Four different media were used in this study, air measurements were performed using mini-Phantom for collimator scatter factor “S_c”, Water Phantom (1 g/cm³), Wood Phantom (0.63 g/cm³), and Acrylic phantom (1.136 g/cm³). All measurements were accomplished by three extraordinary ion chambers, see Table 1.

The linear accelerator (Elekta Synergy, model number (151150)) was utilized, it produces three photon beam energies; 6, 10, and 15 MV. Linear accelerator was attached by Apex Collimator, is an add-on MLC device that is capable of delivering very small conformal fields such as 0.98 × 0.98 cm². For the validation process, CT-Simulator, Brilliance Big Bore-Phillips model, 64-Slice with 85 cm aperture bore, was used. The anthropomorphic phantom was handled as a patient. It was

the first CT scan, and then its image was transferred to the treatment planning system (Monaco) where the radiation treatment plan was done before it was measured on the linac.

In this part, the ion chambers sizes and media densities effect were considered extensively using three different ion chambers.

Figure 1 shows the S_c,p’s at the large fields (>4.41 cm²) for different media and ion chambers are similar with less than 1% due to the presence of (CPE) at the point of measurement [18]. At the small field size (<4.41 cm²), the S_c,p readings in the water for Farmer-Semiflex and Farmer-Pinpoint ion chambers were deviated

by 37% and 43% respectively, due to loss of CPE at the point of interest [2] [14].

S_c,p for the small field that measured by large chamber will be underestimated “i.e., less resolution”, this in turn, will lead to an overestimation in the Monitor Unit calculation (MU), and thus over-dosing the patient [9]. Figure 1 shows the total scatter factor readings as a function of field size.

For the Estimating Formula, the total scatter factor that was measured, where the Pinpoint ion chamber shall only consider. So; the phantom scatter factor (S_p) was calculated based on the total scatter factor measurements using Pinpoint ion chamber through wood, water, and acrylic phantom according to Equation (2):

S p ( r ) ∗ ε ( s ; z ; d ref ) = S ( r ) c , p S ( r ) c (2)

where: ε ( s ; z ; d ref ) = β ( s ; z ; d ) / β ( s ; z ; d ref ) is the electron dis-equilibrium factor ( ε = 1 for adequate depths and positions adequately far from the edges of the field). We assumed that d_ref is sufficiently large to establish electron equilibrium and shield from contamination electrons [7] [8]. Figure 2 shows the S_p values of the three media (wood, water, and acrylic) as a function of field size.

Each curve was fitted by means of an exponential relation due to the attenuation nature of photons in the media [19], and represented by Equation (3):

S ( r ) p , open = y 0 + A 1 ( 1 − e − k 1 ∗ r ) + A 2 ( 1 − e − k 2 ∗ r ) (3)

where: A₁, A₂, k₁, k₂, and y₀ are fitting parameters that depend on the density

(medium).

The first term of Equation (3), y₀, represents the phantom scatter factor at the zero field size, and the second term represents an exponential fitting to small field size < 4.41 cm², while the last term represents an exponential fitting to the large field size > 4.41 cm².

The water medium was used for the total output factor measurement as a base value to all different media with ±2% deviation for the field sizes ≥ 2.94 × 2.94 cm², and with 3% deviation for the field sizes 1.47 < r < 2.94 cm², whilst the ultra small field sizes < 1.47 cm² were deviated by ±5% from the baseline value emphasis that the medium density impact appears clearly at the small radiation beams and should be taken into our account.

Now, the total scatters factor ratio or let say, the total output factor ratio (OFR) is introduced, which is a ratio of total scatters factor S_c,p in any medium relative to that in water media, see Equation (4):

OFR water medium = ( S c , p ) medium ( S c , p ) water ≈ ( S c ( r ) × S p ( r ) ) medium ( S c ( r ) × S p ( r ) ) water (4)

As noticed, the ratio of the phantom scatter factor S p ( r ) medium / S p ( r ) water , is directly proportional to the ratio of charge collected providing that the same chamber and the same irradiation geometry between water and medium. OFR was plotted as a function of field size for two different media and two special ion chambers (see Figure 3).

OFR curves from Figure 3 are seen to be straight with value of 1 for broad beams and to slightly curve upward for low-density medium or downward for high-density medium to ±5% variation; consequently, the exponential function was utilized to fit this behavior and relate the OFR to the field size, r, and to the density in the form of OFR w m = Q ± P e − K r , where Q, P, andK are fitting parameters, where P is negative for high-density medium.

The presence of a foreign object like the ion chamber through the homogenous medium will perturb the electron flux, especially near the ion chamber location [1] [2] [3] [11] [14]. Figure 4 showed that the estimated quantity of the perturbation produces by ion chambers cavity through a homogeneous medium. The chamber volume versus the OFR was plotted for different media and the extrapolation to zero ion chamber size was carried out, see in Figure 4(a) and Figure 4(b).

Extrapolating each curve in Figure 4(a) and Figure 4(b) until it intersects with the Y-axis, yields the OFR ratio with zero chamber size, i.e., independent of ion chamber size, in the form of OFR W M = ( A + B ∗ ionchambervolume ) fieldsize ,

where the A and B are fitting parameters; therefore, the OFR values could be found for small fields without chamber perturbations impact.

Starting from Equation (5), the relative output factor could be used to estimate the phantom scatter value as follow:

S ( r ) p , medium = S ( r ) p , water ⋅ ( O .F .R ) w m (5)

By substituting in the analytical equations for the water scatter phantom S ( r ) p , water and the output factor ratio, OFR, in Equation (5), we get:

S ( r ) p , m = ( y 0 + A 1 ( 1 − e − k 1 ∗ r ) + A 2 ( 1 − e − k 2 ∗ r ) ) p , w ⋅ ( Q ± P e − K ∗ r ) w m (6)

where y₀, A₁, A₂, k₁, k₂, Q and P are values depending on medium density.

Or:

S ( r ) p , m = ( y 0 + A 1 ( 1 − e − k 1 ∗ r ) + A 2 ( 1 − e − k 2 ∗ r ) ) p , w ⋅ ( A ± B ⋅ ionchambervol ) r (7)

Starting with the low-density material, lung density is equal to 0.21 g/cm³, the lung dose was planned and measured at two sites; at the center and peripherial sites of the lung, each site had three beams irradiating as mentioned previously.

Table 2 shows the calculated phantom scatter factor (S_p,calculated) by EF compared to measured phantom scatter factor (S_p,measured) by Pinpoint ion chambers, the central lung site.

As shown, the EF yield just a single estimation of S_p and it matches very well with the value obtained for the Lt Lat and PA beams, and furthermore it ought to be noticed that the fitting parameters were obtained from wood phantom (ρ = 0.63 g/cm³) that utilize to represent the lung medium (ρ = 0.21 g/cm³), however,

it still applies well in this particular case. For the periphery lung site Table 3 was introduced.

As noticed, the ion chamber was located exceptionally close to the soft tissue. As such, the phantom scatter factor that chamber measures “sees” is greatly influenced by the presence of the soft tissue. This scenario is not taken into account with EF, which predicts the phantom scatter factor when the chamber is encompassed by enough homogeneous medium.

The mediastinum site was represented with a normal density equal to 1.01 g/cm³, further measurements were accomplished via the anthropomorphic phantom and pinpoint ion chamber. The S_p,calc was carried out using equation 7 and compared with measured one, see Table 4.

Table 4 shows that the expected values of S_p,calc was within 1% from measurement values because of the ion chamber was completely surrounded by the soft tissue which validates equation 7 for S_p in the soft-tissue medium.

Last but not the least, the S_p for the high-density material, vertebral bone with density equal to 1.59 g/cm³, was measured under the same previous condition and the S_p,calc was calculated by Equation (7). Both values were compared as shown in Table 5.

The measuring values are relatively close to the calculated one with a maximum deviation of 4.92% for the worst case scenario. The EF was based on the fitting parameters that extracting from acrylic media, density is equal to 1.136 g/cm³, to represent the bone media, which might be a main source of this deviation. Furthermore, the AP beam showed a small deviation of 0.04% from the PA beam. The phantom scatters factor that the chamber measured from the AP field “sees” is highly affected by the presence of lung tissue in front of the vertebra tissue.

The measurements with the anthropomorphic phantom have validated the EF to calculate S_p. The values agreed within less than 1% when the chamber was surrounded by a homogeneous medium (all lung, all tissue, or all bone). When the beam arrangements and/or the location of the chamber resulted in the beam passing through variable tissues the results were not as good. This is expected because the EF has been extracted from homogeneous composition media.