Linear attenuation coefficients of tissues from 1 keV to 150 keV

Radiation Physics and Chemistry 102 (2014) 49–59

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Linear attenuation coefficients of tissues from 1 keV to 150 keV Aysun Böke n Balıkesir University, Faculty of Arts and Sciences, Department of Physics, Çağış Campus, Balıkesir, Turkey

H I G H L I G H T S


The inclusion of molecular interference effects will provide new information.
Results are compared with previously reported experimental and theoretical values.
Calculation is achieved in the presence of small concentrations of different atoms.

art ic l e i nf o

a b s t r a c t

Article history: Received 28 August 2013 Accepted 3 April 2014 Available online 13 April 2014

The linear attenuation coefficients and three interaction processes have been computed for liver, kidney, muscle, fat and for a range of x-ray energies from 1 keV to 150 keV. Molecular photoelectric absorption cross sections were calculated from atomic cross section data. Total coherent (Rayleigh) and incoherent (Compton) scattering cross sections were obtained by numerical integration over combinations of F2m(x) with the Thomson formula and Sm(x) with the Klein–Nishina formula, respectively. For the coherent (Rayleigh) scattering cross section calculations, molecular form factors were obtained from recent experimental data in the literature for values of x o1 Å  1 and from the relativistic modified atomic form factors for values of x Z1 Å  1. With the inclusion of molecular interference effects in the coherent (Rayleigh) scattering, more accurate knowledge of the scatter from these tissues will be provided. The number of elements involved in tissue composition is 5 for liver, 47 for kidney, 44 for muscle and 3 for fat. The results are compared with previously published experimental and theoretical linear attenuation coefficients. In general, good agreement is obtained. The molecular form factors and scattering functions and cross sections are incorporated into a Monte Carlo program. The energy distributions of x-ray photons scattered from tissues have been simulated and the results are presented. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Linear attenuation coefficient Tissue

1. Introduction Absorption and scattering cross sections are useful in diagnostic radiology, computerized tomography and for nondestructive testing of materials (Rao et al., 2002). Low angle x-ray scatter for tissue characterization is based on the differences which result from the interference of photons coherently scattered from molecules of each sample (Castro et al., 2005). Molecular interference effects are referred by some researchers (Bradley et al., 1989; Chan and Doi, 1983; Kosanetzky et al., 1987; Tartari, 1999; Tartari et al., 1997, 1998). The atomic photoelectric cross sections are calculated by Scofield (1973) and Chantler (1995). The incoherent (Compton) scattering is the dominating process for most biologically interesting matter in the diagnostic energy range (Neitzel et al., 1985). Values of incoherent scattering function (ISF) are given by Hubbell et al. (1975).

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Current m/ρ compilations such as the Berger-Hubbell (1987) XCOM PC program, the Hubbell-Seltzer (1995) tabulation, and the Cullen et al. (1997) LLNL data base, still rely on the incoherent scattering function S(x,Z) approach (Hubbell, 1999). The x-ray photon transport in amorphous materials needs knowledge of impact parameter data well beyond those obtained from the Independent Atomic Model (IAM) or ‘mixture rule’, which assumes that each atom in the sample scatters independently from the others. While for incoherent scattering IAM can be used to calculate cross sections for compounds from reference values for pure elements, for coherent scattering it cannot. At larger momentum transfer values, which corresponds to small distances and hence to intra-atomic interference, the molecular form factor approaches IAM form factor and, hence mixture rule can be used (Berger and Hubbell, 1987; Hubbell and Seltzer, 1995; Tartari et al., 1997; Theodorakou and Farquharson, 2008). At low values of momentum transfer, corresponding to molecular and intermolecular interference, molecular form factors must be taken from experimental data. The molecular scattering cross-section

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A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

data in a wide interval of momentum transfer have been obtained both from measurements (King et al., 2011; Peplow and Verghese, 1998) and from those calculated from IAM theoretical frame (Morin, 1982; Narten and Levy, 1971, 1972; Tartari et al., 1997). The atomic form factor has the most important impact on the actual cross section (Cesareo et al., 1992). Tabulated atomic form factors are available for all elements from Z¼1 to 100 (Hubbell and Øverbø, 1979; Hubbell et al., 1975). Detailed tabulation of atomic form factors for Z¼30–36, Z¼ 60–89 and for E¼0.1–10 keV are presented by Chantler (2000). The relativistic Hartree–Fock–Slater modified atomic form factor (MFF) corrected by an empirical factor is tabulated by Schaupp et al. (1983) for Z¼1–100 and 0rxr100 Å  1. The finding suggesting the superiority of MFF theory have been reported (Bö ke, 2011; Bradley and Ghose, 1986; Bradley et al., 1999a, 1999b; Eichler et al., 1983; İçelli and Erzeneoğlu, 2001; Kane, 2005; Kane et al., 1983; Kissel, 2000; Kissel et al., 1980; Nayak and Siddappa, 2004; Roy et al., 1983, 1993, 1999; Schaupp et al., 1983; Siddappa et al., 1989; Zhou and Pratt, 1990). It has been established both from the comparison of theoretical results and from experiments that MFF, in general, produces better results than other choices of form factors (Roy et al., 1993). Peplow and Verghese (1998) extracted molecular form factors from experimental measurements for 16 different materials including animal tissues, plastics, human breast tissue and water. The most recent measurements of x-ray coherent scattering form factors were presented by King et al. (2011), covering a large interval of momentum transfer (0.363–9.25 nm  1) for biological tissues (fat, muscle, liver, kidney and bone), four plastics and water using energy dispersion. This study also presents measured differential linear scattering coefficients and linear attenuation coefficients. Attenuation coefficients for x-ray interactions are commonly employed in Monte Carlo calculations and in many other areas of medical physics, both for diagnostic and therapeutic applications (Boone and Chavez, 1996). Radiographic techniques, such as mammography, radiology and bone densitometry, are based on differences of attenuation coefficient between the biological tissues that compose the body or specific organ. Therefore, the accurate knowledge of the attenuation properties of tissues is fundamental to optimize the image quality, and at same time minimizing the absorbed dose in the patient (Geraldelli et al., 2013). The attenuation coefficients of tissues are reported by researchers (Akar et al., 2006; Bradley et al., 1986; Hubbell, 1982; Johns and Yaffe, 1987; Joyet et al., 1974; King et al., 2011; Kosanetzky et al., 1987; McCullough, 1975; Phelps et al., 1975; Rao and Gregg, 1975; Tartari et al., 1997; White et al., 1980). Important papers in the use of Monte Carlo (MC) for medical problems include Chan and Doi (1983, 1985), Dance (1990) and Johns and Yaffe (1983). The MC calculations are significantly in error when interference effects are ignored in the model for coherent scattering (Leliveld et al., 1996). Some researchers (Chan and Doi, 1983; Elshemey et al., 1999; Tartari et al., 2000, 2001, 2002) have proposed to include the molecular interference effects in a model for MC simulation of low angle photon scattering in biological tissues. In this work, theoretical calculations have been performed to obtain the linear attenuation coefficients which contain the photoelectric absorption, coherent and incoherent scattering cross sections for liver, kidney, muscle, fat and for energies covering from 1 keV to 150 keV. Comparison between theoretical and experimental linear attenuation coefficients is carried out. The sum rule is used to obtain molecular form factors which cover values of xZ1 Å  1 and molecular scattering functions which cover values of 0rxr109Å  1. For values of xo1 Å  1, the molecular form factors are taken from the recent experimental data (King et al., 2011) which include molecular interference effects. Compositions and densities of tissues are provided from ICRP (1975) for kidney, muscle, fat and Kosanetzky et al. (1987)

for liver in the presence of small concentrations of different elements, inclusive of high atomic number elements. The energy distributions of x-ray photons scattered from tissues by using MC simulation program are also presented for 50 keV, 100 keV and 150 keV energies. So far, there has been no theoretical study of linear attenuation coefficients on tissues including the molecular interference effects. Therefore, this work will provide new information in literature.

2. Methods 2.1. Interactions of x-rays with matter In the energy range of 1–150 keV, three interaction processes between photon and matter must be taken into account: the photoelectric absorption, incoherent (Compton) and coherent (Rayleigh) scattering. 2.1.1. Photoelectric absorption Because of the lack of data on the molecular photoelectric cross-section (sph) for a compound or a mixture, it was assumed that sph (cm  1) (appropriate conversion between barn and cm was made for each element) could be calculated according to the formula (Zaidi, 2000)

sph ¼ ρNA ∑ i

Wi s Ai i

ð1Þ

where NA is Avogadro's number, ρ is the density of the material, Ai, wi and si are the atomic mass, the fraction by weight, and the atomic cross section of the ith component of the medium, respectively. The molecular photoelectric absorption cross sections are obtained by using Eq. (1) according to compositions and densities of tissues. The atomic photoelectric cross sections are taken from the theoretical data of Scofield (1973). 2.1.2. Incoherent (Compton) scattering Incoherent (Compton) x-ray scattering of a photon from a free stationary electron is well described by the Klein–Nishina (KN) (1929) differential scattering cross section for unpolarized x-rays. At low incident energies the electron binding energy reduces the probability of incoherent scatter interactions. This fact leads to a modification in the KN formula (1929) at low momentum transfer values by a correction factor, the ISF S(x,Z) which accounts for binding effects of the electron (Harding et al., 1987; Johns and Wismayer, 2004). Taking into account the ISF, the total incoherent scattering cross section per atom for inelastic scattering can be written as Z θ¼π sinc ¼ dsKN ðθÞSðx; ZÞ ð2Þ θ¼0

The ISF Sm(x) for a molecule or a mixture can be approximated by the IAM, since there is no interference between the scattered waves and the effect of the molecular binding on the S(x,Z) is very small. Therefore the sum rule seems to be valid for the calculation of the ISF for a molecule or mixture. The molecular ISF Sm(x) is calculated for all values of x according to compositions and densities of tissues from the atomic ISF Si(x,Zi) using Sm ðxÞ w ¼ ∑ i Si ðx; Z i Þ W i Mi

ð3Þ

where wi is the mass fraction of element i, Mi is the atomic mass of element i and W the molecular weight. Zi is the atomic number of element i. Si(x,Zi) is taken from the tables of Hubbell et al. (1975). 2.1.3. Coherent (Rayleigh) scattering The coherent (Rayleigh) scattering cross section can be found by integrating the differential cross section over all possible scatter

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

51

Table 1 The molecular ISF values (Sm(x)) calculated by using atomic ISF values of Hubbell et al. (1975) according to elemental compositions of ICRP (1975) for kidney, muscle, fat and Kosanetzky et al. (1987) for liver are presented for four tissues. x ¼sin (θ/2)/λ(Å  1)

Sm(x)Liver

Sm(x)Kidney

Sm(x)Muscle

Sm(x)Fat

0 1.00E  02 2.00E  02 3.00E  02 4.00E  02 5.00E  02 6.00E  02 7.00E  02 8.00E  02 9.00E  02 1.00E  01 1.10E  01 1.20E  01 1.30E  01 1.40E  01 1.50E  01 1.60E  01 1.70E 01 1.80E  01 1.90E  01 2.00E  01 2.20E  01 2.40E  01 2.50E  01 2.60E  01 2.80E  01 3.00E  01 3.20E  01 3.40E  01 3.50E  01 3.60E  01 3.80E  01 4.00E  01 4.20E  01 4.40E  01 4.50E  01 4.60E  01 4.80E  01 5.00E  01 5.50E  01 6.00E  01 6.50E  01 7.00E  01 8.00E  01 9.00E  01 1.00E þ 00 1.10E þ00 1.20Eþ 00 1.30Eþ 00 1.40Eþ 00 1.50Eþ 00 1.60Eþ 00 1.70Eþ 00 1.80Eþ 00 1.90Eþ 00 2.00E þ00 2.20E þ 00 2.40E þ 00 2.50E þ 00 2.60E þ 00 2.80E þ 00 3.00E þ00 3.30E þ 00 3.50E þ 00 3.60E þ 00 3.90E þ 00 4.00E þ00 4.20E þ 00 4.60E þ 00 5.00E þ00 5.40E þ 00 5.50E þ 00 5.80E þ 00

0 8.4320E  02 3.3839E  01 7.5430E  01 1.3204 2.0234 2.8994 3.7753 4.8188 5.8623 6.9863 8.1447 9.3032 10.4604 11.6164 12.7724 13.8596 14.9469 15.9860 16.9771 17.9682 19.6313 21.2945 22.1260 22.7596 24.0266 25.2937 26.1145 26.9353 27.3457 27.7561 28.5769 29.3976 29.8545 30.3141 30.5399 30.7683 31.2252 31.6821 32.3787 33.0752 33.5668 34.0583 34.8535 35.5450 36.1750 36.7041 37.2333 37.6956 38.0911 38.4867 38.7184 38.9500 39.1817 39.4134 39.6451 39.8447 40.0442 40.1440 40.1850 40.2671 40.3491 40.3998 40.4336 40.4409 40.4628 40.4701 40.4750 40.4846 40.4942 40.4966 40.4972 40.4989

0 4.4771 17.9755 40.0546 70.1095 107.4081 153.9041 200.4001 255.8221 311.2440 370.9972 432.7067 494.4162 556.1131 617.7974 679.4817 737.6841 795.8866 851.5977 904.8177 958.0376 1047.7265 1137.4153 1182.2598 1216.6308 1285.3727 1354.1147 1398.9942 1443.8737 1466.3135 1488.7532 1533.6327 1578.5122 1603.6592 1628.8062 1641.3798 1653.9533 1679.1003 1704.2473 1742.5511 1780.8549 1807.7266 1834.5984 1877.8446 1915.3610 1949.5422 1978.3429 2007.1435 2032.3769 2054.0431 2075.7092 2088.5063 2101.3035 2114.1006 2126.8977 2139.6948 2145.2652 2161.9765 2167.5469 2169.8627 2174.4945 2179.1262 2182.0266 2183.9602 2184.3857 2185.6623 2186.0879 2186.3769 2186.9551 2187.5332 2187.6826 2187.7200 2187.8320

0 3.8798 15.5838 34.7227 60.7861 93.1309 133.4889 173.8470 221.9916 270.1363 322.0835 375.8048 429.5261 483.2543 536.9895 590.7246 641.5102 692.2958 740.9424 787.4499 833.9575 912.4784 990.9993 1030.2598 1060.4339 1120.7821 1181.1302 1220.6727 1260.2153 1279.9865 1299.7578 1339.3003 1378.8428 1401.0543 1423.2659 1434.3716 1445.4774 1467.6890 1489.9005 1523.6618 1557.4231 1581.0021 1604.5810 1642.4048 1675.1831 1705.0734 1730.3301 1755.5869 1777.7590 1796.8463 1815.9336 1827.2656 1838.5976 1849.9295 1861.2615 1872.5935 1882.5122 1892.4309 1897.3903 1899.4621 1903.6059 1907.7496 1910.3577 1912.0964 1912.4817 1913.6374 1914.0227 1914.2871 1914.8159 1915.3447 1915.4839 1915.5187 1915.6230

0 4.1595E  02 1.6544E  01 3.6935E 01 6.4398E  01 9.8510E  01 1.4014 1.8178 2.3059 2.7940 3.3116 3.8288 4.3459 4.8599 5.3708 5.8817 6.3446 6.8076 7.2430 7.6510 8.0590 8.7163 9.3735 9.7021 9.9356 10.4025 10.8695 11.1411 11.4128 11.5486 11.6844 11.9561 12.2277 12.3682 12.5087 12.5789 12.6492 12.7897 12.9302 13.1683 13.4065 13.6062 13.8060 14.1665 14.4915 14.7812 15.0050 15.2287 15.4119 15.5545 15.6972 15.7649 15.8326 15.9004 15.9681 16.0358 16.0814 16.1271 16.1499 16.1577 16.1732 16.1887 16.1972 16.2029 16.2040 16.2072 16.2083 16.2090 16.2103 16.2117 16.2119 16.2120 16.2122

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A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

Table 1 (continued ) x ¼sin (θ/2)/λ(Å  1)

Sm(x)Liver

Sm(x)Kidney

Sm(x)Muscle

Sm(x)Fat

6.00E þ00 6.20E þ 00 6.60E þ 00 7.00E þ 00 7.40E þ 00 8.00E þ00 9.00E þ00 1.00E þ 01 1.10E þ01 1.20E þ 01 1.40E þ 01 1.60E þ 01 1.80E þ 01 2.00E þ01 2.20E þ 01 2.50E þ 01 2.80E þ 01 3.10E þ 01 3.50E þ 01 4.00E þ01 4.50E þ 01 5.00E þ01 6.00E þ01 7.00E þ 01 8.00E þ01 9.00E þ01 1.00E þ 02 1.00E þ 03 1.00E þ 06 1.00E þ 09

40.5001 40.5005 40.5012 40.5020 40.5023 40.502823 40.502800 40.502777 40.502780 40.502783 40.502789 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792 40.502792

2187.9067 2187.9322 2187.9831 2188.0339 2188.0545 2188.0854 2188.0907 2188.0959 2188.0968 2188.0977 2188.0995 2188.1005 2188.1008 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011 2188.1011

1915.6926 1915.7165 1915.7644 1915.8123 1915.8317 1915.8610 1915.8664 1915.8718 1915.8726 1915.8734 1915.8750 1915.8759 1915.8760 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761 1915.8761

16.212366 16.212423 16.212538 16.212652 16.212743 16.212879 16.212785 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692 16.212692

angles per atom given as Z θ¼π scoh ¼ dsT ðθÞ½Fðx; ZÞ2 θ¼0

ð4Þ

where dsT is the classical or Thomson (1906) cross section for a single free electron, the atomic form factor F(x,Z) is a function of the variable x, which is x ¼ λ  1sin(θ/2). At large values of momentum transfer, there is an asymptotic convergence between the experimental data and those obtained by the use of IAM, and therefore, the sum rule can be applied. For a molecule, a simple sum rule is used to find molecular form factor F 2m ðxÞ w ¼ ∑ i F i 2 ðx; Z i Þ W i Mi

ð5Þ

where wi is the mass fraction of element i, Mi is the atomic mass of element i and W the molecular weight. Zi is the atomic number of element i. The Fm(x), molecular form factor, is calculated by using Eq. (5) according to compositions and densities of tissues for values of x Z1 Å  1 from MFF of Schaupp et al. (1983). For values of x o1 Å  1, the Fm(x) is taken from experimental data of King et al. (2011). 2.2. The linear attenuation coefficient The linear attenuation coefficient m (cm  1) can be decomposed into contributions from each mode of photon interaction as m ¼ mph þmC þmR

ð6Þ

where ph, C and R designate photoelectric absorption, incoherent (Compton) and coherent (Rayleigh) scattering, respectively. 2.3. Monte Carlo modeling Let x be a random variable with a probability density function f(x) and a cumulative distribution function F(x). For realization xn

of random variable x follows by definition Z xn f ðxÞ dx q ¼ Fðxn Þ ¼ 1

ð7Þ

The range of the cumulative distribution function is restricted, 0r F(xn)r1 with xn in [  1,1]. If it is possible to calculate the inverse of the distribution function, F  1, then a random sample xn can easily be obtained by substituting an uniformly distributed random number q (q A [0,1]) into Eq. (7) and calculating xn ¼F  1(q). 2.3.1. Simulation of the interaction process Once it is determined that an interaction occurs in the medium, one of the three possible interaction processes are selected by random sampling. The probability that a given type of interaction occurs, p(i), is proportional to its cross section si pðiÞ ¼

si st

ð8Þ

where st is the total cross section. Then i the number of the interaction which occurs, is a random variable with a cumulative distribution function given by PðiÞ ¼

∑ij ¼ 1 sj

st

i

¼ ∑ pðiÞ j¼1

ð9Þ

The number i is selected by generating an uniformly distributed random number q on [0,1] and finding the i which satisfies Pði  1Þ r q r PðiÞ

ð10Þ

The i-th interaction process is thus chosen as the interaction that occurs. Construction of a probability density function from the differential and total cross sections gives the possibility to apply the MC method.

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

53

Table 2 The calculated coherent (Rayleigh), incoherent (Compton) scattering, photoelectric absorption cross sections and linear attenuation coefficients (m) are presented for liver tissue. E (keV)

Coh (b/mol)

Incoh (b/mol)

Photoelectric (1/cm)

m (1/cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

45.9100 90.7593 98.9185 71.5013 64.6425 54.7956 45.8690 38.2995 35.1545 30.1246 26.4085 24.2080 22.2136 19.0746 17.1790 15.7587 14.7464 13.8924 12.5018 11.4329 10.6014 9.8603 9.2882 8.7852 8.3505 7.8469 7.1944 6.7031 6.3435 5.9706 5.6505 5.3868 5.1508 4.9259 4.7238 4.5351 4.2852 4.0233 3.7894 3.6243 3.4785 3.3257 3.1857 3.0651 2.9579 2.8577 2.7620 2.6708 2.5840 2.5098 2.0389 1.7374 1.5157 1.3421 1.1681 1.0128 0.9036 0.8163 0.7445 0.6835 0.6179 0.5508 0.5053 0.4655 0.4316 0.4032 0.3774 0.3544 0.3334 0.3069

1.6821 5.2384 8.7592 11.6021 13.7200 15.3279 16.5467 17.4821 18.2178 18.8119 19.3030 19.7164 20.0698 20.3726 20.6359 20.8680 21.0721 21.2509 21.4093 21.5485 21.6664 21.7671 21.8545 21.9305 21.9977 22.0570 22.1065 22.1456 22.1759 22.1992 22.2166 22.2285 22.2361 22.2387 22.2360 22.2290 22.2187 22.2048 22.1881 22.1690 22.1474 22.1232 22.0968 22.0685 22.0386 22.0071 21.9741 21.9399 21.9047 21.8681 21.6727 21.4621 21.2430 21.0198 20.7957 20.5721 20.3506 20.1323 19.9178 19.7073 19.5013 19.2997 19.1028 18.9107 18.7232 18.5403 18.3617 18.1875 18.0176 17.8521

3601.0737 538.7539 169.8141 72.5212 37.0728 21.2880 15.0308 8.7736 6.5733 4.3730 3.7407 3.1085 2.4763 1.8441 1.2119 1.0658 0.9197 0.7736 0.6275 0.4814 0.4462 0.4110 0.3758 0.3405 0.3053 0.2701 0.2348 0.1996 0.1644 0.1292 0.1213 0.1134 0.1055 9.7653E  02 8.9773E  02 8.1894E  02 7.4014E  02 6.6134E  02 5.8254E  02 5.0375E  02 4.7755E 02 4.5136E  02 4.2517E  02 3.9898E  02 3.7278E  02 3.4659E  02 3.2040E  02 2.9421E  02 2.6801E  02 2.4182E  02 1.8719E  02 1.3256E  02 1.1224E  02 9.1920E  03 7.1600E  03 5.1280E  03 4.4601E  03 3.7922E  03 3.1242E  03 2.4563E  03 2.2757E  03 2.0952E  03 1.9146E  03 1.7340E 03 1.5534E  03 1.3728E  03 1.1922E  03 1.0117E  03 8.3108E  04 6.5049E  04

3.6014E þ 03 5.3952E þ 02 1.7068E þ 02 73.1876 37.7012 21.8504 15.5313 9.2209 7.0013 4.7654 4.1073 3.4608 2.8154 2.1604 1.5097 1.3595 1.2069 1.0554 0.8995 0.7459 0.7050 0.6646 0.6255 0.5868 0.5487 0.5099 0.4698 0.4310 0.3931 0.3551 0.3448 0.3349 0.3251 0.3155 0.3060 0.2965 0.2865 0.2765 0.2666 0.2572 0.2532 0.2492 0.2453 0.2414 0.2377 0.2340 0.2304 0.2268 0.2232 0.2197 0.2089 0.1993 0.1937 0.1885 0.1833 0.1782 0.1749 0.1718 0.1688 0.1660 0.1636 0.1613 0.1591 0.1571 0.1552 0.1533 0.1515 0.1497 0.1480 0.1463

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A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

Table 3 For kidney (see Table 1 for explanation).

Table 4 For muscle (see Table 1 for explanation).

E (keV)

Coh (b/mol)

Incoh (b/mol)

Photoelectric (1/cm)

m (1/cm)

E (keV)

Coh (b/mol)

Incoh (b/mol)

Photoelectric (1/cm)

m (1/cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

1365.7745 3322.2920 3729.4045 2663.7407 2233.2617 1833.0977 1494.6498 1243.3279 1114.0001 893.6739 784.0148 723.8503 652.5674 610.3444 568.9478 531.1359 486.8480 451.7392 411.1974 390.3119 366.5297 339.2596 326.4503 300.2886 282.0509 261.1843 244.8740 234.8606 226.4972 213.0562 200.4403 193.5017 184.1870 172.6147 164.0538 155.2481 146.2048 139.7278 134.6369 131.5256 127.6947 122.1349 116.2143 111.6428 108.5019 105.5171 100.5497 95.8605 92.1483 88.6854 73.8939 65.0474 57.0722 48.5550 41.8016 38.2303 34.3587 31.0030 27.8262 24.9459 22.2727 20.5640 19.5314 17.8956 16.4788 15.5323 14.1838 13.1111 12.0873 11.2162

89.2948 278.5071 466.8499 619.7922 734.3278 821.6018 887.9333 938.9098 979.0242 1011.4214 1038.2001 1060.7410 1080.0167 1096.5362 1110.9114 1123.5966 1134.7574 1144.5474 1153.2327 1160.8768 1167.3628 1172.9161 1177.7489 1181.9665 1185.6982 1189.0057 1191.7187 1193.7611 1195.2583 1196.4702 1197.4860 1198.2881 1198.8437 1199.1068 1199.0726 1198.7881 1198.3193 1197.6442 1196.8164 1195.8519 1194.7444 1193.4988 1192.1244 1190.6473 1189.0784 1187.4253 1185.6879 1183.8816 1182.0215 1180.0808 1169.6932 1158.4534 1146.7330 1134.7696 1122.7463 1110.7384 1098.8358 1087.0915 1075.5498 1064.2184 1053.1243 1042.2685 1031.6598 1021.3034 1011.1967 1001.3342 991.7047 982.3121 973.1498 964.2196

3991.7256 601.8353 191.9446 83.8258 43.0563 24.7984 17.5332 10.2892 7.7192 5.1587 4.4150 3.6712 2.9278 2.1848 1.4411 1.2689 1.0957 0.9226 0.7495 0.5763 0.5342 0.4921 0.4501 0.4080 0.3660 0.3239 0.2832 0.2411 0.1990 0.1568 0.1473 0.1378 0.1283 0.1187 0.1092 9.9682E  02 9.0154E  02 8.0627E  02 7.1099E  02 6.1571E  02 5.8384E  02 5.5198E  02 5.2011E  02 4.8824E 02 4.5638E  02 4.2451E  02 3.9264E  02 3.6077E  02 3.2891E  02 2.9704E  02 2.3026E  02 1.6348E  02 1.3852E  02 1.1355E  02 8.8592E  03 6.3630E  03 5.5561E  03 4.7350E 03 3.9070E  03 3.0789E 03 2.8534E  03 2.6278E  03 2.4022E  03 2.1767E 03 1.9511E  03 1.7256E  03 1.5000E  03 1.2744E  03 1.0488E  03 8.2326E  04

3991.9592 602.4133 192.6182 84.3529 43.5327 25.2245 17.9156 10.6395 8.0552 5.4645 4.7075 3.9577 3.2059 2.4588 1.7108 1.5345 1.3560 1.1788 1.0006 0.8253 0.7804 0.7349 0.6915 0.6460 0.6016 0.5567 0.5138 0.4704 0.4272 0.3831 0.3717 0.3612 0.3503 0.3389 0.3280 0.3170 0.3060 0.2953 0.2848 0.2746 0.2707 0.2664 0.2620 0.2579 0.2539 0.2500 0.2457 0.2415 0.2374 0.2334 0.2226 0.2127 0.2071 0.2013 0.1958 0.1908 0.1875 0.1842 0.1810 0.1779 0.1755 0.1732 0.1711 0.1690 0.1669 0.1650 0.1630 0.1611 0.1592 0.1574

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

1583.7444 3540.0151 3719.1367 2789.3699 2510.2747 2076.0049 1761.5729 1524.1592 1356.0664 1200.5112 1112.2992 1006.3943 916.0007 784.2508 717.9576 673.9797 619.9518 581.8066 525.0430 478.8343 449.7231 429.5532 404.4778 378.0934 358.6916 335.6121 308.5683 286.5012 270.6245 259.0601 249.6143 238.7159 226.9767 215.3366 206.2909 197.5541 186.3290 175.0411 164.7436 156.5206 150.1918 145.0778 140.8829 136.8366 132.3716 127.6631 122.7666 118.0721 114.1025 110.7352 89.5604 76.8233 68.2061 59.6801 51.7094 44.4287 40.2145 36.9591 33.4096 30.5243 27.4842 24.4608 22.1961 20.7716 19.5712 18.2826 16.9446 15.8835 14.8869 13.6867

77.4597 241.9969 406.2601 540.0058 640.4108 717.0414 775.3443 820.1636 855.4253 883.8914 907.4114 927.2034 944.1255 958.6315 971.2573 982.4020 992.2106 1000.8212 1008.4641 1015.1963 1020.9144 1025.8154 1030.0861 1033.8173 1037.1233 1040.0572 1042.5238 1044.4883 1046.0327 1047.2389 1048.1619 1048.8269 1049.2820 1049.4946 1049.4551 1049.2037 1048.7944 1048.2089 1047.4921 1046.6572 1045.6987 1044.6196 1043.4269 1042.1446 1040.7832 1039.3464 1037.8363 1036.2650 1034.6466 1032.9585 1023.9078 1014.1017 1003.8680 993.4176 982.9110 972.4126 962.0038 951.7327 941.6382 931.7250 922.0192 912.5201 903.2377 894.1763 885.3321 876.7012 868.2724 860.0513 852.0317 844.2158

3946.9844 594.1126 190.5530 84.1738 43.2630 24.9456 17.6462 10.3582 7.7736 5.2000 4.4509 3.7019 2.9528 2.2043 1.4551 1.2811 1.1063 0.9317 0.7571 0.5823 0.5399 0.4974 0.4549 0.4125 0.3700 0.3275 0.2851 0.2426 0.2001 0.1577 0.1481 0.1385 0.1289 0.1194 0.1098 0.1002 9.0613E  02 8.1032E  02 7.1450E  02 6.1869E  02 5.8666E  02 5.5462E  02 5.2259E  02 4.9055E  02 4.5852E  02 4.2648E  02 3.9445E  02 3.6242E 02 3.3038E  02 2.9835E 02 2.3124E  02 1.6413E  02 1.3905E  02 1.1398E  02 8.8902E  03 6.3828E  03 5.5560E  03 4.7278E  03 3.8992E  03 3.0706E  03 2.8454E  03 2.6201E  03 2.3949E  03 2.1697E  03 1.9445E  03 1.7193E  03 1.4940E  03 1.2688E  03 1.0436E  03 8.1840E  04

3947.2808 594.7875 191.2892 84.7679 43.8253 25.4440 18.0990 10.7766 8.1683 5.5720 4.8114 4.0469 3.2848 2.5153 1.7566 1.5766 1.3940 1.2141 1.0307 0.8490 0.8023 0.7571 0.7109 0.6644 0.6191 0.5730 0.5262 0.4801 0.4351 0.3908 0.3797 0.3683 0.3567 0.3451 0.3339 0.3227 0.3110 0.2993 0.2878 0.2766 0.2721 0.2678 0.2636 0.2594 0.2552 0.2509 0.2466 0.2422 0.2380 0.2339 0.2218 0.2111 0.2052 0.1993 0.1935 0.1878 0.1844 0.1812 0.1779 0.1748 0.1723 0.1698 0.1675 0.1654 0.1634 0.1614 0.1595 0.1576 0.1557 0.1539

The program accounts for coherent, incoherent scattering cross sections in addition to photoelectric absorption cross sections. Input photon energy is controllable. For each photon energy, the

MC simulation is repeated 5000 times. Interaction type of the photon is sampled using random number q by means of the following conditions:

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

Table 5 For fat (see Table 1 for explanation).

55

Table 6 Comparison between calculated and measured values of linear attenuation coefficients (cm  1) for liver tissue.

E (keV)

Coh (b/mol)

Incoh (b/mol)

Photoelectric (1/cm)

m (1/cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150

19.2836 34.5045 29.4963 20.6337 19.8904 16.0775 13.4362 10.4144 10.2220 9.0943 7.6860 7.1704 6.5364 5.7489 5.1994 4.4662 4.3811 4.1014 3.6712 3.4211 3.1704 2.8445 2.6038 2.5619 2.4417 2.2512 2.0788 1.9620 1.8589 1.7472 1.6134 1.4923 1.4457 1.4214 1.3670 1.3025 1.2202 1.1510 1.0909 1.0571 1.0106 0.9691 0.9165 0.8663 0.8194 0.7887 0.7758 0.7646 0.7440 0.7211 0.5827 0.5010 0.4161 0.3836 0.3298 0.2916 0.2585 0.2239 0.2087 0.1940 0.1732 0.1556 0.1443 0.1326 0.1196 0.1096 0.1057 0.1002 9.3426E  02 8.5838E  02

0.8109 2.4394 3.9552 5.1082 5.9165 6.5048 6.9393 7.2728 7.5396 7.7603 7.9472 8.1076 8.2466 8.3657 8.4694 8.5608 8.6407 8.7096 8.7698 8.8218 8.8647 8.9002 8.9301 8.9552 8.9766 8.9948 9.0091 9.0194 9.0263 9.0306 9.0325 9.0323 9.0305 9.0269 9.0214 9.0144 9.0063 8.9969 8.9865 8.9752 8.9631 8.9502 8.9365 8.9223 8.9075 8.8922 8.8764 8.8602 8.8438 8.8269 8.7387 8.6464 8.5521 8.4572 8.3628 8.2693 8.1772 8.0869 7.9985 7.9120 7.8276 7.7452 7.6648 7.5866 7.5103 7.4360 7.3635 7.2929 7.2241 7.1571

2030.5930 284.0182 85.3958 35.7288 17.9972 10.2143 7.1749 4.1355 3.0848 2.0342 1.7377 1.4412 1.1446 0.8481 0.5516 0.4845 0.4174 0.3503 0.2832 0.2162 0.2002 0.1843 0.1684 0.1525 0.1366 0.1207 0.1048 8.8858E  02 7.2945E  02 5.7031E  02 5.3529E 02 5.0027E  02 4.6525E  02 4.3023E  02 3.9521E  02 3.6019E  02 3.2517E  02 2.9015E  02 2.5513E  02 2.2011E 02 2.0859E  02 1.9706E  02 1.8554E  02 1.7402E  02 1.6250E  02 1.5098E  02 1.3946E  02 1.2794E  02 1.1642E  02 1.0490E  02 8.1045E  03 5.7189E  03 4.8379E  03 3.9569E  03 3.0759E 03 2.1950E  03 1.9077E 03 1.6204E  03 1.3331E  03 1.0458E  03 9.6871E 04 8.9163E  04 8.1455E  04 7.3747E 04 6.6038E  04 5.8330E 04 5.0622E  04 4.2914E  04 3.5206E  04 2.7498E  04

2030.9767 284.7236 86.0345 36.2202 18.4899 10.6455 7.5639 4.4732 3.4240 2.3560 2.0362 1.7329 1.4269 1.1176 0.8125 0.7332 0.6660 0.5949 0.5208 0.4499 0.4300 0.4086 0.3886 0.3724 0.3546 0.3354 0.3165 0.2985 0.2808 0.2628 0.2568 0.2510 0.2465 0.2425 0.2379 0.2330 0.2278 0.2228 0.2179 0.2135 0.2113 0.2091 0.2067 0.2043 0.2020 0.1999 0.1982 0.1965 0.1947 0.1928 0.1861 0.1804 0.1761 0.1727 0.1690 0.1656 0.1630 0.1603 0.1580 0.1558 0.1537 0.1517 0.1499 0.1481 0.1463 0.1446 0.1431 0.1416 0.1401 0.1386

if 0o qoPinc, it is decided that the interaction is incoherent (Compton) scattering. The scattering angle should then be sampled by re-generating random number and so the scattered photon energy should be determined.

E (keV)

This study

King et al. (2011)

6.935 27

15.942 0.4698

30 35 40 45 50 55 60

0.3551 0.3060 0.2572 0.2377 0.2197 0.2089 0.1993

0.410 0.346 0.299 0.275 0.256 0.243 0.233

65 70 75 80 85 90 95 100 105 110 122

0.1937 0.1885 0.1833 0.1782 0.1749 0.1718 0.1688 0.1660 0.1636 0.1613 0.1563

0.229 0.227 0.212 0.208 0.203 0.202 0.208 0.200 0.200 0.165

References 15.32 (pig) (Kosanetzky et al., 1987) 0.441 (human) (Rao and Gregg, 1975) 0.470 (steer) (Rao and Gregg, 1975)

0.221 (human) (Rao and Gregg, 1975) 0.222 (steer) (Rao and Gregg, 1975) 0.219 (human) (McCullough, 1975)a

0.165 (human) (Rao and Gregg, 1975) 0.170 (steer) (Rao and Gregg, 1975)

a Calculated value of m is given by McCullough (1975) by multiplying the mass attenuation coefficient (Hubbell, 1969) by the most appropriate specific gravity of Rao and Gregg (1975).

Table 7 Comparison between calculated and measured values of linear attenuation coefficients (cm-1) for kidney tissue. E (keV)

This study

King et al. (2011)

27 30 35 40 45 50 55 60

0.5138 0.3831 0.3280 0.2746 0.2539 0.2334 0.2226 0.2127

0.389 0.320 0.279 0.257 0.246 0.232 0.218

65 70 75 80 85 90 95 100 105 110 122

0.2071 0.2013 0.1958 0.1908 0.1875 0.1842 0.1810 0.1779 0.1755 0.1732 0.1682

References 0.454 (steer) (Rao and Gregg, 1975)

0.218 (steer) (Rao and Gregg, 1975) 0.211 (human) (McCullough, 1975)a

0.207 0.199 0.220 0.201 0.182 0.193 0.189 0.175 0.166 0.197 0.165 (steer) (Rao and Gregg, 1975)

a Calculated value of m is given by McCullough (1975) by multiplying the mass attenuation coefficient (Hubbell, 1969) by the most appropriate specific gravity of Rao and Gregg (1975).

if Pinc oq o(Pinc þPcoh), it is decided that the interaction is coherent (Rayleigh) scattering. The scattered photon retains its original energy and no energy is deposited. if (Pinc þPcoh)oq o1, it is decided that the interaction is photoelectric and the photon is absorbed.

3. Results and discussion In this paper, we determine the linear attenuation coefficients for liver, kidney, muscle and fat. The physical processes treated are

56

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

Table 8 Comparison between calculated and measured values of linear attenuation coefficients (cm  1) for muscle tissue. E (keV)

This study

King et al. (2011)

1 2 3 4 5 6 6.935 8 9.88 10 15 17.44 20 27 30 35 39.91 40 42

3947.2808 594.7875 191.2892 84.7679 43.8253 25.444 18.5764 10.7766 5.8835 5.572 1.7566 1.3148 0.849 0.5262 0.3908 0.3339 0.2776 0.2766 0.2678

45 50 55 59.32 60

0.2552 0.2339 0.2218 0.2125 0.2111

65 68

0.2052 0.2017

0.230

70 75 80 85 90 95 100

0.1993 0.1935 0.1878 0.1844 0.1812 0.1779 0.1748

0.235 0.241 0.218 0.211 0.222 0.210 0.222

105 110 122

0.1723 0.1698 0.1646

0.247 0.239

140 150

0.1576 0.1539

Hubbell (1982)b

References

3924.96 588.952 190.112 84.084 43.4096 25.1784 16.76 (pig) (Kosanetzky et al., 1987) 10.6496 6.085 (human) (White et al., 1980) 5.4954 1.7347 1.046 (human) (White et al., 1980) 0.8423 0.446 (steer) (Rao and Gregg, 1975) 0.407 0.341

0.3904

0.300

0.2781

0.2623(human) (White et al., 1980) 0.300 (human) (Joyet et al., 1974) 0.271 (monkey) (Phelps et al., 1975) 0.312 (animal) (Bradley et al., 1986) 0.284 0.263 0.251

0.2347

0.242

0.2127

0.2115(human) (White et al., 1980) 0.212 (human) (Rao and Gregg, 1975) 0.219 (steer) (Rao and Gregg, 1975) 0.212 (human) (McCullough, 1975)a 0.192 (human) (Joyet et al., 1974) 0.203 (monkey) (Phelps et al., 1975) 0.229 (animal) (Bradley et al., 1986)

0.1895

0.1761

0.141 (human) (Joyet et al., 1974) 0.179 (monkey) (Phelps et al., 1975) 0.186 (animal) (Bradley et al., 1986)

0.165 (human) (Rao and Gregg, 1975) 0.166 (steer) (Rao and Gregg, 1975) 0.157 (cow) (Akar et al., 2006) 0.1551

a Calculated value of m is given by McCullough (1975) by multiplying the mass attenuation coefficient (Hubbell, 1969) by the most appropriate specific gravity of Rao and Gregg (1975). b The mass attenuation coefficients tabulated by Hubbell (1982) are converted to linear attenuation coefficients using density of 1.04 g cm  3 from ICRP (1975) for muscle.

limited to the photoelectric absorption, coherent (Rayleigh) and incoherent (Compton) scattering since the photon energies are in the range from 1 to 150 keV. Table 1 offers the molecular ISF values in the range of 0 rx r109Å  1. Properly, this table can be used in the incoherent scattering cross section calculations. Tables 2–5 show results of the calculated linear attenuation coefficients. It has been observed that the linear attenuation coefficients decrease with increasing photon energy. Our results are compared in Tables 6–9 with previously reported experimental values (Akar et al., 2006; Bradley et al., 1986; Johns and Yaffe, 1987; Joyet et al., 1974; King et al., 2011; Kosanetzky et al., 1987; Phelps et al., 1975; Rao and Gregg, 1975; Tartari et al., 1997; White et al., 1980) and theoretical values (Hubbell, 1982; McCullough, 1975). The theoretical results calculated in this study agree with theoretical results of Hubbell (1982) within 0.10–1.39% for muscle tissue and theoretical results of McCullough (1975) within 0.42–8.99% for liver, kidney, muscle tissues. When comparing experimental values in literature with theoretical values of this work, the difference was found within 0.20–6.49%

with results of Johns and Yaffe (1987), within 0.04–25.7% with results of the other references and within 0–30.24% with results of King et al. (2011). The experimental error reported by King et al. (2011) was within 11% for the tissues of interest. Also, it is expected that the experimental linear attenuation coefficients decrease with increasing photon energy, but the experimental values of King et al. (2011) increase with the rising photon energy (for 95 keV of liver, 75, 90, 110 keV of kidney, 70, 75, 90, 100, 105 keV of muscle and 60, 70, 85, 100, 105, 110 keV of fat). These can explain the variation up to 30.24% between this study and experimental value of King et al. (2011). However, it must be noted that the samples (steer, pig, monkey, cow, animal, human) of experimental results in literature are compared with theoretical results of this study which examines only human tissue samples. Besides, the reason for such a discrepancy can come from differences in the molecular weight and the elemental composition fractions of tissue. Deviations between experimental and theoretical results may be attributed to the distortion of the electron charge distribution in the molecule due to the effect of binding among different types of atoms (Abdel-Rahman and Kamel, 1998).

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

57

Table 9 Comparison between calculated and measured values of linear attenuation coefficients (cm  1) for fat tissue. E(keV)

This study

King et al. (2011)

6.935 8.04 18 20 25 27

7.7642 4.4312 0.5949 0.4499 0.3546 0.3165

30 35 40 42

0.2628 0.2379 0.2135 0.2091

0.282 0.242 0.219

45 50 55 59.54 60

0.2020 0.1928 0.1861 0.1809 0.1804

0.201 0.189 0.183

65 68

0.1761 0.1741

0.182

70 75 80 85 90 95 100

0.1727 0.1690 0.1656 0.1630 0.1603 0.1580 0.1558

0.183 0.169 0.168 0.172 0.164 0.162 0.167

105 110 122

0.1537 0.1517 0.1474

0.171 0.173

Johns and Yaffe (1987)

References 8.430 (pig) (Kosanetzky et al., 1987) 4.520 (pig) (Tartari et al., 1997)

0.585 0.456 0.333 0.328 (human) (Rao and Gregg, 1975) 0.304 (steer) (Rao and Gregg, 1975) 0.264 0.212 0.227 (human) (Joyet et al., 1974) 0.217 (monkey) (Phelps et al., 1975) 0.223 (animal) (Bradley et al., 1986) 0.191 0.181 (pig) (Tartari et al., 1997) 0.190 (human) (Rao and Gregg, 1975) 0.188 (steer) (Rao and Gregg, 1975)

0.193

0.168 (human) (Joyet et al., 1974) 0.177 (monkey) (Phelps et al., 1975) 0.184 (animal) (Bradley et al., 1986)

Fig. 1. The energy distribution of x-ray photons scattered from liver tissue by Monte Carlo simulation program.

A computer program has been written for the determination of energy distribution of photons in tissues. Molecular photoelectric cross sections, molecular form factors including interference effects and molecular ISF are incorporated in a Monte Carlo simulation program. For each incident photon, absorption and scattering probabilities are determined for interaction types. The results of simulations are shown in Figs. 1–4 to give additional information to the reader. The energy distribution of photons scattered is dominated by incoherent with a ratio of 76.68–79.86% for 50 keV, 92.4–94% for 100 keV and 96.04–96.84% for 150 keV. The contribution in the distribution of coherent scattering is in the range of 19.92–23.24% for 50 keV, 5.96–7.6% for 100 keV and

0.165

0.137 (human) (Joyet et al., 1974) 0.161 (monkey) (Phelps et al., 1975) 0.160 (animal) (Bradley et al., 1986) 0.152 0.153 (human) (Rao and Gregg, 1975) 0.150 (steer) (Rao and Gregg, 1975)

Fig. 2. For kidney (see Fig. 1 for explanation).

3.12–3.96% for 150 keV. The photoelectric absorption is the major factor responsible for the total attenuation coefficient at the photon energies below  40 keV. Therefore the participation in the distribution of photoelectric absorption is very low with a ratio of 0–0.22% for 50 keV, 0–0.06% for 100 keV and 0–0.04% for 150 keV. The accuracy of the linear attenuation coefficients and Monte Carlo calculations depends on the accuracy of the total cross section values, relate three individual processes. In medical x-ray imaging, up to 90% of the photons approaching the image receptor have been coherently or incoherently scattered. Coherent scattering is particularly interesting because the x-ray diffraction cross sections of various tissues can be completely different. The differences in

58

A. Böke / Radiation Physics and Chemistry 102 (2014) 49–59

References

Fig. 3. For muscle (see Fig. 1 for explanation).

Fig. 4. For fat (see Fig. 1 for explanation).

coherent (Rayleigh) scattering cross sections are noticeable at small values of momentum transfer where interference effects are significant. The integration range used in the coherent cross section calculations was from 1 cosθ ¼10  16–2.0 (θ ¼0.00000085381– 1801), divided into intervals in the logarithm of 1 cosθ with formula developed by Bö ke (2011). In the studies of Hubbell et al. (1975) and Hubbell and Øverbø (1979), the integration range was taken from 1 cosθ ¼10-12–2.0. Thus, with taken into account of the interference effects and smaller scattering angles, the more accurate data were obtained for coherent scattering. Additionally, all calculations which contain molecular form factor, molecular ISF, molecular photoelectric absorption, molecular coherent and incoherent scattering cross sections were performed with the inclusion of small concentrations of different elements, involved of high atomic number elements. Therefore, possible errors in calculation of total cross section have been reduced.

4. Conclusions When molecular interference effects are neglected, coherent cross sections are highly overestimated for very low photon energies. As far as it is known, to date there has not been any theoretical prediction of the attenuation coefficients which have been computed including molecular interference effects. The linear attenuation coefficients, incoherent (Compton), coherent (Rayleigh) and photoelectric cross sections presented here add valuable knowledge to the field of medical x-ray scattering research for modeling in MC codes and tissue characterization. The present values may be interpolated to obtain other desired energy values.

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