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Betatron electron beam characterisation for dosimetry calculations
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 299 (http://iopscience.iop.org/0031-9155/24/2/006) View the table of contents for this issue, or go to the journal homepage for more
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PHYS. MED. BIOL., 1979, Vol. 24, No. 2, 299-309.
Printed in Great Britain
Betatron Electron Beam Characterisation for Dosimetry Calculations JOHN D. STEBEN, PH.D., K. AYYANGAR, and N. SUNTHARALINGAM, PH.D.
PH.D.
Department of Radiation Therapy, Thomas Jefferson University, Philadelphia, Pennsylvania, U.S.A. Received 6 J u n e 1977, in j n a l form 3 October 1978
ABSTRACT. Parameters havebeen specified for electron beams with energies 5-45 MeV from a Brown Boveri betatron for use incomputeriseddosimetry calculations. A semi-empirical equation is given for the dose a t any point a t various depths in water. This equation is a modification to Kawachi’s predictive model which was based on solutions of a general age-diffusion equation. The depth doses and isodose curves are predicted as a function of the practical range, source skin distance (SSD) and field size. Depth dose accuracy requirements of f 2y0 above 50% depth dose and + _ 5 % at lower doses, relative to maximum dose, have been set and achieved. Also, the shape of the isodose curves with the constrictions a t higher doses and bulging a t lower values are accurately predicted. Computer calculated beams have been used t o generate summed isodose distributions for certain clinical situations.
1. Introduction Ever since the earliest days of the betatron, electrons have found uniqueand complementary uses in the treatment of cancer (Skaggs, Almy, Kerst, Lanzl andUhlmann 1948, Haas,Harvey, Laughlin,Beattie and Henderson 1954, Tapley 1973). I n recentyears,telecentricelectronrotations(Rassow 1970, 1972, Poser, NBmeth and Kuttig 1973, Fehrentz 1976) and multiple electron fields (Simpson,Borger,Ovadia and Cohen 1974) have also been used, and sometimes electron beams have been mixed with photon beams (Tapley 1973). Dose distribution calculations for complicated irradiation techniques can best be donewithacomputer,provided that sufficiently accuratecomputer representations of the individual beams can be made. While adequate computer representations of 45 MV photon beams for the BBC Asklepitron-45 a t Thomas Jefferson University Hospital (TJUH)have previously been generated through the implementation of tissue-maximum-ratio andscatter-maximum-ratio concepts(SuntharalingamandSteben 1977), the electron beams,with ten electron energies, ten electron cones, open-beam modalities and five scattering foils correlated to the energies, presented a greater challenge. Various methods have been proposed in the past to characterise electron beams of therapy machines. These have varied from numerical solutions of analytic electron transportequations(BergerandSeltzer 1969) tostrict recording of experimental results in a form such as ray-line and constant-depth profiles (Milan and Bentley1974), with various other approachesto characterise groups of electronbeams(Dahler,Baker and Laughlin 1969, Pohlit 1969, 0031-9155/79/020299+11$01.00
@ 1979 The Institute of Physics
300
John D.Steben et al.
Leetz 1976). The analytical extreme requires practically nodata but excessive computer processing, while theother involves simple interpolationswith quantities of data judged to be excessive for the Asklepitron-45 betatron for and which there would be over 200 beams to store.Thecomputertime resources, respectively, in these two cases were judged to be too large to be useful for the routine treatmentplanning calculations, which were to be carried out on a minicomputer. It was thus desirable to consider some semi-empirical approach which would rely on a small amount of stored beam information to generate a relatively largenumber of beams. One method for doing this has been discussed by Lillicrap, Wilson and Boag (1975), and consists of measuring ‘pencil beams’ or treatmentarea. For a andintegratingthem over afinitedistribution scanned-beammachine,such as aSagittairelinearaccelerator,a ‘pencil beam’ could closely approximate the physical situation. Forelectron beamslike those in the Asklepitron-45 which are flattened by scattering techniques, a calculationalmethod which includes the presence of scattering or diffusion would seem to be a better starting point. A parametric method of this latter type was proposed by Kawachi (1975), and uses solutions of the age-diffusion equation (Glasstone and Edlund 1952) to characterise the electronbeams. Diffusion equationsgenerally describe beamdensities, but for high energyelectrons the delivered dose pattern closely resembles the beam density diffusion pattern. Hence, by starting with age-diffusion solutions, one may minimise the number of independent parameters which must be varied. This is the basic method of the present study, as is discussed below. Slightly simpler parametric methods have been tried by others (Simpson et al. 1974, Leetz 1976), but seem to lack the great versatility or predictive accuracy needed for a general characterisation of the electron beams from the Asklepitron-45. Method In the present beam computerisation study, the age-diffusion model proposed by Kawachi (1975) has been applied, with some necessary modification, to parameterise the Brown Boveri Asklepitron-45 electron beams. The modified dose equationis given below, and the nature of the changes are discussed in the paragraphs that follow. 2.
x exp { - [(2.rr/3Rp) ( K T ) ’ ] ~ )
Betatron Electron Beam Characterisation
301
where (KT))
F
=
[(Cz/R,) +PIN
= SSD
Xo(z)= &(width+ ke) [(F+ z ) / F ]
+
+
Yo(z)= &(length he) [(F z ) / F ] .
R, is the practical range, and G,, G,, G,, C, N , P are constants for a given energy. F designates source-to-skin distance, z designates depth in patient, while x and y are the lateral coordinates; length and width of the field are defined a t the patient’s skin ; k, represents a small field size correction for edge effects; 7 represents ‘age’ of the beam and KT is the age-diffusion parameter. R, is the practical electron range (incm), determined experimentally by extrapolatinga line through measured 70% and 40% percentage dose distribution to intersect an empirically determined bremsstrahlung build-up of (energy/5)yo. The energy, E , in MeV, is related to R, by the Markus equation (Markus 1961), that is R, = 0.521E - 0.376. Parameters G,, G,, G,, C, N and P must be varied to determine the optimum fit to measured central axis depth dose and isodose curves. The equation reduces to Kawachi’s form for G, = 0, G, = 2713, G , = -716 and Yolarge. 2.1. Percentage depth dose Initial studies have shown that Kawachi’s cosine factor needed to be modified and that the three depth parameters, G,, G,, G,, need to be varied to achieve acceptable fits to our experimental data. Accuracy of better than & 2% at doses above 50%, and k 5% at lowerdoseswas required. Retaining the cosine dependence, but with the addition of a quadratic term
2.2. Field size effects The variation of percentage depth dose with field sizeis essentially predicted by the error function factors. Kawachi’s treatment is two-dimensional and assumes the beam to be infinite in the third dimension. Hence, Kawachi’s equations predict depth doses for a 16 cm x 4 cm beam significantly different 12
302
John D. Steben et al.
from those for a 4 cm x 16 cm beam. Generalising the formula to include the second lateral dimension was clearly necessary to permit close fitting of the measured depth doses. Themethod used was tointroducethe sameerror function of the second transverse coordinate as a factor. This also enables dose predictions to be made for any point in any plane, not just the central planes, for rectangular beams. Circular beams are, however, predicted correctly only in the central planes. 2.3. Isodose shapes The ‘beam-shape’ parametersare C, N and P. They are strongly interrelated; where P characterisesbeam diffuseness at the surface, C characterises the increase in diffuseness with depth andN is an overall shape-changing parameter. The function relating these three parameters in our equations is identical to the one used by Kawachi (1975). At all energies, a small modification to the physical width and length of the rectangularelectronbeamsis necessary to give proper isodose fits to the experimentaldata. This ‘correction’ tototalwidth, a small constant, k,, represents the fact that the measured radiation width of the beam, taken as the lateral distance between 50% dose points in the beam profile, at the depth of maximum dose, is slightly greater than the nominal width or ‘cone size’. For the betatron measurements, as evident in figs 3, 4 and 5, this constant, k,, was found to be 0.5 cm. 2.4. Calculation of parameters A computerprogramhas been developed which unscrambles the dose equation to calculate G,, G, and G, from other parameters listed below: (1) measured central axis depth dose data, z, D ( z ) ; (2) measured range, R, ; (3) field width and length, X and Y ; (4)the source to surface distance, F, and (5) constants, C, N , P. The program calculates the argument of the cosine factor for each depth fit t o thearguments it generates the andby a quadratic leastsquares parameters G,, G, and G,. Once G,,G, and G, are known, thenextstep was to generate isodose distributions andsee how they fit the experimental data. Parameters C, N and P were to be readjusted when necessary. G,, G, and G, must be recalculated if major changes are seen in C, N and P (the process is thus iterative). 3. Results Using the method described above, the parameters were determined a t each energy for one field size, namely a 10 cm x 14 cm rectangular beam and were then used to predict depth doses and isodoses for all field sizes. The parameters are shown in table 1. It may be noted that the quadratic termcoefficient G, in
Betatron Electron
Beam Characterisation
303
Table 1. Parameters characterising an electron beam Energy (MeV) 45 40 35 30 30 25 20 20 15 10 10
5
Beam type
Q,
G2
Q3
C
N
P
Range (cm) ( i n water)
Cones Cones Cones Open
0.0529 0.236 0.444 0.208
1.804 1.682 1.416 1.879
- 0.4326 - 0.4328 - 0.3664 - 0.5905
1.80 1.00 1.80 1-00 1.80 1.00 1-80 0.50
0.10 0.10 0.10 0.25
21.7 19.56 17.5 14.9
4 x 16 Cones Cones Cones Cones Cones 11 cm 211 cm Cones
0.601 1.339 1.512 1.342 1.998 1.821 1.917 2.458
1.263 0.527 0.369 0.542 0.0064 0.269 0.226 - 0.472
- 0.3798 -0.3118 - 0.3559 - 0.3430 -0.3195 - 0.5752 0.5554 - 0.8153
1.80 1-80 1.80 1.00 1e o 0 1.00 1.00 1.00
1.00 1.00 1.00 1.50 1.50 1.50 1.50 1.50
0.10 0.10 0.10 0.25 0.25 0.25 0.25 0.25
14.9 12.75 9.9 9.9 7.7 4.9 5.2 2.55
-
-
thedepthdependent cosine argument becomes important a t low energies, whereas the argument becomes essentially a linear function of depth at high energies.
3.1. Percentage depth doee The resulting percentage depth dose distributions for the 10 cm x 14cm beams are shown in fig. 1 for 10, 20, 30 and 40 MeV. For other energies, they cases, and a good are listed in table 2. Excellent fits are obtained in these feature of the model is its ability, using these same parameters, to fit the doses a t all field sizes. This is tabulated for 30 MeV in some detail. The comparison between measured and predicted data for 30 MeV is given in table3. Similarly,
Depth Icrnl
Fig. 1. Percentage depth dose for 10 cm x 14 cm electron beams at several energies. Measuredvalues represented by heavy dots, calculated values by dashed and dotted lines.
304
John D. Steben et al. Table 2. Percentage depth doses a t several energies. Electron cone 10 cm x 14 cm. Target surface distance 110 cm Energy (MeV)
35 ~
~~~~~
45 ~
~~~
~
~~
~
~
25 ~
l5
~~
Depth (cm)
Meas.
Calc.
Meas.
Calc.
Meas.
Calc.
Meas.
Calc.
1 2 3 4 5
98.2 99.9 99.9 99.0 97.6
99.0 99.8 99.9 99.3 98.0
98.5 99.5 100.0 98.0 96.0
99.8 100.0 99.5 98.2 96.1
99.0 99.5 98.5 96.5 93.5
99.8 99.3 98.5 96.9 94.2
99.5 100.0 98.0 92.5 79.0
99.7 99.8 98.7 93.2 79.6
6 93.096.095.8 7 8 89.5 9 10 82.0
11 12 13 14 15
27.5
72.6
66.5 58.0 49.0
72.2
60.5 38.0 60.6
20.1 16 19.0 48.1 17 18 34.8 19 20 16.5
90.1 65.0 82.1
89.5 85.076.5 80.0 74.0
93.1 54.8 89.5 53.0 90.0 89.3 20.083.0 83.5 84.6 74.8 79.1 63.6 72.4 49.6 49.5 65.3 30.5 57.3 15.2 15.0 48.7 39.6 30.0
19.2
33.1
48.0 33.5 26.5 21.0
21
28.1 21.5 14.9
Table 3. Percentage depth doses for 30 MeV electrons ~~
Field sizes (cm) Depth (cm) 1 2 3 97.5 98.0 98.1 98.0 4 5
6 7 8 9 10 l1
12 13 14
20 20 x
Calc.
8 x 16 Meas.
Calc.
Meas.
99.6 98.5 100.0 100.0 99.5 99.5 99.5
99.6 100.0
99.0 100.0 100.0 99.5 99.5
95.4 91.6 86.6 80.2 72.5 64.0 63.6 53.5 42.4 30.5 18.0
95.5 92.3 87.5 81.5 73.7 53.2 40.2 26.7 17.0 16.8
8 x 10
Calc.
Meas.
99.6 99.6 98.8 100.0 100.0 99.0 98.0 97.0 95.5 94.5 93.0 95.1 94.0 95.4 91.7 91.0 91.6 89.7 86.3 86.0 86.2 84.0 79.9 79.5 79.8 77.8 72.1 72.0 71.6 69.3 70.0 62.9 62.4 62.0 60.7 57.0 52.5 51.9 48.6 41.2 39.5 36.0 40.3 29.1 26.0 23.4 28.5 17.5 14.7 16.4
6 x 10
4x8
Calc.
Meas.
Calc.
Meas.
99.5 98.0
99.0 100.0 99.5 97.0
99.6 100.0 99.1 96.9 92.3 85.8 77.1 67.8 57.6
98.5 99.5 98.0 94.6 89.1 83.0 74.8 65.6 55.8 45.0 33.9 24.3
91.2 85.6 78.4 59.6 48.9 37.7 26.2 14.9
89.0 83.5 76.5 68.0 47.5 58.0 47.0 27.7 36.0 25.0 12.0 10.3 16.0
37.5 16.9 18.6
Betatron Electron
Beam Characterisation
305
the field-size dependence is shown graphically a t highenergy, 45MeV, in fig. 2 where the symbols represent measurements and the lines represent the theoretical predictions, and a corresponding low energy comparison for 15 MeV is also shown. Predictions were within the design goals of & 2% in depth doses above 50% of maximum and & 5% in the lower dose regions. For only one energy, 10 MeV, it was found preferable to use two sets of parameters, one if the calculated effective field size was 11cm square or greater and the other for smaller fields.
L
8
12
16
Depth [cm)
Fig. 2. Electron beam central axis percentage depth dose plotted against fieldsize. Measured values represented by symbols, calculations by dashed lines.
3.2. Isodose shapes
Thebeam-shapeparameters C, N and P were adjusted for acceptability before thedepth dose parameters were finalised to the values intable 1. Kawachi’s values of these shape parameters (C = 1.0, N = 1.5, P = 0.25) were good only a t low energies ( < 20 MeV) and predicted less than half of the measured constriction a t 40 and 45 MeV. For the higher energies ( 2 20 MeV) the set of parameters C = 1.8, N = 1.0, P = 0.1 have given adequate fits to measured isodoses with much better constriction predictions. The single-beam isodoses resulting from the complete parameter optimisations were typically asshown in fig. 3 (30 MeV) and fig. 4 (10 MeV) for an 8 cm wide x 10 cm rectangular cone. I n these figures, the calculated isodoses (shown solid) are compared with the measured isodoses (shown dotted) and theagreement is good. The departures of the measured beams from perfect symmetry would not permit a much closer fit without a major increase in the work involved. A single beam 4 cm wide x 16 cm at 30 MeV without a cone was also studied for the purposes of generating telecentric and isocentric moving beam dose distributions. Its parameters, representing a more diffuse beam, are also given in table 1 and itsisodose distributions are shown in fig. 5. Measured doses with film in a polystyrene phantom are shown dotted and the computer fit using parameters from table 1 is again shown as a solid line.
306
John D. Steben et al. 30 MeV
8cmxlOcm
Fig. 3. Central plane isodose distribution for a 30 MeV electron beam of 8 cm width by 10 cm. Measured doses are shown as dashed lines, calculated doses as solid lines.
10 MeV
8cm xlOcm
Fig. 4. Central plane isodose distribution for a 10 MeV electron beam of 8 cm width by 10 cm. Dashed lines denote measurements, while solid lines denote calculations.
30MeV electrons
16cmxLcm no cone
Fig. 5. Central plane isodose distribution for open beam of 30 MeV electrons of 4 cm width by 16 cm. Measuredisodoses are shown dashed; calculated isodoses are shown solid.
Betatron Electron
B e a m Characterisation
307
Computerisedbeamgeneration The parameters of table 1 are stored in the computer. Electron doses are calculated by a computer program, CBEAME, developed a t Thomas Jefferson University as a generalised modification of Cunningham’s (Cunningham 1974, private communication) CBEAMF program (originally used forstored doses located on ray lines). Calculating the doses from the formula above has been found to be as fast as retrieval and addition of stored beams, partially because a fast and efficient error function subroutine was developed with an adequate accuracy of about 0.3%. It should be possible to apply this method to characterise beamsfrom other electron accelerators, as well. Potential users of the method should, however, be cautioned thattheactualparameters of table 1 shouldnot be used to represent other electron machines, but rather are tobe re-determined using the method outlined in section 2.4 for a given machine. It is hoped, however, that constants C, N and P may not vary considerably for machines with comparable scattering foils as they influence primarily the shape and constriction of isodose lines.
4.
5. Applications
Theelectron beams defined bythe aboveparametersare used for the computerised generation of treatment plans, alone and in mixed beam configurations. Treatment planning computer programs are now set up to allow for any mixtures of electron beams with rectangular photon beams (Suntharalingam and Steben 1977). One fairly simple situation in which it has proved desirable to mix electrons of appropriate energy (generally25 or 30 MeV) with 45 MV photon beams is in treating the spinal region (see fig. 6). The depth of maximum dose in mixed beams is adjustable by varying the electron energy. Another useful application Mixed beam treatment Sptne L5 MV X-rays 30 MeV electrons 13cm x7cm 13crnx8cm
I
Fig. 6. Isodose distribution for clinical treatment of spine using mixed beams of 45 MV photons, 7 cm wide by 13 cm, and 30 MeV electrons, 8 cm wide by 13 cm. The target area is shaded.
John D . Xteben et al.
308
of mixed beams has been the Jefferson three-beam technique for treating the pancreas (Dobelbower, Strubler and Suntharalingam 1975). Telecentric rotations of electron beams havebeen shown (Rassow 1970, 1972) to enhance the usefulness of electronbeams at the higher energies, as the source-axis distance (SAD) can be varied to achieve a maximum dose two or three times deeper in the patient than with a single beam. The generating beam in this example is a 4cm wide x 16 cm beam of 30 MeV electrons without a cone, a t 130 cm SAD (see fig. 5 ) . The dose distribution for a bladder treatment with an arc of 60' is shown in fig. 7. Predicted doses are shown as solid lines,
r0tation.50~ arc Telecentrlc plan Treatment for bladder _""measurements 15cm Fllm calculation -Computer
30 MeV electrons x
Lcm
SAD I:130Cm
Fig. 7 . Isodose distribution for telecentric rotation with 30 MeV electrons through 60" arc, at 130 cm SAD. The target area is shaded.
while measured doses are shown dotted.The calculated doses havebeen summed in 5' steps. The agreement is reasonably good. A slight waviness of calculated isodose lines near the surface would result when a cone is used, due to the 'sharper' beam edges; it could be removed by summing smaller angular increments. 6.
Conclusions
The Kawachi parametric method has been generalised to the extent necessary in order to fit the measured isodoses a t energies from 5 MeV to 45 MeV for rectangular beams from the Asklepitron-45 betatron. The generation of electron beams by this method has been successful. It is greatly superior to storing over 200 individual beams for a betatron with 10 energies, 10 cones, each with two orientations and with additional open field sizes when implementing minicomputer-based treatment planning. The met'hod has proved to be fast, convenient and adequately accurate for clinical uses. It should be possible to apply this method to characterise beams from other electron accelerators. The ability to generate electron treatment plans by computer may provide an impetus to derive fuller benefit from the newer generation of mechanically and electronicallyversatileelectron treatment machines. This work was supportedbyNationalInstitutes of HealthGrant NO. 5-POZ-CA-11602 (National Cancer Institute Radiation Therapy Center Grant).
Betatron Electron
Beam Characterisation
309
RE SUM^^ On a demand6 des parametres pour des faisceaux Blectroniques dont les Bnergies allaient de 5 A45 MeV e t qui provenaient d’un betatronBrown Boveri, de manibre a les adapter 8, des calculs de dosimbtrie sur ordinateur. Une Bquation semi-empirique est donnee pour la dose en tout point e t A diverses profondeurs sous l’eau. Cette Bquation est une modification du modble predictif de Kawachi qui Btait fond6 sur des solutions B une Bquation gBnerale d’age-diffusion. Les doses en profondeur e t les courbes d’isodoses sont prBdites sous forme de fonctions de la plage pratique, de l’intervalle entre la source e t la peau (SSD) et dela taille du champ. Les nBcessitBs en matiere de prbcision des doses en profondeur ont Bt6 fixBes e t obtenues par rapport B la dose maximum; ces nBcessit6s correspondent 8, f2% au-dessus d’une dose en profondeur de 50%, et B 5% B des doses plus faibles. Par ailleurs, la forme des courbes d’isodoses et les constrictions B des doses plus BlevBes, de m6me que les renflements aux valeurs plus faibles, sont prBdits avec precision. On a utilise des faisceaux calculBs sur ordinateurpour obtenir des repartitions d’isodoses cumulBes pour certaines situations cliniques.
ZUSAMMENFASSUNG Es werden diejenigen Parameter im einzelnen festgelegt, diebei der Berechnung der Elektronenstrahlendosiswerte mit Hilfe eines Rechners gelten und fur Elektronenstrahlen mit Energiewerten von 5 bis 45 MeV, dievon einem Brown Boveri-Betatron erzeugt werden, bestimmt sind. Angefuhrtwird eine halbempirische Gleichung f u r die Dosiswerte an beliebigen Punkten bei verschiedenen Tiefenwerten in Wasser. Die Gleichung stellt eine Umformung des Voraussagemodells von Kawachi dar,das seinerseits auf Losungsansatzen fur eine allgemeine AltersDiffusions-Gleichung basierte. Die Kurvenverlaufe fur die Dosis- und Isodosiswerte werden in Abhangigkeit vom praktischen Bereich, vom Abstand zwischen der Strahlenquelle und der Haut und von der Feldgrosse vorausgesagt. Die verlangte Genauigkeit fur die Tiefendosis betrug f 2 yo oberhalb der Tiefendosis von 50% und If: 5% bei niedrigeren Dosiswerten (bezogen auf die Maximaldosis). Diese Sollwerte fur die Genauigkeit wurden auch erzielt. Weiter wurde auch der Verlauf der Isodosenkurven mit demAuftreten von Einschnurungenim Gebiet hoherer Dosiswerte und Ausweitungen bei niedrigeren Dosiswerten genau vorausgesagt. Zur Erzeugung von summiertenIsodosisverteilungskurven wurden fur bestimmte klinische Verhaltnisse Strahlen verwandt, die mit Hilfe eines Rechners errechnet worden waren.
REFERENCES BERGER, M. J., and SELTZER, S . M., 1969, Ann. N.Y. Acad. Sci., 161,8. DAHLER, A., BAKER,A. S., and LAUGHLIN, J. S., 1969, Ann. N . Y . A d . Sci., 161,198. DOBELBOWER, R. R.,STRUBLER, K. A., and SUNTHARALINGAM, N., 1975, I n t . J . Radiat. Oncol. Biol. Phys., 1, 141. FEHRENTZ, D., 1976, i n HighEnergyPhotonsandElectrons:ClinicalApplications in Cancer Management, Eds. S. Kramer, N. Suntharalingam and G. Zinninger (New York: Wiley) p. 285. GLASSTONE, S., a n d EDLUND,M. C., 1952, TheElements of Nuclear Reactor Theory (Princeton : Van Nostrand) ch. 6. HAAS,L. L., HARVEY, R. A., LAUGHLIN, J. S., BEATTIE,J. S., and HENDERSON, W. J., 1954, Am. J . Roentg., Rad. Ther., Nucl. Med., 72, 250. KAWACHI, K., 1975, Phys. Med. Biol., 20, 571. LEETZ,H. K . , 1976, Physics in Canada, 32, 23.5 (digest), and private communication. LILLICRAP, S . C., WILSON,P., and BOAG,J. W., 1975, Phys. Med. Biol., 20, 30. MARKUS,B . , 1961, Strahlentherapie, 116,280. MILAN, J., and BENTLEY, R.E., 1974, Br. J . Radiol., 47, 115. POHLIT,W., 1969, Ann. N . Y . A c a d . Sci., 161, 189. POSER,H . , NEMETH,G., and KUTTIG,H., 1973, Strahlentherapie, 145,390. RASSOW, J., 1970, Strahlentherapie, 140, 156. RASSOW, J., 1972, Electromedica, 1, 1 . SIMPSON, L. R., BORGER, F., OVADIA,J., and COHEN, L., 1974, paper presented at RSNAI A A P M Meeting, paper 239. SKAGGS, L. S., ALMY,G. M., KERST,D. W., LANZL,L. H., and UHLMANN, E. M., 1948, Radiology, 50, 167. SUNTHARALINGAM, N., and STEBEN,J. D., 1977, Med. Phys., 4, 134. TAPLEY,N., 1973, in Textbook of Radiation Therapy, Ed. G. H . Fletcher (Philadelphia: Lea Febiger) p. 45.
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Betatron electron beam characterisation for dosimetry calculations
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1979 Phys. Med. Biol. 24 299 (http://iopscience.iop.org/0031-9155/24/2/006) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.69.4.4 This content was downloaded on 21/08/2015 at 05:56
Please note that terms and conditions apply.
PHYS. MED. BIOL., 1979, Vol. 24, No. 2, 299-309.
Printed in Great Britain
Betatron Electron Beam Characterisation for Dosimetry Calculations JOHN D. STEBEN, PH.D., K. AYYANGAR, and N. SUNTHARALINGAM, PH.D.
PH.D.
Department of Radiation Therapy, Thomas Jefferson University, Philadelphia, Pennsylvania, U.S.A. Received 6 J u n e 1977, in j n a l form 3 October 1978
ABSTRACT. Parameters havebeen specified for electron beams with energies 5-45 MeV from a Brown Boveri betatron for use incomputeriseddosimetry calculations. A semi-empirical equation is given for the dose a t any point a t various depths in water. This equation is a modification to Kawachi’s predictive model which was based on solutions of a general age-diffusion equation. The depth doses and isodose curves are predicted as a function of the practical range, source skin distance (SSD) and field size. Depth dose accuracy requirements of f 2y0 above 50% depth dose and + _ 5 % at lower doses, relative to maximum dose, have been set and achieved. Also, the shape of the isodose curves with the constrictions a t higher doses and bulging a t lower values are accurately predicted. Computer calculated beams have been used t o generate summed isodose distributions for certain clinical situations.
1. Introduction Ever since the earliest days of the betatron, electrons have found uniqueand complementary uses in the treatment of cancer (Skaggs, Almy, Kerst, Lanzl andUhlmann 1948, Haas,Harvey, Laughlin,Beattie and Henderson 1954, Tapley 1973). I n recentyears,telecentricelectronrotations(Rassow 1970, 1972, Poser, NBmeth and Kuttig 1973, Fehrentz 1976) and multiple electron fields (Simpson,Borger,Ovadia and Cohen 1974) have also been used, and sometimes electron beams have been mixed with photon beams (Tapley 1973). Dose distribution calculations for complicated irradiation techniques can best be donewithacomputer,provided that sufficiently accuratecomputer representations of the individual beams can be made. While adequate computer representations of 45 MV photon beams for the BBC Asklepitron-45 a t Thomas Jefferson University Hospital (TJUH)have previously been generated through the implementation of tissue-maximum-ratio andscatter-maximum-ratio concepts(SuntharalingamandSteben 1977), the electron beams,with ten electron energies, ten electron cones, open-beam modalities and five scattering foils correlated to the energies, presented a greater challenge. Various methods have been proposed in the past to characterise electron beams of therapy machines. These have varied from numerical solutions of analytic electron transportequations(BergerandSeltzer 1969) tostrict recording of experimental results in a form such as ray-line and constant-depth profiles (Milan and Bentley1974), with various other approachesto characterise groups of electronbeams(Dahler,Baker and Laughlin 1969, Pohlit 1969, 0031-9155/79/020299+11$01.00
@ 1979 The Institute of Physics
300
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Leetz 1976). The analytical extreme requires practically nodata but excessive computer processing, while theother involves simple interpolationswith quantities of data judged to be excessive for the Asklepitron-45 betatron for and which there would be over 200 beams to store.Thecomputertime resources, respectively, in these two cases were judged to be too large to be useful for the routine treatmentplanning calculations, which were to be carried out on a minicomputer. It was thus desirable to consider some semi-empirical approach which would rely on a small amount of stored beam information to generate a relatively largenumber of beams. One method for doing this has been discussed by Lillicrap, Wilson and Boag (1975), and consists of measuring ‘pencil beams’ or treatmentarea. For a andintegratingthem over afinitedistribution scanned-beammachine,such as aSagittairelinearaccelerator,a ‘pencil beam’ could closely approximate the physical situation. Forelectron beamslike those in the Asklepitron-45 which are flattened by scattering techniques, a calculationalmethod which includes the presence of scattering or diffusion would seem to be a better starting point. A parametric method of this latter type was proposed by Kawachi (1975), and uses solutions of the age-diffusion equation (Glasstone and Edlund 1952) to characterise the electronbeams. Diffusion equationsgenerally describe beamdensities, but for high energyelectrons the delivered dose pattern closely resembles the beam density diffusion pattern. Hence, by starting with age-diffusion solutions, one may minimise the number of independent parameters which must be varied. This is the basic method of the present study, as is discussed below. Slightly simpler parametric methods have been tried by others (Simpson et al. 1974, Leetz 1976), but seem to lack the great versatility or predictive accuracy needed for a general characterisation of the electron beams from the Asklepitron-45. Method In the present beam computerisation study, the age-diffusion model proposed by Kawachi (1975) has been applied, with some necessary modification, to parameterise the Brown Boveri Asklepitron-45 electron beams. The modified dose equationis given below, and the nature of the changes are discussed in the paragraphs that follow. 2.
x exp { - [(2.rr/3Rp) ( K T ) ’ ] ~ )
Betatron Electron Beam Characterisation
301
where (KT))
F
=
[(Cz/R,) +PIN
= SSD
Xo(z)= &(width+ ke) [(F+ z ) / F ]
+
+
Yo(z)= &(length he) [(F z ) / F ] .
R, is the practical range, and G,, G,, G,, C, N , P are constants for a given energy. F designates source-to-skin distance, z designates depth in patient, while x and y are the lateral coordinates; length and width of the field are defined a t the patient’s skin ; k, represents a small field size correction for edge effects; 7 represents ‘age’ of the beam and KT is the age-diffusion parameter. R, is the practical electron range (incm), determined experimentally by extrapolatinga line through measured 70% and 40% percentage dose distribution to intersect an empirically determined bremsstrahlung build-up of (energy/5)yo. The energy, E , in MeV, is related to R, by the Markus equation (Markus 1961), that is R, = 0.521E - 0.376. Parameters G,, G,, G,, C, N and P must be varied to determine the optimum fit to measured central axis depth dose and isodose curves. The equation reduces to Kawachi’s form for G, = 0, G, = 2713, G , = -716 and Yolarge. 2.1. Percentage depth dose Initial studies have shown that Kawachi’s cosine factor needed to be modified and that the three depth parameters, G,, G,, G,, need to be varied to achieve acceptable fits to our experimental data. Accuracy of better than & 2% at doses above 50%, and k 5% at lowerdoseswas required. Retaining the cosine dependence, but with the addition of a quadratic term
2.2. Field size effects The variation of percentage depth dose with field sizeis essentially predicted by the error function factors. Kawachi’s treatment is two-dimensional and assumes the beam to be infinite in the third dimension. Hence, Kawachi’s equations predict depth doses for a 16 cm x 4 cm beam significantly different 12
302
John D. Steben et al.
from those for a 4 cm x 16 cm beam. Generalising the formula to include the second lateral dimension was clearly necessary to permit close fitting of the measured depth doses. Themethod used was tointroducethe sameerror function of the second transverse coordinate as a factor. This also enables dose predictions to be made for any point in any plane, not just the central planes, for rectangular beams. Circular beams are, however, predicted correctly only in the central planes. 2.3. Isodose shapes The ‘beam-shape’ parametersare C, N and P. They are strongly interrelated; where P characterisesbeam diffuseness at the surface, C characterises the increase in diffuseness with depth andN is an overall shape-changing parameter. The function relating these three parameters in our equations is identical to the one used by Kawachi (1975). At all energies, a small modification to the physical width and length of the rectangularelectronbeamsis necessary to give proper isodose fits to the experimentaldata. This ‘correction’ tototalwidth, a small constant, k,, represents the fact that the measured radiation width of the beam, taken as the lateral distance between 50% dose points in the beam profile, at the depth of maximum dose, is slightly greater than the nominal width or ‘cone size’. For the betatron measurements, as evident in figs 3, 4 and 5, this constant, k,, was found to be 0.5 cm. 2.4. Calculation of parameters A computerprogramhas been developed which unscrambles the dose equation to calculate G,, G, and G, from other parameters listed below: (1) measured central axis depth dose data, z, D ( z ) ; (2) measured range, R, ; (3) field width and length, X and Y ; (4)the source to surface distance, F, and (5) constants, C, N , P. The program calculates the argument of the cosine factor for each depth fit t o thearguments it generates the andby a quadratic leastsquares parameters G,, G, and G,. Once G,,G, and G, are known, thenextstep was to generate isodose distributions andsee how they fit the experimental data. Parameters C, N and P were to be readjusted when necessary. G,, G, and G, must be recalculated if major changes are seen in C, N and P (the process is thus iterative). 3. Results Using the method described above, the parameters were determined a t each energy for one field size, namely a 10 cm x 14 cm rectangular beam and were then used to predict depth doses and isodoses for all field sizes. The parameters are shown in table 1. It may be noted that the quadratic termcoefficient G, in
Betatron Electron
Beam Characterisation
303
Table 1. Parameters characterising an electron beam Energy (MeV) 45 40 35 30 30 25 20 20 15 10 10
5
Beam type
Q,
G2
Q3
C
N
P
Range (cm) ( i n water)
Cones Cones Cones Open
0.0529 0.236 0.444 0.208
1.804 1.682 1.416 1.879
- 0.4326 - 0.4328 - 0.3664 - 0.5905
1.80 1.00 1.80 1-00 1.80 1.00 1-80 0.50
0.10 0.10 0.10 0.25
21.7 19.56 17.5 14.9
4 x 16 Cones Cones Cones Cones Cones 11 cm 211 cm Cones
0.601 1.339 1.512 1.342 1.998 1.821 1.917 2.458
1.263 0.527 0.369 0.542 0.0064 0.269 0.226 - 0.472
- 0.3798 -0.3118 - 0.3559 - 0.3430 -0.3195 - 0.5752 0.5554 - 0.8153
1.80 1-80 1.80 1.00 1e o 0 1.00 1.00 1.00
1.00 1.00 1.00 1.50 1.50 1.50 1.50 1.50
0.10 0.10 0.10 0.25 0.25 0.25 0.25 0.25
14.9 12.75 9.9 9.9 7.7 4.9 5.2 2.55
-
-
thedepthdependent cosine argument becomes important a t low energies, whereas the argument becomes essentially a linear function of depth at high energies.
3.1. Percentage depth doee The resulting percentage depth dose distributions for the 10 cm x 14cm beams are shown in fig. 1 for 10, 20, 30 and 40 MeV. For other energies, they cases, and a good are listed in table 2. Excellent fits are obtained in these feature of the model is its ability, using these same parameters, to fit the doses a t all field sizes. This is tabulated for 30 MeV in some detail. The comparison between measured and predicted data for 30 MeV is given in table3. Similarly,
Depth Icrnl
Fig. 1. Percentage depth dose for 10 cm x 14 cm electron beams at several energies. Measuredvalues represented by heavy dots, calculated values by dashed and dotted lines.
304
John D. Steben et al. Table 2. Percentage depth doses a t several energies. Electron cone 10 cm x 14 cm. Target surface distance 110 cm Energy (MeV)
35 ~
~~~~~
45 ~
~~~
~
~~
~
~
25 ~
l5
~~
Depth (cm)
Meas.
Calc.
Meas.
Calc.
Meas.
Calc.
Meas.
Calc.
1 2 3 4 5
98.2 99.9 99.9 99.0 97.6
99.0 99.8 99.9 99.3 98.0
98.5 99.5 100.0 98.0 96.0
99.8 100.0 99.5 98.2 96.1
99.0 99.5 98.5 96.5 93.5
99.8 99.3 98.5 96.9 94.2
99.5 100.0 98.0 92.5 79.0
99.7 99.8 98.7 93.2 79.6
6 93.096.095.8 7 8 89.5 9 10 82.0
11 12 13 14 15
27.5
72.6
66.5 58.0 49.0
72.2
60.5 38.0 60.6
20.1 16 19.0 48.1 17 18 34.8 19 20 16.5
90.1 65.0 82.1
89.5 85.076.5 80.0 74.0
93.1 54.8 89.5 53.0 90.0 89.3 20.083.0 83.5 84.6 74.8 79.1 63.6 72.4 49.6 49.5 65.3 30.5 57.3 15.2 15.0 48.7 39.6 30.0
19.2
33.1
48.0 33.5 26.5 21.0
21
28.1 21.5 14.9
Table 3. Percentage depth doses for 30 MeV electrons ~~
Field sizes (cm) Depth (cm) 1 2 3 97.5 98.0 98.1 98.0 4 5
6 7 8 9 10 l1
12 13 14
20 20 x
Calc.
8 x 16 Meas.
Calc.
Meas.
99.6 98.5 100.0 100.0 99.5 99.5 99.5
99.6 100.0
99.0 100.0 100.0 99.5 99.5
95.4 91.6 86.6 80.2 72.5 64.0 63.6 53.5 42.4 30.5 18.0
95.5 92.3 87.5 81.5 73.7 53.2 40.2 26.7 17.0 16.8
8 x 10
Calc.
Meas.
99.6 99.6 98.8 100.0 100.0 99.0 98.0 97.0 95.5 94.5 93.0 95.1 94.0 95.4 91.7 91.0 91.6 89.7 86.3 86.0 86.2 84.0 79.9 79.5 79.8 77.8 72.1 72.0 71.6 69.3 70.0 62.9 62.4 62.0 60.7 57.0 52.5 51.9 48.6 41.2 39.5 36.0 40.3 29.1 26.0 23.4 28.5 17.5 14.7 16.4
6 x 10
4x8
Calc.
Meas.
Calc.
Meas.
99.5 98.0
99.0 100.0 99.5 97.0
99.6 100.0 99.1 96.9 92.3 85.8 77.1 67.8 57.6
98.5 99.5 98.0 94.6 89.1 83.0 74.8 65.6 55.8 45.0 33.9 24.3
91.2 85.6 78.4 59.6 48.9 37.7 26.2 14.9
89.0 83.5 76.5 68.0 47.5 58.0 47.0 27.7 36.0 25.0 12.0 10.3 16.0
37.5 16.9 18.6
Betatron Electron
Beam Characterisation
305
the field-size dependence is shown graphically a t highenergy, 45MeV, in fig. 2 where the symbols represent measurements and the lines represent the theoretical predictions, and a corresponding low energy comparison for 15 MeV is also shown. Predictions were within the design goals of & 2% in depth doses above 50% of maximum and & 5% in the lower dose regions. For only one energy, 10 MeV, it was found preferable to use two sets of parameters, one if the calculated effective field size was 11cm square or greater and the other for smaller fields.
L
8
12
16
Depth [cm)
Fig. 2. Electron beam central axis percentage depth dose plotted against fieldsize. Measured values represented by symbols, calculations by dashed lines.
3.2. Isodose shapes
Thebeam-shapeparameters C, N and P were adjusted for acceptability before thedepth dose parameters were finalised to the values intable 1. Kawachi’s values of these shape parameters (C = 1.0, N = 1.5, P = 0.25) were good only a t low energies ( < 20 MeV) and predicted less than half of the measured constriction a t 40 and 45 MeV. For the higher energies ( 2 20 MeV) the set of parameters C = 1.8, N = 1.0, P = 0.1 have given adequate fits to measured isodoses with much better constriction predictions. The single-beam isodoses resulting from the complete parameter optimisations were typically asshown in fig. 3 (30 MeV) and fig. 4 (10 MeV) for an 8 cm wide x 10 cm rectangular cone. I n these figures, the calculated isodoses (shown solid) are compared with the measured isodoses (shown dotted) and theagreement is good. The departures of the measured beams from perfect symmetry would not permit a much closer fit without a major increase in the work involved. A single beam 4 cm wide x 16 cm at 30 MeV without a cone was also studied for the purposes of generating telecentric and isocentric moving beam dose distributions. Its parameters, representing a more diffuse beam, are also given in table 1 and itsisodose distributions are shown in fig. 5. Measured doses with film in a polystyrene phantom are shown dotted and the computer fit using parameters from table 1 is again shown as a solid line.
306
John D. Steben et al. 30 MeV
8cmxlOcm
Fig. 3. Central plane isodose distribution for a 30 MeV electron beam of 8 cm width by 10 cm. Measured doses are shown as dashed lines, calculated doses as solid lines.
10 MeV
8cm xlOcm
Fig. 4. Central plane isodose distribution for a 10 MeV electron beam of 8 cm width by 10 cm. Dashed lines denote measurements, while solid lines denote calculations.
30MeV electrons
16cmxLcm no cone
Fig. 5. Central plane isodose distribution for open beam of 30 MeV electrons of 4 cm width by 16 cm. Measuredisodoses are shown dashed; calculated isodoses are shown solid.
Betatron Electron
B e a m Characterisation
307
Computerisedbeamgeneration The parameters of table 1 are stored in the computer. Electron doses are calculated by a computer program, CBEAME, developed a t Thomas Jefferson University as a generalised modification of Cunningham’s (Cunningham 1974, private communication) CBEAMF program (originally used forstored doses located on ray lines). Calculating the doses from the formula above has been found to be as fast as retrieval and addition of stored beams, partially because a fast and efficient error function subroutine was developed with an adequate accuracy of about 0.3%. It should be possible to apply this method to characterise beamsfrom other electron accelerators, as well. Potential users of the method should, however, be cautioned thattheactualparameters of table 1 shouldnot be used to represent other electron machines, but rather are tobe re-determined using the method outlined in section 2.4 for a given machine. It is hoped, however, that constants C, N and P may not vary considerably for machines with comparable scattering foils as they influence primarily the shape and constriction of isodose lines.
4.
5. Applications
Theelectron beams defined bythe aboveparametersare used for the computerised generation of treatment plans, alone and in mixed beam configurations. Treatment planning computer programs are now set up to allow for any mixtures of electron beams with rectangular photon beams (Suntharalingam and Steben 1977). One fairly simple situation in which it has proved desirable to mix electrons of appropriate energy (generally25 or 30 MeV) with 45 MV photon beams is in treating the spinal region (see fig. 6). The depth of maximum dose in mixed beams is adjustable by varying the electron energy. Another useful application Mixed beam treatment Sptne L5 MV X-rays 30 MeV electrons 13cm x7cm 13crnx8cm
I
Fig. 6. Isodose distribution for clinical treatment of spine using mixed beams of 45 MV photons, 7 cm wide by 13 cm, and 30 MeV electrons, 8 cm wide by 13 cm. The target area is shaded.
John D . Xteben et al.
308
of mixed beams has been the Jefferson three-beam technique for treating the pancreas (Dobelbower, Strubler and Suntharalingam 1975). Telecentric rotations of electron beams havebeen shown (Rassow 1970, 1972) to enhance the usefulness of electronbeams at the higher energies, as the source-axis distance (SAD) can be varied to achieve a maximum dose two or three times deeper in the patient than with a single beam. The generating beam in this example is a 4cm wide x 16 cm beam of 30 MeV electrons without a cone, a t 130 cm SAD (see fig. 5 ) . The dose distribution for a bladder treatment with an arc of 60' is shown in fig. 7. Predicted doses are shown as solid lines,
r0tation.50~ arc Telecentrlc plan Treatment for bladder _""measurements 15cm Fllm calculation -Computer
30 MeV electrons x
Lcm
SAD I:130Cm
Fig. 7 . Isodose distribution for telecentric rotation with 30 MeV electrons through 60" arc, at 130 cm SAD. The target area is shaded.
while measured doses are shown dotted.The calculated doses havebeen summed in 5' steps. The agreement is reasonably good. A slight waviness of calculated isodose lines near the surface would result when a cone is used, due to the 'sharper' beam edges; it could be removed by summing smaller angular increments. 6.
Conclusions
The Kawachi parametric method has been generalised to the extent necessary in order to fit the measured isodoses a t energies from 5 MeV to 45 MeV for rectangular beams from the Asklepitron-45 betatron. The generation of electron beams by this method has been successful. It is greatly superior to storing over 200 individual beams for a betatron with 10 energies, 10 cones, each with two orientations and with additional open field sizes when implementing minicomputer-based treatment planning. The met'hod has proved to be fast, convenient and adequately accurate for clinical uses. It should be possible to apply this method to characterise beams from other electron accelerators. The ability to generate electron treatment plans by computer may provide an impetus to derive fuller benefit from the newer generation of mechanically and electronicallyversatileelectron treatment machines. This work was supportedbyNationalInstitutes of HealthGrant NO. 5-POZ-CA-11602 (National Cancer Institute Radiation Therapy Center Grant).
Betatron Electron
Beam Characterisation
309
RE SUM^^ On a demand6 des parametres pour des faisceaux Blectroniques dont les Bnergies allaient de 5 A45 MeV e t qui provenaient d’un betatronBrown Boveri, de manibre a les adapter 8, des calculs de dosimbtrie sur ordinateur. Une Bquation semi-empirique est donnee pour la dose en tout point e t A diverses profondeurs sous l’eau. Cette Bquation est une modification du modble predictif de Kawachi qui Btait fond6 sur des solutions B une Bquation gBnerale d’age-diffusion. Les doses en profondeur e t les courbes d’isodoses sont prBdites sous forme de fonctions de la plage pratique, de l’intervalle entre la source e t la peau (SSD) et dela taille du champ. Les nBcessitBs en matiere de prbcision des doses en profondeur ont Bt6 fixBes e t obtenues par rapport B la dose maximum; ces nBcessit6s correspondent 8, f2% au-dessus d’une dose en profondeur de 50%, et B 5% B des doses plus faibles. Par ailleurs, la forme des courbes d’isodoses et les constrictions B des doses plus BlevBes, de m6me que les renflements aux valeurs plus faibles, sont prBdits avec precision. On a utilise des faisceaux calculBs sur ordinateurpour obtenir des repartitions d’isodoses cumulBes pour certaines situations cliniques.
ZUSAMMENFASSUNG Es werden diejenigen Parameter im einzelnen festgelegt, diebei der Berechnung der Elektronenstrahlendosiswerte mit Hilfe eines Rechners gelten und fur Elektronenstrahlen mit Energiewerten von 5 bis 45 MeV, dievon einem Brown Boveri-Betatron erzeugt werden, bestimmt sind. Angefuhrtwird eine halbempirische Gleichung f u r die Dosiswerte an beliebigen Punkten bei verschiedenen Tiefenwerten in Wasser. Die Gleichung stellt eine Umformung des Voraussagemodells von Kawachi dar,das seinerseits auf Losungsansatzen fur eine allgemeine AltersDiffusions-Gleichung basierte. Die Kurvenverlaufe fur die Dosis- und Isodosiswerte werden in Abhangigkeit vom praktischen Bereich, vom Abstand zwischen der Strahlenquelle und der Haut und von der Feldgrosse vorausgesagt. Die verlangte Genauigkeit fur die Tiefendosis betrug f 2 yo oberhalb der Tiefendosis von 50% und If: 5% bei niedrigeren Dosiswerten (bezogen auf die Maximaldosis). Diese Sollwerte fur die Genauigkeit wurden auch erzielt. Weiter wurde auch der Verlauf der Isodosenkurven mit demAuftreten von Einschnurungenim Gebiet hoherer Dosiswerte und Ausweitungen bei niedrigeren Dosiswerten genau vorausgesagt. Zur Erzeugung von summiertenIsodosisverteilungskurven wurden fur bestimmte klinische Verhaltnisse Strahlen verwandt, die mit Hilfe eines Rechners errechnet worden waren.
REFERENCES BERGER, M. J., and SELTZER, S . M., 1969, Ann. N.Y. Acad. Sci., 161,8. DAHLER, A., BAKER,A. S., and LAUGHLIN, J. S., 1969, Ann. N . Y . A d . Sci., 161,198. DOBELBOWER, R. R.,STRUBLER, K. A., and SUNTHARALINGAM, N., 1975, I n t . J . Radiat. Oncol. Biol. Phys., 1, 141. FEHRENTZ, D., 1976, i n HighEnergyPhotonsandElectrons:ClinicalApplications in Cancer Management, Eds. S. Kramer, N. Suntharalingam and G. Zinninger (New York: Wiley) p. 285. GLASSTONE, S., a n d EDLUND,M. C., 1952, TheElements of Nuclear Reactor Theory (Princeton : Van Nostrand) ch. 6. HAAS,L. L., HARVEY, R. A., LAUGHLIN, J. S., BEATTIE,J. S., and HENDERSON, W. J., 1954, Am. J . Roentg., Rad. Ther., Nucl. Med., 72, 250. KAWACHI, K., 1975, Phys. Med. Biol., 20, 571. LEETZ,H. K . , 1976, Physics in Canada, 32, 23.5 (digest), and private communication. LILLICRAP, S . C., WILSON,P., and BOAG,J. W., 1975, Phys. Med. Biol., 20, 30. MARKUS,B . , 1961, Strahlentherapie, 116,280. MILAN, J., and BENTLEY, R.E., 1974, Br. J . Radiol., 47, 115. POHLIT,W., 1969, Ann. N . Y . A c a d . Sci., 161, 189. POSER,H . , NEMETH,G., and KUTTIG,H., 1973, Strahlentherapie, 145,390. RASSOW, J., 1970, Strahlentherapie, 140, 156. RASSOW, J., 1972, Electromedica, 1, 1 . SIMPSON, L. R., BORGER, F., OVADIA,J., and COHEN, L., 1974, paper presented at RSNAI A A P M Meeting, paper 239. SKAGGS, L. S., ALMY,G. M., KERST,D. W., LANZL,L. H., and UHLMANN, E. M., 1948, Radiology, 50, 167. SUNTHARALINGAM, N., and STEBEN,J. D., 1977, Med. Phys., 4, 134. TAPLEY,N., 1973, in Textbook of Radiation Therapy, Ed. G. H . Fletcher (Philadelphia: Lea Febiger) p. 45.
