Electronic stopping power for proton in amino acids and protein in 0.05–10 MeV range

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NIM B Beam Interactions with Materials & Atoms

Nuclear Instruments and Methods in Physics Research B 266 (2008) 1938–1942 www.elsevier.com/locate/nimb

Electronic stopping power for proton in amino acids and protein in 0.05–10 MeV range Zhenyu Tan a,*, Yueyuan Xia b, Mingwen Zhao b, Xiangdong Liu b a

School of Electrical Engineering, Shandong University, Jinan, 250061 Shandong, PR China b Department of Physics, Shandong University, Jinan, 250100 Shandong, PR China Received 18 February 2008; received in revised form 28 February 2008 Available online 10 March 2008

Abstract The stopping powers for 0.05–10 MeV protons in a group of 15 amino acids and a protein have been systematically calculated. The calculations are based on Ashley’s dielectric model. An approach of evaluating the optical energy loss function is incorporated into the Ashley’s model because no experimental optical data are available for these bioorganic compounds. The Barkas-effect correction and Bloch correction are included and an empirical minimum impact parameter a is used for the Barkas-effect correction. The proton stopping powers for the 15 amino acids and the protein in the energy range from 0.05 to 10 MeV are presented here for the first time and might be useful for studies of various radiation effects in these materials. Ó 2008 Elsevier B.V. All rights reserved. PACS: 34.50.Bw; 61.82.Pv Keywords: Stopping power; Amino acid; Protein; Bioorganic compounds; Dielectric theory; Radiation biology

1. Introduction Knowledge of the stopping power (SP) of bioorganic materials for protons is of importance in many fields of fundamental and applied research, such as radiation biology, biomedical applications and radiotherapy. Especially, the use of protons in radiotherapy offers a good potential to spare healthy tissues due to their characteristic depth dose curve [1], where the proton SP in biological tissues is required particularly. Proteins are very important bioorganic compounds. The functions and physical structures of proteins are the subjects of longstanding experimental and theoretical interest [2,3]. Amino acids are basic constituent units of proteins and the amino acid sequences contribute to protein’s functions. Therefore, the prediction of the SP

*

Corresponding author. Tel.: +86 0531 88392806; fax: +86 0531 82955999. E-mail address: [email protected] (Z. Tan). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.03.043

of amino acids or proteins for protons is of essential significance for radiation biology and biomedical applications. The Bethe’s theory [4,5] can be applied for the calculation of SP for fast charged particles in matter. From this theory the prediction of SP requires a material-dependent parameter, i.e. the mean excitation energy, and additional theoretical calculations for shell corrections. Another commonly used theory to describe the interaction between the charged particle and matter is based on the dielectric response model, which was first developed by Lindhard and Dan [6] and Ritchie [7]. An important feature of SP calculation due to this theory is that the mean excitation energy is not required, and the shell corrections are included in a self-consistent way. In the framework of the dielectric response theory, many methods [8–15] have been presented to evaluate the SPs of various materials for low energy electrons. However, the reports on the calculations of the SPs of organic compounds for protons by means of the dielectric model are sparse. Ashley [16] has developed an optical data model to calculate the proton SP of the

Z. Tan et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 1938–1942

condensed matters. Ashley’s calculations included the two higher-order corrections, i.e. the Barkas-effect correction and Bloch correction. Akkerman et al. [17] have also described a method for estimating the SPs of 10 organic compounds for protons in the energy range of 50– 500 keV, based on the extension of Ashley’s model [16]. Recently, using the dielectric response theory Emfietzoglou et al. [18] have presented new SP calculations for protons in liquid water. A common feature of these works is that the numerical evaluation of optical energy loss function (OELF) needs to make use of experimental optical data, i.e. the refractive index and the extinction coefficients. On the other hand, to our knowledge, there are only 14 organic compounds, for which the optical data have been determined experimentally and these organic compounds were listed in the work of Tanuma et al. [11]. Thus, the requirement for experimental optical data hinders the application of Ashley’s optical data model in other organic compounds without experimental optical data. In a previous work [19], an algorithm for the calculation of proton SP in a group of bioorganic compounds has been described. This algorithm is based on the optical data model of Ashley [16] and on an evaluation approach for the OELF [14] and includes the higher-order corrections, i.e. the Barkas-effect correction and Bloch correction. Especially, in this algorithm an empirical parameter a involved in the Barkas-effect correction has been given for bioorganic compounds. In this work, systematic calculations of the SPs for 0.05– 10 MeV protons in a group of 15 amino acids and a protein have been performed. The calculations are based on the method described previously [19]. For these 15 amino acids and the protein the mean ionization potentials calculated by using the evaluated OELFs are in good agreement with those given by Bragg’s rule. The aim of this work is to present the proton SPs of these bioorganic compounds because of their importance in radiation biology. At the same time the present work also shows that the approach to evaluate the OELF extends Ashley’s model into those organic materials without available optical data. The data presented here are the first results of proton SPs for the 15 amino acids and the protein over the energy range of 0.05– 10 MeV. 2. Calculation method The stopping power of a medium for a proton, including the Barkas-effect correction and Bloch correction, can be written in the form: SP ¼

dE þ SP1 þ SP2 ; dS

ð1Þ

where dE/dS is referred to as basic stopping power and SP1 and SP2 correspond to the Barkas-effect correction and Bloch correction, respectively. The detailed calculation principles of these three terms have been presented in previous work [16,19], so only a brief description will be given here.

1939

According to Ashley [16], the basic stopping power can be calculated by Z E0 dE 1 ¼ ðhxÞIm½1=eðxÞvðaÞ dðhxÞ; ð2Þ dS pa0 E0 0 0

where E is the proton kinetic energy, E = E/r with r the ratio of proton mass to electron mass, a0 is the Bohr radius, hx is the energy transfer, Im[1/e(x) is the optical energy loss function and v(a) is vðaÞ ¼ ln½ð1  a þ sÞ=a;

ð3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi with a = hx/2E and s ¼ 1  2a. From Eq. (2), it is clear that a key problem of the SP calculation is to derive the OELF. Because for the bioorganic compounds under consideration there are no available experimental optical data, their OELFs will be evaluated by using the approach given in previous work [14], in which the OELF is modeled by a Drude function at low energy transfers and by photoabsorption cross sections at high energy transfers and an interpolation function is used to evaluate the OELFs between low energy transfers and high energy transfers. In addition, it should be emphasized that the obtained OELF satisfies the following f-sum rule due to dielectric response theory: Z 1 2 Z¼ 2 2 ðhxÞIm½1=eðxÞ dðhxÞ; ð4Þ ph XP 0 0

where Xp = (4pne2/m)1/2, n = Navq/M is the density of molecule, Nav is Avogadro’s number, q is the bulk density and M is the molecule weight, Z is the total numbers of electrons per molecule and m is the mass of electron. For the calculation of SP1, following the principles described in Ashley [16] and using Eq. (4) especially, which the evaluated OELFs satisfy, the SP1 can be expressed in a concise form Z 4E0 1 ðhxÞ dðhxÞIm½1=eðxÞL1 ðx; nÞ; ð5Þ SP1 ¼ pa0 E0 0 where the function L1(x;n) contains the free parameter n, which will be further described later and L1(x;n) can be approximately expressed in a analytic form [16] for easy numerical calculation. For the calculation of SP2 corresponding to the Bloch correction, the following formula [20,21] is used, i.e. 2

SP2 ¼

ðhxP Þ fy 2 ½1:20206  y 2 ð1:042  0:8549y 2 2a0 E0 þ 0:343y 4 Þg;

ð6Þ

where xP is the plasma angular frequency, y = a/b with a the fine structure constant and b the ratio of proton velocity to light velocity. As mentioned above, the free parameter n is involved in the calculation of the SP1.The parameter n can be written as n ¼ 5:811 phxffiffiE a with a the minimum impact parameter and here hx and E are in eV. For convenience the  hx and E are also in eV in following discussions about the

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Z. Tan et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 1938–1942

parameter a. There are, in general, two choices for pffiffiffidifferent ffi the parameter a. One is a ¼ 88:748= Epas given by Lindffiffiffiffiffiffi hard [22] and the other is a ¼ 3:688=  hx suggested by Jackson and pffiffiffiffiffiffi With the use of pffiffiffiffi McCarthy [23]. a ¼ 88:748= E and a ¼ 3:688= hx, respectively, the proton SPs in organic material polystyrene has been evaluated by Ashley [16] and by Akkerman et al. [17] and compared with the tabulated values [24,25] based on experimental data. On the basis of the calculations of Ashley [16] and Akkerman et al. [17], the dependence of SP calculations on the two choices of the minimum impact parameter a has been analyzed in detail in previous work [19] and thus an empirical parameter a are suggested as follows: 88:748 3:688 a ¼ k1 pffiffiffiffi þ k2 pffiffiffiffiffiffi : E hx

ð7Þ

In Eq. (7) k1 and k2 are two weight coefficients to obey k1 + k2 = 1 and k1 = 0.6 and k2 = 0.4 are determined for organic compounds. With the different parameter a, for organic material polystyrene the evaluated proton SPs as a function of proton energy are presented in Fig. 1 as an example of our calculations, whereas the evaluations from Akkerman et al. [17] and the tabulated values [25] are also displayed in this figure for the purpose of comparison. It is clear that with pffiffiffiffi the choice of a ¼ 88:748= E (corresponding to k1 = 1 and k2 = 0) the calculated SPs, compared to the tabulated values, overestimate the SPs at the energies of pEffiffiffiffiffiffi 6 200 keV, the SPs evaluated by adopting a ¼ 3:688= hx (i.e. k1 = 0 and k2 = 1) are consistent with those from Akkerman et al. [17] and both are smaller than the tabulated values in the regions of E 6 200 keV. However, It is shown clearly that with the parameter a due to k1 = 0.6 and k2 = 0.4, the evaluated proton SPs are in

polyethylene

8

4

0 10

3

In present work, we will adopt the empirical a value given by Eq. (7) with k1 = 0.6 and k2 = 0.4 for the calculations of the SPs of 15 amino acids and a protein for protons. As components of proteins, there are 20 kinds of amino acids. Here, these 15 amino acids are selected because their mass densities are available. The 15 amino acids and a protein, which includes one of each of the 20 amino acids, as well as their composition, mass density q and molecule weight M, are presented in Table 1. For simplicity, we use the code number, from 1 to 16, to represent the 16 compounds, respectively, as shown in Table 1. Applying our approach of evaluating the OELF [14], the obtained OELFs for these 15 amino acids and the protein satisfy the f-sum rule naturally. The f-sum rule expects such result that the effective number, Zeff, of electrons per molecule should be equal to Z, i.e. the total number of electron per molecule, when hx ? 1. As an example, Fig. 2 presents an effective number, Zeff, of electrons per molecule as a function of energy loss hx for arginine, methionine and glycine, respectively. It is clear that with the evaluated

Material

12

2

3. Proton stopping powers in 15 amino acids and a protein

Table 1 The composition, parameters and I values for: 1 – alanine, 2 – arginine, 3 – asparagine, 4 – aspartic acid, 5 – cysteine, 6 – glutamic acid, 7 – glutamine, 8 – glycine, 9 – leucine, 10 – lysine, 11 – methionine, 12 – phenylalanine, 13 – serine, 14 – tyrosine, 15 – valine, 16 – protein

16

10

close agreement with the tabulated values [25] except for those at energies below 50 keV. It should be pointed out that at the energies lower than 100 keV, as stated by Ashley [16], the two correction terms begin to dominate the behavior of the stopping power and thus the validity of this approach is questionable. Hence, in present work all calculated proton SPs at the energies below 100 keV might be considered only as rough data.

10

4

Fig. 1. Stopping power (SP) results as a function of proton energy for polyethylene. (- - -) calculated with a from Lindhard [22]; (     ) calculated with a from Jackson and McCarthy [23]; () tabulated values from ICRU report [25]; (M) calculated by Akkerman et al. [17]; (—) present work by use of the empirical a.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Composition

I (eV)

C

H

Calculated Expected

3 6 4 4 3 5 5 2 6 6 5 9 3 9 5 107

7 1 2 1.401a 14 4 2 1.100a 8 2 3 1.540a 7 1 4 1.660a 7 1 2 1 1.740b 9 1 4 1.460a 10 2 3 1.460a 5 1 2 1.607a 13 1 2 1.191a 14 2 2 1.360b 11 1 2 1 1.340a 11 1 2 1.340b 7 1 3 1.537a 11 1 3 1.465a 11 1 2 1.230a 197 29 49 2 1.300c

q M 3 N O S (g/cm ) 89.09 174.20 132.12 133.1 121.15 147.13 146.15 75.07 131.17 146.19 149.21 165.19 105.09 181.19 117.15 2738.1

70.4 67.7 73.8 77.0 90.7 74.2 71.7 74.0 65.1 65.8 81.9 69.0 74.0 71.2 66.3 71.4

71.9 68.6 75.5 79.4 93.3 76.7 73.4 75.5 66.4 66.4 84.4 71.1 76.0 73.1 67.8 73.3

In this table, (a) from Nahway [26], (b) from Khawas [27], (c) from Ronan et al. [28] and the expected I values are calculated using the Bragg rule.

Z. Tan et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 1938–1942

From Eq. (8), the I values for these 15 amino acids and the protein have been calculated and listed in Table 1 whereas the I values expected by Bragg’s additive rule are also displayed in Table 1 for comparison. Obviously, the calculated results show a good agreement with expected data. According to the method described in Section 2, the systematic calculations of the SPs of the 15 amino acids and the protein for protons in the energy range from 0.05 to 10 MeV have been carried out. The calculated results are listed at selected energies in numerical form in Tables 2. From the calculated results, as a group these bioorganic compounds show some similarities in their SP properties. The SPs dependences on energy are very similar from compound to compound and the maxima occur at energies located in 90–100 keV for all the SPs versus proton energy. Plots of SP versus proton energy are shown in Fig. 3 for arginine, methionine, protein and serine as examples of our results. Basically, the SP is large for large mass density because of the similar constituents of these amino acids and the protein. For example, methionine and protein are of density values 1.34 and 1.3 g/cm3, respectively and hence present almost identical SPs.

arginine, Z=94 methionine, Z=80 glycine, Z=40

120

80

40

0 1

2

10

3

10

4

10

10

Fig. 2. The effective number, Zeff, of electrons per molecule as a function of photon energy loss  hx for arginine, methionine and glycine.

OELFs for arginine, methionine and glycine, their Zeff are equal to respective Z, under the limit hx ? 1. Commonly, another check for the accuracy of the approach to evaluate OELF is to calculate the mean ionization potential I by using the OELF, i.e.

ln I ¼

Z

2 pZð hX P Þ

2

4. Summary Based on the incorporation of our approach for evaluating the OELFs [14] in Ashley’s optical data model [16], the systematic calculations of the SPs for 0.05–10 MeV protons in a group of 15 amino acids and a protein have been per-

1

ð hxÞIm½1=eðxÞ lnð hxÞ dð hxÞ:

1941

ð8Þ

0

Table 2 ˚ ) for the 15 amino acids and a protein, in same order of the compounds as in Table 1 SPs (in eV/A E (keV)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

50 60 70 80 90 100 150 200 300 400 500 600 700 800 900 1000 1500 2000 3000 4000 5000 6000 7000 8000 9000 10000

8.405 10.75 11.83 12.20 12.30 12.21 11.02 9.750 7.851 6.580 5.681 5.028 4.539 4.148 3.829 3.562 2.694 2.192 1.620 1.298 1.089 0.942 0.832 0.746 0.678 0.622

7.072 8.928 9.771 10.07 10.10 10.00 8.912 7.843 6.284 5.250 4.524 3.999 3.607 3.295 3.044 2.834 2.143 1.742 1.286 1.029 0.863 0.746 0.659 0.591 0.537 0.493

8.260 10.85 12.07 12.55 12.69 12.65 11.54 10.26 8.301 6.973 6.027 5.338 4.820 4.405 4.068 3.786 2.870 2.339 1.731 1.388 1.165 1.008 0.891 0.799 0.727 0.667

8.108 10.89 12.24 12.78 12.98 12.99 11.98 10.71 8.708 7.335 6.351 5.631 5.088 4.653 4.296 3.996 3.032 2.474 1.833 1.471 1.236 1.070 0.945 0.849 0.771 0.708

7.061 10.11 11.66 12.37 12.67 12.70 11.69 10.40 8.399 7.105 6.204 5.541 5.037 4.628 4.291 4.006 3.059 2.500 1.857 1.491 1.245 1.087 0.962 0.865 0.787 0.722

7.780 10.24 11.40 11.85 11.99 11.95 10.90 9.702 7.851 6.597 5.704 5.055 4.567 4.177 3.857 3.588 2.719 2.216 1.640 1.315 1.104 0.955 0.844 0.758 0.689 0.632

8.361 10.81 11.96 12.39 12.49 12.42 11.24 9.961 8.031 6.734 5.814 5.148 4.648 4.249 3.924 3.653 2.766 2.252 1.665 1.334 1.120 0.969 0.856 0.768 0.698 0.640

8.653 11.34 12.59 13.08 13.23 13.20 12.07 10.75 8.709 7.323 6.335 5.611 5.063 4.625 4.269 3.971 3.008 2.451 1.814 1.455 1.222 1.057 0.934 0.838 0.762 0.699

8.299 10.30 11.18 11.47 11.48 11.33 10.03 8.808 7.027 5.861 5.045 4.461 4.029 3.684 3.402 3.166 2.388 1.939 1.429 1.143 0.958 0.828 0.731 0.656 0.596 0.546

9.314 11.58 12.58 12.92 12.95 12.81 11.38 10.01 7.996 6.673 5.746 5.081 4.586 4.191 3.871 3.602 2.719 2.208 1.628 1.303 1.092 0.944 0.833 0.748 0.679 0.623

6.770 9.108 10.24 10.72 10.87 10.82 9.761 8.622 6.910 5.812 5.051 4.498 4.084 3.749 3.473 3.240 2.464 2.009 1.488 1.193 1.002 0.868 0.768 0.689 0.627 0.575

8.184 10.46 11.52 11.91 11.97 11.85 10.57 9.315 7.449 6.216 5.352 4.739 4.290 3.931 3.636 3.388 2.565 2.086 1.540 1.233 1.034 0.894 0.790 0.709 0.644 0.590

8.281 10.85 12.06 12.52 12.66 12.63 11.55 10.28 8.328 7.003 6.058 5.367 4.845 4.427 4.085 3.800 2.878 2.345 1.736 1.392 1.169 1.011 0.894 0.802 0.729 0.669

8.345 10.81 11.97 12.42 12.52 12.43 11.18 9.888 7.940 6.641 5.725 5.072 4.591 4.206 3.889 3.623 2.745 2.234 1.651 1.323 1.110 0.960 0.848 0.761 0.691 0.634

8.277 10.34 11.26 11.57 11.59 11.46 10.19 8.963 7.166 5.984 5.154 4.559 4.117 3.764 3.475 3.234 2.441 1.983 1.463 1.170 0.981 0.848 0.749 0.672 0.610 0.560

7.643 9.844 10.87 11.25 11.33 11.25 10.11 8.934 7.177 6.010 5.188 4.595 4.154 3.800 3.512 3.270 2.476 2.015 1.489 1.193 1.001 0.866 0.765 0.686 0.624 0.572

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References serine methionine arginine protein

10

1

10

2

10

3

10

4

Fig. 3. Stopping power (SP) results as a function of proton energy for arginine, methionine, protein and serine.

formed. The calculations include the Barkas-effect correction and Bloch correction and use an empirical parameter a for the Barkas-effect correction, which is important for the calculations of the proton SPs in organic compounds. The dependences of the calculated proton SPs on energy are very similar from compound to compound and the maxima occur at energies located in 90–100 keV for all the SPs versus proton energy. The presented results of the proton SPs for the 16 bioorganic compounds in the energy range from 0.05 to 10 MeV might be useful for studies of various radiation effects in these materials.

Acknowledgements This work was supported by the Foundation of Ministry of Education of China under Grant No. 20050422038 and by the Natural Science Foundation of Shandong Province of China under Grant No. Y2005D02.

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