Robust calculation of effective atomic numbers: The Auto-Zeff software

Robust calculation of effective atomic numbers: The Auto-Zeff software M. L. Taylor, R. L. Smith, F. Dossing, and R. D. Franich Citation: Medical Physics 39, 1769 (2012); doi: 10.1118/1.3689810 View online: http://dx.doi.org/10.1118/1.3689810 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/39/4?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Quantification of differences in the effective atomic numbers of healthy and cancerous tissues: A discussion in the context of diagnostics and dosimetry Med. Phys. 39, 5437 (2012); 10.1118/1.4742849 A basic insight to FEM_based temperature distribution calculation AIP Conf. Proc. 1448, 164 (2012); 10.1063/1.4725451 Calculation of conversion factors for effective dose for various interventional radiology procedures Med. Phys. 39, 2491 (2012); 10.1118/1.3702457 Segmentation of pulmonary nodules in three-dimensional CT images by use of a spiral-scanning technique Med. Phys. 34, 4678 (2007); 10.1118/1.2799885 The symposium commemorating the centennial of the discovery of radium by Maria (Marie) Sklodowska-Curie Med. Phys. 26, 1760 (1999); 10.1118/1.598679

Robust calculation of effective atomic numbers: The Auto-Zeff software M. L. Taylora) School of Applied Sciences and Health Innovations Research Institute, RMIT University, Melbourne 3000, Australia; Physical Sciences, Peter MacCallum Cancer Centre, East Melbourne 3000, Australia; and Medical Physics, WBRC, The Alfred Hospital, Melbourne 3000, Australia

R. L. Smith School of Applied Sciences and Health Innovations Research Institute, RMIT University, Melbourne 3000, Australia and Medical Physics, WBRC, The Alfred Hospital, Melbourne 3000, Australia

F. Dossing Cimbex Software, Holmegaard, Denmark

R. D. Franich School of Applied Sciences and Health Innovations Research Institute, RMIT University, Melbourne 3000, Australia

(Received 6 September 2011; revised 7 February 2012; accepted for publication 9 February 2012; published 12 March 2012) Purpose: The most appropriate method of evaluating the effective atomic number necessitates consideration of energy-dependent behavior. Previously, this required quite laborious calculation, which is why many scientists revert to over-simplistic power-law methods. The purpose of this work is to develop user-friendly software for the robust, energy-dependent computation of effective atomic numbers relevant within the context of medical physics, superseding the commonly employed simplistic power law approaches. Method: Visual Basic was used to develop a GUI allowing the straightforward calculation of effective atomic numbers. Photon interaction cross section matrices are constructed for energies spanning 10 keV to 10 GeV and elements Z ¼ 1–100. Coefficients for composite media are constructed via linear additivity of the fractional constituents and contrasted against the precalculated matrices at each energy, thereby associating an effective atomic number through interpolation of adjacent cross section data. Uncertainties are of the order of 1–2%. Results: Auto-Zeff allows rapid (0.6 s) calculation of effective atomic numbers for a range of predefined or user-specified media, allowing estimation of radiological properties and comparison of different media (for instance assessment of water equivalence). The accuracy of Auto-Zeff has been validated against numerous published theoretical and experimental predictions, demonstrating good agreement. The results also show that commonly employed power-law approaches are inaccurate, even in their intended regime of applicability (i.e., photoelectric regime). Furthermore, comparing the effective atomic numbers of composite materials using power-law approaches even in a relative fashion is shown to be inappropriate. Conclusion: Auto-Zeff facilitates easy computation of effective atomic numbers as a function of energy, as well as average and spectral-weighted means. The results are significantly more accurate than normal power-law predictions. The software is freely available to interested readers, who are C 2012 American Association of Physicists in Medicine. encouraged to contact the authors. V [http://dx.doi.org/10.1118/1.3689810] I. INTRODUCTION Since Moseley,1 the atomic number Z has been associated with fundamental properties of the elements, and an “effective” atomic number, Zeff, is necessarily employed for composite media. Being fundamentally connected with radiation interaction processes, the latter has direct applications in the characterization of dosimeters and surrogate materials and biological tissues and the calculation of particle interactions. There are a number of ways to evaluate the effective atomic number, the most common of which involves the use of a simple power law of the form: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X fi Zim ; (1) Zeff ¼ m whereby the relativeP electron fraction of the ith element Zi is given by fi, such that fi ¼ 1. Mayneord2 used a value of 2.94 1769

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for the exponent m, and this method may be found in contemporary radiotherapy textbooks.3 This dated approach, however, is overly simplistic for many applications and was derived for a particular x-ray source; the prevalence of the approach is for historic reasons and simplicity. For a more detailed discussion, the interested reader is referred elsewhere.4 Although the limitations of the simplistic means of Zeff calculation are well-accepted, many researchers nonetheless return to the power-law approach to avoid the relative complexities of the more rigorous energy-dependent computations. Recognizing that this practice has arisen as a result of the time pressures in the contemporary research environment, we have developed a user-friendly graphical user interface (GUI) program that computes the effective atomic number using a robust, energy-dependent approach. One may request

0094-2405/2012/39(4)/1769/10/$30.00

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Zeff for a broad range of precomputed materials relevant within the context of medical physics (such as biological tissues, surrogate materials, and dosimeters) or, optionally, one may enter the mass fractions for a particular material of interest. For those dealing with polyenergetic radiation fields, the program also incorporates a database of spectra corresponding to a range of radioisotopes, kilovolt and megavolt photon sources, against which Zeff values may be weighted. This allows the user to compute single-valued effective atomic numbers for their particular application of interest. The program is freely available to interested persons by contacting the authors. II. METHODOLOGY II.A. Calculation of the effective atomic number

In this work, the calculation of effective atomic number draws fundamentally on photon attenuation data, the relationship between atomic number and which is somewhat irregular. However, there is a smooth correlation between cross section and atomic number,5 which may be exploited to yield Zeff values, as described below. For a detailed description of the determination of the mass attenuation coefficients used in calculations presented here, the reader is referred elsewhere.6,7 For a composite material, one may additively write the total mass attenuation coefficient in terms of those of the constituent elements, ðl=qÞi , considering their fractional weighting, wi, such that the total for the mixture is given by:     X l l ¼ ðwi Þ : (2) q mixt q i i The atomic cross section may be defined as   l q mixt rmixt ¼ X wi ; NA Ai i

(3)

where NA is Avogadro’s number and Ai is the atomic weight of element i within the composite. The total cross section is the sum of the photoelectric cross section, coherent (Rayleigh) and incoherent (Compton) scattering cross sections, and the cross sections for pair production in the field of the nucleus and triplet production in the field of the atomic electrons. Some studies involve the use of Hubbell’s earlier data8 with renormalized cross sections9; however, it has been shown that agreement with experiment is improved without renormalization.10 As a result, the present work draws on the x-ray mass attenuation data without normalization provided by Hubbell and Seltzer.6 In summary, for energies up to 1.5 MeV, the photoelectric cross section is determined via the Dirac equation for orbital electrons moving in a static Hartree–Slater central potential.9 For the (low probability) photoelectric effect above 1.5 MeV, Hubbell and Seltzer semiempirically extend the data to the high energy values of Pratt.11 Coherent scattering cross sections are from Hubbell and Øverbø,12 determined by integration of the Thomson formula13 weighted by the (relativistic) squared Hartree–Fock Medical Physics, Vol. 39, No. 4, April 2012

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atomic form factor. Incoherent scattering cross sections are from Hubbell et al.14 and are determined via integration of the Klein–Nishina formula15 weighted by the incoherent scattering function. The Hubbell and Seltzer data use the pair and triplet production cross sections from Hubbell et al.16 It is important to note that the photonuclear cross section is not incorporated in this study. In this process, a photon is absorbed by the nucleus and a nucleon(s) is ejected. While measurable, its incorporation is inhibited by sensitivity to isotopic abundances, irregular dependence on A and Z, gaps in available information, and a lack of theoretical models comparable to those for atomic cross section computations.17 Its effect is a small increase of the total cross section at the peak of the giant resonance of the target nuclide. The contribution to the total cross section due to this effect is likely to be small—of the order of several percent or for many radiotherapy applications about half a percent or less.8,18 As stated, Zeff may be determined via exploitation of the smooth correlation between atomic cross section and atomic number. Mass attenuation coefficient data were obtained for the 100 elements and the corresponding cross section values were calculated. A matrix of cross sections was constructed spanning atomic numbers Z ¼ 1–100 for photon energies ranging between 10 keV and 1 GeV. The cross sections for polyelemental media are calculated via linear additivity. These cross section values are then contrasted with the cross section matrix as a function of Z, and an effective Z number for each energy is obtained by interpolation (b-spline) of Z values between the adjacent cross section data. In the region of the absorption edges (in particular, the K-shell), discrete jumps in Zeff may be apparent that correspond to photoelectric absorption at shell binding energies; these will be more pronounced for compounds of higher atomic number. Uncertainties at higher energies (far from the absorption edges) are of the order of 1–2%. If users request it, we are also able to provide software that incorporates photon energies down to 1 keV, though in this case uncertainties may range up to 25–50%. II.B. The Auto-Zeff software

Auto-Zeff was written using the Microsoft Visual Basic .NET programming language. The program was developed and compiled within the Microsoft Visual Studio 2010 development suite (Version 10.0.30319.1). At runtime, the program loads precalculated data files from an internal resource, including mass attenuation coefficients and cross section matrices as a function of photon energy for the first 100 elements. Predefined material files and photon source spectra files are also loaded. The user can select from one of these predefined materials or create a user defined material by adding elements and the corresponding fractional components. The user can also load a previously saved material definition or save the current custom material for use later. The system confirms the fractional sum is 1.0 before allowing the user to define (“set”) the material for Zeff calculation. Once the material has been defined, the system immediately begins the Zeff calculation. The program first

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TABLE I. The various media for which precomputed effective atomic numbers are provided, with the data source. For most applications, the user may define their own material of interest using the GUI by inputting the fractional elemental content (by mass) of their particular compound. Medium Water Adipose tissue Brain tissue Breast tissue Cortical bone Soft tissue Tissue-equivalent plastic Borosilicate glass Polyethylene Fricke solution PAG gel dosimeter TLD-100 (LiF:Mg,Ti) TLD-100 H (LiF:Mg,Cu,P)

Typical composition

Source

H: 11.1898%, O: 88.8102% H: 11.4%, C: 59.8%, N: 0.7%, O: 27.8%, Na: 0.1%, S: 0.1%, Cl: 0.1% H: 10.7%, C: 14.5%, N: 2.2%, O: 71.2%, Na: 0.2%, P: 0.4%, S: 0.2%, Cl: 0.3%, K: 0.3% H: 10.6%, C: 33.2%, N: 3.0%, O: 52.7%, Na: 0.1%, P: 0.1%, S: 0.2%, Cl: 0.1% H: 3.4%, C: 15.5%, N: 4.2%, O: 43.5%, Na: 0.1%, Mg: 0.2%, P: 10.3%, S: 0.3%, Cl: 22.5% H: 10.2%, C: 14.3%, N: 3.4%, O: 70.8%, Na: 0.2%, P: 0.3%, S: 0.3%, Cl: 0.2%, K: 0.3% H: 10.133%, C: 77.5498%, N: 3.35057%, O: 5.52315%, F: 1.7423%, Ca: 1.8377% B: 4.0066%, O: 53.9559%, Na: 2.8191%, Al: 1.1644%, Si: 37.7220%, K: 0.3321% H: 14.3716%, C: 85.6284% H: 10.8376%, O: 87.8959%, Na: 0.0022%, S: 1.2553%, Cl: 0.0035%, Fe: 0.0055% H: 10.7%, C: 4.7%, N: 1.7%, 82.9% Li: 26.748%, F: 73.231%, Mg: 0.02%, Ti: 0.001% Li: 26.5065%, F: 72.9895%, Mg: 0.2%, P: 0.004%, Cu: 0.3%

ICRU Report 44 (Ref. 19)

selects the mass attenuation coefficient data files for each corresponding element in the selected material. The total attenuation coefficient is calculated via linear additivity (for each energy in the mass attenuation coefficient data files), based on the mass fraction of each element in the material. The cross section for each energy is calculated, and the effective atomic number is determined by using this calculated cross section (for each energy) by using the preloaded cross section matrix for each element as a lookup table. At each energy, the program compares the calculated cross section against the cross section matrix data and the corresponding atomic number is determined via interpolation. The calculated effective atomic number data are then displayed graphically under the Zeff (E) tab and as an exportable table. A broad range of predefined materials of relevance to the medical physics community are provided with the software. These media are summarized (with their sources) in Table I. Interested readers are referred to the respective references for uncertainties in the mass fraction values for these materials. The option for user-specified media is also provided. For computation of spectral-weighted single-valued effective atomic numbers relevant to particular applications, a range of photon spectra are incorporated into the software. These are summarized in Table II. There is also the option for user-specified energies and spectra. The single-valued effective atomic number is evaluated simply using Eq. (4) below, where W(E) is the energy spectrum (normalized to unity): ð Emax X Zeff ðEi ÞWðEi Þ ffi Zeff ðEÞWðEÞdE: (4) Zeff ¼ i

dependent effective atomic numbers are calculated for the material in the spectrum of energies spanning 10 keV to 1 GeV. In some cases, the user may be interested in a singlevalued Zeff relevant to a particular application. The user is given the option of either choosing from a list of predefined TABLE II. The spectral-weighted mean effective atomic numbers are available for a broad range of spectra, as described in this table. Source references are g\iven; where the spectrum is generated for a dosimetrically matched Monte Carlo model of a local machine, this is denoted as “local.” Note that radioisotope spectra include x-rays in addition to gamma transitions. AutoZeff also allows for user-input spectra. Note that the subscript FFF refers to flattening filter free. Photon spectra 60

Co Tc 103 Pd 125 I 131 I 192 Ir 226 Ra kV imager (80 kVp) kV imager (100 kVp) kV imager (125 kVp) Intrabeam XRS (30 kVp) Intrabeam XRS (40 kVp) Intrabeam XRS (50 kVp) 4 MV 99

4FFF MV 6 MV

Emin

III. RESULTS

6FFF MV

III.A. The Auto-Zeff program

18 MV

The functionality of the Auto-Zeff software is illustrated in Fig. 1. The user is first prompted to either choose from a list of predefined media (see Table I) or define their own material of interest by specification of the fractional (by mass) elemental composition. After the medium is defined, energy-

18FFF MV 25 MV

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Taylor et al. (Ref. 20) Taylor et al. (Ref. 21)

Source Tuli (Ref. 40) Tuli et al. (Ref. 41) De Frenne (Ref. 42) Katakura (Ref. 43) Khazov et al. (Ref. 44) Baglin (Ref. 45) Akovali (Ref. 46) Varian G242, local (egsnrc) Varian G242, local (egsnrc) Varian G242, local (egsnrc) Biggs et al. (Ref. 47) Biggs et al. (Ref. 47) Biggs et al. (Ref. 47) Mohan et al. (Ref. 48) Elekta Synergy-II, local (egsnrc) Elekta Synergy-II, local (egsnrc) Mohan et al. (Ref. 48) Varian 600 C, local (egsnrc) Varian 21-iX, local (egsnrc) Elekta Synergy-II, local (egsnrc) Varian 21-iX, local (egsnrc) Elekta Synergy-II, local (egsnrc) Mohan et al. (Ref. 48) Varian 21-iX, local (egsnrc) Elekta Synergy-II, local (egsnrc) Elekta Synergy-II, local (egsnrc) Mohan et al. (Ref. 48) Elekta Synergy-II, local (egsnrc)

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FIG. 1. The basic functionality of Auto-Zeff. (a) Splash screen. (b) One specifies the material of interest, whether from the set of predefined media or by manual input. (c) The program checks to ensure the fractions sum to unity and the medium is defined. (d) Effective atomic number as a function of energy is defined for 1 keV to 1 GeV. (e) If a single-valued effective atomic number is desired, the user may specify (again, from a predefined set or by manual input) a spectrum relevant to the application. (f) A single-valued Zeff is generated.

spectra (for radioisotopes relevant to nuclear medicine, brachytherapy and Cobalt therapy, kilovolt treatment and diagnostic devices, and megavolt radiotherapy devices) or defining their own spectrum of interest. This then allows calculation of a single-valued, spectrum-weighted effective atomic number. Effective atomic numbers for monoenergetic photons can be evaluated in the same way. The program was optimized for accuracy rather than speed, but the computation is nonetheless rapid. We have evaluated the calculation speed on a Dell Latitude E4310 notebook computer with an Intel i5Core Processor running Medical Physics, Vol. 39, No. 4, April 2012

at 2394 MHz, with 4 Gb installed memory (OS: Microsoft Windows XP Professional SP3). Ultimately, the calculation time may be given by t ¼ (2.85  102)NZ þ 0.46 s (65%), where NZ is the number of constituent media. Typical calculation times are of the order of 0.5–0.7 s. III.B. Effective atomic numbers for biological, surrogate, and dosimeter materials

To demonstrate the capabilities of the Auto-Zeff software, some example datasets have been generated for biological,

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tissue-surrogate, and dosimeter materials. These are shown in Fig. 2. Figure 2(a) shows the effective atomic number as a function of energy for soft tissue. Also shown are the theoretical values of Kumar and Reddy22 and Shivaramu,23 both of which demonstrate good agreement. For medical physics applications, media incorporating elements spanning roughly Z ¼ 1–30 are typically of most interest. To illustrate the behavior of composites at the upper end of this regime, Fig. 2(b) shows Zeff for brass (a Cu–Zn alloy with trace amounts of Pb and Fe). To validate the results, recent experimental data from Kaewkhao et al.24 and Kurudiek et al.25 are also shown—both of which agree well with the Auto-Zeff calculated values. Oleic acid (C18H34O2) is a monounsaturated fatty acid common in both animal and vegetable fats, and numerous investigators have reported effective atomic numbers for this substance. Figure 2(c) shows Zeff as a function of energy, plotted alongside experimental data from Goswami and Chaudhari,26 El-Kateb and Abdul-Hamid,27 Manjunathagaru and Umesh,28 and Bandhal et al.29 as well as theoretical predictions from Manohara et al.30 For the most part, these agree well. The data of Manjunathagaru and Umesh indicate slightly lower effective atomic numbers, which could be attributed to sample purity. The data of Bhandal et al. is notably higher and appears to disagree with not only the results presented in this work but also the other experimental data (by about 20–30%). This could be attributed to differences in experimental arrangement. An additional biological material, leucine—an amino acid (C6H13O2N), is presented in Fig. 2(d) with experimental data from Gowda et al.,31 demonstrating very good agreement. A common thermoluminescent dosimeter material, calcium sulfate dihydrate (CaSO42H2O), is shown in Fig. 2(e) with experimental data from Kaginelli et al.32 and Gowda et al.,33 as well as theoretical predictions from Shivaramu.34 These demonstrate good agreement with the exception of the very low energy value of Kaginelli et al.; one may surmise that this relates to the difficulty of experimental measurements involving such low energy photons. To demonstrate how Auto-Zeff may be readily employed for straightforward radiological comparison of materials, the effective atomic numbers of water, Plastic Water (CIRS, Norfolk) and Solid Water (Gammex, Middleton) are overlayed in Fig. 2(f). Of these, water demonstrates the lowest Zeff, followed by Solid Water, with Plastic Water having the highest effective atomic number. From this, one can expect dosimetric differences between the three—and indeed, this is what has been shown experimentally.35 Data from Kumar and Reddy22 as well as Parthasaradhi et al.36 are shown to support the Auto-Zeff calculations. III.C. High-Z composites

Although it is anticipated that the software will primarily find use within the medical physics context and, thus, be applied to composite materials typically having Z < 30, the software is nonetheless capable of evaluating Zeff for high-Z media. The reason for considering these two regimes someMedical Physics, Vol. 39, No. 4, April 2012

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what separately is that the complex behavior (specifically, the discontinuous nature of the cross sections in the vicinity of K-edges) results in quite abrupt changes in Zeff over small energy intervals for Z > 30 (where K-edges exceed 10 keV). As such, it is worthwhile considering these independently, comparing calculated values with experimental values, and discussing some of the interesting features, such as multivalued solutions. Consider Fig. 3, which shows the effective atomic number calculated with Auto-Zeff for two high-Z composites, compared to other published values. Tungsten has desirable properties but is very brittle and is consequently typically alloyed with other metals. Figure 3(a) shows Zeff for a tungsten–copper alloy (65% and 35% by weight, respectively). Also plotted are data from HPGe detector transmission measurements undertaken by Murty et al.37 and Murty,38 as well as theoretical predictions undertaken by Kurudirek et al.25 The theoretical data exhibit notable disagreement to the results presented here and in experimental studies; the authors of that study acknowledged this disagreement and attributed it to possible errors in their interpolation process. The experimental data, however, match the Auto-Zeff calculations very well, even in the vicinity of the tungsten K-edge at 69.53 keV. Figure 3(b) shows the effective atomic number for a lead– antimony alloy (96.69% and 3.31%, respectively). Also plotted are the data from NaI detector transmission measurements undertaken by El-Kateb et al.39 Again, the agreement with published experimental measurements is very good. The reader will notice the interesting features in the plots of Zeff for these high-Z composites that arise from shell structure, which are less apparent in the previous plots (Fig. 2) where the K-edges are typically