

ORIGINAL ARTICLE 



Year : 2014  Volume
: 39
 Issue : 1  Page : 2431 

Effective atomic numbers of some tissue substitutes by different methods: A comparative study
Vishwanath P Singh, NM Badiger
Department of Physics, Karnatak University, Dharwad, Karnataka, India
Date of Submission  13May2013 
Date of Decision  04Nov2013 
Date of Acceptance  04Nov2013 
Date of Web Publication  20Jan2014 
Correspondence Address: Vishwanath P Singh Health Physics Section, Kaiga atomic Power Station3&4, NPCIL, Karwar, 581400, Karnatak India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09716203.125489
Abstract   
Effective atomic numbers of some human organ tissue substitutes such as polyethylene terephthalate, red articulation wax, paraffin 1, paraffin 2, bolus, pitch, polyphenylene sulfide, polysulfone, polyvinylchloride, and modeling clay have been calculated by four different methods like AutoZ _{eff,} direct, interpolation, and power law. It was found that the effective atomic numbers computed by AutoZ _{eff} , direct and interpolation methods were in good agreement for intermediate energy region (0.1 MeV < E < 5 MeV) where the Compton interaction dominates. A large difference in effective atomic numbers by direct method and AutoZ _{eff} was observed in photoelectric and pairproduction regions. Effective atomic numbers computed by power law were found to be close to direct method in photoelectric absorption region. The AutoZ _{eff} , direct and interpolation methods were found to be in good agreement for computation of effective atomic numbers in intermediate energy region (100 keV < E < 10 MeV). The direct method was found to be appropriate method for computation of effective atomic numbers in photoelectric region (10 keV < E < 100 keV). The tissue equivalence of the tissue substitutes is possible to represent by any method for computation of effective atomic number mentioned in the present study. An accurate estimation of Rayleigh scattering is required to eliminate effect of molecular, chemical, or crystalline environment of the atom for estimation of gamma interaction parameters.
Keywords: Atomic numbers, Compton scattering, gamma, mixture rule, tissue substitutes, Rayleigh scattering
How to cite this article: Singh VP, Badiger N M. Effective atomic numbers of some tissue substitutes by different methods: A comparative study. J Med Phys 2014;39:2431 
How to cite this URL: Singh VP, Badiger N M. Effective atomic numbers of some tissue substitutes by different methods: A comparative study. J Med Phys [serial online] 2014 [cited 2019 Jan 19];39:2431. Available from: http://www.jmp.org.in/text.asp?2014/39/1/24/125489 
Introduction   
Simulation of depthdose distribution inside human organs and tissues is made by tissue equivalent materials. The tissue equivalent materials are called as tissue substitutes for various tissues, organs of the human body, and tissue components having similar properties with respect to ionizing radiation. Average soft tissues are mainly composed of lowatomic number (Z) such as H, C, N, O, and so on. ^{[1]} International Commission on Radiation Units (ICRU) report ^{[2]} describes various types of tissue substitutes which are used in medicine, radiation protection, and radiolobiology to calibrate the radiation detectors and application in nuclear engineering for realistic body phantom. ^{[3],[4]} The effective atomic number is most vital parameters for tissue equivalence, radiation absorption, radiation scattering, and shielding effectiveness for gamma and neutron for compound materials.
Waxes are organic compounds which consist of longalkyl chains. Polymers are made from the monomer propylene which is rugged and highly resistant to the chemicals. Polymers are lowZ, nonflammable, light weight, high durability, ease processing, economical, and stable against environment. Nowadays, radiation shielding for gamma rays as well as neutron ismade of polymer matrix lead shielding materials. Various types of waxes, plastics, and polymers are being used as tissue substitutes in field of medical, dosimetry, and radiological protection.
Investigators have studied effective atomic numbers of gaseous mixtures, ^{[5]} composite materials, ^{[6]} solutions, ^{[7]} dosimetric materials, ^{[8],[9]} and biological materials. ^{[10],[11],[12],[13]} Several photon interaction studies are reported for lowZ materials ^{[14],[15],[16],[17]} for X and gammaray photons. Rubbers containing varying degree of carbon show a wide range of effective atomic numbers. ^{[18]} Mass attenuation coefficients of few common tissue substitutes have been reported. ^{[19]} Recently, alcohol tissue substitutes for human organs have been investigated. ^{[20]} The equivalence of tissue substitutes for experimental radiation physics has been reported by (⊠/ρ) _{substitute} /(⊠/ρ) _{tissue} for energy 0.01100 MeV and linear attenuation coefficients at 6080 keV for radiation characteristic of tissue substitutes. ^{[21]}
The bolus tissue substitute is easily made, inexpensive, and moldable during treatment. ^{[22],[23]} The paraffin is lipid equivalent material ^{[2]} and polyethylene terephthalate is a good skeletoncartilage substitute. ^{[2]} Red articulation wax is used for dosimetry phantom preparation and pitch is very good skin equivalent material. The effective atomic number dependency upon photon energy is evaluated by various methods like AutoZ _{eff} , ^{[24]} direct method, ^{[25]} and logarithmic interpolation method. These methods use different input parameters (e.g. atomic crosssections, atomic numbers, and attenuation coefficients) for computation of effective atomic numbers. The direct and interpolation methods use mixture rule ^{[26],[27]} for computation of mass attenuation coefficients of the compound or mixture. The mixture rule is simple additive law for weighted sum of mass attenuation of constituent elements. The photon interaction processes are energy dependent and photoelectric effect (E <0.1 MeV) is largely energy dependent. Appreciable differences between experiment and theoretical mass attenuation coefficients of polymers (59.5 keV), vitamins (30.8259.54 keV), amino acids (8.0480.99 keV), Au alloys (59.5 keV), and brass alloys (81 keV) are noted. ^{[13],[28],[29],[30],[31]} Effective atomic numbers of highZ compounds like magnesium ferrite and borosilicate glass are observed to be following the mixture rule in lowenergies. ^{[32],[33]} It is expected that the effective atomic numbers computed by different theoretical methods should be identical in magnitude for the compound materials at selected energies. Presently, such type of comparative study is not found in the literature and was studied first time.
In view of above, we have chosen some important tissue substitutes (1 ≤ Z < 18) given in [Table 1] for computation of effective atomic numbers by AutoZ _{eff} , direct, interpolation, and power law methods for various medical applications. This study will benefit for readily available effective atomic numbers of the tissue substitutes and choice for appropriate method for computation of effective atomic numbers.
Computational work and theoretical background
Mass attenuation coefficients and effective atomic numbers of compound materials are derived by mixture rule using mass attenuation coefficients and atomic crosssections of the elements. In our study, we have computed the mass attenuation coefficients and effective atomic numbers of the tissue substitutes given in [Table 1]. These materials have been taken from the references. ^{[2],[34],[35]} The attenuation crosssection data can be found for 100 of elements in energy range of 1 keV100 GeV by XCOM program. ^{[36]} The XCOM data have been transformed to userfriendly software package WinXCom ^{[37]} for the window platform which is easily exportable in the excel files. Using WinXCom, mass attenuation coefficients and attenuation crosssection data were generated for the elements in photon energy region 10 keV20 MeV. The atomic numbers and atomic masses of the elements are taken from atomic weight of elements 2009, International union of Pure and Applied Chemistry. ^{[38]}
Mass attenuation coefficients
AutoZ _{eff}
AutoZ _{eff} is userfriendly software in visual basic for rapid computation of the energydependent effective atomic numbers, average atomic numbers, and spectralweighted mean atomic numbers. AutoZ _{eff} surpasses dubious powerlaw approach. In this method, Z _{eff} , _{Auto} is determined via exploitation of the smooth correlation between atomic crosssection and atomic number. A matrix of crosssections was constructed spanning atomic number Z = 1100 for photon energies ranging between 10 keVand1 GeV and crosssections of polyelemental media are calculated by linear additivity. The crosssectional values are constructed with the crosssection matrix as a function of Z and an effective atomic number for any energy is obtained by interpolation of Z values between adjacent crosssection data. ^{[24]}
Direct method
Computation of the effective atomic number, Z _{eff, PI} of the selected tissue substitutes for total gamma photon interaction has been carried out by practical formula. ^{[25]} The mass attenuation coefficients of the elements have been obtained from WinXcom computer program. The effective atomic number, Z _{eff, PI} is given by
where ∫_{1} and ∫_{2} are the elemental crosssection (barn/atom) in between which the atomic crosssection ∫ of the tissue substitutes and Z _{1} and Z _{2} are atomic numbers of the elements (dimensionless) corresponding to the cross sections ∫_{1} and ∫_{2} , respectively.
Power law method
The effective atomic number, Z _{eff} , _{PL} , of a tissue substitute material by power law can be calculated according to the following equation:
where a _{1} , a _{2} ,… are the fractional contents of electron belonging to element Z _{1} , Z _{2} ,.………., respectively, n _{i} is the numberof electrons, in one mole, belonging to each element Z _{i} and N _{A} is the Avogadro's number. The x values are in 2.94 ^{[39]} and 3.5 ^{[40]} ranges.
Errors
AutoZ _{eff} software calculates effective atomic numbers with errors of 1%2% for photon energies 10 keV1000 MeV. This errors increase to 25%50% below10 keV. ^{[24]} The errors in mass attenuation coefficients of the elements is about 1% for lowZ (1 < Z < 8) in the energy region where Compton scattering dominates (30 keV100 MeV). ^{[41]} Below 30 keV and above 100 MeV, the errors are as much as 5%10%. For mediumZ elements sodium through copper, the errors are 1%2% for energies 10 keV1 MeV and 2%3% for energies 1100 MeV. Medical, biological, and industrial, applications and transportation tend to use sources with photon energies above 5 keV, so that the errors in our results may not have any practical impact. The power law method is an inaccurate method as the exponent of 2.94 relates to an empirical formula for the photoelectric process which incorporates a ''constant'' of 2.64 × 10^{−26} , which is in fact not a constant but rather a function of the photon energy. A linear relationship between Z ^{2.94} has been shown for a limited number of compounds for lowenergy xrays. ^{[42]}
Result and Discussion   
Variation of mass attenuation coefficients of selected tissue substitutes is shown in [Figure 1] for photon energy range 1 keV100 GeV. The effective atomic numbers of the selected tissue substitutes by AutoZ _{eff} , direct, interpolation methods, and power law is shown in [Figure 2]aj. The variations of these parameters were explained in detail in the next section.  Figure 1: Mass attenuation coefficients of tissue substitutes in photon energy range 1 keV100 GeV
Click here to view 
 Figure 2: Effective atomic numbers of Red Articulation Wax, Polyethylene Terephthalate, Paraffin 1, Paraffin 2, Bolus, Pitch, Poly  Phenylene Sulfide, PolySulfone, Polyvinylchloride and Modeling Clay for photon energy ranges from 10 keV to 20 MeV by AutoZeff, direct, interpolation, and power Law
Click here to view 
Mass attenuation coefficient
[Figure 1] shows the mass attenuation coefficient, ∝/ρ variation of tissue substitutes in photon energy range 1 keV100 GeV. The [Figure 1] shows that the ∝/ρ values of the tissues substitutes decrease with increase in the photon energies in lowenergy, minimum in intermediateenergy, and constant in highenergy region. The variation in ∝/ρ values with energy can be explained by partial photon interaction processes as the dependence of the total atomic crosssection on atomic number and photon energy. The interaction crosssection is directly proportional to Z ^{45/} E ^{3.5} for photoelectric absorption in lowenergy; therefore, ∝/ρ values of the tissues substitutes reduces sharply. In Compton scattering region (intermediateenergy region), the interaction crosssection is dependent upon Z/E, whereas crosssection is directly proportional to Z ^{2} for pairproduction (highenergy). The ∝/ρ values of all the selected tissues substitutes merges in Compton scattering (100 keV < E < 10 MeV) region.
The ∝/ρ values of Griffith breast tissue substitutes and breast of human body tissue [2] were compared with the selected tissue substitutes. It was found that the Polyethylene Terephthalate polyphenylene sulfide, and polysulfone, can replace Griffith tissue in photon energy region 0.1010 MeV, whereas Polyethylene Terephthalate is most suitable for entire energy region
Effective atomic number
Effective atomic numbers of the selected tissue substitutes by different four methods (AutoZ _{eff} , direct, interpolation, and power law) is shown [Figure 2]aj. From [Figure 2], it is observed that the effective atomic numbers computed by AutoZ _{eff} , direct and interpolation methods are in good agreement and almost identical in the energy region 0.15 MeV where the Compton interaction dominates. The effective atomic number values were found constant in the intermedium photon energy region, whereas significant variation was observed in the lower (0.010.1 MeV) as well as in the higherenergy regions (520 MeV). The effective atomic numbers computed by direct method were higher in photoelectric absorption and pairproduction regions as compared with interpolation method. The effective atomic numbers calculated by AutoZ _{eff} were 6.016.23, 4.556.07, 5.906.27, 5.846.21, 5.966.28, 6.877.22, 5.099.07, 4.306.88, 5.3411.02, and 7.638.14 for RAW, PETE, PF1, PF2, BOLUS, PITCH, PPS, PSU, PVC and MC respectively, by direct method were 6.016.46, 4.556.81, 5.906.6, 5.846.52, 5.966.58, 6.877.63, 5.1113.64, 4.309.71, 5.3416.05, and 7.628.96 for RAW, PETE, PF1, PF2, BOLUS, PITCH, PPS, PSU, PVC and MC, respectively; whereas by interpolation method 6.006.29, 4.576.09, 5.916.34, 5.856.27, 5.976.35, 6.887.28, 5.119.08, 4.406.91, 5.3711.02, and 7.648.18 for RAW, PETE, PF1, PF2, BOLUS, PITCH, PPS, PSU, PVC, and MC, respectively. The effective atomic numbers by power law is shown in the graphs for x equals to 2.94 and 3.5. The independency of effective atomic numbers on photon energy in Compton dominant region can be found in various literatures for materials containing low and highZ elements; however, photoelectric absorption and pairproduction region are not found experimentally. ^{[6],[7],[9]} The effective atomic numbers computed by power law were found to be in the photoelectric absorption region close to direct method. In Compton scattering region, it was found that the effective atomic numbers by all the three methods were in good agreement.
The variation in effective atomic numbers of the tissue substitutes by AutoZ _{eff} , direct, and interpolation methods may be due to basic concept and input parameters for computation. In the AutoZeff effective atomic numbers are determined via exploitation of the smooth correlation between atomic crosssections, atomic numbers, and mass attenuation coefficients. The crosssections of polyelement compounds are calculated by linear additivity. The effective atomic numbers of the tissue substitutes was calculated by interpolation of Z of adjacent crosssection data of crosssection matrix as function of Z. Large variation in effective atomic number by direct method and interpolation method is observed because of different weight fractions are given as input in computation.
For tissue equivalence of the tissue substitutes, a graph for substitute (PVC) and tissue (cortical bone) has been plotted in [Figure 3]. The ratio of effective atomic numbers by direct, AutoZ _{eff} , and interpolation for substitute to tissue along with (∝/ρ)_{substitute}/ (∝/ρ)_{tissue} reported by White for photon energies 0.01, 0.1, 1, 10, and 100 MeV ^{[21]} are shown on abscissa; whereas radiation characteristics (effective atomic numbers and mass attenuation coefficients) are shown in the ordinate. It is evident from the [Figure 3] that the tissue equivalence of PVC is far away for (∝/ρ)_{substitute}/ (∝/ρ)_{tissue} method in low energy (<0.1 MeV) compared with effective atomic number method. In Compton scattering region, a good correlation is to be noted. With increase in the photon energy (>10 MeV), the deviations by both the methods were observed of same order. The ratio of Z _{eff, PL} of PVC to cortical bone was found to be 1.05 and 1.03 for x values of 2.94 and 3.5 respectively. Therefore, the doublevalued Z _{eff, PL} using power law method shows insignificant variation of the radiation characteristics of PVC and cortical bone. The difference between the ratios of effective atomic numbers of tissue substitutes to tissues using above methods (direct, AutoZ _{eff} , and interpolation) was noted insignificant. Therefore, it can be concluded that the tissue equivalence of the tissue substitutes is possible to represent by using above methods (AutoZ _{eff} , direct, interpolation) for computation of effective atomic numbers.  Figure 3: Comparison of radiation characteristics (mass attenuation coefficients and effective atomic number) of poly  vinyl  chloride and cortical bone at photon energies 0.01, 0.1, 1, 10, and 100 MeV
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In case of photoelectric absorption region, the interaction process is affected near absorption edges of the elements which may perturb the wavefunction of the compound or mixture. The results support the conclusion elaborated in earlier several reports ^{[41],[42]} that the mixture rule is valid for the evaluation of photon attenuation coefficients for compounds in Compton scattering region. Therefore, mixture rule is limited to Compton scattering region and demand for further experimental data for smoothening of the interaction cross sections and it's applicability in photoelectric (10 keV < E < 100 keV) and pairproduction regions (E > 10 MeV). From [Figure 2], it is evident that in photoelectric absorption region the effective atomic numbers by interpolation method and AutoZ _{eff} are far away from direct method. The experimental data for effective atomic numbers of various compounds/mixtures ^{[13],[28],[29],[30],[31]} and power law method results were found to be closest to the direct method; therefore, it is concluded that the direct method may be appropriate for computation of effective atomic numbers in photoelectric absorption region.
Discrepancies
The mass attenuation coefficients of chemical compound or mixture is evaluated by weighted sum of mass attenuation coefficients of the constituent elements using mixture rule. The mixture rule is not valid near kabsorption edge of the compounds and mixtures. ^{[43],[44],[45],[46]} The mixture rule doesnot consider the molecular, chemical, or crystalline environment of the atom which results in change in the atomic wavefunction. With the exception of the fine structure region above absorption edge (>10 keV) errors from these sources are expected to be a few percent, whereas at low energies (10100 eV), errors of as much as a factor of two can occur. In case of AutoZ _{eff} , the crosssections of polyelemental media are also calculated by linear additive method.
The study upon molecular, chemical, or crystalline environment of the atom ^{[43],[44]} reveals that there is requirement for revision of the crosssection and attenuation coefficients libraries in photoelectric absorption region. The reason behind is that the actual energy at which a particular photoelectric absorption edge of an element occurs is dependent on the chemical state of the absorbing atom and the nature of the chemical environment: For example, a chemical shift in the position of the kedge in iron can be seen as the oxidation state of the atom is changed. ^{[47]} Highresolution studies show that within the edge region the structure depends on the electronic structure of the absorbing atom. The actual height of the edge is also sensitive to the atomic environment and hence there may be uncertainty in determining the kjump ratios. ^{[48]} The errors in theoretical estimates of Rayleigh scattering may be the cause of part of the deviation in absorption coefficient are observed. ^{[48]} It may be concluded that the theoretical estimates for Rayleigh scattering for the photon energies very close to the Kedge are not accurate. An accurate estimation of Rayleigh scattering is required to eliminate effect of molecular, chemical, or crystalline environment of the atom. Therefore, the discrepancies in the effective atomic number may be diminished by considering the molecular, chemical, or crystalline environment of the atom for deriving the atomic crosssection, interaction and attenuation coefficients.^{[49]}
Conclusions   
In the present study, we have compared effective atomic numbers of some important tissue substitutes by four methods (i.e., AutoZ _{eff} , direct method, interpolation method, and power law). The direct and power law methods are applicable in lowphoton energy (10 keV < E < 100 keV) where photoabsorption dominate and intermediateenergy (0.1 MeV < E < 10 MeV) where Compton scattering interaction dominates. A large difference in effective atomic numbers by direct method and AutoZ _{eff} methods was observed in photoelectric and pairproduction regions. An accurate estimation of Rayleigh scattering is required to eliminate effect of molecular, chemical, or crystalline environment of the atom. Therefore, the discrepancies in the effective atomic number may be diminished by considering the molecular, chemical, or crystalline environment of the atom for deriving the atomic crosssection, interaction, and attenuation coefficients. Direct method was found to be an appropriate method for deriving effective atomic numbers in photoelectric region (10 keV < E < 100 keV).
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[Figure 1], [Figure 2], [Figure 3]
[Table 1]
