Reconsideration of the Iwasaki–Waggener iterative perturbation method for reconstructing high-energy X-ray spectra

Abstract

We have reviewed applicable ranges for attenuating media and off-axis distances regarding the high-energy X-ray spectra reconstructed via the Iwasaki–Waggener iterative perturbation method for 4–20 MV X-ray beams. Sets of in-air relative transmission data used for reconstruction of spectra were calculated for low- and high-Z attenuators (acrylic and lead, respectively) by use of a functional spectral formula. More accurate sets of spectra could be reconstructed by dividing the off-axis distances of R = 0–20 cm into two series of R = 0–10 cm and R = 10–20 cm, and by taking into account the radiation attenuation and scatter in the buildup cap of the dosimeter. We also incorporated in the reconstructed spectra an adjustment factor (f _adjust ≈ 1) that is determined by the attenuating medium, the acceleration voltage, and the set of off-axis distances. This resulted in calculated in-air relative transmission data to within ±2 % deviation for the low-Z attenuators water, acrylic, and aluminum (Al) with 0–50 cm thicknesses and R = 0–20 cm; data to within ±3 % deviation were obtained for high-Z attenuators such as iron (Fe), copper (Cu), silver (Ag), tungsten (W), platinum (Pt), gold (Au), lead (Pb), thorium (Th), and uranium (U) having thicknesses of 0–10 cm and R = 0–20 cm. By taking into account the radiation attenuation and scatter in the buildup cap, we could analyze the in-air chamber response along a line perpendicular to the isocenter axis.

Appendices

Appendix 1: The ${\varvec{{\mu}_{\bf{{cap}}}^{\bf{{eff}}}} }$ factor

The $\mu_{\text{cap}}^{\text{eff}}$ factor (Eq. 14) for a given X-ray beam should be determined around the region forming the maximum cap-absorbed dose under electronic equilibrium. The chamber generally detects the primary and scattered doses generated in the buildup cap. When f _cap = 1 in Eq. 14, $\mu_{\text{cap}}^{\text{eff}} (E) = (\mu (E))_{\text{cap}} ,$ with the result that any scattered photons produced in the buildup cap do not contribute to the chamber readings. When f _cap = 0 in Eq. 14, $\mu_{\text{cap}}^{\text{eff}} (E) = (\mu_{\text{en}} (E))_{\text{cap}} ,$ with the result that all scattered photons produced in the buildup cap do contribute to the chamber readings.

Let ρ_cap denote the mass density of the buildup cap material. Around the region forming the maximum cap-absorbed dose, the value of $\mu_{\text{cap}}^{\text{eff}} (E)H_{\text{cap}}^{\text{eff}}$ in Eq. 13 can be relatively small (also by taking into account the relative magnitude of Ψ₀ regarding E). Then, with the use of Eq. 14, we can formulate the following approximate expression:

$$ {\text{e}}^{{ - \mu_{\text{cap}}^{\text{eff}} (E)H_{\text{cap}}^{\text{eff}} }} = 1 - \left( {\frac{{\mu_{\text{en}} (E)}}{\rho }} \right)_{\text{cap}} \rho_{\text{cap}} H_{\text{cap}}^{\text{eff}} - f_{\text{cap}} \times \left[ {\left( {\frac{\mu (E)}{\rho }} \right)_{\text{cap}} - \left( {\frac{{\mu_{\text{en}} (E)}}{\rho }} \right)_{\text{cap}} } \right]\rho_{\text{cap}} H_{\text{cap}}^{\text{eff}} . $$

(27)

On the right side of Eq. 27, it can be understood that, for a unit of energy fluence, the second term expresses the energy loss as the primary cap collision kerma component, and the third term the energy loss as the scattered photon dose component. It should also be noted that the f _cap factor should vary with the size and shape of the buildup cap and with the X-ray beam size and position for the buildup cap (Fig. 1). Accordingly, it can be understood that Eq. 14 is reasonable around the region forming the maximum cap-absorbed dose under electronic equilibrium.

Appendix 2: The correlation factor

For a representative energy E _N (N = 1 − N _max), we take Y _K  = log[Φ_disc (E _N , R _K )] as a function of off-axis distance R _K (K = K _min − K _max), where Φ_disc (E _N , R _K ) is the normalized photon fluence as a function of E _N at R _K (see Eq. 12). For E _N , the square of the correlation factor (r _N ) between Y _K and R _K values is calculated as

$$ r_{N}^{2} = \frac{{\left( {\frac{1}{{K_{\max } - K_{\min } + 1}}\sum\nolimits_{{K = K_{\min } }}^{{K_{\max } }} {(R_{K} - \overline{R} )(Y_{K} - \overline{Y} )} } \right)^{2} }}{{\frac{1}{{K_{\max } - K_{\min } + 1}}\sum\nolimits_{{K = K_{\min } }}^{{K_{\max } }} {(R_{K} - \overline{R} )^{2} } \times \frac{1}{{K_{\max } - K_{\min } + 1}}\sum\nolimits_{{K = K_{\min } }}^{{K_{\max } }} {(Y_{K} - \overline{Y} )^{2} } }}, $$

(28)

where r ² _N  ≤ 1 with

$$ \overline{R} = \frac{1}{{K_{\max } - K_{\min } + 1}}\sum\limits_{{K = K_{\min } }}^{{K_{\max } }} {R_{K} } , $$

$$ \overline{Y} = \frac{1}{{K_{\max } - K_{\min } + 1}}\sum\limits_{{K = K_{\min } }}^{{K_{\max } }} {Y_{K} } . $$

When the denominator of Eq. 28 has a value less than 10⁻³, the calculation of r ² _N is unstable. Such values are excluded from the calculation of Index-1 (Eq. 23). Therefore, Index-1 is expressed with use of the number $\left( {N_{\max }^{\prime } } \right)$ of the adopted r ² _N data $\left( {N_{\max }^{\prime } \le N_{\max } } \right)$.