036CL3016/79/02014263/$02.00/0

Int. .I. Radiation Oncology Biol. Phys., Vol. 5. pp. 263-267 @ Pergamon Press Inc.. 1979. Printed in the U.S.A.

??Technical Innovation

and Note

AN ANALYTIC ELECTRON

EXPRESSION FOR CENTRAL AXIS DEPTH DOSE DISTRIBUTIONS?

LEONARD SHABASON, Ph.D. Department

of Radiology,

University

and WILLIAM R. HENDEE, Ph.D.

of Colorado Medical Center, 4200 East Ninth Avenue, CO 80262, U.S.A.

Denver,

A new analytic expression is given for the central axis depth dose distribution for electron beams; its accuracy is verified by comparison with experimental data from a linear accelerator and a betatron for a variety of incident energies and field sizes. In addition, comparisons are provided with Laughlin et al’s” exponential model for the electron depth dose distribution. Electron dose; Analytic expression;

Central axis.

INTRODUCTION

gling of a charged particle beam by a normal distribution.3 Laughlin et al’s equation for” the central axis per cent depth dose (PDD) as a function of depth, d, is given by

widely used analytic expression for the central axis depth dose (DD) distribution beyond the build-up region is Laughlin et al’s simple exponential model.” This expression is useful beyond the buildup region for electron beams of relatively low energy, but is not very satisfactory for electron beams of higher energy. One approach to improving the generality of Laughlin et af.‘s” expression is Bagne’s addition of an extra variable to the original exponential model.2 Other models for the electron depth dose distributions such as Kawachi’s modified diffusion theory’ require a complex set of equations to extract the central axis DD. For the central axis depth dose distribution for electron beams, we propose an equation of the form of the Fermi-Dirac distribution function. Qualitative arguments follow to justify this functional form as appropriate for simulating the electron DD.

The

most

PDD(d) = 110 - 10 exp (~(d - dm))

(1)

where CLis an attenuation coefficient and dm is the depth of maximum dose. Upon examining electron depth dose data, the authors noticed the remarkable similarity of these data to those described by the Fermi-Dirac distribution function specifying the level spacing of a Fermi gas.’ This functional form has been used to describe the density distribution of the nucleus’ and the attractive nuclear potential.” The similarity of this function and the normal distribution has long been recognized in nuclear physics, as evidenced by its use to describe the surface of the nuclear potential which often also is simulated by a normal distribution. The expression we propose for the central axis depth dose distribution has the form

ORIGIN OF EXPRESSION One would expect that the central axis DD would be approximately proportional to the electron flux remaining along the central axis of the beam as a function of depth. The number of transmitted electrons is fairly constant for small thicknesses and then begins to decrease.7 The primary interaction of electrons in low atomic number materials is electronelectron collisions and the statistical nature of these interactions permits description of the range-strag-

100 PDD(d) = (1 + exp (d - dso)/a))

(2)

where dso is the 50% DD point and a is a measure of the diffusivity of the distribution.5 Another possible function is the modified Gaussian which has many features in common with equation

tThis investigation was supported by Grant Number 5 T32 CA09073 awarded by the National Cancer Institute, Department of Health, Education and Welfare.

Acknowledgement-The authors thank G. Kenney Barnes for their assistance. Accepted for publication 12 June 1978. 263

and J. E.

264

Radiation Oncology 0 Biology 0 Physics

February

1979, Volume 5, Number 2 -

IOOr 90

-

CLINAC-I8

60 70 2

60

a

JO-

-

40

-

30

-

20

-

IO 1 1

I 2

I 3

I 4

I 5

I 6

DEPTH

I

I 6

7

I 9

(cm)

CLINAC-16

DEPTH (cm) 100 90 80 70 0

60

k?

50

__.kMev

I

\

\

Lllllllrl,,ll,,l,,,,,, I 2 3

4

5

6

7

8

\

9

IO

II

12

13 I4

DEPTH

100 -

15

I6

BROWN-BOVERI

90 -

I7

I8

19 20

21

(cm)

x 20

20

45

cm2

60 70 60 50 40 30 20 IO I

I

I

I

I

I

2

3

4

5

1 I 6

7

IIll 6

9

IO

II

I2

DEPTH

1

l

I

l

l

13

I4

I5

16

17

l , , , I8

I9

20

21

1, 22

23

(cm)

FIG. 1. Correlation of Clinac-18 and Brown-Boveri electron per cent depth dose (PDD) data to equation 2 for (a) 6 X 6 (b) 25 X 25 and (c) and (d) 20 x 20 cm* fields for nominal incident energies ranging from 6 to 45 MeV.

An analytical expression

for central axis electron depth dose distributions

(2). We chose to deal only with equation (2) because it has a simpler form and yields similar results.’ Arguments above in support of this function are not proposed as a derivation. Instead, they are offered as evidence that since the proposed functional form has features similar to the expression used to describe range-straggling, it is reasonable to expect that the expression will be successful in simulating electron DD data. Our approach is to adopt a functional form which has some physical relationship to electron beam attenuation, and to empirically adjust variables to obtain the best possible fit to experimental data.

EXPERIMENTAL

VERIFICATION

Central axis electron depth dose data from a Varian Clinac-18 electron accelerator4 and a Brown-Boveri 45 MeV betatron were obtained for a variety of field sizes and energies. Between the depth of maximum dose and the 10% DD level, these data were fit to both equations (1) and (2) with all data points weighted equally. In the fitting procedure, variables of both equations (1) and (2) (dm, CL,d5,, and a) were adjusted to obtain the best overall correlation with the data. In all illustrations of this correlation the stated energy is the nominal electron beam energy incident on the medium.

0 L.

ano w. R.

SHABASON

Figure 1 illustrates the correlation of the proposed equation (2) to experimental data for both accelerators for electron energies from 6 to 45 MeV and for various field sizes. In general, depth doses predicted by equation (2) differed from experimental data by ~5% between the depth of maximum dose and the 20% DD point. In Fig. 2 experimental data are compared both to equations (1) and (2). These data reveal that the correlation of both equations with experimental data are comparable at lower electron but differ significantly for electrons of energies, higher energy. In all cases studied, the correlation of equation (2) with experimental data is superior to that with equation (1). Bagne’s2 modification of Laughlin et al.‘s equation for unit density material has the form HID(d)

= lOO-

- dm) (1 - h(d - drr~)~)

10 exp (~(d

(3) where A is an empirically adjusted variable.2 The application of equations (2) and (3) to 45 MeV data is illustrated in Fig. 3 together with the fit of equation (1) to the initial portion of the DD curve. The correlations obtained with equations (2) and (3) were comparable although the extra free variable of Bagne’s’ equation offered a slight improvement.

100

BROWN-BOVERI

90

35 MaV 12cm

00 70

70

??

\

60

\ -

45

Round EXPERIMENTAL

---EQUATION

60

265

r,oly~~~

EQUATION

I 2

\

50

40 0 0 a 30

20

20-

; IO

I I

I

I

I

I

I

2

3

4

5

6

DEPTH

(cm)

I I .

7

IO

l

2

I

4

’

6

’

8

’

IO

’

12

DEPTH

I (cm)

Fig. 2. Comparisons of fit obtained for per cent depth dose (PDD) from equations 1 and 2 for nominal incident energies and field sizes of (a) 15 MeV and 6 x 6 cm*; and (b) 35 MeV and 12 cm round.

Radiation

266

Oncology

0 Biology

0 Physics

February

1979, Volume

5, Number

2

1.5

BROWN-BOVERI

\

‘.

12 cm

‘$

6C

-

-.

YS

45

Round

1.2 EQUATION

1

I 2

---EQUATION

3 0.9

‘\

5c

z

9 \

F

t

1.3

-x-EQUATION ‘;‘$

1.4

2

‘t

n-----\

I6 MeV

0.8 0.7

0

0.6 \

4(

f i

f

I

‘. ?

\

0 P n

i 3c

5

IO I5 SIDE OF SOUARE

20 FIELD

25

(a)

2c 6 7

IO

I

I

I

I

I

2

4

6

8

IO

I

I

Ii,

12 14 16 DEPTH (cm)

I

I

18 20

22

/P.

16 MeV

11 24

26 5 SIDE

Fig. 3. Correlation of per cent depth dose from equations 1, 2 and 3 for a 12 cm round field of 4.5 MeV incident electron energy.

FIELD SIZE DEPENDENCE

d5,, = 0.8Rp 4.4a = 0.4RP

20 FIELD

25 (cm)

(b)

Fig. 4. Variation of (a) dso and (b) a, as a function of field size for nominal incident energies ranging from 6 to 18 MeV from a Clinac-18 linear accelerator. and for energies

In the use of equation (2), the variables, &, and, a, become essentially constant at larger field sizes. For the Clinac-18 accelerator, these variables are plotted in Fig. 4 as a function of field size. In all cases there is little change in the variables for field sizes larger than 8 x 8 cm’. For larger field sizes the following expressions were found to empirically yield reasonably good first estimates of d5,, and a. In general, variation of the variables will enhance the fit to the data. For energies (25 MeV the variables are given by

IO I5 OF SQUARE

~25 MeV by dSo = 0.7Rp 4.4a = 0.6Rp

where

RP is the practical

range.h

CONCLUSION An analytic expression is proposed which characterizes the central axis electron depth dose with reasonable accuracy and simplicity. The expression may be easily adapted for heterogeneity correction techniques such as the modified absorption coefficient of Bagne’ or the coefficient equivalent thickness of Almond et al.’

REFERENCES Almond, P.R., Wright, A.E., Boone, M.L.M.: High energy electron dose perturbations in regions of heterogeneity. Radiology 88: 1146-l 153, 1967. Bagne, F.: Electron treatment planning system. Med. Phys. 3: 31-38, 1976. Bethe, H.A., Ashkin, J.: Passage of radiations through

matter. In Experimental Nuclear Physics, ed. by Serge, E. New York, Wiley, 1953, Vol. 1, pp. 166-357. 4. Gooden, D.S., Goede, M.R., Grogg, B.C.: Calibration of the Clinac-18 Linear Accelerator, Natalie Warren Bryant Cancer Center, Saint Francis Hospital, Tulsa, OK, Internal Report, pp. 48-87.

An analytical

expression

for central

axis electron

depth

5. Hahn, B., Ravenhall, P.G., Hofstader, R.: High energy electron scattering and the charge distribution of selected nuclei. Phys. Rev. 101: 1131-1142, 1953. 6. ICRU, Report 21, Radiation dosimetry: electrons with energies between 1 and 50 MeV, International commission on Radiation Units and Measurements, Washington, DC, 1972. 7. Katz, L. Penfold, A.S.: Range-energy relations for electrons and determinations of beta-ray end-point energies by absorption. Rev. Mod. Phys. 24: 28-44, 1952.

dose distributions

0

L. SHABASON and

W. R. HENDEE

26-l

8. Kawachi, K.: Calculation of electron dose distribution Phys. Med. Biol. for radiotherapy treatment planning. 20: 571-577, 1975. 9. Kittel, C.: Introduction to Solid State Physics, 3rd Edn. New York, Wiley, 1968, Chap. 7, pp. 199-223. 10. Laughlin, J.S., Ovadia, J., Beatlie, W.J., Henderson, W.J., Harvey, R.A., Haas, L.L.: Some physical aspects of electron beam therapy. Radiology 60: 165-185, 1953. 11. McCarthy, I.E., Introduction to Nuclear Theory. New York, Wiley, 1%8, Chap. 11, pp. 352-386.