Evaluation of the scatter field for high-energy photon beam attenuators

M E Castellanos ^1,2 and J C Rosenwald ¹
1 Institut Curie, Service de Physique Médicale, Paris, France
2 Instituto Nacional de Cancerologia, Bogota, Colombia

Abstract : Based upon sector integration, a method has been developed to evaluate the scatter from attenuating beam modulators at any point in the field for arbitrarily shaped fields and attenuators with variable thickness. The method requires preliminary measurements of narrow and broad beam transmission fractions as a function of filter thickness and field size. The ratio S, of the contribution from photons scattered by the attenuator to the non-attenuated primary, contribution was derived from these measurements. Sp was determined for x-ray beam energies between 4 and 23 MV with brass and lead attenuators. This quantity was found to be practically independent of beam energy for a given field site and material. The variation of Sp as a function of slab thickness for attenuators covering the entire beam showed a maximum for a thickness of approximately one mean free path. This maximum represents about 6.0% of the transmitted primary dose for an extreme case of a very heavily (1.6 cm thick lead slab) attenuated 15 cm x 15 cm field. The 'scatter field', corresponding to the scatter contribution from the attenuator across the field, was calculated for different partial attenuators and wedges. The results show that this component has a limited influence on calculation of dose distribution. but should be taken into account in absolute dosimetry analysis for large fields and thick wedge filters.

Wedges, attenuators and compensators are frequently used in radiation therapy to modify the dose distribution in external photon beams. These devices not only attenuate but also scatter the primary beam and the scatter contribution must be taken into account for calculation of the dose distribution.

This problem has been recognized and studied by several authors. Huang et al (1986) reported a scatter contribution on the centre line of about 6% of the transmitted primary dose at 4 MV for a 20 cm x 20 cm field, using a I cm thick copper sheet. Van Dyk (1986) performed an experimental and theoretical study of attenuation coefficients for broad beam geometries approximating typical treatment conditions, and found a deviation of as much as 16% from the narrow beam data for a 40 cm x 40 cm field. El-Khatib et al (1986) investigated the broad beam attenuation in a homogeneous phantom for various radiation field sizes, photon beam energies and depths in phantom, and the results obtained for various radiation fields indicated very complicated behavior dependence on beam spectral distribution, radiation field sizes and depths in phantom. Papiez and Froese (1990) developed a semiempirical method to account for scattered photons produced by beam modifying absorbers and reported a reduction of computation errors from 8 to 3% in a 4 MV photon beam. Islam and Van Dyk (1995) developed a method based upon Compton first scatter to calculate effective broad beam transmission. They reported deviations of 17% and 5.5% for Co-60 and 6 MV respectively when narrow beam transmission coefficients were used. Ahnesjö et al (1995) developed a model based on first principles for calculation of scatter and transmission for photon beam attenuators, using the beam spectrum as basic data

The objective of this study was to quantify the effect of scatter from beam modifying devices on the primary field, for the range of high-energy x-ray beams used in radiotherapy (from 4 MV to 23 MV) and to evaluate the sector integration method for calculation of the scatter contribution across the field for static modulators of arbitrary shapes and thickness. The final objective was the modification of a dose calculation algorithm used in clinical practice la order to accurately take into account the modification of primary and scatter dose components, for devices such as compensating filters, partial attenuators or wedges.

Narrow beam transmission measurements were carried out using five x-ray beam energies, 4, 6, 12, 15 and 23 MV, using three different GE-Medical Systems linear accelerators. The corresponding quality indices (ratio of tissue phantom ratio in water at 20 and 10 cm depth) were 0.617, 0.683, 0.747, 0.765 and 0.785. The narrow beam geometry (figure 1) was achieved by choosing a small chamber (0.14 cm³ WeIlhöffer IC10 ion chamber) and a large distance (2.0 m) between source and detector. The collimator opening was 5 cm x 5 cm, 1 m from the source. The photon field was further collimated to a small 1 cm x 1 cm square field with a thick lead block located on the shadow tray of the accelerator treatment head, 68 cm from the source. A second thick lead collimator was used to shield the detector from scattered radiation. The attenuating slabs were placed directly on the first collimator. Both lead and brass attenuators were studied.

Measurements were made along the central ray and the chamber was placed in a columnshaped polystyrene miniphantom with a square cross section of 2, 4 and 5 cm large enough to ensure lateral electronic equilibrium (Zefkili et al 1994) for 4 and 6 MV, 12 and 15 MV, and 23 MV respectively. The chamber was situated at a depth of 5 cm for 4 and 6 MV and 10 cm for 12, 15 and 23 MV, as recommended in dosimetry protocols for clinical beams. These measurements will subsequently be referred to as 'in air' measurements. In every case, the reproducibility of measurements was better than 1%. Slight changes in the experimental set-up (i.e. modification of the position of the secondary collimator) did not induce variations greater than 0.8%.

Although the beams were non-monoenergetic, the ratio of detector signals measured without (L₀) and with (L) attenuator was used to determine the narrow beam transmission T₀ and derive an effective attenuation linear coefficient µa where the subscript 'a' stands for the attenuator material, according to the following expression

(1)

where t is the attenuator thickness. It is well recognized that the spectral changes have an influence on the transmission factors if defined as the ratio of energy fluences rather than ratio of readings. The reasons for this influence are modifications of the penetration properties and dose deposition in the miniphantom material. In addition, the scatter from the miniphantom should also be considered (Zefkili et al 1994). These corrections are not taken into account, and will be discussed later.

As the radiation beams were not monoenergetic µ_a may change with thickness t and with depth in the miniphantom. The term 'effective attenuation coefficient' is intended to reflect those variations, due to the polychromatic spectrum of the x-ray beam.

The scatter contribution from the attenuator does not lend itself to direct measurements. However, the scatter from peripheral parts of the attenuator can be estimated separately by differentiation of broad transmission with respect to the field size. Broad beam transmission measurements were therefore performed for all beam energies along the central axis with the chamber at the isocentre (figure 1).

In order to isolate the primary photon fluence, we used the same miniphantom and depth of measurement as for narrow beam measurements. Lead and brass absorbers of different thickness covering the entire beam were located at the usual blocking tray position. For each field size (c), we obtained the transmission curve as a function of attenuator thickness (t), according to the following expression

(2)

where L(t, c) and L(0, c) are the detector signals for a collimator opening c x c, with and without attenuator of thickness t respectively.

Based on the assumption that the difference between narrow and broad beam transmissions (T₀ and T_b respectively) is due to the scatter in the attenuator (Van Dyk 1986), we calculated the 'scatter to non-attenuated primary dose ratio' S_p(t, c):

S_p(t,c) = T_b(t,c) - T₀(t)                    (3)

which represents the ratio between the dose due to the scatter from the attenuator reaching a point of measurement and the primary at the same point without attenuator. This is the scatter component of the total transmitted fraction, as defined by Papiez and Froese (1990). As is well known (Huang et al 1986, Islam and Van Dyk 1995) this contribution decreases with distance. That means that Sp values are valid only for a given distance between attenuator and calculation plane.

The scatter reaching any point of interest can then be determined by integration over the field, using Clarkson-Cunningham sector integration (Cunningham 1972, Papiez and Froese 1990): the radiation field is divided into angular sectors centered at the point, and each angular sector is further divided radially into small elements (figure 2).

Each field element is related to an attenuator element by scaling up. The total 'scatter to non-attenuated primary dose ratio' from the whole attenuator is then

(4)

where S_p(t, r) represents the 'scatter to non-attenuated primary dose ratio' for a thickness t and a circular field of radius r. It is directly derived from equation (3) considering the relationship between the side c of a square field and the radius r of its equivalent circular field assumed to generate the same amount of scatter on its central axis if it has the same area, so:

The beam modifying device is depicted as a matrix containing the thickness t_ij at each pixel element (i, j) of a two-dimensional grid located at the source-attenuator distance. This is used to separately estimate the scatter from each modulator element of thickness t_ij at distance r_ij from the calculation point in the plane passing through the isocentre, and to calculate the 'scatter field' defined as the distribution of the scatter contribution from the attenuator across the field.

The basic assumptions behind this approach were that the amount of scatter from an element of each irradiated ring of radius r and width Dr was independent of spectral variations across the beam, as well as of the scatter in neighboring elements.

To evaluate the performance of the sector integration method, 'in air' beam profiles were measured for a 20 cm x 20 cm field modulated by a 45° brass wedge filter, a 1.0 cm thick lead slab covering the whole field and slabs only partially covering the field. Total transmitted 'primary' profiles ('narrow beam transmitted primary' plus 'scatter field'), computed with the scatter integration method, were compared with the corresponding measured profiles.

Figure 3 represents the narrow beam mass effective attenuation coefficient, versus lead and brass filter thickness, derived from measurements on the central axis using equation (1). The filter-detector distance was 32 cm for all beam energies. It demonstrates that, for 4 and 6MV, the narrow beam effective attenuation coefficient decreased as the filter thickness increased. This means that a substantial beam hardening effect was present for the lower energies. For these energies, the decrease in effective attenuation coefficient was greater for lead than for brass. The relevance of this effect in radiation therapy was emphasized by Knöös and Wittgren (1991).

As expected. for higher energies the filtration effect was practically negligible. For 15 and 23 MV, a small increase of the effective attenuation coefficient was observed for lead and for thickness less than 2.5 cm. This increase can he attributed to a 'softening effect' due to pair production. For the same reason the attenuation coefficient for lead filters was found to be greater for 23 MV than for 15 MV

The validity of these conclusions when the changes in penetration and energy absorption are taken into account will be discussed later.

An example of narrow and broad beam transmission variation as a function of lead thickness is shown in figure 4. Due to scatter contribution, the transmission value increased with increasing field size for ail thickness. These results were used to calculate the scatter contribution for different thickness and field sizes. Using equation (4), we then calculated the scatter contribution from any lead or brass beam modifier.

Values for the scatter-to-non-attenuated primary ratio as a function of field size are shown in figure 5, for different lead attenuator thickness t, for the 23 MV beam. The behavior of this function was the same for ah beam energies. As shown in this figure, an increase of attenuator scatter with field size was observed for all thickness, reflecting the fact that the number of primary photons scattered in the attenuator and contributing to the dose at the central beam axis is influenced by the volume of attenuator actually exposed to radiation. However, Sp depends not only on how many photons were scattered by the irradiated volume, but also on the proportion of these scattered photons which reach the point. The position of each ring of scatterer referred to the point defines scattering angles, so S_p values are not linear functions of area (parabolic fitting as a function of the field side).

The scatter contribution S_p is shown in figure 6 for two field sizes as a function of attenuator thickness. The scatter contribution initially increased with attenuator thickness, then subsequently decreased as the scattered radiation generated near the entrance side was attenuated by the larger thickness of the attenuator. The number of scattered photons generated on the exit side of the attenuator was also reduced due to the lower intensity of the primary photons. The maximum contribution was reached for a thickness corresponding to approximately one mean free path, defined as the thickness equal to the inverse effective attenuation coefficient, leading to a transmission T₀ of 36.8%. Using this representation, we can also sec that, for a given attenuator material, S_p was almost independent of beam energy in the range 4 MV to 23 MV. We can therefore retain the same value for any off-axis position where the spectrum may be different. Finally, as theoretically predictable and as already shown by Ahnesjö et al (1995), S_p was lower for attenuators with a higher atomic number.

Alternatively, Sp can be plotted as a function of the narrow beam transmission T₀ (figure 7). The scatter contribution is expected to be 0 in the absence of attenuator (T₀ = 1), but also for an infinitely thick attenuator (T₀ = 0), 'which would block all primary and scattered radiation. The intermediate values are consistent with this behavior, and the maximum contribution was reached for a value of T₀ around 1/e = 36.8%, corresponding to about 4% of the primary non-attenuated beam for a 20 cm x 20 cm field. Fitting of S_p versus T₀ values shows that the curve shape can be approximated by S_p = - kT₀ ln(T₀), as pointed out by Ahnesjö et al (1995) on the basis of first scatter theory.

Since it represents only a small fraction of the contribution to the dose, S_p, has a much larger percentage of uncertainty than the transmission factors, especially for small field sizes and thickness. The maximum uncertainty was estimated to be ± 15%.

Figure 8 presents the ratio of S_p to transmitted primary as a function of transmitted primary T₀, for different field sizes and all beam energies. The full curves are polynomial fits. To clearly show the order of magnitude of the scatter from the attenuator expressed as a percentage of the transmitted primary, we have listed in table I the numerical values found from the polynomial fit. For instance, for a brass attenuator covering a 10 cm x 10 cm field, when the transmitted primary was 30% the scatter from the attenuator was about 5.8% of the residual dose at a distance of 32 cm, regardless of the energy used. Similar tables could be produced for other distances which would be larger in clinical practice, resulting in a lower scatter contribution (Ahnesjö et al 1995).

The sector integration method was applied to various situations. Figure 9 shows a distribution of the scattered contribution across a 20 cm x 20 cm field at 4 MV for four different lead filters and one brass wedge filter. The plane of calculation was through the isocenter, 1 m from the source. For a 1 cm thick lead filter covering the whole field, S_p varied from 2.8% on the beam axis to about 2% at the field edge. This parameter was when the lateral dimensions of the attenuator were reduced to an 8 cm x 8 cm square. The shape of the Sp profile clearly illustrates the influence of the size of the attenuator with the predominantly forward emission of scattered photons A further decrease of Sp was obtained of the lead when the thickness of the lead attenuator was reduced to 5 mm. When the 1 cm thick lead attenuator covered only half of the field, the maximum value of Sp was about 1.5% with a reduction of about 0.5 to 1.0% in the non-attenuated area. The scatter contribution from the 45° brass wedge filter was more marked. The role of the wedge thickness was clearly demonstrated with an increasing contribution in the direction corresponding to the thicker edge of the filter.

The comparison between calculations and measurements is shown in figure 10 for the 1 cm thick lead slab covering the whole 20 cm x 20 cm field (figure 10(a)) and for the 45° brass wedge filter in the same field (figure 10(b)). The profiles are normalized to 1 on the beam axis, considering the measured dose and the total calculated dose (primary + scatter from the attenuator). The relative contributions of the transmitted primary and the scatter component are clearly seen. The scatter component amounts to about 5% of the total dose on the beam axis for both the lead slab and the wedge filter. It would be larger in the case of lower attenuator transmission (see table I).

Despite the good agreement between measurements and calculations, no definite advantage was observed for calculation of the scatter from each individual element of the attenuator. In practice, the same result would have been obtained if the transmitted primary had been corrected by a constant 'effective transmission factor' over the entire field area. This is due to the fact that the 'scatter field' is rather homogeneous and represents only a small proportion of the total transmitted dose. However, for thicker attenuation devices such as those used for partial shielding or for compensators with large thickness variations, it could be important to properly account for the actual scatter generated from the attenuator.

As pointed out previously, ail these results are based on the ratio of readings without any correction for spectral changes. The quantitative influence of these changes can be calculated using the methodology described in the appendix. The variation of the product k₁ k₂ with lead thickness from 0 to 5 cm is at most around 4%. It should normally be applied to the measured transmissions T₀ and Tb resulting in slight changes in the values of the effective attenuation coefficients. Considering the uncertainty on S_p values, this correction, which is only significant for thick filters, can be neglected.

A complete set of measurements for energies between 4 and 23 MV involving brass and lead attenuators has been acquired, leading to quantitative evaluation of the scatter from these materials for a standard distance of 32 cm between the filter and the point of measurement. It was found that the 'scatter to non-attenuated beam primary ratio' for a given filter thickness was practically independent of the beam energy. This was also true when the scatter was expressed as a fraction of the transmitted primary and plotted as a function of primary transmission. This allows the use of 'universal' curves for the quantitative assessment of the scatter component and for generation of tables used as an input for a Clarkson integration model. The influencé of the distance of the attenuator was not investigated.

This sector integration was found to be quite easy to perform for calculation of the scatter contribution from static beam modulators at any point in the field, provided that a double decomposition (angular + radial) was applied. Calculations for different types of attenuators showed that this contribution was only small in the case of small compensators or partial attenuators. However, this method could be useful for dose distribution calculations for large-angle wedge filters and large field sizes where the scatter contribution is important and heterogeneous across the field. In other cases, attenuator scatter distributions can be ignored without any significant loss of accuracy of dose distribution calculation, provided the scatter is included in an 'effective transmission' factor used for determination of the number of monitor units.

Changes in beam spectrum lead to altered attenuation and scatter conditions in the miniphantoms as well as differences in energy absorption coefficients. If we define the transmission factor as the ratio of energy fluences , the dose measurements should be corrected according to the following expression:

D₀ and D(t) are the doses in depth in the miniphantom respectively without and with attenuator of thickness t. k₁ is a correction factor to be applied to the dose ratio in the miniphantom to account for the change in beam penetration and energy absorption. k₂ is a correction factor to be applied to the dose ratio to account for the change in scatter from the miniphantom.

The correction coefficients k₁ and k₂ can be expressed as

and

In this last expression C_d represents the fraction of scatter from the miniphantom relative to the total dose. We used as C_d values those obtained as a function of depth, z, and primary attenuation coefficients,(µ/r)_p by Zefkili (1995) for the same miniphantoms. For this we calculated (µ/r)_p as

In these expressions µ_p and µ_a are monoenergetic attenuation coefficients for polystyrene and attenuator material respectively and Y_E is the differential energy fluence at energy E, without filter.

Using photon beam spectra taken from the literature (Mohan and Chui 1985) and monoenergetic attenuation coefficients (Hubbell 1982), we obtain for lead attenuators the results presented in table 2. If we neglect the changes in stopping power ratio between water and air, k₁ and k₂ can be applied directly to the ratio of readings, T₀.