Water equivalent phantom materials for ¹⁹²Ir brachytherapy

The radial dose functions, g_L(r), of the absorbed dose to water in water phantoms with various values of radius, R, and height, H, are shown in figure 1, hereby illustrating the influence of the phantom size. The small, downward curved 'hooks' of the full and the dashed curves appearing near the outer boundaries of the phantoms illustrate the absence of backscattering from the outside. Furthermore, in figure 1, the dashed curve family with logarithmic ordinate scale shows the radial variation of the absorbed dose to water, D(r), normalized to its value at r  =  1 cm. The open circles illustrate that the dose profiles closely follow a 1/r ² law from r  =  1 cm up to large source distances. Near r  =  10 cm, D(r) has decayed to 1% of its value achieved at r  =  1 cm; therefore, source distances up to 10 cm can be roughly regarded as the zone of clinical interest. According to Perez-Calatayud et al (2012), the dose profile in a substitute material should closely approximate the dose profile in water within this zone. These results compare well with the literature (Ballester et al 2001a, Pérez-Calatayud et al 2004). Ballester et al (cited in Taylor and Rogers 2008) performed their calculations using the Monte Carlo code Geant 3. These results were presented using a third degree polynomial fit.

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The left side panels of figures 2–5 show a comparison between the computed radial dose functions, g_L(r), in water and in the tested substitute materials. The right side panels of these figures represent the ratios of the computed absorbed doses to water, D(r), in phantoms of the tested substitute materials and in water phantoms according to equation (2). Figures 2–5 refer to phantom sizes ranging from R  =  5 cm, H  =  10 cm up to R  =  30 cm, H  =  60 cm.

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The maximum dose deviations between the substitute and water phantoms, appearing in figures 2–5 for radial source distances up to 10 cm are summarized in table 2. In small phantoms with radii up to 10 cm and heights up to 20 cm, the absorbed doses to water in RW1, RW3, Solid Water, HE Solid Water, Plastic Water LR, Plastic Water DT, and Virtual Water deviate from those in water not more than 0.8%. For Plastic Water (as of 1995), Original Plastic Water (as of 2015), Blue Water, polyethylene, and polystyrene the deviations remain within 2.6%. In larger phantoms with radii exceeding 10 cm and heights above 20 cm, the deviations of RW1, RW3, Solid Water, HE Solid Water, Virtual Water, Plastic Water DT, and Plastic Water LR remain below 1.4%, while the deviations of Plastic Water (as of 1995), Original Plastic Water (as of 2015), Blue Water, polyethylene, and polystyrene reach 8.11%. PMMA phantoms up to R  =  10 cm show deviations up to 4.25%, whereas the deviations in the larger PMMA phantoms remain below 1.0%.

In order to supply a framework for the discussion, the spectral fluence of the photons was calculated for points of interest at varying distances from the source center (figure 6), showing a strong relative increase of the continuous, low-energy contribution by scattered photons with increasing distance. The total fluence as well as the relative fluence contributions by primary and scattered photons were calculated for varying distances from the source center and for water phantoms of variable size, as shown in figures 7(a)–(d). Notably, the relative fluence contribution by scattered photons at r  =  1 cm from the source is about 25%, and at r  =  2 cm it is near 40%, whatever the phantom radius. The point of equality between the relative fluence contributions due to primary and scattered photons is regularly found near a source distance of about r  =  3.5 cm, slightly moving towards the radiation source with increasing phantom radius. These results match those by Russel and Ahnesjö (1996), who predict the point of equality to be near 6 cm, however, by including an air-filled applicator in the simulation effectively reducing the portion of scattered radiation near the radiation source. Thus the low-energy fluence contribution by the scattered photons dominates in regions with source distances exceeding 3.5 cm, i.e. well within the range of the clinically relevant distances up to r  =  10 cm.

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This important role of the scattered photons is the reason why the photon attenuation coefficient over a large energy range (figure 8) is of central importance for the ability of a water substitute to mimic a water absorber. Some of the substitute mixtures have been designed in such way that their linear attenuation coefficient deviates from that of water by less than 1% over a large energy region.

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In order to further elucidate this subject by Monte-Carlo simulations, the assumed mass density, ρ, of a polystyrene phantom was varied between 1.02 g cm⁻³ and 1.10 g cm⁻³. Table 3 shows the corresponding electron densities, ρ_e, and relative electron densities, ρ_e/ρ_{e, water}. The photon fields and dose-to-water profiles in the material surrounding the ¹⁹²Ir source were computed for each phantom size (figures 9(a)–(d)). The shapes of the resulting radial dose profiles vary with the mass density and electron density, but neither the polystyrene with electron density close to water (ρ  =  1.03 g cm⁻³) nor with any higher electron density is able to closely simulate the radial dose-to-water profile for all phantom sizes.

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The effect of admixed inorganic additives on the radial dose-to-water profile according to the Monte-Carlo results is shown in figure 10. The variation of the titanium-dioxide admixture to polystyrene (table 4), including RW3 (which is polystyrene with 2% of TiO₂ by weight, see table 1), has a considerable influence upon the shape of the lateral dose profile. In table 4 we have assumed that the producer can vary the TiO₂ content, but keep the mass density of the mixture constant at 1.045 g cm⁻³. This assumption has been made to illustrate the influence of the amount of the inorganic admixture, but is based on the actually existing possibility of a producer to vary the density of polystyrene. To visualize this effect, figure 10(a) displays the dose profiles relative to that in water. The reason behind the effect is shown in figure 10(b), where the attenuation coefficients of the mixtures are shown in comparison to those of water for energies between 10 keV and 1 MeV.

Table 4. Elementary composition of polystyrene with different admixtures of titanium dioxide.

Material	ρ/g cm−3	ρe/1023 cm−3	$\frac{{{\rho}_{e}}}{{{\rho}_{e,\text{Water}}}}$	Mass fraction	Chemical composition
RW3 (polystyrene with 2% TiO2)	1.045	3.376	1.010	H: 0.0759; C: 0.9041; O: 0.0080 Ti: 0.0120	0.98 C8H8
					0.02 TiO2
Polystyrene with 2.5% TiO2	1.045	3.374	1.009	H: 0.0754; C: 0.8995; O: 0.0100 Ti: 0.01499	0.975 C8H8
					0.025 TiO2
Polystyrene with 3% TiO2	1.045	3.372	1.009	H: 0.0751; C: 0.8949; O: 0.0120 Ti: 0.0180	0.97 C8H8
					0.03 TiO2
Polystyrene with 3.5% TiO2	1.045	3.370	1.008	H: 0.0747; C: 0.8903; O: 0.0140 Ti: 0.0210	0.965 C8H8
					0.035 TiO2
Polystyrene with 4% TiO2	1.045	3.368	1.007	H: 0.0743; C: 0.8857; O: 0.0160 Ti: 0.0240	0.96 C8H8
					0.04 TiO2

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The purpose of this paper was to check the suitability of some water substituting phantom materials, known from ICRU 44, ICRU 48, DIN 6800–1 and the publications mentioned in the introduction, in the energy region of the HE photon sources for brachytherapy. ¹⁹²Ir was taken as the example.

The first parameter to be considered was the size of the clinically relevant source distance region. As seen in figure 1 by comparing the radial profiles of the absorbed dose to water (dashed curves) with the open circles, these dose profiles closely follow the 1/r ² law up to large source distances. For the primary photons, the 1/r ² behavior of the lateral dose profile is known from the geometry functions, G_L(r,θ), for arbitrary values of θ (Rivard et al 2004), which all converge towards 1/r ² for values of r $\gg$ L /2. For the scattered photons, the 1/r ² behavior of the lateral dose distribution is the expression of their diffusion-like, spherically symmetric outward transport from the near-source region where they are released by the primary photons. Therefore, as doses exceeding 1% of the dose at 1 cm distance can occur at distances up to 10 cm, the region r  ⩽  10 cm is considered as the clinically relevant distance region (Perez-Calatayud et al 2012).

The second important parameter is the size of the phantom. The downward curved 'hooks' of the radial dose functions shown in figure 1, which occur near the outer phantom surfaces, are characteristic of the dose contribution by scattered photons. They can be explained by the absence of photons backscattered from the outside of the phantoms. Moreover, the Monte Carlo simulation of the radial profiles of the relative fluence contributions by primary and scattered photons (figures 7(a)–(d)) illustrates that the fluence contribution by scattered photons is already considerable at a few centimeters from the source center and is becoming the prevailing contribution at distances exceeding 3.5 cm. In accordance with that, Anagnostopoulos et al (2003) have shown that the dose contribution by the scattered photons becomes predominant at source distances exceeding 6 cm. Therefore, it is appropriate to judge the water equivalence of the various phantom substances by their dose-to-water profiles for the clinically relevant source distances up to 10 cm and for various phantom sizes (figures 2–5).

Figures 2–5 and table 2 provide a survey of the deviations of the radial dose profiles in the investigated substitutes from those in water for cylindrical phantoms of varying size. The commercially available phantom materials RW1, RW3, Solid Water, HE Solid Water, Virtual Water, Plastic Water DT, and Plastic Water LR have satisfying water-equivalent properties for all sizes of the investigated cylindrical phantoms with radii varying from 5 to 30 cm. Other substances such as Original Plastic Water (as of 2015), Plastic Water (as of 1995), and Blue Water, although closely agreeing with water in the electron density (see table 1), as well as the pure plastics such as polyethylene and polystyrene produce larger deviations of their radial dose-to-water profiles from water. Moreover, there is the surprising effect that PMMA, in spite of its largely deviating electron density (table 1), can approximately be called water equivalent in the case of large phantom radii, but is not water equivalent at smaller phantom sizes (table 2). These results confirm those obtained by Tedgren and Carlsson (2009) for Solid Water, polystyrene and PMMA.

In an attempt to understand the physical background for the shapes of these dose profiles, it is useful to consider the spectral distribution of the photons contributing to the dose deposition in phantoms. Figure 6 shows plots of the spectral fluence at radial distances of 0 cm, 1 cm, 2 cm, 5 cm, 10 cm, and 15 cm. While near the source surface, the peaks of the primary photon spectrum at energies of 295–320 keV, 465–485 keV, and 585–615 keV are clearly expressed, the continuous spectrum of the scattered radiation, extending down into the energy region of 25 keV, becomes dominant with increasing source distance (compare Chofor et al 2015). These results are consistent with the spectra shown by Tedgren and Carlsson (2009) for the radial distance of 7 cm. It is evident that in the case of ¹⁹²Ir and, more generally, of the HE photon sources for brachytherapy, the contributions to the total dose at energies below 100 keV is becoming non-negligible and may even become preponderant at the larger source distances. Therefore, the key features to be considered in the choice of water simulating phantom substances are their photon interaction coefficients over a large range of energies (figure 8), covering the energy regions where either the photoelectric effect or the Compton effect prevails.

Figure 8 permits the comparison of the linear attenuation coefficient, μ, of various phantom materials with that of water over a large range of photon energies. It is evident from this comparison that the same substances which distinguish themselves by a good reproduction of the dose profiles in water can as well be identified as those whose linear attenuation coefficients closely resemble that of water. Thus, the close simulation of the linear attenuation coefficient at all relevant photon energies is the property enabling a phantom material to mimic the dose profiles in water. This confirms the conclusion drawn by Tedgren and Carlsson (2009). This general conclusion is not limited to a special geometrical setup or radiation source, but can be regarded as valid for other HE brachytherapy sources and geometries as well.

The approximate equality of the electron density of the phantom material with that of water has frequently been used as the criterion for the suitability of the material to replace water in practical dosimetry. However, this criterion is typical for accelerator-produced high-energy photons with energies in the MeV region. In that range of the primary photon spectrum, the Compton effect is the preponderant interaction with water not only of the primary photons, but also of the scattered photons released in the phantom by Compton interactions. But the spectral situation is different from the case of brachytherapy photon sources, where the spectra of the Compton-scattered photons are significantly shifted to energies reaching into the region of dominant photoelectric effect.

These conclusions have been further underpinned by a series of dose calculations for model polystyrene phantoms with various mass densities (table 3). Supporting calculations have been performed for model polystyrene phantoms with variable admixtures of TiO₂ (table 4).

Figure 9 shows the dose distribution in polystyrene with various mass densities. Close simulation of water over at least 10 cm from the source is never achieved. As shown in figure 8(a), polystyrene shows a reduced attenuation coefficient compared to water in the relevant range below 100 keV.

However, by adding additives with enhanced atomic numbers, e.g. titanium dioxide, the attenuation in the energy range of the photoelectric effect can be increased. The addition of 2% titanium dioxide as in RW3 produces overall acceptable results (see figures 2–5 and table 2). By further increasing the portion of titanium dioxide up to 2.5% and 3%, the photoelectric attenuation approaches that of water, but in the region beyond 100 keV, i.e. of prevailing Compton effect, the absorption is now becoming too strong (figure 10).

Finally, the surprising fit of PMMA for the larger phantom sizes (table 2) appears as the result of a fortuitous compensation of attenuation and scattering effects (compare figure 8). However, PMMA is not suitable as a water substitute for the smaller phantom sizes and can therefore not be relied upon in studies of the dosimetric effect of detailed organ or body surface contours.