Water equivalence of some 3D dosimeters: A theoretical study based on the effective atomic number and effective fast neutron removal cross-section

Water equivalence of some 3D dosimeters: A theoretical study based on the effective atomic number and effective fast neutron removal cross-section 1 2 corresp first-author Corresponding author, Ahmed_el_khayatt@yahoo.com, Corresponding author, Ahmed_el_khayatt@yahoo.com, Ahmed_el_khayatt@yahoo.com Physics Department, College of Science, Ibn Saud University (IMSIU), Saudi Arabia. Physics Department, College of Science, Ibn Saud University (IMSIU), Saudi Arabia. Reactor Physics Department, NRC, Atomic Energy Authority, 13759 Cairo, Egypt. Reactor Physics Department, NRC, Atomic Energy Authority, 13759 Cairo, Egypt. Effective atomic numbers for photon energy absorption ( <math id="M1"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> ) and their corresponding electron numbers <math id="M2"><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow></math> , and effective macroscopic removal cross-sections of fast neutrons (Σ R ) were calculated for 27 different types of three-dimensional (3D) dosimeters, four types of phantom materials, and water. The values of <math id="M3"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> and <math id="M4"><mrow><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> were obtained using the direct method for energies ranging from 10 keV to 20 MeV. Results are presented relative to water, for direct comparison over the range of examined energies. The effect of monomers that are used in polymer gel dosimeters on the water equivalence is discussed. The relation between Σ R and hydrogen content was studied. Micelle gel dosimeters are highly promising because our results demonstrate perfect matching between the effective atomic number, electron density number, and fast neutron attenuation coefficient of water. 3D dosimeters Water equivalence Effective atomic number Photon energy absorption Removal cross-section Fast neutrons 1. Introduction 1 Modern radiation treatment techniques require three-dimensional (3D) dosimeters that can accurately measure dose distributions in three dimensions with high spatial resolution. Literature review suggests that 3D dosimeters are widely used in many radiotherapy applications, such as photon-beam intensity-modulated radiation therapy (IMRT) [ 1 1 ], stereotactic radiosurgery (SRS), X-knife and γ-knife radiosurgery, and computed tomography-based (CT-based) brachytherapy, where steep dose gradients exist for conforming the prescription isodose to the target volume only [ 2 2 ]. In addition, developments in charged particle therapy allow radiation distributions to be tightly tailored to irregular 3D tumor volumes; as a result, 3D dosimeters are needed [ 3 3 ]. The need for 3D dose measurements is not limited to radiotherapy applications. Diagnostic radiology also requires measuring the distribution of radiation in patients undergoing medical imaging for a range of clinical diagnoses [ 4 4 ]. Two types of 3D dosimeters are commercially available: 1) gel dosimeters and 2) polyurethane radiochromic plastic dosimeters, which are known as “PRESAGE” dosimeters. Gel-based 3D dosimeters were first suggested in 1950 [ 5 5 ]. Gel dosimetry systems can in turn be divided into three types, based on: 1) the Fricke gel (featuring ferrous sulphate), 2) polymer gels, and 3) micelle gels. Micelle gel systems are a hybrid of PRESAGE and gel dosimeters. The Fricke gel utilizes radiation-induced transformation of ferrous (Fe 2+ ) ions into ferric (Fe 3+ ) ions. This radiation-induced chemical change can be quantified by performing nuclear magnetic resonance (NMR) relaxation measurements [ 6 6 ], and can be used to obtain information on the 3D spatial dose using magnetic resonance imaging (MRI) [ 7 7 , 8 8 ]. Although the Fricke gel is easy to fab ricate and handle, its post-irradiation stability is poor [ 9 9 ]. The first polymer dosimetry system was developed and reported by Alexander et al., in 1954 [ 10 10 ]. Polymer gels are chemical dosimeters based on dose-dependent radiation-induced polymerization and cross-linking of monomers in an irradiated volume. When a polymer gel is exposed to radiation, it becomes opaque via polymerization. Their optical density of these gels increases with increasing the absorbed dose, which is utilized by NMR [ 11 11 ] or optical CT [ 12 12 ]. Other, less established, readout techniques have been introduced, such as the X ray CT [ 13 13 ], ultrasound tomography imaging [ 14 14 ], and vibrational spectroscopy [ 15 15 ]. Polymer gel dosimeters have several advantages, including tissue equivalence, high spatial resolution, and good post-irradiation stability. Polymer gels can be also potentially used for dosimetry in mixed neutron-gamma radiation fields [ 16 16 , 17 17 ]. However, many polymer gel dosimeters have significant limitations and shortcomings; for example, they require external containers, which leads to edge artifacts, which in turn reduce the useful region of these dosimeters [ 18 18 ]. Many of these drawbacks were overcome following the development of plastic PRESAGE dosimeters [ 19 19 ]. This is an entirely new class of polymer dosimeters – radiochromic optically transparent 3D dosimeters based on polyurethane combined with leuco-dye leucomalachite green. Upon exposure to radiation, radiochromic material changes its color owing to the oxidation of leuco-dyes by halogen radicals [ 19 19 ]. PRESAGE has a number of potential advantages over both conventional polymer and Fricke gels. It is a transparent material with excellent properties for dosimetry, such as insensitivity of the dose response to oxygen and environmental conditions, a solid texture reducing edge effects by negating the need for an external container, and a radiochromic response that is well suited for accurate optical CT owing to the very low scattering fraction [ 20 20 ]. In addition, the polyurethane matrix prevents the diffusion of the dose distribution image [ 21 21 ]. Unfortunately, these dosimeters suffer from poor tissue equivalence and cannot be easily manufactured or molded into anthropomorphic phantoms [ 22 22 ]. Aiming to overcome the limitations of PRESAGE, Jordan and Avvakumov [ 23 23 ] and Babic et al. [ 24 24 ] developed radiochromic micelle gel dosimeters for optical readout. In their proposed approach, the color dye and halogen are dissolved in a gelatin gel. Because the color dye and halogen do not readily dissolve in the gelatin hydrogel, the dye and halogen are embedded in micelles [ 25 25 ]. A micelle gel changes its color upon irradiation [ 26 26 ]. These novel gel dosimeters have specific advantages compared with polyurethane dosimeters (such as PRESAGE dosimeters). The former exhibit better spatial stability and good water/soft tissue equivalence, over a wide range of photon energies. At the same time, the fabrication procedure of gelatin-based chemical dosimeters is less complicated than that of polyurethane-based dosimeters. The 3D dosimeters evaluated in the present work can be divided into three main categories: 1) “conventional” (polymer and Fricke) gels, 2) “modern” (micelle) gels, and 3) polyurethane radiochromic plastic (PRESAGE) dosimeters. Conventional polymer gel dosimeters may be generally classified in terms of hypoxic, reduced toxic, or normoxic gels. Different types of hypoxic polymers have been suggested, such as polyacrylamide gelatin (PAG) [ 27 27 ], and BIS acrylamide nitrogen gelatin (BANG) gel formulations such as BANG-1. The term BANG is trademarked and a patent was acquired for this gel type [ 28 28 ]. BANG-2 uses acrylic acid as a monomer and NaOH to buffer the pH [ 29 29 ]. BANG gels have evolved to the third product from MGS research, known as BANG-3. The BANG-3 gel consists of BIS, methacrylic acid, sodium hydroxide, nitrogen, and gelatin. This new formulation exhibits stronger optical and NMR responses [ 8 8 ]. The monomers of these polymers are highly toxic; thus, they were replaced with reduced toxic monomers such as polyethylene glycol diacrylate bis-gelatin (PABIG) [ 30 30 ] and N-vinyl pyrolidone argon (VIPAR) gels [ 31 31 ]. Even though these monomers are less toxic, all of these gel dosimeters have to be prepared under the hypoxic condition. Because these gel dosimeters are inhibited by oxygen, free oxygen has to be removed from the gel. The term “normoxic” refers to a gel that can be fabricated under normal atmospheric conditions. In 2001, Fong et al. developed the first normoxic gel dosimeter [ 32 32 ]. This novel polymer gel dosimeter features a gel known as MAGIC, which is an acronym for the methacrylic acid, ascorbic acid, gelatin initiated by copper. The MAGIC gel utilizes the ascorbic acid oxygen scavenger, which binds free oxygen within the aqueous gelatin matrix into metallo-organic complexes, in a process that is initiated by copper sulphate. Replacing the ascorbic acid and copper sulphate by tetrakis in the MAGIC formulation yields a new formulation that consists of the methacrylic acid in gelatin and tetrakis (MAGAT) [ 33 33 ]. Gel manufacturers provided many types of such normoxic dosimeters, including MAGAS (which consists of the methacrylic acid and gelatin gel with ascorbic acid), HEAG (which consists of the hydroxy-ethyl-acrylate gel) [ 34 34 ], nPAG (which consists of the normoxic polyacrylamide gel), nMAG (which consists of the normoxic methacrylic gel) [ 35 35 ], and ABAGIC (which consists of the ascorbic acid, bis-acrylamide, in gelatin initiated by copper) [ 33 33 ]. In addition, some efforts were made to modify hypoxic gel dosimeters to normoxic ones. For example, the hypoxic PAG gel was combined with tetrakis (hydroxymethyl) phosphonium chloride (THPC) as an anti-oxidant, to form a normoxic gel dosimeter that utilizes the PAGAT gel (which consists of polyacrylamide, gelatin, and tetrakisphosphonium chloride) [ 36 36 ]. As another example, Senden et al. replaced the highly toxic acrylamide monomer in the PAGAT gel with N-isopropylacrylamide, obtaining NIPAM [ 37 37 ]. VIPAR polymer gel dosimeters were also modified, by Kantemiris et al. [ 38 38 ], to eliminate the need for de-oxygenation in the manufacturing process. The new formulation, VIP, consists of N-Vinylpyrrolidone, gelatine, N, N / -methylenebisacrylamide, as well as of copper sulfate and ascorbic acid. The effective atomic number, <math id="M5"><mrow><msub><mi>Z</mi><mrow><mtext>eff</mtext></mrow></msub><mo>,</mo></mrow></math> and the electron density, <math id="M6"><mrow><msub><mi>N</mi><mrow><mtext>eff</mtext></mrow></msub><mo>,</mo></mrow></math> are particularly valuable parameters for characterizing interactions in various multi-element materials. Many studies on Z eff and N eff of different materials for interactions of photons [ 39 39 - 40 40 39-40 ], electrons [ 41 41 ], and heavy charged particles [ 42 42 - 44 44 42-44 ] are available. For a 3D dosimeter to be useful in radiation dosimetry, it should have water-equivalent radiological properties. The radiological properties such as the effective atomic number, Z eff, effective electron density, <math id="M7"><mrow><msub><mi>N</mi><mrow><mtext>eff</mtext></mrow></msub><mo>,</mo></mrow></math> and mass density, should ideally be the same as the radiological properties of water or tissue [ 45 45 ]. In a number of studies, a single Z eff was calculated to support water equivalence of 3D dosimeters used in radiotherapy dosimetry (for an example, see [ 46 46 ]). In addition, energy-dependent data for the effective atomic number of photon interaction, <math id="M8"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PI</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo>,</mo></mrow></math> which is equivalent to taking into account the variation in the mass attenuation coefficient <math id="M9"><mrow><mrow><mrow><mo>,</mo><mtext> </mtext><mi>μ</mi></mrow><mo>/</mo><mrow><mi>ρ</mi><mo>,</mo></mrow></mrow></mrow></math> with photon energy, have been presented elsewhere for some 3D dosimeters [ 47 47 ]. On the other hand, only a few studies dealing with variations in the effective atomic number for photon energy absorption, <math id="M10"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> , which is equivalent to taking into account the variation in the mass energy absorption coefficient, <math id="M11"><mrow><mrow><mrow><msub><mi>μ</mi><mrow><mi>e</mi><mi>n</mi></mrow></msub></mrow><mo>/</mo><mrow><mi>ρ</mi><mo>,</mo></mrow></mrow></mrow></math> with photon energy, have been reported [ 48 48 ]. The dose is more strongly related to the mass absorption coefficient. It is therefore generally accepted that <math id="M12"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> is more appropriate than <math id="M13"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PI</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> for water (or tissue) equivalence. This motivated us to conduct the studies described herein. Attenuation of fast neutrons by hydrogenous materials may be approximately calculated using the empirical Albert–Welton kernel and removal cross-sections [ 49 49 ]. The macroscopic effective cross-section for removal of fast neutrons, for simplicity referred to as removal cross-section, ∑ R (cm -1 ), is the probability that a fast or fission-energy neutron undergoes one collision that removes it from the group of penetrating, non-collided neutrons [ 50 50 ]. Here, attenuation or "removal" implies removal from the group of fast neutrons. Because 3D dosimeters in general have sufficient hydrogen content, they can be considered as ideal substances for application of the removal cross-section concept. Moreover, fast neutrons are used for treating certain types of cancer. These particles are also likely to be advantageous over other particles that are used in radiation therapy, such as photons, electrons, and protons, owing to their high linear energy transfer (LET) radiation and because damage is inflicted primarily by nuclear interactions [ 51 51 ]. However, to the best of the author’s knowledge, no reports have been published regarding water equivalence of 3D dosimeters with respect to the attenuation of fast neutrons based on the concept of removal cross-section. In this study, water equivalence of 3D dosimeters is discussed from the point of view of photon energy absorption and fast neutron attenuation coefficient. The calculated values of effective atomic numbers for photon energy absorption, <math id="M14"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo>,</mo></mrow></math> over a wide range of energies (10 keV to 20 MeV) as well as the removal cross-sections of fast neutrons, Σ R , were considered for 27 different types of 3D dosimeters. For comparison, these parameters were also evaluated for water, soft tissue, brain tissue, muscle, and bone. 2. Materials and Methods 1 Table 1 Table 1 lists the elemental compositions, expressed as percentage by mass, of the 3D polymers and micelle gels that were considered in this study. a Ref. [ 16 16 ] b Ref. [ 38 38 ] c Ref. [ 37 37 ] d Ref. [ 26 26 ] e Ref. [ 61 61 ] 1 1 Abbreviationor acronym 1 1 Meaning 1 1 wH 1 1 wC 1 1 wN 1 1 wO 1 1 wNa 1 1 wP 1 1 wS 1 1 wCl 1 1 wCu 1 1 Hypoxic 1 1 PAG 1 1 Polyacrylamide gel 1 1 10.7367 1 1 6.2009 1 1 2.1804 1 1 80.8820 1 1   1 1   1 1 1.54E-4 1 1   1 1 5.06E-4 1 1 BANG-1 1 1 BIS, acrylamide, nitrogen, and gelatin 1 1 10.7685 1 1 5.6936 1 1 2.0063 1 1 81.5316 1 1 BANG-2 1 1 Refers to successor of BANG-1 1 1 10.6369 1 1 5.6728 1 1 1.4152 1 1 81.7004 1 1 0.5748 1 1 BANG-3 a 1 1 Refers to successor of BANG-2 1 1 10.5100 1 1 5.6400 1 1 1.3500 1 1 81.7300 1 1 0.5800 1 1   1 1   1 1   1 1   1 1 Reduced toxic 1 1 VIPAR 1 1 n-Vinylpyrrolidone argon 1 1 10.7321 1 1 7.1825 1 1 2.0638 1 1 80.0217 1 1 PABIG 1 1 Polyethylene glycol diacrylate bis-gelatin 1 1 10.6454 1 1 6.8373 1 1 1.5649 1 1 80.9524 1 1   1 1   1 1   1 1   1 1   1 1 Normoxic 1 1 MAGIC 1 1 Methacrylic ascorbic acid ingelatin initiated by copper 1 1 10.5473 1 1 9.2231 1 1 1.3916 1 1 78.8373 1 1   1 1   1 1 3.00 E-4 1 1   1 1 5.00 E-4 1 1 MAGAS 1 1 Methacrylic acid gelatin withascorbic acid 1 1 10.5087 1 1 9.3591 1 1 1.3799 1 1 78.7523 1 1 ABAGIC 1 1 Ascorbic acid, bis-acrylamide, in gelatin initiated by copper 1 1 10.5263 1 1 8.963 1 1 3.105 1 1 77.4054 1 1   1 1   1 1 3.00 E-4 1 1   1 1 5.00 E-4 1 1 MAGAT 1 1 Methacrylic acid, gelatin, and tetrakis 1 1 10.522 1 1 9.5417 1 1 1.366 1 1 77.6988 1 1   1 1 0.4064 1 1   1 1 0.4651 1 1 PAGAT 1 1 Polyacrylamide gel and tetrakis 1 1 10.7257 1 1 6.2174 1 1 1.9688 1 1 80.2166 1 1   1 1 0.4064 1 1   1 1 0.4651 1 1 VIP b 1 1 Refers to normoxic formulation of VIPAR 1 1 10.4521 1 1 11.6534 1 1 2.9215 1 1 74.9725 1 1 nPAG 1 1 Normoxic polyacrylamide gel 1 1 10.7107 1 1 6.5251 1 1 2.1814 1 1 80.1385 1 1   1 1 0.5748 1 1   1 1 0.2371 1 1 nMAG 1 1 Normoxic methacrylic acid based gel 1 1 10.6775 1 1 7.5066 1 1 1.3868 1 1 80.2527 1 1   1 1 0.0822 1 1   1 1 0.0941 1 1 NIPAM1 c 1 1 N-Iso propyl acrylamide 1 1 10.8055 1 1 6.5998 1 1 1.7531 1 1 79.9702 1 1   1 1 0.4064 1 1   1 1 0.4651 1 1 NIPAM2 d 1 1 N-Iso propyl acrylamide 1 1 10.6400 1 1 13.6400 1 1 3.1660 1 1 72.2300 1 1   1 1 0.1534 1 1   1 1 0.1756 1 1 NIPAM3 d 1 1 N-Iso propyl acrylamide 1 1 11.1700 1 1 29.9400 1 1 3.2820 1 1 55.2800 1 1   1 1 0.1534 1 1   1 1 0.1756 1 1 HEAG 1 1 Hydroxy-ethyl-acrylate gel 1 1 10.7641 1 1 5.7243 1 1 1.4152 1 1 82.0964 1 1 Micelle e 1 1 MGDF1 1 1 Refers to micelle gel dosimeter formulation 1 1 1 11.0400 1 1 2.2900 1 1 0.0100 1 1 86.4900 1 1   1 1   1 1   1 1 0.1700 1 1 MGDF2 1 1 Refers to micelle gel dosimeter formulation 2 1 1 10.8700 1 1 3.7900 1 1 0.0100 1 1 84.1400 1 1 0.1600 1 1   1 1 0.1100 1 1 0.9100 1 1   Table 3 Table 3 lists the chemical formulae and fractional weights of six PRESAGE formulations that were studied here. Excluding PRESAGE 6, which does not contain any halogens, the other formulations have different halogen contents. PRESAGE formulations 3–6 contain very small concentrations (≤0.03 wt%) of metal compounds. Metal compounds accelerate the polymerization process, improve the post-irradiation response stability, and yield good sensitivity to radiation [ 52 52 ]. a Ref. [ 25 25 ] b Ref. [ 22 22 ] c Ref. [ 52 52 ] d Ref. [ 60 60 ] 1 1 Material 1 1 Formula 1 1 w H 1 1 w C 1 1 w N 1 1 w O 1 1 w S 1 1 w Cl 1 1 w Zn 1 1 w Br 1 1 w Sn 1 1 PRESAGE1 a 1 1 C 1758 N 121 H 3000 O 442 S 4 Cl 30 Br 1 1 1 8.8474 1 1 61.7815 1 1 4.9589 1 1 20.6912 1 1 0.3753 1 1 3.1119 1 1   1 1 0.2338 1 1 PRESAGE2 b 1 1 C 481 H 842 N 30 O 129 Cl 9 Br 1 1 1 8.925 1 1 60.7555 1 1 4.419 1 1 21.7048 1 1   1 1 3.3555 1 1   1 1 0.8403 1 1 PRESAGE3 c 1 1 C 64951 N 4391 H 113401 O 15791 C l1414 Sn 1 1 1 9.08 1 1 61.9725 1 1 4.8858 1 1 20.07 1 1   1 1 3.9823 1 1   1 1   1 1 0.0094 1 1 PRESAGE4 c 1 1 C 35904 N 2426 H 62685 O 8728 C l819 Zn 1 1 1 9.063 1 1 61.858 1 1 4.8742 1 1 20.0305 1 1   1 1 4.1649 1 1 0.0094 1 1 PRESAGE5 c 1 1 C 50455 N 3417 H 88064 O 12295 C l775 Br 17 Sn 1 1 1 9.1669 1 1 62.5851 1 1 4.9428 1 1 20.3152 1 1   1 1 2.8375 1 1   1 1 0.1403 1 1 0.0123 1 1 PRESAGE6 d 1 1 C 18746 N 1239 H 32825 O 4455 C 360 Sn 1 1 1 9.4176 1 1 65.3203 1 1 4.9398 1 1 20.2885 1 1   1 1   1 1   1 1   1 1 0.0338 The Fricke gel dosimeter and some phantom materials are listed in Table 2 Table 2 . The bone, muscle, and tissue definitions used data that were obtained from the international commission of radiation units and measurements (ICRU) [ 53 53 ]. * Ref. [ 53 53 ] 2 1 Material 1 1 w H 1 1 w C 1 1 w N 1 1 w O 1 1 w Na 1 1 w Mg 1 1 w P 1 1 w S 1 1 w Cl 1 1 w K 1 1 w Ca 1 1 w Fe 1 1 w Cu 1 1 w Zn 2 1 Fricke 1 1 10.736 1 1 2 1 1 0.67 1 1 85.736 1 1 0.0021 1 1   1 1   1 1 0.85 1 1 0.0033 1 1   1 1   1 1 0.0026 1 1   1 1 Phantom* 2 1 Bone 1 1 4.7234 1 1 14.4330 1 1 4.1990 1 1 44.6096 1 1   1 1 0.2200 1 1 10.4970 1 1 0.3150 1 1   1 1   1 1 20.9930 1 1   1 1   1 1 0.0100 1 1   1 1 (cortical) 2 1 Tissue 1 1 10.2000 1 1 14.3000 1 1 3.4000 1 1 70.8000 1 1 0.2000 1 1   1 1 0.3000 1 1 0.3000 1 1 0.2000 1 1 0.3000 1 1   1 1 (soft) 2 1 Muscle 1 1 11.000 1 1 12.3000 1 1 3.5000 1 1 72.900 1 1 0.0800 1 1 0.0200 1 1 0.2000 1 1 0.3000 1 1   1 1 (striated) 2 1 Water 1 1 11.1898 1 1   1 1   1 1 88.8102 1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   2.1. Calculations of the effective atomic number, Z eff , and effective electron density, N eff , over a wide range of energies 2 In composite materials, for photon interactions, the atomic number cannot be represented uniquely across the entire range of energies, as in the case of elements, by a single number. The procedure for calculating the effective atomic number using the direct method has been described elsewhere [ 54 54 ]. The effective atomic numbers of the studied samples were calculated using the following practical formula [ 55 55 ] ‎: <math display="block" id="M15"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo>=</mo><msub><mi>N</mi><mtext>A</mtext></msub><mfrac><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>f</mi><mi>i</mi></msub><msub><mi>A</mi><mi>i</mi></msub><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>μ</mi><mrow><mtext>en</mtext></mrow></msub></mrow><mi>ρ</mi></mfrac><mo stretchy="false">)</mo></mrow><mi>i</mi></msub></mrow></mstyle></mrow><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>f</mi><mi>i</mi></msub><mfrac><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mrow><msub><mi>Z</mi><mi>i</mi></msub></mrow></mfrac><msub><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>μ</mi><mrow><mtext>en</mtext></mrow></msub></mrow><mi>ρ</mi></mfrac><mo stretchy="false">)</mo></mrow><mi>i</mi></msub></mrow></mstyle></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M16"><mrow><msub><mi>N</mi><mtext>A</mtext></msub></mrow></math> is the Avogadro constant and <math id="M17"><mrow><msub><mi>f</mi><mi>i</mi></msub></mrow></math> is the molar fraction of the i th constituent element (normalized, so that <math id="M18"><mrow><mstyle displaystyle="true"><mo>∑</mo><mrow><msub><mrow/><mi>i</mi></msub></mrow></mstyle><msub><mi>f</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow></math> ). Here, <math id="M19"><mrow><mrow><mrow><msub><mi>μ</mi><mrow><mtext>en</mtext></mrow></msub></mrow><mo>/</mo><mi>ρ</mi></mrow></mrow></math> is the mass energy absorption coefficient, which was obtained from the tabulation of Hubbell and Seltzer [ 56 56 ]. Each material was considered to be a mixture of different compounds in various proportions ( Table 1 Table 1 ). The effective electron density, <math id="M20"><mrow><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo>,</mo></mrow></math> expressed in the number of electrons per unit mass, is closely related to the effective atomic number, and is given by <math display="block" id="M21"><mrow><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo>=</mo><msub><mi>N</mi><mtext>A</mtext></msub><mfrac><mrow><mi>n</mi><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow><mrow><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>n</mi><mi>i</mi></msub><msub><mi>A</mi><mi>i</mi></msub></mrow></mstyle></mrow></mfrac><mo>=</mo><msub><mi>N</mi><mtext>A</mtext></msub><mfrac><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow><mrow><mo>〈</mo><mi>A</mi><mo>〉</mo></mrow></mfrac><mo stretchy="false">(</mo><mtext>electrons/g</mtext><mo stretchy="false">)</mo></mrow></math> where <math id="M22"><mrow><mo>〈</mo><mi>A</mi><mo>〉</mo></mrow></math> is the mean atomic mass, which is given by <math display="block" id="M23"><mrow><mo>〈</mo><mi>A</mi><mo>〉</mo><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>f</mi><mi>i</mi></msub><msub><mi>A</mi><mi>i</mi></msub></mrow></mstyle><mo>,</mo></mrow></math> Where A i is the atomic mass. The effective atomic numbers and effective electron densities for photon energy absorption as a function of photon energy (for energies ranging from 10 keV to 20 MeV) were calculated for 27 3D dosimeters, four biological materials, and water. 2.2. Calculation of the effective macroscopic cross-section for removal of fast neutrons, Σ R 2 The concept of the removal cross-section is valid for fast neutrons with energies in the 2–12 MeV range, because in this range the removal cross-section is considered to be nearly constant [ 57 57 ]. The method assumes that collisions with hydrogen atoms are equivalent to absorption events. The removal cross-section for a given compound may be calculated from the value of Σ R or Σ R / ρ for various elements in the compound or mixture, using the mixture rule [ 57 57 ]: <math display="block" id="M24"><mrow><msub><mi>Σ</mi><mtext>R</mtext></msub><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>i</mi></munder><mrow><msub><mi>ρ</mi><mi>i</mi></msub><msub><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi><mtext>R</mtext></msub><mo>/</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mi>i</mi></msub></mrow></mstyle><mo>,</mo></mrow></math> where <math id="M25"><mrow><msub><mi>ρ</mi><mi>i</mi></msub></mrow></math> and <math id="M26"><mrow><msub><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi><mtext>R</mtext></msub><mo>/</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><mi>i</mi></msub></mrow></math> are the partial density (the density as it appears in the mixture) and mass removal cross-section of the i th constituent, respectively. The NXcom computer program [ 50 50 ] was employed for calculating the effective removal cross-sections for the studied 3D dosimeters, water, and phantom materials. 3. results and Discussion 1 3.1. Energy dependence of <math id="M27"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> and <math id="M28"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> ratios 2 To evaluate the degree of water equivalence, we have calculated, for each material, the ratio, <math id="M29"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> , of <math id="M30"><mrow><msub><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>m</mtext></msub></mrow></math> (characterizing the material) to <math id="M31"><mrow><msub><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext></msub></mrow></math> (characterizing water), over the full range of energies (from 1 keV to 20 MeV). These ratios, <math id="M32"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> , are plotted in Figs. 1 Figs. 1 , 2 2 , 3 3 , 4 4 , 5 5 , 6 6 , 7 7 , 8 8 , and 9 9 . The effective electron density N PEAeff is closely related to the effective atomic number, as shown by Eq. 2. Consequently, the energy dependence of <math id="M33"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> is similar to that of <math id="M34"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> . Therefore, only one example for the variation in <math id="M35"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>N</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> with photon energy is given in Fig. 7b Fig. 7b . 3.1.1. Water equivalence of conventional and modern gels 3 Fig. 1 Fig. 1 shows the <math id="M56"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> ratio curves for some 3D polymer gel dosimeters based on hypoxic and reduced toxic monomers for photon energy absorption, as it varies with photon energy. Because all polymer dosimeters contain water as the major constituent, they exhibit similar variations with energy; in most cases, these curves differ from that for water by no more than 2% ( Fig. 1 Fig. 1 ). Considering the mean disparity, the effective atomic number for the photon energy absorption of BANG-1 (as an example of an hypoxic polymer) is most similar to that of water, as shown in Fig. 1a Fig. 1a , with no constituents with Z > 8. The results for the VIPAR formulation, shown in Fig. 1b Fig. 1b , also closely match those for water. As for normoxic gels based on the methacrylic acid as monomers ( Fig. 2a Fig. 2a ), nMAG is the most similar to water in terms of <math id="M57"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> , as shown in Fig. 2a Fig. 2a . Moreover, it was shown that replacing the ascorbic acid by tetrakis, as in MAGAT, makes the <math id="M58"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> matching less tight. For NIPMA normoxic formulations, Fig. 2b Fig. 2b shows that for NIPAM 3, the gel that matches water the least, the <math id="M59"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> values are systematically lower than those for water, especially for photon energies above 400 keV. The low matching may be attributed to the lower oxygen content compared with water. On the other hand, Fig. 2c Fig. 2c shows that <math id="M60"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> change with energy for some other normoxic gel dosimeters that are based on other monomer types. The results in this figure suggest that the formulations based on hydroxyl-ethyl-acrylate (HEAG) and bis-acrylamide (ABAGIC) monomers are the most similar to water. The ratios <math id="M61"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> for the hypoxic PAG dosimeter and its normoxic formulation, nPAG, are plotted in Fig. 3 Fig. 3 . A better similarity to water is observed for the hypoxic formulation compared with its normoxic gel, throughout the entire range of energies. Similar behaviors were observed for the hypoxic BANG-3 gel that is based on the methacrylic acid as a monomer, and those normoxic gels that are based on the same monomer (methacrylic acid with an oxygen scavenger), as shown in Fig. 4 Fig. 4 . Finally, Fig. 5 Fig. 5 shows the ratios <math id="M62"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> for the reduced toxic VIPAR and its normoxic formulation, VIP. The results indicate that the reduced toxic formulation matches water better than its normoxic edition. The results in Fig. 6 Fig. 6 suggest that the <math id="M63"><mrow><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub></mrow></math> values for the Fricke gel are systematically higher than that of water, especially at low energies. Nevertheless, for energies above 100 keV, the results for the Fricke gel dosimeter were in close agreement (2%) with those for water. The <math id="M64"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> curves for two micelle formulations are shown in Fig. 7 Fig. 7 . The MGDF1 formulation exhibits a perfect match to the effective atomic and electron density numbers of water. 3.1.2. Water equivalence of PRESAGE dosimeters 3 The <math id="M65"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Z</mi><mrow><msub><mrow><mtext>PEA</mtext></mrow><mrow><mtext>eff</mtext></mrow></msub></mrow></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> ratio curves for PRESAGE dosimeters ( Fig. 8 Fig. 8 ) show a maximal difference at ~35 keV, indicating that the effective atomic numbers differ from that of water by factors as large as 1.76 (PRESAGE 2) and 1.37 (PRESAGE 1) and as small as 0.74 (PRESAGE 4 and PRESAGE 6). However, all PRESAGE formulations come close to matching water for energies higher than 100 keV. Moreover, PRESAGE 3 and PRESAGE 5 formulations exhibit good matching for the entire range of energies, with the corresponding maximal differences, at low energies, being under 6%. Conventional and modern gels and PRESAGE dosimeters typically match water better than water matches some tissues ( Fig. 9b Fig. 9b ), and in most cases, slight differences in effective atomic number between water and dosimeters may be considered insignificant, especially over the therapeutic range of energies, 1–20 MeV. 3.2. Effective mass removal cross-section of fast neutrons, Σ R /ρ 2 Fig. 10 Fig. 10 shows the ratios, <math id="M66"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi><mtext>R</mtext></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> , of Σ R for the various 3D dosimeters, to Σ R of water. Because all dosimetric materials are predominantly composed of water, the results show that, except for PRESAGE formulations, the removal cross-sections of fast neutrons of 3D dosimeters and water agree well (1.5–3%). Owing to their low hydrogen and oxygen concentrations, differences (~7.5%) from the values for water were observed for PRESAGE 1 and PRESAGE 2. Excluding the NIPMA3 formulation, all 3D dosimeter materials had lower Σ R values compared with water. It is also obvious that the considered micelle gel dosimeters exhibit excellent matching to water. Table 4 Table 4 lists the calculated values of Σ R for the studied materials. From Table 4 Table 4 and Fig. 11 Fig. 11 , it can be seen that the removal cross-section values for all phantom materials are quite close (~ ±1.5%) to that of water - except the bone tissue, which varied by ~38%, which was owing to a lower hydrogen content. 1 1 Material 1 1 w H 1 1 Σ R 1 1 <math id="M69"><mrow><msubsup><mrow><mo stretchy="false">(</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><msub><mrow/><mtext>R</mtext></msub></mrow></mstyle><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mi>m</mi></msubsup></mrow></math> 1 1 Water 1 1 0.11190 1 1 0.10288 1 1 1.000 1 1 FRICKE 1 1 0.00107 1 1 0.10047 1 1 0.977 1 1 PAG 1 1 0.10737 1 1 0.10105 1 1 0.982 1 1 BANG-1 1 1 0.10769 1 1 0.10117 1 1 0.983 1 1 BANG-2 1 1 0.10637 1 1 0.10038 1 1 0.976 1 1 BANG-3 1 1 0.10510 1 1 0.09958 1 1 0.968 1 1 PABIG 1 1 0.10645 1 1 0.10058 1 1 0.978 1 1 VIPAR 1 1 0.10732 1 1 0.10112 1 1 0.983 1 1 ABAGIC 1 1 0.10526 1 1 0.10019 1 1 0.974 1 1 HEAG 1 1 0.10764 1 1 0.10113 1 1 0.983 1 1 MAGAS 1 1 0.10509 1 1 0.10113 1 1 0.983 1 1 MAGAT 1 1 0.10522 1 1 0.10002 1 1 0.972 1 1 MAGIC 1 1 0.10547 1 1 0.10026 1 1 0.974 1 1 NIPAM1 1 1 0.10806 1 1 0.10134 1 1 0.985 1 1 NIPAM2 1 1 0.10640 1 1 0.10123 1 1 0.984 1 1 NIPAM3 1 1 0.11170 1 1 0.10577 1 1 1.028 1 1 nMAG 1 1 0.10678 1 1 0.10079 1 1 0.980 1 1 nPAG 1 1 0.10711 1 1 0.10098 1 1 0.982 1 1 PAGAT 1 1 0.10726 1 1 0.10086 1 1 0.980 1 1 VIP 1 1 0.10452 1 1 0.10003 1 1 0.972 1 1 MGDF1 1 1 0.11040 1 1 0.10224 1 1 0.994 1 1 MGDF2 1 1 0.10870 1 1 0.10130 1 1 0.985 1 1 PRESAGE1 1 1 0.08847 1 1 0.09545 1 1 0.928 1 1 PRESAGE2 1 1 0.08925 1 1 0.09563 1 1 0.929 1 1 PRESAGE3 1 1 0.09080 1 1 0.09673 1 1 0.940 1 1 PRESAGE4 1 1 0.09063 1 1 0.09660 1 1 0.939 1 1 PRESAGE5 1 1 0.09167 1 1 0.09742 1 1 0.947 1 1 PRESAGE6 1 1 0.09418 1 1 0.09954 1 1 0.968 1 1 Bone(Cortical) 1 1 0.047234 1 1 0.06367 1 1 0.619 1 1 Brain tissue 1 1 0.10700 1 1 0.10147 1 1 0.986 1 1 Tissue(soft) 1 1 0.10200 1 1 0.10466 1 1 1.017 1 1 Muscle(Striated) 1 1 0.10997 1 1 0.10320 1 1 1.003 Considering the values of <math id="M71"><mrow><msubsup><mrow><mo stretchy="false">(</mo><msub><mi>Σ</mi><mtext>R</mtext></msub><mo stretchy="false">)</mo></mrow><mtext>w</mtext><mtext>m</mtext></msubsup></mrow></math> , the 3D dosimetric materials were found to match water better than water matches some tissues (as indicated in Fig. 11 Fig. 11 and Table 4 Table 4 ). As mentioned above, the concept of the removal cross-section is based on the presence of hydrogen. Therefore, Figs. 12 Figs. 12 and 13 13 show the variation of the removal cross-section of fast neutrons with the hydrogen content. The results presented in these figures show that Σ R systematically increases with increasing the dosimetric material’s hydrogen content ( w H ). In addition, it has been found that the variation for most polymer gel dosimeters and different PRESAGE formulations can be captured by a simple linear regression equation, with excellent correlation coefficient R 2 , as shown in Figs. 12 Figs. 12 and 13 13 , respectively. 3.3. Accuracy of calculations 2 Eq.1 prescribes that the accuracy of the effective atomic number calculation is basically determined by the accuracy of the elemental mass attenuation coefficient, ( μ/ρ ) i . For energies in the range of interest to medical and biological applications, from 5 keV to a few MeV, Hubbell showed that the uncertainty of ( μ/ρ ) i is on the order of 1–2%. Discrepancies, between experimental results and theoretical calculations, of 25–50%, are known to occur for low energies, in the 1–4 keV range [ 58 58 ]. Therefore, our calculated Zeff values are accurate to within a few percent, for energies above 5 keV. On the other hand, the values of Σ R that were obtained from Eq. 4 are usually accurate to within ~10% of the corresponding experimentally determined values [ 59 59 ]. 4. Conclusion 1 Here, we have presented effective atomic numbers, and effective electron densities for photon energy absorption as well as the removal cross-sections for 31 dosimetric and phantom materials. The results are presented relative to water, to allow direct comparisons over a range of energies. Regarding the mean disparity over a wider range of energies our results suggest, broadly, that highly toxic and reduced toxic polymer gels typically match water better than water matches normoxic gels, and replacing the ascorbic acid by tetrakis yields worse matching. More specifically, the results show that the 3D dosimeters that exhibit the closest radiological water equivalence are PAG, VIPAR, nMAG, NIPAM2, HEAG, PRESAGE3, PRESAGE5, and MGDF1 formulations. PRESAGE1 and PRESAG2, Fricke and NIPAM3 dosimeters, on the other hand, were found to be the least water equivalent over all energies. With regard to the removal cross-sections, which were calculated here for the first time, it was found that the MGDF1 micelle gel, conventional (polymer and Fricke) gels and PRESAGE formulations typically very closely, closely, and considerably match the values for water, respectively. Differences in Σ R between water and dosimeters were attributed to hydrogen content. 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