WEDGE FILTERS FOR MEGAVOLTAGE ROENTGEN RAY BEAMS

Transcription

1 Acta Radiologica Oncology 23 (1984) Fusc. 6 FROM THE RADIATION THERAPY DEPARTMENT, ANTON1 VAN LEEUWENHOEK HOSPITAL, THE NETHERLANDS CANCER INSTITUTE, AMSTERDAM, THE PHILIPS MEDICAL SYSTEMS DIVISION, BEST, AND THE LABORATORY OF MEDICAL PHYSICS, UNIVERSITY OF AMSTERDAM, AMSTERDAM, THE NETHERLANDS. WEDGE FILTERS FOR MEGAVOLTAGE ROENTGEN RAY BEAMS R. VAN DER LAARSE, P. A. M. VAN OVERBEEK and J. STRACKEE Abstract The aim of this investigation, to construct a range of fixed wedge filters and to simulate these with a motorized wedge, led to the derivation of 5 equations. These equations can be used to construct and test a consistent set of fixed wedge filters, eliminating elaborate trial and error experiments. The fixed wedge filters already in existence for the Philips SL75-10 and SL75-20 linear accelerators fitted these equations rather well. After adapting the motorized wedge of the SL75-14 according to these equations, it simulated the fixed wedges accurately. In external beam radiation therapy, wedge filters are used to tilt the isodose lines of megavoltage roentgen ray beams over the so called wedge isodose angle in order to homogenize the dose distribution in the target volume (1). The shape of a wedge filter required for a given tilt is usually designed by trial and error. A first try is often to tilt graphically the isodose lines of the open beam over the desired angle and to calculate the resulting off axis dose ratios along a line normal to the central axis at a given depth (4). Treatment set-up parameters such as beam energy, beam incidence directions, wedge filters and beam weights must be adapted for each target volume to be treated. Usually only a limited number of wedge filters is available. TATCHER (3) combined wedged and open beams in order to obtain wedge isodose angles smaller than the one obtained by the wedged beam only. Thus it has long been known that the combination of an unwedged beam with a wedged beam gives a resultant wedge angle of an intermediate value determined by the relative weightings of the wedged and unwedged fields. A motorized wedge filter insertion mechanism greatly simplifies the practical implementation of this principle. Recently, a linear accelerator with a motorized wedge filter (Philips SL75-14) was installed at the Antoni van Leeuwenhoek Hospital. As soon as the wedged field has been delivered, the wedge filter is immediately motored out of position, and delivery of the unwedged field follows automatically in one treatment session. This paper examines the relationship between the shape of a fixed wedge and the corresponding wedge isodose angle, the relationship between the wedged fraction of the beam and the resultant wedge angle, and the compatibility between the use of the motorized wedge filter system and the use of the fixed wedge angles provided by the SL75-10 and SL75-20 already in operation. Fixed wedge filters The influence of a wedge filter on an isodose curve well inside the field edge of a megavoltage roentgen ray beam is indicated by the wedge isodose angle (wedge angle), i.e. the angle between the isodose line and the normal to the central axis. This angle depends on the irradiation conditions. Therefore the wedge angle is defined for a specified field Accepted for publication 26 April

2 478 R. VAN DER LAARSE, P. A. M. VAN OVERBEEK AND J. STRACKEE size at standard SSD at a specified depth on the central axis. In this section two relations between the shape of a wedge and the corresponding wedge angle in the phantom are given. Also the determination of the wedge angle from measured off axis dose ratios is presented. Wedge isodose angle with respect to a given setup. Consider a ray from the focus through the wedge to point Q on the central axis at depth d (Fig. 1). When passing a distance u, through the wedge filter, the ray is attenuated by a fraction exp (-pw u,),,uw being the linear attentuation coefficient of the wedge material for the radiation quality used. The path length d through water decreases the ray intensity further with a fraction exp (-,up(d-t)),,up being the practical linear attenuation coefficient in water and t the dimension of the build-up region;,up is a function of the radiation quality, of the field size and also of the focus-skin-distance F. Its dependence on F is caused by the decrease of dose due to the inverse square law. A ray to P on the isodose line through Q travels a distance u,+s, through the wedge (Fig. 1). Now assume that the unwedged beam is flat. Then the shift s of the isodose line through Q is caused by the additional wedge path sw, thus exp (-,ups)= exp(-p,s,) or pps=,uwsw. From Fig. I, with W being the focus-wedge base-distance, follows that b/(f+d) = bw/( W+uo) Combining these equations and approximating s/b by tan (o) and sw/b, by gives In order to obtain a straight isodose line through the whole beam, wedge angles a and are usually different to correct for the curvature of the isodose lines of the non-filtered beam (inset Fig. 1). Let y be the effective open beam isodose angle, which in fact represents the curvature, then tan(w+y) =Ztan(a) tan (o - y) = Z (1) w+uo P, with Z = - F+d Pup The linear attenuation coefficient of the wedge material pw can be calculated from semi-logarithmic graphs of the measured attenuation versus the thick- locus Fig. 1. Quantities used in the description of wedged filters and wedged beams. W = focus wedge distance, a = distance of wedge tip to central axis, u,= wedge thickness at central axis, a, /3 = wedge geometrical angles,,u, = linear attenuation coefficient of wedge material, F = focus skin distance, 2A = maximum wedged field dimension, d = depth in phantom of point Q on the central axis, b =distance from P to central axis (point Q), w = wedge isodose angle of the isodose line through Q, y = open beam isodose angle at Q,,up = practical linear attenuation coeficient of water for the beam set-up considered. For s, s, b,, Ax and Ay see text. ness of slabs of wedge material. The percentage depth dose (PDD) for depth d exceeding the depth of peak absorbed dose, can be approximated by PDD(d)= 100.exp (-pp(d-t)); hence pup can easily be calculated from PDD tables by pp=ln (01/D2)/ (d2-dl) with D1 the PDD at depth dl and D2 the PDD at depth d2. The coefficient,up depends on photon energy, source subject distance, and field size. Equation (1) cannot be used directly to calculate a or /3 for a requested wedge isodose angle o, because o depends also on W+u,. However, on the central axis at a SSD of F one has uo = - AW tan(a) F with 2A the maximum wedged field dimension at SSD F. Inserting this relation into (1) and using the

3 WEDGE FILTERS FOR MEGAVOLTAGE ROENTGEN RAY BEAMS 4 79 conditions w+y=o for a=o and o-y=o gives tan (a) = 4a C 2a - w + J w+- tan(w+y) for p=o, and being equal to the same expression but with tan (w-y); with Pw AW C= and a=- (F+ 4 Pp F With equation 2 the full shape of a wedge filter for a given wedge isodose angle can be calculated, once the position of the wedge in the beam and its material is known. Calculation of the wedge isodose angle from measured offaxis dose ratios. If Q is a point on the central axis at depth d, the dose DQ in Q can be related to the peak absorbed dose DM (Fig. 1) by DQ=DMeXp (-Pp(d-t)). The off axis dose ratio rp in point P, lying on the same depth as Q, is defined as the ratio DpIDQ; Dp being the dose per monitor unit in P and DQ the dose per monitor unit in Q. It can be seen from Fig. 1 that for the high side of the wedge one has to a first order of approximation rp= l-p,s= l-p,btan(w-y); tan(w-y) must be taken instead of tan(w) because the effective isodose angle of the open beam isodose line is y instead of 0. This can be generalized to r -1 tan (ofy) = f P PPb with the + sign for the tip side of the wedge, and the - sign for the high side. Equation (3) is used to calculate the wedge isodose angle from an off axis ratio measurement in a point at the depth at which the wedge isodose angle is defined. Motorized wedge filters Wedged fraction versus wedge angle. In the SL75-14 a motorized wedge is inserted into the beam during a fraction f, of the total number of monitor units (MU s) to be delivered: &=(number of MU s with wedge)/(total number of MU s) with O<f,<l. (2) (3) Let w, be the wedge isodose angle in Q (Fig. 1) for the fully wedged beam (f,= l), i.e. the maximum wedge angle possible with the motorized wedge. Let T be the effective transmission factor of the motorized wedge, i.e. the dose per MU measured in point Q on the central axis with fully wedged beam, divided by the dose per MU in Q for the open beam. Then the fraction of monitor units to be delivered with wedge in order to get a wedge angle w is given by (c.f. Appendix): tan (o) Ttan(wo)+(l-T)tan(o) Calculation of the wedged fraction f, from measured off axis dose ratios. Consider a point P at a given distance from the central axis and let Q be the point on the central axis at the same depth (Fig. 1). Now perform the following relative measurements in the beam with a sufficiently large field size: In the open beam, the reading at P will be u, and in Q u,, digits per monitor unit. For the motorized wedge in the beam during the whole irradiation, f,=l, the reading at P will be uw and in Q uw.= digits/mu. For the fraction f, given with the motorized wedge, the reading in P will be mp and in Q mq digits/mu, thus the off axis ratio obtained with f, will be r,=mp/mq. The reading mp obtained with f, in P equals f,u,+(l-f,)u, and the reading mq in Q equals f,uw,.+(l-f,)u,,c digits/mu. Thus This equation can be used to calculate f, from measured off axis ratios. Simulating fixed wedges. When a motorized wedge is used clinically for simulating fixed wedges, the following remarks are pertinent. 1) In order to simulate a series of fixed wedges by a motorized one, the maximum wedge angle obtainable with the motorized wedge must be equal to or larger than the largest wedge angle of the fixed ones. 2) Fixed wedge filters are usually designed so that the isodose lines within the geometrical field edges are straight. The lines obtained with the mo- torized wedge inserted (f,=l), must then also be straight. In the Appendix it is explained why an isodose line obtained with any wedged fraction f, is straight when the isodose lines obtained with f,=o and with f,=1 are straight. (4)

4 480 R. VAN DER LAARSE. P. A. M. VAN OVERBEEK AND J. STRACKEE 3) The total number of monitor units for a given peak absorbed dose must be the same for a fixed wedge and for the motorized wedge having the appropriate wedged fraction. Therefore, all wedges-the fixed ones and the motorized one-must in principle define the same maximum wedged field width. The isodose lines will then have the same dose values, because at the wedge tip there is no attenuation by wedge material in both wedge modalities. However, any difference in backscatter to the monitor chamber from the fixed wedge and from the motorized one, should be taken into account by adapting the thickness u, (Fig. 1) of the fixed or motorized wedge. Fixing the wedged fraction. Equation (5) can be used to simulate a fixed wedge on linac 1 by the motorized one on linac 2 with both linacs having the same field flatness. Take any point P and let Q be again its projection on the central axis (Fig. 1). Now measure on linac 1 the off axis ratio rp in point P with the fixed wedge inserted. Perform on linac 2 the other dose measurements in P and Q for the open and the wedged beam, as described previously. Using these measurements and substituting rp for rw, equation (5) gives the fw required to obtain the off axis ratio rp on linac 2. Re s u 1 t s Already in 1972 we made special fixed wedges for our SL75-10, designed by trial and error. They give straight isodose lines for field size 14 cm x 14 cm at SSD 100 cm. The wedge angles of these fixed wedges at a depth of 8 cm are respectively 15", 27", 43.5" and 55". All wedges define the same maximum field dimension, 20 cm, at SSD 100 cm. These fixed wedges were used to check the equations 1-5. Determination of the shape of the motorized wedge using extrapolation. The shape of our motorized wedge was already determined before equation 2 was derived. This shape was obtained graphically from the shapes of the fixed wedges by linear extrapolation of the graphs of tan(a) and tan@) against tan(w) (Fig. 2). The extrapolated values for tan(a) and tan@) were taken at w=59" and brass as material &,=0.317 cm-' for 8 MV roentgen rays). For the lead alloy of the motorized wedge &,=0.460 cm-' for 8 MV roentgen rays) these values were adapted and our motorized wedge should have tan(a)=1.51x0.317/0.460=1.041 and = 0.95 X /0.460= For technical rea I / Fig. 2. Extrapolation of the wedge geometrical angles for the motorized wedge from the geometrical angles of the existing fixed wedges. The isodose wedge angles o of the fixed wedges are 15", 27", 43.5" and 55", respectively. o actual tangents of the geometrical angles of the fixed wedges, made of brass. The extrapolated tangents of the wedge geometrical angles for o=59" are given for brass as wedge material. + calculated tangents of the same angles using equation 2 with the lead alloy as wedge material w,=59", y=7.5", tan (a)= I.OO and sons they were machined with tan (a)= 1.O and =O.667. These experimentally found wedge material angles can be used to verify equation 2. For a wedge isodose angle w=59" and y=7.5", equation 2 gives for lead tan (a)= and using W= cm, A= 10 cm and,uu,=0.046 cm-i. Calculation of the wedge isodose angle of the motorized wedge. With the motorized wedge (f,= 1) the relative dose values along the normal to the central axis at 8 cm depth for a field 17 cm X 17 cm were measured in 13 points. These measurements give the off axis ratios in these points and, using equation 3, suffice to calculate w,; y=7.5" was used. Table 1 gives the result for some typical points: w,=58.6". Comparison of existing fixed wedges with calculated ones. Using 0,=59" in equation 2, tan(a) and tan@) were calculated as a function of tan(o+y)

5 WEDGE FILTERS FOR MEGAVOLTAGE ROENTGEN RAY BEAMS 48 1 Table 1 Determination of maximum wedge isodose angle w, from measured off axis ratios rp, (ug for the motorized wedge. The off axis dose ratios rp, wg are calculated from the dose measurements u,. See section on Results for more details Point UW TP, (I," 00 (-7. 8) " (-5, 8) " (0, 8) 325 I - (5, 8) " (7, 8) " oj0 = 58.6" Table 2 Determination off, values. Values off, obtained with equation 5 using the dose measurements with the motorized wedge inserted and with the open beam, and the off axis dose values for the fixed wedges. At the end of each column the average value, resulting from 12 off axis points distributed over the whole beam dimension at depth 8 cm, is given with its standard deviation. The calculated fw values are based on equation 4 for the value 58.6" of w, given in Table I Point fsy h3.s f27" fw Measured (-7, 8) (-5, 8) (5, 8) (7, 8) Averaged over points f0.045 k0.051 Calculated and tan (w- y) for respectively the wedge angles w=15", 27", 43.5" and 55" with y=7.5". The resulting tan (a) and for each of these wedge angles are compared in Fig. 2 with the corresponding ones of the experimentally determined fixed wedges. Their differences are negligible. Comparison of measured and calculated f, values. Values for f, to simulate fixed wedges can be obtained with equation 5 from dose measurements with a motorized wedge and with an open beam, combined with off axis ratio measurements with the fixed wedges. The resulting values off, are given in Table 2. In the same table, values of f, calculated by equation 4 for the wedge angles of the fixed wedges are compared with the measured ones for each of the two values of w, as given previously. Comparison of measured and calculated wedge transmission factors. When the fixed wedges and the motorized one are all defining the same maximum wedged field, then, disregarding the backscatter into the monitor chamber, the total number of monitor units given with a fixed wedge should equal that given with the corresponding fraction of motorized wedge plus open beam. See section 'Motorized wedge filters'. However, both the fixed wedges and the motorized one scatter radiation back into the monitor chamber. The influence of backscatter from the motorized wedge appeared to be 13 per cent higher than that from the fixed wedges, due to differences in material (lead and brass, respectively) and the construction of the monitor chambers of the different Philips linacs. To correct for the corresponding decrease of the wedge transmission, the thickness of the motorized wedge along the central axis was decreased with 0.3 cm, reducing the maximum field dimension 2A from 20 cm to 17 cm at 100 cm SSD. The fixed wedges all define a maximum wedged field of 20 cm. After this output correction, an agreement within 1 per cent was found between the total number of monitor units for a dose of at least 100 cgy on the central axis at the depth of peak absorbed dose, either obtained with a fixed wedge or with the corresponding f, of the motorized wedge. Comparison of measured and calculated dose values. In Table 3 measured dose values obtained with the motorized wedge on the SL75-14 are compared with dose values for the fixed wedges. In order to exclude any difference in field flatness between the linacs, the dose values for the fixed wedges were calculated by our planning program (2), because in this program the calculated dose values are based directly on measured beam profiles. The difference between a measured and calculated dose value in Table 3 was related to 100 per cent at the depth of peak absorbed dose on the central axis (2 cm with 8 MV roentgen rays) by dividing it by the transmission factor of the fixed wedge considered. Discussion From rewriting equation 1 as tan (w+y)= Z/((F+d)p,), with Z depending only on the wedge

6 482 R. VAN DER LAARSE. P. A. M. VAN OVERBEEK AND J. STRACKEE Table 3 Comparison of the doses measured using the motorized wedge with the final values for f,,, with the corresponding doses calculated with the planning program for the fixed wedges Point Wedge Motorized wedge Fixed wedges Difference x 100/ Measured angle Measured dose Calculated dose wedge output fixed wedge for 100 MU for 100 MU factor output factor (~GY) WY) (0, 8) I5 (-7, 8) (7, 8) shape and position in the beam, it follows that for a given wedged field, the wedge isodose angle o depends on the depth d in water, the distance F from focus to the surface and the practical linear attenuation coefficient pup. In the next paragraph only the thin side of the wedge will be considered. For the high side (0-y) should be replaced by (w+y). The decrease of the wedge isodose angle with depth is thus not primarily caused by scatter as suggested in ICRU report 24, section 2.4.1, but is mainly due to the decreasing difference in path length through the wedge with increasing depth in water when moving laterally over the same distance. The influence of d on o+y is illustrated by considering a wedge with w+y=45" at 10 cm depth for a field of 14 cm x 14 cm at F=100 cm. From equation 1 it follows that Z/p,=F+d= 110, resulting for 2 cm depth in w+y=47" and for 20 cm depth in o+y=42.5". The dependence of w+y on the field size is related to the inverse of the practical attenuation coefficient pp in water. In above example for 8 MV roentgen rays with w+y=45" for field size 14 cm x 14 cm, the angle o+y for field size 6 cm x 6 cm becomes 42". The equations 1 to 5 are all based on straight isodose lines, with or without wedge inserted. Although in reality isodose lines are curving towards the surface near the beam edges, it appears that fixed wedges can be remarkedly well simulated by the motorized one. This is explained by the existence of only one single bend in the oblique side of the wedges, at the central axis of the beams. The influence of curving towards the surface near the beam edges is therefore equally present in a wedged beam whatever the field size. The original SL75 wedges all had several bends in the oblique side in order to straighten the isodose line as far as possible, accordingly, simulation of these wedges with a motorized one would be less successful, especially for the smaller wedge angles where the unstraightened open beam profile would prevail. In the section 'Fixed wedge filters' the experimental determination of the parameters needed for the calculation of a whole range of wedge filters is described. Only two parameters need additional attention: the attenuation coefficient of the wedge pw and the open beam isodose angle y. We determined pw in an indirect way as follows. First, simple brass wedges were constructed without a bend along the oblique side. Then, for each of these wedges, measured off axis ratios at several depths in water were compared with calculated ones, changing pw until the best fit was obtained. By using off axis ratios instead of dose values any backscatter from the wedges into the monitor chamber was excluded. The open beam isodose angle y can be used to overtilt slightly the wedged isodose in order to extend the straight part of the isodose line somewhat

7 WEDGE FILTERS FOR MEGAVOLTAGE ROENTGEN RAY BEAMS 483 to the beam edges. When simulating fixed wedges with a motorized one, care must be taken not to overtilt too much. The use of equation 5 to obtain measured f, values for the simulation of fixed wedges on linac 1 by a motorized wedge on linac 2 presumes identical field flatness on both linacs. The susceptibility of equation 5 for even small local differences in field flatness between the linacs is quite high. For example, it appeared from our measurements that a 2 per cent difference between the linacs in field flatness at a given point off axis, changed fz7" into for that point when calculated with equation 5. Therefore, the final f, values obtained with equation 5 (Table 2, f, averaged), were based on the values in a number of points along the normal to the central axis at the depth considered. In this way the influence of such local differences onf, is reduced. According to ICRU 24 (l),'the superposition of a beam with wedge isodose angle 45" and an unmodified beam would produce a wedge isodose angle of approximately 22.5", each beam contributing 50 per cent of the absorbed dose to the reference point'. This statement is quite crude. Suppose a wedge filter has a wedge transmission factor T. When the wedged beam and the open one both contribute 50 per cent of the absorbed dose to the reference point at the depth of peak absorbed dose on the central axis, then the number of monitor units with wedge MU, and with open beam MU, satisfy the relation TMU,=MU,. Now fw=mu,/ (MU,+MU,)= l/(l+z). Inserting this in equation 4 gives tan(w)=0.5 tan(w,) instead of w=o.5w0. For 0,=60" one has 0=40.9" and 0,=45" gives w=26.6". The angle w is thus rather far off from the corresponding w,/2. Clinical aspects The use of a motorized wedge in clinical practice provides a wider degree of freedom when optimizing a treatment plan, as any wedge angle between zero and the maximum angle is available. The setting up work associated with wedged treatments. is simplified as the need to insert and remove wedges manually is eliminated. The flexibility in selecting wedge angles has its disadvantages. First, it is in some cases impossible with the SL74-14 linear accelerator used, to rotate with either a large or a small wedge isodose angle. With a large wedge angle, the number of monitor units for the open beam fraction is then too small to be delivered over a large angle of rotation. With a small wedge angle the same happens for the wedged beam fraction. Secondly, when delivering small doses, rounding to the nearest number of MU's, as well for the wedged fraction as for the open beam fraction, leads to a corresponding error in delivered dose. For example, suppose 50 MU's are to be given with w=55", f,=0.952 and the motorized wedge transmission factor T=0.407, then the error in dose on the central axis will be 1 per cent for any field size. For the maximum wedged field size, the error in dose near the beam edge at the tip side of the wedge will of course be 0 per cent but near the beam edge at the thick side of the wedge the error will be 2 per cent relative to the correct dose on the central axis. APPENDIX Relation between the wedged fraction f, and its wedge isodose angle Define f, as the number of monitor units delivered with wedge, divided by the total number of monitor units given. Now suppose a hypothetic wedge filter with thickness zero along the central axis (u,=o in Fig. 1). If f,=1 the whole dose contribution is given with wedge inserted and the wedge angle obtained will be 0,. With fo=o the open beam isodose line is obtained. An isodose line is defined by the differential equation dd dd -dx + -dy = 0 dx 8Y (Fig. 1.) If the isodose line is a straight line, dyldr is constant and for f,= 1 dyldx= - tan (w,); thus: For f,=o, we have w,=o and thus ==O. (2) ax If the fractionf, of the total number of monitor units is given with the hypothetic wedge, only this fraction will contribute to the dose gradient in the x-direction. From (1) and (2) it follows that: This line will again be a straight isodose line because f,,tan(w,) is constant for a given f,. Defining w as the wedge isodose angle obtained with fraction fo, one gets (3) tan (w) = f, tan (w,) (4)

8 484 R. VAN DER LAARSE, P. A. M. VAN OVERBEEK AND J. STRACKEE For a real wedge the dose along the central axis delivered by the fraction f, is attenuated by the wedge transmission factor T. So with wedge only a fraction Tf, of the number of monitor units is effective, the remaining fraction I-f, given with open beam is of course fully effective. Therefore, f, in (4) should be replaced by Tf,l(Tf,+ 1 -f,), which gives again a straight isodose line with a wedge angle w given by or tan (w) = Tf~ tan (w,) Tf,+ 1 -f, tan (w) f, = Ttan (a,)+( 1 -T) tan (w) (5) (6) REFERENCES 1. ICRU Report 24: Determination of absorbed dose in a patient irradiated by beams of X or gamma rays in radiotherapy procedures, Washington, USA VAN DER LAARSE R.: Computerized radiation treatment planning. Ph.D. Thesis, University of Amsterdam, TATCHER M.: A method for varying the effective angle of wedge filters. Radiology 97 (1970), TRANTER F. W.: The design of wedge filters for use with a 4 MeV linear accelerator. Brit. J. Radiol. 30 (1957), 329. Equation 6 can be used to calculate f, for a given wedge isodose angle w provided that w<o,.