0360-3016/79/0201-0271/$02.00/0

Int. 1. Radiation Oncology Biol. Phys., Vol. 5, pp. 271-275 6$ Pergamon Press Inc.. 1979. Printed in the U.S.A.

??Technical Innovation

CONCEPT

and Note

OF DEPTH DEFINITION

DOSE FUNCTIONS AND VALUES PRAKASH

Allegheny

General

Hospital,

FOR RADIOTHERAPY

FOR DIFFERENT

ENERGY

BEAMS:

BEAMS

N. SHRIVASTAVA, Ph.D. 320 East North

Avenue,

Pittsburgh,

PA 15212, U.S.A.

Depth dose functions are defined and based on empirically observed interrelation of per cent depth doses for different energy beams. By using this function, if the depth dose for a given source-surface distance (SSD), field size and depth are known for a standard beam energy, the corresponding depth dose for another beam energy can be derived. The results obtained are accurate to within ?l% in a considerable range of SSD, field size and depths used in routine radiotherapy. The formulation can be extended to hold at depths between the surface and depth of maximum but is somewhat less accurate than *I% for depths up to two times the depth of maximum. This method is of considerable usefulness in computer dosimetry algorithms to save memory space and/or to increase speed by avoiding multiple table look up procedures. It is the particular importance in irregular field dose calculations for high energy beams where specific measured data for separation of primary and scatter components of dose are not available. Radiation

dosimetry,

Radiation

Treatment

planning,

Depth dose function, Computers

INTRODUCTION

DEFINITION

for publication

29 August

OF DEPTH

DOSE

FUNCTION

The definition of the DDF is based on the graph in Fig. 1 where the per cent depth dose data P4(d, A, f) for 4 MV beams’ where d, A, and f are depth, field area and SSD respectively, are plotted against the corresponding per cent depth doses PC(d, A, f) for 6oCo beams.3 It is seen that in a large range of depths (3-26 cm) for field areas of 5 x 5 to 30 X 30 cm2 and SSD’s of 80 and 100 cm, the two variables are related by a linear equation.

Present computer dosimetry algorithms require storage of extensive measured data on depth dose, isodose distributions, tissue air ratios, scatter air ratios, tumor maximum ratios, and field flatness. With small memory dedicated computers, many empirical equations and interpolations have been employed. Empirical relations have considerable value in interpolating data for irradiation conditions in which measurements are not available and for increasing computational speed by reducing long multiple table look up procedures or for saving memory space. This empirically another observed presents paper relationship between depth doses for different energy beams. The depth dose functions (DDF) defined and presented have some simple characteristics which make it attractive for use in converting the present boCo beam algorithms to other higher energy beams. As with all empirical formulations, extreme care must be exercised in understanding the limits within which the relationships are valid. The need for verification with actual measurements of the applicability of empirical formulations in one’s particular setup cannot be overemphasized. Accepted

in radiotherapy.

P4(d, A,

f) = PC(d,

A,

f) + K4C

(1)

At first sight it seems surprising that the line has unit slope and that the intercept K is independent of d, A or f; however, as we will show later, this is consistent with the other measurements available for the two beams. It is important to realize that equation (1) holds only for depths greater than 3 cm, i.e. in the range where P4 is 90% or less. Also, K4C which is given by [P4(d, A, f) - PC(d, A, f)] is a constant in the range of d, A and f where equation (1) holds, but outside this range, for example at depths less than 3 cm, it is a continuously varying function.

1978. 271

272

Radiation

Depth doss P4ld.A.f) 100

-

for 4MV

Oncology

0

Biology 0 Physics

Vs

Depth doss PC(d.A,f) for Co-60

9080 -

2 *

70-

8 LL 60M 0 500 c’ 40: 0 8

30zoIO IO

20

30 40 50 60 70 80 % DEPTH DOSE FOR Co-60

90

100

Fig. 1. Plot of per cent depth dose for 4MV Clinac beam vs that for ?Zo beam in corresponding geometrical setup (data from references).3.5 SSD = source-surface distance. We use the concept of “K4C” to define a new quantity called “depth dose function,” which is the difference of per cent depth doses under identical geometry for two different energy beams. A convention of using ordered postscripts 4C to denote the difference in this paper. P4 - PC is employed The variation of points about the smooth line in Fig. 1 is approximately 1%. Assuming that any measurement of depth dose can have errors of +l%, the error in K4C (which is the difference of two measurements) may be up to 1.4%. As determined from Fig. 1, for d > 3 cm, the value of K4C = 4.5 -C 1. The implications of the DDF (K4C) being constant in the range of d, A and f where equation (1) is valid can be realized from the following considerations. In Fig. 2, the conventions and rules of the mnenomic diagram approach6 are followed. The meaning of the symbols is as follows: DC, and D4d represent the

//I

February

1979, Volume

2

doses at depth d for cobalt and 4 MV beams under similar geometry. DC,, and D4s4 are the peak doses at the corresponding depths of maxim 6c and 64. Doa and Dd’ (with superscript a for air) denote doses to a small mass of tissue in air placed at point corresponding to the surface (d = 0) and depth d respectively. The air doses are related by the inverse square law for either beam energy. The convention and rules in using this diagram are: (1) Multiply the quantity at the origin of an arrow by the arrow label to obtain the value of the quantity at the arrow tip, i.e. while going along the arrow multiply by the arrow label. (2) If going against the arrow, divide by the arrow label. (3) If two quantities are related by different routes along arrows, the product of arrow labels of one route equals the product of arrow labels on the other when products are taken according to rules (1) and (2) above. Applying these rules, equation (1) is equivalent to D4d -DCd ~K4C 04,, = DC,, + 100 .

(2)

Consider the dose D4d to be made up of a component D4h such that D4; -=-_-_ D4s4

DC, _ PC DC,, 100

and a component F(d, A, f) which K4C. We then have D4d = D4; + F(d,

would

be related

A, f)

to

(3)

and 04~4 D4& = --DCd=M4C.DCd DC,, where maximum denoted by M4C.

SSD-f

S, Number

to maximum From equations

ratio D4JDCk (2) and (3)

is

COBALT 111 4MV SURFACE

F(d, A, f) = z

T4

air

(4)

In the region where the DDF (K4C) is constant, this equation implies that F(d, A, f) is a constant proportion of the peak dose D4,, for all depths, field area and SSD.

P4/100

Fig. 2. A mnemonic representation of interrelationships various doses and functions corresponding to irradiation

. 0464.

of

in or tissue with 6oCo and 4MV beams. Meaning of symbols and rules to be followed are described in text.

CONVERSION OF EQUATION (1) TO TISSUE AIR RATIO FORMULA Let T4 and TC represent and DCJD,” respectively.

tissue air ratios D4JDdo It is assumed here that

Concept of depth dose functions 0 P. N.

273

SHRIVASTAVA

irradiation times for the two beams is controlled to yield the same doses in air and that tissue air ratios (TAR) can be defined for 4 MV beams. Then from equations (3) and (4) above

t

(5) Since Dda changes with depth d, equation (5) indicates that T4 and TC are not linearly related. However, if t4 = T4 - [f/(f + d)]* and tc = one writes TC . [f/(f + d)]*, equation (5) can be written in the form

t4=

[

tc+G.

fC&].

.5

(6)

M4C

where tCG, is the value of tC at d = 6~. From Fig. 2, it may be seen that t4 and tC are quantities somewhat similar in definition to the “scatter function” in that they are ratios of dose at depth to the dose in air at the surface. As shown below, equation (6) indicates a linear relationship for quantities denoted by t4 and tC. For 4 MV and 6oCo beams, the maximum to maximum ratio M4c is plotted as a function of field area in Fig. 3 and is seen to remain essentially constant for 5 x 5 to 30 x 30cm* field sizes and for SSDs of 80 and 100 cm. The parameter tCs, contains a back scatter factor term which increases with field size. However, the BSF values for 5 x 5 to 30 x 30 cm* fields of cobalt beam vary between 1 and 1.07, respectively. Therefore, the effect of BSF in the rather small intercept term is negligible. A plot of t4 vs tC in Fig. 4 plotted from data in references (3 and 5) confirms this point. The slope and intercept of the line in Fig. 4 yield values of M4c and K4C in close agreement with those obtained above. A similar type of equation for tissue maximum ratios (TMRs) can also be derived and verified. Equation (6) represents the TAR corollary of equation (1). It could be derived independently from standard definitions, however,

I

tC =TC .(f/f+dj2

Fig. 4. A plot of T4 * (f/f+ d)’ vs Tc *(f/f + d)* for 10 x 10 cm* and 30 x 30 cm2 fields. The straight line relationship observed was predicted from the straight line curve in Fig. 1.

the rationale for such a derivation would not be apparent without first realizing the usefulness of the depth dose function concept. Equation (6) will also offer convenience similar to equation (1) when designing computer algorithms for isocentric treatment planning for which TAR or TMR are the appropriate data.

GENERALIZATION FUNCTIONS FOR

OF DEPTH DOSE HIGHER ENERGIES

As Fig. 5 shows, the straight line relationship observed for per cent depth doses for 4 MV and 6oCo

Depth 100

dose

PE(d,A,f

) for Ouohty E

VS

Depth

dose

PCld,A,f

1for Co-60

90007060g

80cm SSD A 100 cm SSD ??

50403020IO-

IOXIO FIELD

20x20 SIZE (cm21

,I

30x30

Fig. 3. Ratio of doses (D4JDCs,) at depth of maxima for 4 MV and cobalt beams as a function of field size for 80 and 100 cm SSD’s.

’ IO

’ 20

’ 30

’ 40

’ 50

’ 60

’ 70

’ 80

’ 90

L

IO

Fig. 5. Plot of per cent depthzses PE for various beam E (13’Cs, @‘Co, 4, 6, 10 and 22 MV) vs the corresponding per cent depth dose PC for 6oCo beam.

energies

274

Radiation Oncology 0 Biology

0 Physics

beams is also true in a certain range of depths for other higher energy beams. Data used in this figure was taken from: “‘Cs, 6oCo and 22 MV beams;’ 4 MV beams;’ 6 and 8MV beams;4 10 MV beams.’ In general, the straight line relation holds for depths where the per cent depth dose PE(d, A, f) for energy E has values less than 90% for most of the high energy X-ray beams used in radiotherapy. Therefore, we can define a constant DDF in this range of depths. For shallower depths where PE has values between 100 and 90%, the DDF can be looked upon as a continuously varying function. The per cent depth doses for 6oCo, 4 MV, 8 MV and 22 MV beams for a 10 x 10 field at 100 cm SSD are plotted against depth in Fig. 6. The DDF which is the difference PE-PC for each depth derived from these curves is plotted in the bottom portion of the figure. It is observed that for any beam of energy E the DDF (KEC) has nearly a constant value for depths greater than about 2 times the depth of maximum for this energy. Figure 6 also indicates that for shallower depths, a simple linear relationship for KEC as a function of depth may be adequate. In order to determine if some small but systematic variations in KEC can be explicitly accounted for, a

100cm SSD

February

1979, Volume

5, Number

2

computer program was written to determine the values of KEC for all field sizes between 6 x 6 and 20 x 20 cm* and for depths between 1 and 20cm in water using measured depth dose data for 6oCo and X-ray beams produced at 4, 6, 8 and 10 MV. The mean values of KEC (averaged over all field sizes) are plotted as a function of depth in Fig. 7. An examination of the minimum and maximum values of KEC at each depth indicated that the variation in KEC around the mean value is less than 2% for 4 and 6 MV beams and up to 3% for 8 and 10 MV beams. The mean values of KEC averaged over depth are plotted as a function of field size in Fig. 8. A systematic decrease in the mean value of KEC as a function of field size is noted. Although this variation increases for higher energy beams, it is noted that for field sizes greater than 8 x 8 cm2 the use of a mean value KEC would yield acceptable accuracy in most cases. If increased accuracy is desired, a simple inKEC as a function of field terpolation to determine size may be used.

I

4MV c

I 5

I

I

IO 15 FIELD SlZE(cm2)

I 20

Fig. 8. Mean values of depth dose function KEC averaged over depth plotted vs field size for 4,6,8 and 10 MV beams. I

I

I

IO

20 DEPTH (cm)

DISCUSSION

30

Fig. 6. Per cent depth doses PE and the difference PE-PC (bottom 3 curves with ordinate scale on right) are plotted as a function of depth in tissue. The deviation from constant depth dose function at small depths for high energy beams is to be noted.

- .

I

I

5

I

I

15 IO DEPTH km)

.

.

4MV

1

20

Fig. 7. Mean values of depth dose function KEC averaged field size plotted vs depth in tissue for 4, 6, 8 and 10 MV beams.

over

AND CONCLUSIONS

The linear relationship of depth dose PE for radiotherapy beams of energy E and the depth dose PC for cobalt beams in identical geometry for a considerable range of field sizes, depths and SSDs is only an empirical observation. Many attempts to derive a physical basis for this relationship so far have proven illusive. Although a physical basis would have been more satisfying to the physicists, the empirical observation can be of value in computerized dosimetry for beams produced with the newly proliferating high energy accelerators. Its particular value would be in calculating doses for irregularly shaped beams based on simple alterations of algorithms presently used for Cobalt beams.* We have used this approach for the 4 MV beam produced with Varian Clinac IV accelerator having a lead flattening filter and found excellent agreement with measured data.’ This approach may be useful for other energy beams but verification with actual measurements must be stressed.

Concept

of depth

dose functions

0 P. N. SHRIVASTAVA

275

REFERENCES 1. Connor, W.G., Hicks, J.A., Boone, M.L., Mayer, E.G., 5. Peterson, M., Golden, R.: Dosimetry of the Varian Miller, R.C.: 10 MV X-ray beam characteristics from a Radiology 103: 675-680, Clinac-4 Linear Accelerator. new 18 MeV linear accelerator. Znt. .Z. Radiat. Oncol. June 1972. Biol. Phys. 1: 705-712. 6. Shrivastava, P.N.: An approach to summarizing interrelations between functions used in radiotherapy dose 2. Cunningham, J.R., Shrivastava, P.N., Wilkinson, J.M.: Program IRREG-calculation of dose from irregularly calculations. Med. Phys. 1: No. 4, 223-225, July/Aug. shaped radiation beams. Comput. Prog. BioMed. 2: No. 1974. 3, 192-199, 1972. 7. Shrivastava, P.N., Samulski, T.V.: Computer Dosimetry, 3. Johns, H.E., Cunningham, J.R.: The Physics of Verification Procedures. Proc. Computers Applications Radiology, 3rd Edn. Thomas Springfield, Illinois, 1%9, in Radiotherapy Symposium, Midwest Center for pp. 75 l-767. Radiological Physics, Univ. of Wisconsin, WI, March 1978. 4. Jones, D.E.A.: 4, 6 and 8 MV X-rays produced by linear electron accelerators. Br. J. Radiol. (Suppl. 11): 63-75, 1972.