian bell, for more than 5 aperture an asymmetry on the peak appears. This asymmetry is due to the variation of the cross-section. This effect is evident also in the Fig. 2b where a scattering angle of 90 has been simulated. In this case the asymmetry of the peak appears at angles larger than 10 and it is minor than in the previous case considering the center of the peak. In Fig. 2c, the scattering angle is 120 and the peaks show a behavior similar to those in Fig. 2a but with a reflection around the center of the peak. In the set of Figs. 3–9, a fixed 60 scattering angle has been simulated with several samples. In Fig. 3a, a A150TEP (tissue equivalent) material has been simulated. The shapes of the peaks in Fig. 3a, are qualitatively similar to those in Fig. 2a. In Fig. 4, a sample of adipose tissue (ICRP) has been simulated. Even in this case the shape of the peaks are similar to the previous one, but small structures can be noticed on the right part in the
5.00E-038
3.00E-038
2.50E-038
5˚
2.00E-038 1.50E-038 1.00E-038
1˚
5.00E-039 0.00E+000 40000
45000
(b)
50000
55000
60000
Energy (eV) 7.00E-038
30˚
6.50E-038 6.00E-038 5.50E-038 5.00E-038
H 2O (70%) K 2HPO460˚ (30%) 120˚
20˚
4.50E-038
DCS (barn)
(1) K2HPO4 diluted in water as bone simulator, according to [12]; (2) A150TEP (tissue equivalent); (3) ICRP reference of adipose tissue; (4) ICRU reference of bone compact; (5) ICRP reference of bone cortical; (6) ICRP reference of skeletal muscle; (7) ICRU reference of striated muscle; (8) Blood.
DCS (barn)
6.00E-038
10˚
4.00E-038 3.50E-038 3.00E-038
5˚
2.50E-038 2.00E-038 1.50E-038 1.00E-038
1˚
5.00E-039 0.00E+000 40000
(c)
45000
50000
55000
60000
Energy (eV)
Fig. 2. Simulation of cross-section inner bone (see text for composition) at different scattering angles and different detector apertures. Monochromatic excitation spectrum (59.54 keV). The detector apertures are the same in each figure and are 1 , 5 , 10 , 20 and 30 , respectively. Scattering angle: (a) 60 ; (b) 90 and (c) 120 .
bottom of the peaks obtained by using 20 and 30 detector apertures. This features are due to the presence or lack of contribution from partial
ARTICLE IN PRESS A. Brunetti et al. / Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
588 1.10E-037 1.00E-037 9.00E-038
1.80E-037
30˚
A150TEP (tissue equivalent) 60˚
1.60E-037
Cortical Bone (ICRP) 60˚
30˚
1.40E-037
8.00E-038
20˚
6.00E-038
10˚
5.00E-038
5˚
4.00E-038
20˚
1.20E-037
DCS (barn)
DCS (barn)
7.00E-038
1.00E-037
10˚
8.00E-038
5˚ 6.00E-038
3.00E-038
4.00E-038
2.00E-038
1˚
1˚
2.00E-038
1.00E-038 0.00E+000
0.00E+000
50000
50000
55000
60000
Energy (eV)
Fig. 3. Simulation of cross-section of A150TEP (tissue equivalent) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
9.00E-038
1.30E-037 1.20E-037
30˚
Adipose tissue (ICRP) 60˚
1.10E-037
20˚
8.00E-038
DCS (barn)
6.00E-038
10˚
5.00E-038
5˚
4.00E-038
7.00E-038
10˚
6.00E-038
5˚
5.00E-038 4.00E-038
3.00E-038
3.00E-038 2.00E-038
2.00E-038
1˚
1˚
1.00E-038
1.00E-038
0.00E+000
0.00E+000 50000
55000
50000
60000
55000
60000
Energy (eV)
Energy (eV)
Fig. 4. Simulation of cross-section of adipose tissue equivalent (ICRP) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
Fig. 7. Simulation of cross-section of skeletal muscle (ICRP) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
1.60E-037
1.30E-037
30˚
Compact Bone(ICRU) 60˚
1.20E-037 1.10E-037
30˚
Striated Muscle (ICRU) 60˚
1.00E-037
1.20E-037
20˚
20˚
9.00E-038
1.00E-037
8.00E-038
DCS (barn)
DCS (barn)
30˚
9.00E-038
20˚
7.00E-038
DCS (barn)
Skeletal Muscle (ICRP) 60˚
1.00E-037
8.00E-038
1.40E-037
60000
Fig. 6. Simulation of cross-section of cortical bone equivalent (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
1.10E-037 1.00E-037
55000
Energy (eV)
10˚
8.00E-038
5˚
6.00E-038
7.00E-038
10˚
6.00E-038
5˚
5.00E-038 4.00E-038
4.00E-038
3.00E-038 2.00E-038
1˚
2.00E-038
1˚
1.00E-038 0.00E+000
0.00E+000 50000
55000
60000
Energy (eV)
Fig. 5. Simulation of cross-section of compact bone equivalent (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
50000
55000
60000
Energy (eV)
Fig. 8. Simulation of cross-section of striated muscle (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
ARTICLE IN PRESS A. Brunetti et al. / Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
589
1.40E-039
6.00E-038
30˚
Blood 60˚
adipose tissue striated muscle skeletal muscle 0.1˚ aperture 60˚ scattering angle
1.20E-039
5.00E-038 1.00E-039
8.00E-040
DCS (barn)
DCS (barns)
20˚ 4.00E-038
10˚ 3.00E-038
5˚
6.00E-040
2.00E-038
4.00E-040
1.00E-038
1˚
2.00E-040
0.00E+000
0.00E+000 50000
55000
53000
60000
Energy (eV)
54000
55000
56000
57000
Fig. 9. Simulation of cross-section of blood at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
59000
adipose tissue striated muscle skeletal muscle 0.1˚ aperture 60˚ scattering angle
1E-39
DCS (barn)
1E-40
1E-41
53000
54000
55000
(b)
56000
57000
58000
59000
Energy (eV)
1E-39
DCS (barn)
shells, or equivalently, to the use of energies larger than the binding energies (see Eq. (2)). The samples in Figs. 5–9 represent Compact bone (ICRU), Cortical bone (ICRP) and blood, respectively. The shape are similar to the previous cases and so the considerations above are still valid. In Figs. 10, a comparison of different materials is shown. A collimator aperture of 0.1 is used. This size of the aperture will cancel the influence of the cross-section variation with the solid angle. In Figs. 10a (linear scale) and b (semi-logarithmic scale) two types of muscle are compared with adipose tissue. In this case, the profile of the adipose tissue is very similar to that of striated muscle while the profile of the skeletal muscle appears to be larger. In Fig. 10c the striated muscle is compared with cortical and compact bones. In this case the differences in shape of the profile of the striated muscle appear to be greater than in the previous case. In order to study the effect of a polychromatic excitation on the capability of discrimination of the Compton profile, a flat excitation (same photon counts rate at each energy channel) from 58 to 62 keV has been used (Fig. 11). The small differences previously noticed in the tails of the profiles in Fig. 10 are practically absent in Fig. 11. Thus, even an energy spread of the incident X-ray beam of few percents hides the difference in the
58000
Energy (eV)
(a)
striated muscle compact bone cortical bone 0.1˚ aperture 60˚ scattering angle
1E-40
1E-41
52000
(c)
53000
54000
55000
56000
57000
58000
59000
Energy (eV)
Fig. 10. Comparison of cross-section of different elements: adipose tissue, striated muscle, skeletal muscle (a and b, linear and semi-logarithmic scale, respectively) and striated muscle, compact bone and cortical bone (c, semi-logarithmic scale). Scattering angle of 60 , Detector aperture of 0.1 . Monochromatic excitation of 59.54 keV.
ARTICLE IN PRESS A. Brunetti et al. / Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
590
4.00E-039
Excitation 58-62 keV 60˚ scattering angle 0.1 aperture Striated muscle Compact bone
1E-37
adipose tissue skeletal muscle striated muscle
3.50E-039 3.00E-039
DCS (barn)
3.00E-039
DCS (barn)
1E-38
2.50E-039 2.00E-039
2.00E-039 1.50E-039
1.50E-039 1.00E-039
1E-39
2.50E-039
1.00E-039
19400
19450
19500
19550
5.00E-040 0.00E+000 50000
52000
54000
56000
58000
60000
18600
62000
Energy (eV)
18800
19000
19200
(a)
Fig. 11. Influence of polychromaticity on the shape of Compton peak for compact bone and striated muscle. Scattering angle of 60 , Detector aperture of 0.1 . Flat excitation spectrum (energies from 58 to 62 keV).
19400
19600
19800
20000
Energy (eV)
Compact Bone Cortical Bone Striated Muscle
5.00E-039
4.00E-039 3.00E-039
DCS (b arn)
Compton profile, at least at the energy used in these examples. The behavior could change with the energy. Thus it is mandatory to check the behavior of the profiles at various energies. Fig. 12a shows the same example of Fig. 10a by simulating a 20 keV monochromatic line. Comparing the two figures, 12a and 10a, it is possible to see differences in the form of small structures on the peaks in Fig. 12a. These differences, as previously seen in other cases, are due to the contribution of different shells to the profile. This result evidences the importance of choosing the optimal energy for these structures. In fact the structures probably represent the main difference that can be found in the shape of the Compton profile peaks. The structures observed in Fig. 12a are due to the shell contribution of those element with binding energy around 500 KeV. Viewing the composition of the material used in the reference samples this structure will principally come from the 1S shell of oxygen. In Fig. 12b the same materials of Fig. 10c are reported. Comparing the two figures it is again possible to observe the presence of small structures in Fig. 12b. The discrimination of the material in Fig. 10a was possible on the basis of the different shape of the three peaks by using a logarithmic scale. Thus, the presence of such structures will give to the experimentalist another tool for identification of these materials.
3.00E-039
2.50E-039 2.00E-039
2.00E-039 1.50E-039
1.00E-039
1.00E-039 19400
19450
19500
0.00E+000 18000 18200 18400 18600 18800 19000 19200 19400 19600 19800 20000
(b)
Ener gy (e V)
Fig. 12. Comparison of cross-section of different elements: adipose tissue, striated muscle, skeletal muscle (a) and striated muscle, compact bone and cortical bone (b). Scattering angle of 60 , Detector aperture of 0.1 . Monochromatic excitation at 20 keV.
In Fig. 13 two different excitation energies has been used. All the graphs in Fig. 13 are in semilogarithmic scale. The sample reported in Figs. 13 are tissue, bone, muscle and blood representing the principal components of human body. A 59 and 20 keV excitation energy have been used in Fig. 13a and b, respectively. In Fig. 13a, the shapes of blood and skeletal muscle are very similar and so their discrimination is difficult. On the contrary the other elements show different shapes and so it is relatively simple to identify each of them. In Fig. 13b a different situation has been found. At 59 keV the blood and the skeletal Compton peaks are not almost completely superimposed as in Fig. 13a, but their shape appears to be quite different. Thus this energy should be better than
ARTICLE IN PRESS A. Brunetti et al. / Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
Adipose Tissue Compact Bone Skeletal Muscle Blood 59.54 keV Excitation 60˚ scattering angle 1˚ aperture angle
DCS (barns)
1E-38
1E-39
1E-40
53000
54000
1E-39
DCS (barns)
55000
56000
57000
58000
59000
Energy (eV)
(a)
Adipose tissue Blood Compact bone Skeletal muscle 20 keV Excitation 60˚ scattering angle 0.1˚ aperture angle
1E-40
1E-41
18600
18800
(b)
19000
19200
19400
19600
19800
20000
Energy (eV)
Fig. 13. Comparison of cross-section of different elements at different energies. The material are: adipose tissue, compact bone, skeletal muscle and blood. Scattering angle of 60 , Detector aperture of 0.1 . (a) 59.54 keV, logarithm scale; (b) 20 keV, logarithm scale.
20 keV for discriminating this elements, even without using of the characteristic structures due to the binding energies.
4. Conclusions In this paper, a study of the influence of the geometrical factor and excitation energy on the Compton cross-section due to the Compton profile has been simulated on reference biological samples. The influence of the X-ray excitation on the
591
Compton shape has been studied at two different energies, 20 and 60 keV, simulating the Mo-Kaline and the Am-g line. Also a polychromatic beam has been simulated, and the polychromaticity has been simulated with an energy spectrum in the range of a few percent around the two above cited energies. The effect of the polychromaticity of the X-ray excitation is a partial disruption of the difference introduced by the Compton profile, and although in principle the discrimination of different elements should be yet possible, it is preferable to use a monochromatic source. The 60 keV line, although easily available as radioactive source (241Am), does not allow to discriminate the elements coming from the different absorption edges of each shell of an element. The use of a lower energy, i.e. 20 keV, should be a better choice at least for the samples analyzed here. The effect of the detector aperture has been studied for different angles ranging from 0.1 to 30 . Up to 5 the shape of the Compton peak appears to be symmetric around its center, while this symmetry is lost for larger aperture angles; in this case the shape will depend on the choice of the scattering angle. About the possibility to discriminate different biological samples, based on our simulations, it cannot be possible for adipose tissue and striated muscle, and probably striated muscle and compact bone. Also the two types of muscles are indistinguishable. Regarding the blood, its shape appears to be different from adipose tissue and skeletal muscle, but it is very similar to the skeletal muscle except at the central energies of the peak where the Compton peak of blood appears to be sharper than the skeletal muscle Compton peak. Future works are foreseen in the direction of introducing the second-order interaction in analytical form, in order to improve the quality of the simulation. This correction will require the study of fast integration routines due to the large number of integrals involved.
References [1] J.J. Battista, L.W. Santon, M.J. Bronskill, Phys. Med. Biol. 22 (1977) 220.
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[2] J.J. Battista, M.J. Bronskill, Phys. Med. Biol. 26 (1981) 81. [3] A.L. Huddleston, D. Bhaduri, Phys. Med. Biol. 24 (1979) 310. [4] M.D. Herr, J.J. McInnerney, D.G. Lamser, G.L. Copenhaver, IEEE Trans. Med. Imaging 13 (1994) 461. [5] F.A. Balogun, N.M. Spyrou, Nucl. Instr. and Meth. B 83 (1993) 533. [6] I.L.M. Silva, R.T. Lopez, E.F.O. de Jesus, Nucl. Instr. and Meth. A 422 (1999) 957. [7] N.J.B. McFarlane, C.R. Bukk, R.D. Tillet, R.D. Speller, G.J. Royle, K.R.A. Johnson, J. Agric. Eng. Res. 75 (2000) 265. [8] R. Guzzardi, G. Licitra, CRC Crit. Rev. Bio. Eng. 15 (1988) 237. [9] G. Harding, Radiat. Phys. Chem. 50 (1997) 91. [10] A. Brunetti, R. Cesareo, B. Golosio, P. Luciano, A. Ruggero, Nucl. Instr. and Meth. B 196 (2002) 161. [11] W.M. Elshemey, A.A. Elsayed, A. Eli-Lakkani, Radiat. Meas. 30 (1999) 715. [12] A. Tartari, E. Casnati, J.E. Fernandez, J. Felsteiner, C. Baraldi, Phys. Med. Biol. 39 (1994) 219. [13] E. Gatti, P. Rehak, J. Kemmer, IEEE Trans. Med. Imaging MI-5 (1986) 207. [14] A. Tartari, E. Casnati, J. Felsteiner, C. Baraldi, B. Singh, Nucl. Instr. and Meth. B 71 (1992) 209. [15] A. Tartari, C. Baraldi, J. Felsteiner, E. Casnati, Phys. Med. Biol. 36 (1991) 567.
[16] A. Tartari, J.E. Fernandez, E. Casnati, C. Baraldi, X-ray Spectrom. 24 (1995) 167. [17] O. Klein, Y. Nishina, Z. Phys. 52 (1929) 853. [18] M.J. Cooper, Rep. Prog. Phys. 48 (1985) 415. [19] M.J. Cooper, Radiat. Phys. Chem. 50 (1997) 63. [20] F. Bell, J. Chem. Phys. 85 (1986) 303. [21] J. Datta, P.K. Bera, B. Talukdar, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) 3261. [22] D. Brusa, G. Stutz, J.A. Riveirs, J.M. Fernandez-Varea, F. Salvat, Nucl. Instr. and Meth. A 379 (1996) 167. [23] Y. Namito, S. Ban, H. Hirayama, Nucl. Instr. and Meth. A (1994) 349. [24] I. Kawrakow, D.W.O. Rogers, http://www.irs.inms.nrs.ca/ inms/irs/EGSnrc/EGSnrc.html. [25] B.G. Williams, P. Pattison, M.J. Cooper, Philos. Mag. 27 (1974) 537. [26] J. Festeiner, P. Pattison, M. Cooper, Philos. Mag. 30 (1974) 537. [27] P. Paatero, S. Manninen, T. Paakkari, Philos. Mag. 30 (1974) 1281. [28] A.L. Hanson, G.E. Gigante, Phys. Rev. A 40 (1989) 171. [29] F. Biggs, L.B. Mendelsohn, J.B. Mann, At. Data. Nucl. Data. Tables 16 (1975) 201. [30] R. Ribberfors, K.F. Berggren, Phys. Rev. A 26 (1982) 3325. [31] A. Brunetti, M. Sanchez del Rio, B. Golosio, D.V. Rao, Commun. Phys. Chem. 2004, submitted.
Nuclear Instruments and Methods in Physics Research A 526 (2004) 584–592
Geometrical factor influence on Compton profile measurement for biological samples A. Brunettia,b,*, R. Cesareoa,b, D.V. Raoa, B. Golosioa,b a
Dipartimento di Matematica e Fisica, Universita" di Sassari, Via Vienna 2, 07100 Sassari, Italy b Sezione I.N.F.N., Cagliari, Italy
Received 1 December 2003; received in revised form 9 February 2004; accepted 19 February 2004
Abstract The Compton profile is a correction of the Klein–Nishina cross-section for the motion of the electrons. This correction modifies the shape of the Compton peak in the spectrum of the scattered photons and it depends on the atomic wave functions of the electrons inside the sample. Thus, the Compton profile can be used as a probe for the electronic structure of atoms or molecules. However, the shape of the Compton peak is also influenced from the geometrical factors or apertures of collimators used in the experimental setup. Since the energy of the Compton scattered photons depends on the scattering angle, in principle, the best choice is to collimate the detector as much as possible, but, as a drawback, this means also a drastic reduction of the photon flux at the detector. This paper deals with a study of the influence of the geometrical factor on the discrimination of different biological elements. The results can be extended to other materials. Some results from reference biological samples are reported and discussed. r 2004 Elsevier B.V. All rights reserved. PACS: 07.05.t; 29.30.Kv; 78.70.Ck Keywords: Scattering; Material discrimination; X-ray
1. Introduction X- and g-rays, in the energy range of interest for medical and biological applications, interact with matter according to three fundamental ways: photoelectric, coherent (or Rayleigh) and incoherent (or Compton) scattering. Usually, the measured signal corresponds to transmitted radiation, i.e. the fraction of initial radiation not interacting *Corresponding author. Tel.: +39-79-229481; fax: +39-79229482. E-mail address: [email protected] (A. Brunetti).
with the sample. This is the case, for example, of mammographic, radiographic and tomographic measurements. In all these cases, the photons interacting with the sample and reaching the detector contribute to background. Among these interactions the Compton scattering is usually the most disturbing one. However, the scattered radiation also contains useful information and these positive properties have been studied in the last 30 years [1–16]. In spite of this, only a few applications have been reported [12–16] using the information due to the ‘‘Compton profile’’.
0168-9002/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2004.02.029
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J(pz)
4 3 2 1
-20
-15
-10
-5
(a)
0 pz
5
10
15
20
3.50E-036 Al, E = 10 keV, = 180˚
3.00E-036 2.50E-036 DDCS (barns)
Compton scattering is produced by the interaction of a photon with a electron usually supposed to be initially at rest and free. The probability to have a Compton interaction is described by the Klein–Nishina formula [17]. However, in real sample the electrons are moving and they are bounded to the atom. These effects must be considered in the cross-section and they constitute an important source of information about the sample composition [18,19]. The electron movement produces an energy spread of the Compton scattered photons, while the binding effect produces edges in the energy spectrum. Thus, even in an experimental condition of very small collimation aperture, the shape of the Compton peak will be different from a delta. An example of a Compton profile and Compton peak shape are reported in Fig. 1. The shape in Fig. 1a is obtained by using the Impulse Approximation (IA), the simplest of the models available. The exact profile exhibits an asymmetry called Compton defect [20,21]. In all our calculations we used the IA framework that can be considered valid for large momentum transfer. This approximation is normally used in the papers and Monte-Carlo codes reported in literature [22–24]. The shape of the Compton peak close to the maximum is essentially due to the interaction of the photon with the more external electrons and this part of the peak will be influenced by the chemical bound of the elements inside the sample. On the contrary, the two tiles (high and low energies) of the peak will be due to the interactions with the most internal electrons. In the tiles could be observed also the effect of the binding. Thus, the two tiles contain the information about the Z number of the scatterer. The incoming photon can also be affected by more than one interaction. The percentage of photons having more than one interaction with respect to those having just one is varying with the energy of the radiation and the nature and thickness of the sample, but usually it is ranging from few percents up to several 10th percent [25,26]. The influence depends also on the part of the Compton peak we are analyzing, because the tile at low energies (left tile) with respect to the center of the peak will be influenced more than the
585
2.00E-036 1.50E-036 1.00E-036 5.00E-037 0.00E+000 9000
(b)
9500
10000
Energy (eV)
Fig. 1. Example of Compton profile: aluminium. (a) Simulated Compton profile obtained by using the approach described in Ref. [30]; (b) Double differential cross-section by using a monochromatic line at 10 keV and a scattering angle of 180 . It is observable the edge due to the electron binding energy from most internal shell.
tile at energies higher than the center energy [14]. However, if the sample is small sized such as found in synchrotron measurements, the multiple scattering can be neglected. In our paper we intend to explore the Compton profile from the point of view of the optimization of the experimental parameters, with particular attention to the optimization of geometrical effects for the discrimination of different biological samples. The geometrical parameters have a great importance in the measurement because they are tightly bounded to the counting rate and therefore to the measurement time and the dose absorbed by the sample. This subject has been studied using different approaches by Paatero et al. [27] and
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Hanson and Gigante [28]. The first paper discusses the possibility to perform a deconvolution of the spectrum based on a priori information and the second paper is just a geometrical study of the energy spreading due to the source and detector aperture. Our approach is different: we start from the double differential cross-section that depends on the scattering angle and on the energy of the scattered photon. Then the aperture of the detector is included by integration of the double differential cross-section. We consider both the case of monochromatic and that of polychromatic incident X-ray excitation. Our approach provides a good approximation of the true shape of the detected spectrum. Moreover, the procedure is faster from a computational point of view than the approach reported in Ref. [28]. Several simulations are reported by using reference biological samples under solid angle apertures and X-ray excitation.
Compton profile, following the approach due to Ribberfors and Berggren [30]. The cross-section described in Eq. (1) does not consider the binding energy of the electrons. The binding effects modify Eq. (1) in the following manner d2 s r20 E 0 E E0 2 1 þ cos y þ ¼ ð1 cos yÞ dO dE 0 2_K E mc2 ! X 0 ni Ji ðpz ÞYðE E Ub Þ ð2Þ i
where ni is the number of electrons in the ith shell, Jðpz Þ has been expressed as a sum of contributions from each shell, Ub is the corresponding binding energy and Y is the Heaviside function (step function). In order to obtain the differential crosssection (DCS) the double differential cross-section can be integrated over the solid angle formed by the detector and the interaction point Z
2. Methods The determination of the influence of the geometrical factors on the Compton peak requires the introduction of the double differential Compton cross-section s; which can be expressed as d2 s r20 E E E0 2 ¼ 1 þ cos y þ dO dE 0 2_K E 0 mc2 ð1 cos yÞ Jðpz Þ ð1Þ where O is the solid angle, E the incident photon energy, E 0 the scattered photon energy, r0 the electron radius, K the modulus of the X-ray scattering vector, y the scattering angle, m the mass of electron, c the light speed, pz the projection of the initial momentum p of the electron on the scattering vector, Jðpz Þ is the Compton profile value at p ¼ pz : The Compton profile Jðpz Þ has been tabulated by Biggs et al. [29] according to the Hartree–Fock model. However, it is possible to simulate with a good approximation the Compton profile values by using just the Compton profile value at pz ¼ 0 and a linear interpolation of the integrated
Odetec
d2 s dO dO dE 0
ð3Þ
where Odetec is the solid angle from the interaction point to the detector. The integral can be also expressed as Z
ymax
2p ymin
d2 s sin y dy: dO dE 0
ð4Þ
The angle y is influenced by the finite aperture of the detector collimator. All these equations have been coded by using the C code described in [31] and they have been modified in this paper in order to introduce the possibility to use a polychromatic X-ray beam. This code is analytical and it allows to simulate every kind of Compton interaction as well as to separate the contribution from each atomic shell. It is also possible to select the energy step both in the X-ray spectra and multi-channel analyzer (MCA). In this way the energy resolution of the detector may be considered. Only first-order interactions may be simulated, because higher-order analytical simulations are difficult and have high computational cost.
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1.00E-037
3. Results
30˚
9.00E-038 8.00E-038
The equations above described have been applied to several biological reference samples:
7.00E-038
H 2O (70%) K 2HPO460˚ (30%) 60˚
20˚
10˚
4.00E-038
5˚
2.00E-038
1˚
1.00E-038 0.00E+000 50000
55000
(a)
60000
Energy (eV)
4.50E-038 4.00E-038 3.50E-038
30˚
H 2O (70%) K 2HPO460˚ (30%) 90˚
20˚ 10˚
3.00E-038
DCS (barn)
For the simulated spectra shown in Figs. 2–10 the monochromatic line shown at 59.54 keV from Am241 has been used, while for Figs. 11 and 12 a source of 20 keV was used. However, a lower energy such as the 20 keV (near to the molybdenum K-line) allows to take an advantage from the information about the absorption edges. All the reported data need to be convoluted for the detector response. Figs. 2 report the results of simulations with K2HPO4. In Fig. 2a, the detector is supposed to be at 60 . Fig. 2a shows the effect of the different aperture of the detector on the shape of the peak. While up to 5 the peak appears to be a symmetric Gaussian bell, for more than 5 aperture an asymmetry on the peak appears. This asymmetry is due to the variation of the cross-section. This effect is evident also in the Fig. 2b where a scattering angle of 90 has been simulated. In this case the asymmetry of the peak appears at angles larger than 10 and it is minor than in the previous case considering the center of the peak. In Fig. 2c, the scattering angle is 120 and the peaks show a behavior similar to those in Fig. 2a but with a reflection around the center of the peak. In the set of Figs. 3–9, a fixed 60 scattering angle has been simulated with several samples. In Fig. 3a, a A150TEP (tissue equivalent) material has been simulated. The shapes of the peaks in Fig. 3a, are qualitatively similar to those in Fig. 2a. In Fig. 4, a sample of adipose tissue (ICRP) has been simulated. Even in this case the shape of the peaks are similar to the previous one, but small structures can be noticed on the right part in the
5.00E-038
3.00E-038
2.50E-038
5˚
2.00E-038 1.50E-038 1.00E-038
1˚
5.00E-039 0.00E+000 40000
45000
(b)
50000
55000
60000
Energy (eV) 7.00E-038
30˚
6.50E-038 6.00E-038 5.50E-038 5.00E-038
H 2O (70%) K 2HPO460˚ (30%) 120˚
20˚
4.50E-038
DCS (barn)
(1) K2HPO4 diluted in water as bone simulator, according to [12]; (2) A150TEP (tissue equivalent); (3) ICRP reference of adipose tissue; (4) ICRU reference of bone compact; (5) ICRP reference of bone cortical; (6) ICRP reference of skeletal muscle; (7) ICRU reference of striated muscle; (8) Blood.
DCS (barn)
6.00E-038
10˚
4.00E-038 3.50E-038 3.00E-038
5˚
2.50E-038 2.00E-038 1.50E-038 1.00E-038
1˚
5.00E-039 0.00E+000 40000
(c)
45000
50000
55000
60000
Energy (eV)
Fig. 2. Simulation of cross-section inner bone (see text for composition) at different scattering angles and different detector apertures. Monochromatic excitation spectrum (59.54 keV). The detector apertures are the same in each figure and are 1 , 5 , 10 , 20 and 30 , respectively. Scattering angle: (a) 60 ; (b) 90 and (c) 120 .
bottom of the peaks obtained by using 20 and 30 detector apertures. This features are due to the presence or lack of contribution from partial
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588 1.10E-037 1.00E-037 9.00E-038
1.80E-037
30˚
A150TEP (tissue equivalent) 60˚
1.60E-037
Cortical Bone (ICRP) 60˚
30˚
1.40E-037
8.00E-038
20˚
6.00E-038
10˚
5.00E-038
5˚
4.00E-038
20˚
1.20E-037
DCS (barn)
DCS (barn)
7.00E-038
1.00E-037
10˚
8.00E-038
5˚ 6.00E-038
3.00E-038
4.00E-038
2.00E-038
1˚
1˚
2.00E-038
1.00E-038 0.00E+000
0.00E+000
50000
50000
55000
60000
Energy (eV)
Fig. 3. Simulation of cross-section of A150TEP (tissue equivalent) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
9.00E-038
1.30E-037 1.20E-037
30˚
Adipose tissue (ICRP) 60˚
1.10E-037
20˚
8.00E-038
DCS (barn)
6.00E-038
10˚
5.00E-038
5˚
4.00E-038
7.00E-038
10˚
6.00E-038
5˚
5.00E-038 4.00E-038
3.00E-038
3.00E-038 2.00E-038
2.00E-038
1˚
1˚
1.00E-038
1.00E-038
0.00E+000
0.00E+000 50000
55000
50000
60000
55000
60000
Energy (eV)
Energy (eV)
Fig. 4. Simulation of cross-section of adipose tissue equivalent (ICRP) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
Fig. 7. Simulation of cross-section of skeletal muscle (ICRP) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
1.60E-037
1.30E-037
30˚
Compact Bone(ICRU) 60˚
1.20E-037 1.10E-037
30˚
Striated Muscle (ICRU) 60˚
1.00E-037
1.20E-037
20˚
20˚
9.00E-038
1.00E-037
8.00E-038
DCS (barn)
DCS (barn)
30˚
9.00E-038
20˚
7.00E-038
DCS (barn)
Skeletal Muscle (ICRP) 60˚
1.00E-037
8.00E-038
1.40E-037
60000
Fig. 6. Simulation of cross-section of cortical bone equivalent (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
1.10E-037 1.00E-037
55000
Energy (eV)
10˚
8.00E-038
5˚
6.00E-038
7.00E-038
10˚
6.00E-038
5˚
5.00E-038 4.00E-038
4.00E-038
3.00E-038 2.00E-038
1˚
2.00E-038
1˚
1.00E-038 0.00E+000
0.00E+000 50000
55000
60000
Energy (eV)
Fig. 5. Simulation of cross-section of compact bone equivalent (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
50000
55000
60000
Energy (eV)
Fig. 8. Simulation of cross-section of striated muscle (ICRU) at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
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589
1.40E-039
6.00E-038
30˚
Blood 60˚
adipose tissue striated muscle skeletal muscle 0.1˚ aperture 60˚ scattering angle
1.20E-039
5.00E-038 1.00E-039
8.00E-040
DCS (barn)
DCS (barns)
20˚ 4.00E-038
10˚ 3.00E-038
5˚
6.00E-040
2.00E-038
4.00E-040
1.00E-038
1˚
2.00E-040
0.00E+000
0.00E+000 50000
55000
53000
60000
Energy (eV)
54000
55000
56000
57000
Fig. 9. Simulation of cross-section of blood at different detector apertures. Scattering angle of 60 . Monochromatic excitation spectrum (59.54 keV).
59000
adipose tissue striated muscle skeletal muscle 0.1˚ aperture 60˚ scattering angle
1E-39
DCS (barn)
1E-40
1E-41
53000
54000
55000
(b)
56000
57000
58000
59000
Energy (eV)
1E-39
DCS (barn)
shells, or equivalently, to the use of energies larger than the binding energies (see Eq. (2)). The samples in Figs. 5–9 represent Compact bone (ICRU), Cortical bone (ICRP) and blood, respectively. The shape are similar to the previous cases and so the considerations above are still valid. In Figs. 10, a comparison of different materials is shown. A collimator aperture of 0.1 is used. This size of the aperture will cancel the influence of the cross-section variation with the solid angle. In Figs. 10a (linear scale) and b (semi-logarithmic scale) two types of muscle are compared with adipose tissue. In this case, the profile of the adipose tissue is very similar to that of striated muscle while the profile of the skeletal muscle appears to be larger. In Fig. 10c the striated muscle is compared with cortical and compact bones. In this case the differences in shape of the profile of the striated muscle appear to be greater than in the previous case. In order to study the effect of a polychromatic excitation on the capability of discrimination of the Compton profile, a flat excitation (same photon counts rate at each energy channel) from 58 to 62 keV has been used (Fig. 11). The small differences previously noticed in the tails of the profiles in Fig. 10 are practically absent in Fig. 11. Thus, even an energy spread of the incident X-ray beam of few percents hides the difference in the
58000
Energy (eV)
(a)
striated muscle compact bone cortical bone 0.1˚ aperture 60˚ scattering angle
1E-40
1E-41
52000
(c)
53000
54000
55000
56000
57000
58000
59000
Energy (eV)
Fig. 10. Comparison of cross-section of different elements: adipose tissue, striated muscle, skeletal muscle (a and b, linear and semi-logarithmic scale, respectively) and striated muscle, compact bone and cortical bone (c, semi-logarithmic scale). Scattering angle of 60 , Detector aperture of 0.1 . Monochromatic excitation of 59.54 keV.
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590
4.00E-039
Excitation 58-62 keV 60˚ scattering angle 0.1 aperture Striated muscle Compact bone
1E-37
adipose tissue skeletal muscle striated muscle
3.50E-039 3.00E-039
DCS (barn)
3.00E-039
DCS (barn)
1E-38
2.50E-039 2.00E-039
2.00E-039 1.50E-039
1.50E-039 1.00E-039
1E-39
2.50E-039
1.00E-039
19400
19450
19500
19550
5.00E-040 0.00E+000 50000
52000
54000
56000
58000
60000
18600
62000
Energy (eV)
18800
19000
19200
(a)
Fig. 11. Influence of polychromaticity on the shape of Compton peak for compact bone and striated muscle. Scattering angle of 60 , Detector aperture of 0.1 . Flat excitation spectrum (energies from 58 to 62 keV).
19400
19600
19800
20000
Energy (eV)
Compact Bone Cortical Bone Striated Muscle
5.00E-039
4.00E-039 3.00E-039
DCS (b arn)
Compton profile, at least at the energy used in these examples. The behavior could change with the energy. Thus it is mandatory to check the behavior of the profiles at various energies. Fig. 12a shows the same example of Fig. 10a by simulating a 20 keV monochromatic line. Comparing the two figures, 12a and 10a, it is possible to see differences in the form of small structures on the peaks in Fig. 12a. These differences, as previously seen in other cases, are due to the contribution of different shells to the profile. This result evidences the importance of choosing the optimal energy for these structures. In fact the structures probably represent the main difference that can be found in the shape of the Compton profile peaks. The structures observed in Fig. 12a are due to the shell contribution of those element with binding energy around 500 KeV. Viewing the composition of the material used in the reference samples this structure will principally come from the 1S shell of oxygen. In Fig. 12b the same materials of Fig. 10c are reported. Comparing the two figures it is again possible to observe the presence of small structures in Fig. 12b. The discrimination of the material in Fig. 10a was possible on the basis of the different shape of the three peaks by using a logarithmic scale. Thus, the presence of such structures will give to the experimentalist another tool for identification of these materials.
3.00E-039
2.50E-039 2.00E-039
2.00E-039 1.50E-039
1.00E-039
1.00E-039 19400
19450
19500
0.00E+000 18000 18200 18400 18600 18800 19000 19200 19400 19600 19800 20000
(b)
Ener gy (e V)
Fig. 12. Comparison of cross-section of different elements: adipose tissue, striated muscle, skeletal muscle (a) and striated muscle, compact bone and cortical bone (b). Scattering angle of 60 , Detector aperture of 0.1 . Monochromatic excitation at 20 keV.
In Fig. 13 two different excitation energies has been used. All the graphs in Fig. 13 are in semilogarithmic scale. The sample reported in Figs. 13 are tissue, bone, muscle and blood representing the principal components of human body. A 59 and 20 keV excitation energy have been used in Fig. 13a and b, respectively. In Fig. 13a, the shapes of blood and skeletal muscle are very similar and so their discrimination is difficult. On the contrary the other elements show different shapes and so it is relatively simple to identify each of them. In Fig. 13b a different situation has been found. At 59 keV the blood and the skeletal Compton peaks are not almost completely superimposed as in Fig. 13a, but their shape appears to be quite different. Thus this energy should be better than
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Adipose Tissue Compact Bone Skeletal Muscle Blood 59.54 keV Excitation 60˚ scattering angle 1˚ aperture angle
DCS (barns)
1E-38
1E-39
1E-40
53000
54000
1E-39
DCS (barns)
55000
56000
57000
58000
59000
Energy (eV)
(a)
Adipose tissue Blood Compact bone Skeletal muscle 20 keV Excitation 60˚ scattering angle 0.1˚ aperture angle
1E-40
1E-41
18600
18800
(b)
19000
19200
19400
19600
19800
20000
Energy (eV)
Fig. 13. Comparison of cross-section of different elements at different energies. The material are: adipose tissue, compact bone, skeletal muscle and blood. Scattering angle of 60 , Detector aperture of 0.1 . (a) 59.54 keV, logarithm scale; (b) 20 keV, logarithm scale.
20 keV for discriminating this elements, even without using of the characteristic structures due to the binding energies.
4. Conclusions In this paper, a study of the influence of the geometrical factor and excitation energy on the Compton cross-section due to the Compton profile has been simulated on reference biological samples. The influence of the X-ray excitation on the
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Compton shape has been studied at two different energies, 20 and 60 keV, simulating the Mo-Kaline and the Am-g line. Also a polychromatic beam has been simulated, and the polychromaticity has been simulated with an energy spectrum in the range of a few percent around the two above cited energies. The effect of the polychromaticity of the X-ray excitation is a partial disruption of the difference introduced by the Compton profile, and although in principle the discrimination of different elements should be yet possible, it is preferable to use a monochromatic source. The 60 keV line, although easily available as radioactive source (241Am), does not allow to discriminate the elements coming from the different absorption edges of each shell of an element. The use of a lower energy, i.e. 20 keV, should be a better choice at least for the samples analyzed here. The effect of the detector aperture has been studied for different angles ranging from 0.1 to 30 . Up to 5 the shape of the Compton peak appears to be symmetric around its center, while this symmetry is lost for larger aperture angles; in this case the shape will depend on the choice of the scattering angle. About the possibility to discriminate different biological samples, based on our simulations, it cannot be possible for adipose tissue and striated muscle, and probably striated muscle and compact bone. Also the two types of muscles are indistinguishable. Regarding the blood, its shape appears to be different from adipose tissue and skeletal muscle, but it is very similar to the skeletal muscle except at the central energies of the peak where the Compton peak of blood appears to be sharper than the skeletal muscle Compton peak. Future works are foreseen in the direction of introducing the second-order interaction in analytical form, in order to improve the quality of the simulation. This correction will require the study of fast integration routines due to the large number of integrals involved.
References [1] J.J. Battista, L.W. Santon, M.J. Bronskill, Phys. Med. Biol. 22 (1977) 220.
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[2] J.J. Battista, M.J. Bronskill, Phys. Med. Biol. 26 (1981) 81. [3] A.L. Huddleston, D. Bhaduri, Phys. Med. Biol. 24 (1979) 310. [4] M.D. Herr, J.J. McInnerney, D.G. Lamser, G.L. Copenhaver, IEEE Trans. Med. Imaging 13 (1994) 461. [5] F.A. Balogun, N.M. Spyrou, Nucl. Instr. and Meth. B 83 (1993) 533. [6] I.L.M. Silva, R.T. Lopez, E.F.O. de Jesus, Nucl. Instr. and Meth. A 422 (1999) 957. [7] N.J.B. McFarlane, C.R. Bukk, R.D. Tillet, R.D. Speller, G.J. Royle, K.R.A. Johnson, J. Agric. Eng. Res. 75 (2000) 265. [8] R. Guzzardi, G. Licitra, CRC Crit. Rev. Bio. Eng. 15 (1988) 237. [9] G. Harding, Radiat. Phys. Chem. 50 (1997) 91. [10] A. Brunetti, R. Cesareo, B. Golosio, P. Luciano, A. Ruggero, Nucl. Instr. and Meth. B 196 (2002) 161. [11] W.M. Elshemey, A.A. Elsayed, A. Eli-Lakkani, Radiat. Meas. 30 (1999) 715. [12] A. Tartari, E. Casnati, J.E. Fernandez, J. Felsteiner, C. Baraldi, Phys. Med. Biol. 39 (1994) 219. [13] E. Gatti, P. Rehak, J. Kemmer, IEEE Trans. Med. Imaging MI-5 (1986) 207. [14] A. Tartari, E. Casnati, J. Felsteiner, C. Baraldi, B. Singh, Nucl. Instr. and Meth. B 71 (1992) 209. [15] A. Tartari, C. Baraldi, J. Felsteiner, E. Casnati, Phys. Med. Biol. 36 (1991) 567.
[16] A. Tartari, J.E. Fernandez, E. Casnati, C. Baraldi, X-ray Spectrom. 24 (1995) 167. [17] O. Klein, Y. Nishina, Z. Phys. 52 (1929) 853. [18] M.J. Cooper, Rep. Prog. Phys. 48 (1985) 415. [19] M.J. Cooper, Radiat. Phys. Chem. 50 (1997) 63. [20] F. Bell, J. Chem. Phys. 85 (1986) 303. [21] J. Datta, P.K. Bera, B. Talukdar, J. Phys. B: At. Mol. Opt. Phys. 25 (1992) 3261. [22] D. Brusa, G. Stutz, J.A. Riveirs, J.M. Fernandez-Varea, F. Salvat, Nucl. Instr. and Meth. A 379 (1996) 167. [23] Y. Namito, S. Ban, H. Hirayama, Nucl. Instr. and Meth. A (1994) 349. [24] I. Kawrakow, D.W.O. Rogers, http://www.irs.inms.nrs.ca/ inms/irs/EGSnrc/EGSnrc.html. [25] B.G. Williams, P. Pattison, M.J. Cooper, Philos. Mag. 27 (1974) 537. [26] J. Festeiner, P. Pattison, M. Cooper, Philos. Mag. 30 (1974) 537. [27] P. Paatero, S. Manninen, T. Paakkari, Philos. Mag. 30 (1974) 1281. [28] A.L. Hanson, G.E. Gigante, Phys. Rev. A 40 (1989) 171. [29] F. Biggs, L.B. Mendelsohn, J.B. Mann, At. Data. Nucl. Data. Tables 16 (1975) 201. [30] R. Ribberfors, K.F. Berggren, Phys. Rev. A 26 (1982) 3325. [31] A. Brunetti, M. Sanchez del Rio, B. Golosio, D.V. Rao, Commun. Phys. Chem. 2004, submitted.