2.1 Theory of the PDD reconstruction method

The PDD in a water phantom at depth d along an off-axis angle θ for a field size \(_\) irradiated with a continuous energy spectrum \(\lambda\), can be calculated as follows:

$$\begin PDD \left(\lambda , _, d, \theta \right)= _}}_}}} }_(\theta )\,\times \,D \left(E, \,_, \,d, \,\theta \right) \textE}_}}_}}}}_(\theta )\,\times \,D \left(E, \,_, \,_}, \,\theta \right) \textE}} \times 100, \end$$

(1)

where \(_(\theta )\) is the differential photon fluence and \(D(E, _, d, \theta )\) is the absorbed dose per photon of monochromatic energy E. Since the source-surface distance (SSD) varies with θ, it has been included as an influencing parameter of \(D(E, _, d, \theta )\). d is considered beyond the depth of maximum dose dmax due to significant dose contributions from the charged-particles in the build-up region. As the PDD varies slightly within a small energy range, depending on the properties of the mass attenuation coefficient, the energy spectrum can be handled as a discrete probability distribution [7, 9] and Eq. (1) can be rewritten as follows:

$$\beginPDD \left(\lambda , _, d, \theta \right)=^_\left(\theta \right)\,\times \,D \left(_,\,_,\,d, \,\theta \right)\Delta E}^_\left(\theta \right)\,\times \,D \left(_,\,_,\,_}, \,\theta \right)\Delta E}} \times 100,\end$$

(2)

where n is the total number of energy bins i of the discrete energy interval \(\Delta E\) and \(_\) is the mid-energy of each energy bin. The relative fluence \(_(\theta )\) can be defined as follows:

$$\begin_\left(\theta \right)= _\left(\theta \right)\,\times\,\Delta E\}}_}_^_\left(\theta \right)\,\times\,\Delta E\}}_}},\end$$

(3)

where \(\Delta _\) is the width of the ith energy bin. \(_(\theta )\) can be obtained by minimizing the variance σ2 of the relative difference δj (%) between the reconstructed and measured PDDs. In this study, σ2 was calculated as

$$\begin^ = }\sum_^_}^= }\sum_^_} - _}}_}}} \times 100\right)}_}^,\end$$

(4)

where m is the total number of sampling depths j. The optimization process employed the generalized-reduced-gradient (GRG), which generalizes the reduced gradient method by allowing nonlinear constraints and arbitrary bounds on the variables [17]. As mentioned above, the PDD varies slightly within small energy range, so the GRG can generate unrealistic energy spectrum. Therefore, several constraints were applied to obtain a realistically shaped energy spectrum (i.e., \(_\le _\) in the energy range lower than energy \(_}\) where the fluence is maximum and \(_\ge _\) in the energy range higher than \(_}\)) [7]. Similarly, to ensure the gradual off-axis softening of energy spectra from the central-axis to the maximum radial distance, the following constraints were imposed:

$$_\right)}_}\le _\right)}_}\le _\right)}_}\cdot \cdot \cdot \cdot \le _\right)}_}[\text \space _<_}]$$

$$\begin_\right)}_}\ge _\right)}_}\ge _\right)}_}\cdot \cdot \cdot \cdot \ge _\right)}_} \left[\text \space _>_}\right].\end$$

(5)

Here, the numeric subscripts indicate radial distance \(_\) on the surface of the water phantom. In addition, to satisfy the condition of relative fluence according to Eq. (3), the constraint \(\sum_^_(\theta )=1\) was applied. In the pCCC algorithm of the Monaco TPS, energy fluence \( _\) is applied for the TERMA calculation [15, 18]. Therefore, \(_(\theta )\) was converted to \(_(\theta )\) by the following equation:

$$\begin_\left(\theta \right)= _\left(\theta \right)\times _.\end$$

(6)

2.2 Measurement of dose distribution

To estimate the energy spectra at radial distance \(_\) ranging from 0 to 18 cm in 2 cm increments, the PDDs were measured along the off-axis angles θ (\(_\)) of 0° (0 cm), 1.15° (2 cm), 2.29° (4 cm), 3.43° (6 cm), 4.57° (8 cm), 5.71° (10 cm), 6.84° (12 cm), 7.97° (14 cm), 9.09° (16 cm), and 10.2° (18 cm) (see Fig. 1). Measurements were conducted using a 6 MV photon beam emitted from a Versa HD linac [Elekta Oncology Systems, Crawley, UK] with a 40 cm × 40 cm field and an SSD of 100 cm. The blue phantom2 with beam-scanning software OmniPro-Accept 7 and CC13 waterproof ionization chambers [IBA dosimetry, GmbH, Germany] were used for the beam-scanning. As the radius of the active chamber cavity was 3.0 mm, the effective point of measurement was set 1.8 mm toward the source from the geometrical center of the field chamber [19]. In the beam-scanning software, the scan type was set to “Fan line” and the PDDs were measured up to depths of 25 cm in 2 mm increments. Although the measurements were performed along off-axis angles, the depths were determined to be parallel to the central-axis to simplify the calculation of MDDs. The PDDs were normalized to the dose at 10 cm depth (d10) to reduce the effects of charged-particle contamination and to avoid the positional uncertainty of dmax. With the same field size and SSD, the OARs were measured at depths of 5, 10, and 20 cm for verification of the method.

Fig. 1

Measurement of percentage depth doses (PDDs) along the off-axis angles θ of 0°, 1.15°, 2.29°, 3.43°, 4.57°, 5.71°, 6.84°, 7.97°, 9.09°, and 10.2° for a 6 MV photon beam emitted from a Versa HD linac with a field of 40 cm × 40 cm and a source-surface distance (SSD) of 100 cm

2.3 Estimation of energy spectra

To generate energy spectra for a 6 MV polyenergetic photon beam, the range of monoenergetic photons were considered from 0.05 to 7.00 MeV [1, 6, 14]. As mentioned in Sect. 2.1, the PDD varies slightly within small energy range, depending on the mass attenuation coefficient, which changes rapidly in the low-energy region and slowly in the high-energy region. To optimize the number of energy bins, the widths of the energy bins \(\Delta _\) were gradually increased in accordance with the mass attenuation coefficient, defined as follows: 0.05–0.15 MeV, 0.15–0.25 MeV, 0.25–0.35 MeV, 0.35–0.50 MeV, 0.50–0.70 MeV, 0.70–1.00 MeV, 1.00–1.40 MeV, 1.40–1.90 MeV, 1.90–2.50 MeV, 2.50–3.25 MeV, 3.25–4.10 MeV, 4.10–5.15 MeV, 5.15–7.00 MeV. The mid-energy \(_\) of these corresponding energy bins were considered for the MDD calculation. MDDs for the energies of 0.100, 0.200, 0.300, 0.425, 0.600, 0.850, 1.200, 1.650, 2.200, 2.875, 3.675, 4.625, and 6.075 MeV were calculated in water along the central-axis and at \(_\) of 1.5–21.5 cm in 1.0 cm increments (see Fig. 2a). The MDDs were calculated in cylindrical geometry using DOSRZnrc, a component of the egs_inprz graphical user interface [20], to reduce statistical uncertainty and to derive energy spectra as functions of \(_\). The calculations assumed a point source with a circular field of radius 22.568 cm (≡ 40 cm × 40 cm square field) at 100 cm SSD and depths up to 25 cm in 2 mm increments. The electron and photon cut-off energies were set to ECUT = 0.521 MeV and PCUT = 0.01 MeV, respectively. The number of histories per calculation was 2 × 109 except for the central-axis, where it was raised to 3 × 109. A higher number of histories was applied to reduce the statistical uncertainty due to smaller calculation area at the central-axis. To convert MDDs from the cylindrical geometry to the angular directions following the measured PDDs as shown in Fig. 2b, linear interpolation method was applied. Required radial distances \(r\left(d\right)\) at depth d along the angular directions were calculated as follows:

Fig. 2

a Cylindrical geometry with a circular field of radius 22.568 cm (≡ 40 cm × 40 cm square field) at 100 cm source-surface distance (SSD) to calculate the monoenergetic depth doses (MDDs) along the central-axis and at radial distances \(_\) of 1.5 to 21.5 cm in 1.0 cm increments using the DOSRZnrc code; b Conversion of MDDs along the off-axis angles θ of 0°, 1.15°, 2.29°, 3.43°, 4.57°, 5.71°, 6.84°, 7.97°, 9.09°, and 10.2° using the linear interpolation method

$$\beginr\left(d\right)= _}}} \times \left(\text+d\right).\end$$

(7)

After formulating a simple mathematical model in Microsoft Excel, varying the MDDs and constraints of the off-axis softening at each \(_\), the GRG of the add-in program “Solver” was iterated until the reconstructed PDD acceptably agreed with the measured PDD. Thus, \(_(\theta )\) was obtained by minimizing \(^\).

2.4 Comparison of estimated energy spectrum at the central-axis with that of Monaco TPS

Energy spectrum at the central-axis was obtained from the commissioning report of the pCCC algorithm (Monaco TPS) and compared with the energy spectrum estimated in this study. For comparison, the energy spectrum of the Monaco TPS was rebinned to match the energy-bin division of the present study. In addition, the mean energy in off-axis angle \(\theta\), \(\overline\left(\theta \right)\) for both energy spectra was calculated as [21]

$$\begin\overline\left(\theta \right)= ^_(\theta )}^_(\theta )}}=\sum_^_(\theta ),\end$$

(8)

since, \(\sum_^_ (\theta )=1\) as mentioned in Sect. 2.1.

2.5 Comparison of PDDs and OARs

To analyze the off-axis softening of the Monaco TPS, PDDs were computed in water using the pCCC algorithm of the Monaco TPS for a 6 MV photon beam with a 40 cm × 40 cm field and an SSD of 100 cm along the off-axis angles θ as mentioned in Sect. 2.2. Calculations were performed on a 50 cm × 50 cm × 40 cm virtual water phantom with a 3 mm grid size. Data were collected using the “Dose plane output” option of the Monaco TPS. The calculated PDDs were compared with both the reconstructed and measured PDDs.

Simultaneously, the OARs were calculated using the Monaco TPS and the estimated energy spectra with their corresponding MDDs at depths of 5, 10, and 20 cm. Fluence ratios required for calculating OARs using the estimated energy spectra were determined by the Microsoft Excel add-in program “Solver” to achieve OARs in agreement with measured OARs. Calculated OARs were compared with measured OARs. The PDD and OAR comparisons were evaluated by the depth-dependent relative difference δj (%) and the radial-dependent relative difference δr (%), both defined according to Eq. (4).