An Investigation of a Model of Percentage Depth Dose for Irregularly Shaped Fields

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1 Int. J. Cancer (Radiat. Oncol. Invest): 96, (2001) 2001 Wiley-Liss, Inc. Publication of the International Union Against Cancer An Investigation of a Model of Percentage Depth Dose for Irregularly Shaped Fields Murshed Hossain, Ph.D.,* Ying Xiao, Ph.D., and M. Saiful Huq, Ph.D. Department of Radiation Oncology, Thomas Jefferson University, Philadelphia, Pennsylvania SUMMARY A significant component of the total dose delivered to tumor and surrounding tissue during a radiation treatment arises from the scattering of the primary beam. Accounting for this component accurately and efficiently is a necessity. In this study we investigate a method for calculating the phantom-scatter contributions to the total dose by simple summation of scatter dose from a set of individual triangles that span an irregular field. The calculation of phantom scatter is based on a two-parameter model, which is applicable to regions where electron equilibrium is established. The two physical parameters are the dose-averaged linear attenuation coefficient and the beam-hardening coefficient. The advantage of this model is that it is a natural method when an irregular field is shaped by a multi-leaf collimator (MLC). Accuracy is not compromised by the triangulation since the irregular field is defined by the straight edges of the MLC leaves. The model predicts the percent depth dose with acceptable accuracy for any arbitrary shape of fields. We report on results for 6- and 18-MV photon beams and for a number of irregularly shaped fields Wiley-Liss, Inc. Key words: percentage depth dose modeling, phantom scatter, multi-leaf collimator (MLC), irregularly shaped field, photon beam INTRODUCTION A significant component of the absorbed dose at a point in radiation therapy is due to the scattering of primary beam. For example, typical ratios of scatter to primary dose at a 10-cm depth for a cm 2 field are 0.24 and 0.15 for 6- and 18-MV photons, respectively. It is essential to include the correct amount of scattering in any dose calculation. For correction-based algorithms, Clarkson integration is usually performed over the irradiated area to account for the scattered component, which involves look-up tables. This process is time consuming, especially when many repeated calculations are needed for beam optimization. A simple model, which would work within an acceptable limit of error and for a reasonable range of parameters, is highly desirable. Attempts have been made to describe the tissue-phantom ratio in terms of a few measured parameters, which are characteristics of the beam [1,2]. A two-parameter model developed by Bjärngard et al. [1] estimates the scattered component as asd/(ws + d), where a and w are the two parameters and s and d are the field size and depth, respectively. For high-energy photons, where lateral and longitudinal electron equilibrium exists, a and w are related to the attenuation coefficient as a and w [1]. Percentage depth dose (), tissue-phantom ratio (TPR), tissue maximum ratio (TMR), tissue to air ratio (TAR), etc. contain in them the scattering effect and their dependence on depth and field size. They consist of one part, primary fluence (adjusted for inverse square, attenuation, and beamhardening effects), and another part, which represents the effects of scattering. and the other quantities listed above are typically measured for simple square-shaped fields for each therapy machine and modality. This information then forms the basis for future dosimetric calculations. When *Correspondence to: Murshed Hossain, Ph.D., Medical Physics Division, Department of Radiation Oncology, Thomas Jefferson University, 111 South 11 th St., Philadelphia, PA Phone: (215) ; Fax: (215) ; Murshed.Hossain@mail.tju.edu Received 13 March 2000; Revised 6 September 2000; Accepted 6 December 2000 Published online 8 March 2001.

2 Hossain et al.: Model of Percentage Depth Dose 141 calculating or any of the other ratios related to the volume scatter for rectangular fields, an approximate equivalent square is used in the computation. Much research has gone into finding better models for determining the equivalent square [3,4]. For an irregularly shaped field, scattering effects are obtained by performing integration over the given field. Additionally, a multi-leaf collimator (MLC) is often used to conform the field to a given shape. In this case the irregular field can exactly be broken into a set of right triangles. Siddon et al. [5] have shown how one can compute the scattering effects exactly for the case of a right triangular field. Xiao et al. [6] have drawn on the works of Siddon et al. and have expressed the central axis phantom scatter as a function in closed form to be summed over appropriate right triangles spanning the given irregular field. In this work we have considered a number of irregularly shaped fields and computed the for each field using two experimentally measured parameters, namely,, and, the linear attenuation and the beam-hardening coefficients, respectively. We have also measured the s for these shapes and compared them with the results obtained from the model computations. MATERIALS AND METHODS The scatter-to-primary dose ratio for a square field of side s at a depth d is given by Bjärngard et al. [1] as: s,d = asd ws + d, (1) where a and w are the two parameters introduced by Bjärngard et al. [1]. For circular fields of radius r the above equation takes the form: r,d = a 0rd w 0 r + d. (2) The new parameters with the subscript 0 are related to the original parameters as a 0 a/0.561 and w 0 w/ These relations are valid within a 0.3% accuracy [7]. For an irregular field (r,d) can be integrated over all angles on a plane perpendicular to the beam, and an expression for (d) can be obtained: d = 1 r,d d = 1 a0 rd w r + d d. (3) Fig. 1. a: An irregular field of N vertices may be decomposed into 2N right triangles, four of which are shown (taken from Xiao et al. [7]). RT stands for right triangle; the + or signs show whether to add or subtract the specified right triangle. Notice that in triangle 023, RT1 is the large right triangle from which RT2 is subtracted. b: An irregular field (circle) is shaped by a multi-leaf collimator (MLC). Triangles and rectangles are used to decompose the scatter contribution of one leaf, as described in the text. The integration simplifies if the area under consideration can be decomposed into a set of right triangles, as shown in Figure 1 [7]. The sum of any function over the irregular shape is the sum over the number of right triangles with proper signs. For

3 142 Hossain et al.: Model of Percentage Depth Dose each of these right triangles, the scatter-to-primary ratio can be obtained by integrating Equation (3). Let us assume that the two short sides of a right triangle are X and Y, and thus the apex angle is arctan(y/x). The integration can be performed in closed form as [7]: X,Y = Q, = a0d arctanh Q 1 Q 2 1 tan w 0 Q 2 1 when Q 1, a 0 d 1 w 0 2 tan when Q = 1, a 0 d arctan 1 Q 1 Q 2 tan w 0 1 Q 2 when Q 1. (4) where the symbol Q d/(w 0 X). Once the scatterto-primary ratio is computed for a given shape, the ratio of at a depth d to that at a reference depth d r can be readily obtained [7]: Fig. 2. ln(dose) versus water-column depth for attenuation coefficient measurement. d,s d r,s = e d 1 d 1 + d,s f + d r 2 e d r (1 d r 1 + d r,s f + d, 2 (5) where f is SSD and s is the field size. Here we have used d r 10 cm. The scatter component can be characterized by two parameters a and w, which in turn can be expressed in terms of the linear attenuation coefficient, as pointed out in the Introduction. As a first step in applying this technique, the attenuation and beam-hardening coefficients for the 6- and 18-MV beams of an Elekta SLi accelerator were measured. This was accomplished by passing a narrow beam (1 cm diameter at 100 cm from the source) through a column of water of various thicknesses. The exit dose was measured using a diode at extended SSD. The data were fitted to the equation I = I 0 exp d 1 d. (6) As shown in Figure 2, the logarithm of the dose was found to closely fit a polynomial d + d 2 (correlation of for both energies). The values of and were found to be cm 1 and cm 2 for 6-MV photons and cm 1 and cm 2 for 18-MV photons, respectively. Using the physical parameters, as discussed above, the model described here can be used to compute for a given field shape. A total of four different shapes of irregular fields and three to four different points of measurement for each were used to check the accuracy of the calculation. The four shapes shown in Figure 3 are identified as Circle, Cross, I, and Irreg. The measurement points are labeled A, B, C, and D. For each point, the shape is translated horizontally such that the central axis of the beam passes through the chosen point. This means that different points of measurement for a given geometric shape effectively represent different field shapes. Each shape is discretized using the Elekta system Multi Leaf Planning System (MLP) for digitizing MLC shape. Entering the shape into the MLP system also defined the MLC leaf positions for the accelerator. The scatter-to-primary ratio is computed for each leaf by considering two right triangles and two rectangles. Figure 1b shows this schematically. The end points of one leaf are marked C and D in Figure 1b. The two right triangles of interest are OAC and OAD, and the point O corresponds to a certain depth on the central axis where the dose is calculated. The scatter component is computed for the rectangles OACE and OADB by using Equation (4) for the triangles OAC and its counterpart OCE and

4 Hossain et al.: Model of Percentage Depth Dose 143 Fig. 3. Four shapes used in this study. a: Circle of 20 cm diameter. b: Cross shape. c: I-shape. d: An arbitrary irregular field. The gross dimension of the shape in d is cm. The points of measurements are labeled A, B, C, and D. OAD and its counterpart ODB, respectively. Then, by subtracting the scatter component for the region OACE from that of the region OADB, the scatter component for the shaded region CDBE is obtained. The scatter-to-primary ratio of a region similar to the one shown shaded in the figure is computed for each leaf. The scatter-to-primary ratio is found by summing over all the leaves using appropriate signs (negative when a leaf crosses the central line perpendicular to the leaves). Then, Equation (5) is used to compute the. This calculated is then compared with the measured for 6- and 18-MV x-rays. A Wellhoffer IC-10 ion chamber (inner dimensions: 3.3 mm length by 6 mm diameter) was used to measure the. The measurements were made in a cm water phantom with a 100-cm SSD. The results of this comparison are given below for both 6- and 18-MV x-rays. RESULTS The model described above is only valid where electron equilibrium is established. Thus, to make a reasonable comparison between the measured and calculated, the depth dose is normalized at a depth where electron equilibrium is established. A depth of 10 cm is chosen for this normalization [7]. Electron contamination at this reference

5 144 Hossain et al.: Model of Percentage Depth Dose Fig. 4. and calculated value of percentage depth dose () for 6-MV photons for the irregularly shaped field (Fig. 3d) at point A. The agreement is better than 1% at all depths greater than 3.5 cm. This is typical among the shapes considered. depth and deeper depths we have considered here are negligible. For 6-MV photons, the agreement between the calculated and measured values is better than 1% for all depths greater than 3.5 cm and better than 2% for all depths greater than 1.0 cm. The same level of accuracy holds for all shapes and points considered. A typical comparison for 6 MV is shown in Figure 4. For 18 MV, the agreement is better than 2% for depths greater than 4.5 cm. Figure 5 shows the best (a) and the worst (b) fit cases for 18-MV photons. The model is not valid in the shallow buildup region and thus departs from the measurements. In Tables 1 and 2, the measured and the difference between it and the model calculation are presented for selected depths for 6- and 18-MV photons, respectively. The results are similar for a number of other dose points, which were included in the study but are not shown here. DISCUSSION Our study provides independent validation of the Bjärngard and Vadash [8] model of describing the beam qualities in terms of attenuation coefficients. The results presented here show that the model Fig. 5. and calculated percentage depth dose () for 18MV photons. a: The best fit (I-shaped field, point A) and (b) the worst fit (cross-shaped field, point C) comparison between the model and the measured data are shown. agrees well with measurements within a couple of percent. Moreover, the method of summing over right triangles used here is a fast and accurate procedure to calculate the scatter components exhibited in, TPR, etc. It makes it possible to compute or any other ratio for any arbitrary shape once and are measured for the given modality (beam and energy) with reasonable accuracy.

6 Table 1. Comparison between and Calculated Percentage Depth Dose () for 6-MV Photons and for Various Field Shapes* Shape Depth 7cm 10cm 16cm 22cm Hossain et al.: Model of Percentage Depth Dose 145 Circle Pt. A Circle Pt. C I Pt. A I Pt. C Cross Pt. A Cross Pt. B Irreg. Pt. A Irreg. Pt. B *The irregular fields are depicted and labeled in Figure 3. The percentage depth doses are normalized to a depth of 10 cm. The percentage difference is given by 100 (calculated value-measured value)/measured value. Table 2. Comparison between and Calculated Percentage Depth Dose () for 18-MV Photons and for Various Field Shapes* Shape Depth 7cm 10cm 16cm 22cm Circle Pt. A Circle Pt. C I Pt. A I Pt. C Cross Pt. A Cross Pt. B Irreg. Pt. A Irreg. Pt. B *The irregular fields are depicted and labeled in Figure 3. The percentage depth doses are normalized to a depth of 10 cm. REFERENCES 1. Bjärngard BE, Vadash P, Ceberg CP. Quality control of measured x-ray beam data. Med Phys 1997;24: Storchi P, van Gasteren JJ. A table of phantom scatter factors of photon beams as a function of the quality index and field size. Phys Med Biol 1996;41: Venselaar JL, Heukelom S, Jager HN, Mijnheer BJ, van Gasteren JJ, van Kleffens HJ, van der Laarse R, Westermann CF. Is there a need for a revised table of equivalent square fields for the determination of phantom scatter correction factors? Phys Med Biol 1997;42: McDermott PN. The physical basis for empirical rules used to determine equivalent fields for phantom scatter. Med Phys 1998;25: Siddon RL, DeWyngaert JK, Bjärngard BE. Scatter integration with right triangular fields. Med Phys 1985; 12: Xiao Y, Altschuler MD, Bjärngard BE. Quality assurance of central axis dose data for photon beams by means of a functional representation of the tissue phantom ratio. Phys Med Biol 1998;43: Xiao Y, Bjärngard BE, Reiff J. Equivalent fields and scatter integration for photon fields. Phys Med Biol 1999;44: Bjärngard BE, Vadash P. Analysis of central-axis doses for high energy x-rays. Med Phys 1995;22: