1975, British Journal of Radiology, 48, 739-748
SEPTEMBER 1975
Empirical equations for the representation of depth dose data for computerized treatment planning By R. O. Kornelsen, Ph.D., and M. E. J. Young, M.Sc. British Columbia Cancer Institute, Vancouver, B.C., Canada {Received October 1974)
value by less than 1 per cent of the maximum dose, or by an amount corresponding to a displacement of less than 1 mm. This paper describes empirical equations which satisfy these criteria and have been used at the British Columbia Cancer Institute. In Part I we consider the representation of central axis data. The equation suggested was originally developed to fit cobalt 60 depth dose data but by fitting data from The British Journal of Radiology, Supplement 11, we have shown that it can also be used at both lower and higher energies. In Part II the representation of dose at points other than those lying on the axis is considered.
ABSTRACT
Equations of the form P=100 ( l - ( l - e x p ( - d / Q ) ) M ) and TAR = S ( 1 - ( 1 - e x p (-d/Q)) M ) have been used to represent the variation of central axis percentage depth dose P or tissue-air ratio (TAR) with depth d below the dose maximum. These equations were originally developed for the representation of cobalt 60 depth dose data but have also been fitted to the central axis depth dose data published in the British Journal of Radiology Supplement 11, for radiations ranging in energy from 1 -5 mm Cu HVT to 8 MV. Values of the constants Q and Mfor standard field sizes are presented together with an estimate of the goodness of fit in each case. Two different approaches have been used in determining the dose at points other than those on the central axis. In the simpler method, used for rotation techniques, the off-axis ratio (OAR) is calculated from the equation. OAR = &i + (l -ki) [1/(1 +exp (k2 (x-0-5 w))) ] where x is the off-axis distance, w the field width at the depth and ki and kz are constants. In the more accurate method, used for fixed field techniques, different equations are used within the main beam, within the geometrical penumbra and outside the beam.
In the preparation of radiation treatment plans by computer, it is necessary to provide information from which depth doses can be obtained. The method will depend upon whether the treatment is to be delivered by standard fields, which have been pre-measured in a phantom, or by fields which are irregular in shape, or unique in some other way. The present paper is concerned only with the representation of pre-measured standard fields. For such fields, dose data may be stored as a point-by-point array but this requires considerable storage if acceptable accuracy is to be maintained. If the data can be fitted by analytical equations, then relatively little storage is required and the calculation of dose may be made both faster and cheaper. In this context, we have arbitrarily defined "acceptable accuracy" to mean the following: (1) on the central axis, a computed dose value differs from the measured or tabulated value by less than 0-5 per cent of the maximum dose. (2) elsewhere in the field (whether within the main beam, within the penumbra or outside the beam) a computed value differs from the measured *Based on a paper read at the Canadian Association of Radiologists' Meeting at Vancouver, Canada, on April 23, 1974.
PART I
Central axis data There is a marked similarity in shape between a graph of percentage depth dose against depth and a graph of the fraction of surviving cells against X- or y-ray dose. One of the equations that has been used successfully to represent the survival curve is surviving fraction=n/no = l —(1 —e D/ Do ) m where D denotes dose and Do and m are constants. By analogy, it might be expected that percentage depth doses (P) could be represented by an equation of the form P/100 = l - ( l - e - d / Q ) M . . (1) where d denotes depth beyond the peak of the dose curve and M and Q are constants. The form of the equation ensures that after an initial shoulder, the curve will gradually become exponential, the extent of the shoulder being determined by M and the gradient of the exponential portion by Q. The fact that only the two constants M and Q have to be determined empirically is an advantage compared with equations giving comparable accuracy which have been used by others (Glover (1966), Sterling, Perry and Weinkam (1967), Van de Geijn (1970), Melski (1970), Patomaki (1971), Weinkam, Kolde and Sterling (1973)). The disadvantage is that the equation cannot be used to describe doses in the build-up region for high energy radiation. For 60Co y radiation, we have fitted the equation . . (2)
739
VOL.
48, No. 573 R. O. Kornelsen and M. E. J. Young
(where y denotes depth below the surface in cm and Q is expressed in units of cm"1) to central axis data for more than 160 field sizes on different treatment machines. Table I illustrates how well our own experimental data are represented by the computed values for three typical fields (a fairly small square, a fairly large square and a medium-sized elongated field). We have calculated the root mean square value of the difference between computed and
measured values for all field sizes (denoted by A in Table I and hereafter) and A has never exceeded 0-25 and is usually less than 0 - 20. Table II shows that the goodness of fit is relatively independent of field size and shape. To test whether the same form of equation could be used at other energies, we have fitted data from The British Journal of Radiology, Supplement 11, for X rays from 1 mm to 4 mm copper half-value
TABLE I COMPARISON OF MEASURED* CENTRAL AXIS PERCENTAGE DEPTH DOSES FOR 6 0 C O y RADIATION WITH VALUES COMPUTED USING THE EQUATION p / 1 0 0 = l —(1 —EXP ( — ( Y — 0 - 5 ) / Q ) ) M
Field size (cm) 15x15
6x6 Depth (cm)
Meas.
Calc.
Diff. -0-1 -0-2 -0-2 -0-2 -0-2
5x20
Calc.
Diff.
100-0 98-2 94-0 89-5 85-0 80-4
1000 98-5 94.4 89-8 85-2 80-6
+ 0-3 + 0-4 + 0-3 + 0-2 + 0-2
100-0 98-1 93-3 88-2 82-9 77-8
100-0 98-2 93-4 88-2 82-9 77-7
761 71-7 67-5 63-5 59-7
0-0 0-0 -0-2 -0-1 -0-2
72-8 68-1 63-4 59-1 55-0
72-7 67-9 63-4 59-1 55-0
-0-1 -0-2 00 00 00 0-0 00 00
Meas.
Calc.
Meas.
Diff.
0-5 1-0 2-0 3-0 40 5-0
100 0 98-1 93-0 87-5 81-9 76-5
100 0 98-0 92-8 87-3 81-7 76-3
60 7-0 8-0 9-0 100
71-2 66-3 61-4 57-1 52-8
71-1 66-2 61-5 571 53-0
+ 0-2
76-1 71-7 67-7 63-6 59-9
11-0 120 130 140 15-0
49 0 45-4 42-0 38-9 36-0
49-2 45-5 42-2 39-1 361
+ 0-2 + 01 + 0-2 + 0-2 + 0-1
56-2 52-7 49-4 46-4 43-5
560 52-6 49-3 46-3 43-3
-0-2 -0-1 -01 -0-1 -0-2
51-2 47-6 44-2 41 0 38-0
51-2 47-6 44-2 411 38-2
160 170 18-0 19-0 20-0
33-3 30-8 28-5 26-4 24-4
33-4 30-9 28-7 26-4 24-4
4-0-1 0-0 0-0
40-7 38-1 35-6 33-3 31-2
40-6 38-0 35-6 33-3 21-3
-0-1 -01 0-0 00 0-0
35-2 32-8 30-4 28-2 26-2
35-4 32-9 30-5 28-3 26-2
+ 01 + 0-1 + 01 + 0-1
21-0 22-0 23-0 24-0 25-0
22-6 20-9 19-3 17-9 16-5
22-6 20-8 19-3 17-8 16-4
00 -0-1 0-0 -01 -01
291 27-2 25-4 23-8 22-2
29-2 27-3 25-5 23-9 22-3
+ 0-1 + 0-1 + 0-1 + 01 + 01
24-3 22-5 20-9 19-4 18-0
24-3 22-5 20-9 19-4 17-9
0-0 00 00 00 -0-1
26-0 27-0 28-0 29-0 30-0
15-3 14-2 131 12-2 11-3
15-2 140 12-9 11-9 11-0
-0-1 -0-2 -0-2 -0-3 -0-3
20-8 19-4 18-1 16-9 15-8
20-8 19-5 18-2 171 16-0
+ 0-1 + 01 + 0-2
16-7 15-5 14-4 13-4 12-4
16-6 15-4 14-3 13-2 12-2
-01 -0-1 -01 -0-2 -0-2
Q = 12-:54 M = 1-:>15 A = 0-1 5
-0-1 -0-1
+ 01 00
+ 0-1 + 0-2
Q =14-47 M = 1-242 A = 0-15
0-0
+ 0-2
+ 01 + 01 00 00 -01
+ 01 + 0-2
0-0
Q = 1 2 *14 M = 1-:132 A = 0-1 0
*Measured values obtained at the British Columbia Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit with 80 cm SSD. Adenotes the root mean square value of the differences between measured and computed values.
740
SEPTEMBER 1975
Empirical equations for the representation of depth-dose data for computerized treatment planning TABLE II The effect of field size and shape on the constantsf Q and M and the root mean square difference A between measured* and computed values Rectangular fields
Square fields Field size
Q
M
A
Field size
Q
M
A
5x5 6x6 7x7 8x8 10x10 12x12 15x15 20x20
12-07 12-34 12-62 12-80 13-39 13-84 14-47 15-31
1-204 1-215 1-224 1-230 1-239 1-242 1-242 1-232
0-22 0-15 012 0-10 0-10 0-12 0-15 0-17
6x4 6x5 6x6 6x8 6x10 6x12 6x15 6x20
1203 12-19 12-34 12-56 12-75 12-85 13-00 1314
1-198 1-208 1-215 1-224 1-229 1-234 1-235 1-236
0-23 0-18 0-15 0-11 0-11 009 010 011
fThe constants refer to the equation P/100 = l - ( 1 -exp(—(y —0-5)/Q) ) M *Measured values obtained at the British Columbia Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit with 80 cm SSD.
5
5
10 EQUIVALENT
IS DIAMETER
10 EQUIVALENT
15 DIAMETER
(cm)
(cm)
FIG. 1. Variation of the empirical constant M with the field size 60 for Co radiation, 80 cm SSD. British Columbia Cancer Institute data.
FIG. 2. Variation of the empirical constant Q with field size for 60 Co radiation, 80 cm SSD. British Columbia Cancer Institute data.
thickness, for 137Cs and 60Co y rays and for megavoltage X rays up to 8 MV.* Values of the constants Q and M and the root mean square value of the difference, A> between computed and tabulated values for the Supplement 11 data are given in Tables III, IV and V. Although the published data are not fitted quite so well as our own 60Co data, the agreement (shown by the value of A) in Tables III and IV is certainly acceptable. For X rays of higher energy (Table V), the equation represents the data quite well for field sizes of 6 X 6 cm2 and larger at both 6 MV and 8 MV, but at 4 MV the fit is not so good. For the sake of completeness, we have included the values of the constants for 4 MV radiation
but when A is of the order of 0-5, individual values of the dose calculated from the equation may differ from the tabulated values by as much as 1 per cent. The major discrepancies appear to occur at dose points in the vicinity of the peak. It should be pointed out, however, that real differences of the order of 1 per cent or more appear to exist between depth dose data measured at different centres. Figures 1 and 2, respectively, show the constants M and Q for the British Columbia Cancer Institute 60 Co fixed field data plotted against equivalent diameter. It is clear that M and Q could be expressed as functions of the equivalent diameter and evaluated when required. We have not considered it useful to do this when depth doses for constant sourceskin distance techniques are being computed, but for the corresponding constants which appear in the equation for tissue-air ratios, we do take this extra step.
*The data were fitted using a Fletcher-Powell minimization routine to solve the least squares curve-fitting problem. We are indebted to Mr. Robert Harrison, M.Sc, of the Biophysics Department of this Institute for this computer program. 741
VOL. 48, No. 573
R. O. Kornelsen and M. E. J. Young TABLE III VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l —(1 — EXP ( —D/Q) ) M FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11
X-rays
Square fields
Closed applicators
Radiation
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
HVT 1 0 mm Cu Q 50 cm SSD M A
5-02 1-381 (0-26)
5-23 1-413 (0-29)
5-46 1-436 (0-30)
5-64 1-470 (0-26)
5-82 1-499 (0-23)
6 09 1-567 (0-20)
6-34 1-619 (0-21)
6-64 1-681 (0-26)
7-12 1-691 (0-37)
HVT 1-5 mm Cu Q 50 cm SSD M A HVT 2 0 mm Cu Q 50 cm SSD M A
5-38 1-353 (0-37)
5-60 1-390 (0-24)
5-79 1-432 (0-22)
5-98 1-469 (0-19)
614 1-506 (0-17)
6-47 1-567 (0-13)
6-72 1-631 (0-16)
7-02 1-703 (0-18)
7-38 1-783 (0-32)
5-46 1-361 (0-31)
5-70 1-398 (0-22)
5-90 1-440 (0-17)
610 1-474 (0-18)
6-28 1-514 (0-17)
6-58 1-591 (0-15)
6-86 1-642 (0-18)
7-19 1-703 (0-23)
7-87 1-732 (0-31)
HVT 2-5 mm Cu Q 50 cm SSD M A HVT 3 0 mm Cu Q 50 cm SSD M A
5-59 1-356 (0-32)
5-82 1-394 (0-25)
603 1-434 (0-19)
6-25 1-464 (0-22)
6-43 1-506 (0-20)
6-75 1-581 (0-22)
6-99 1-646 (0-20)
7-35 1-700 (0-22)
8 00 1-715 (0-31)
5-71 1-353 (0-32)
5-97 1-380 (0-31)
6-20 1-411 (0-28)
6-41 1-445 (0-28)
6-60 1-481 (0-25)
6-90 1-570 (0-20)
7-18 1-629 (0-19)
7-48 1-699 (0-22)
8-07 1-714 (0-34)
HVT4 0mmCu Q 50 cm SSD M A
5-85 1-355 (0-22)
6-07 1-378 (0-22)
6-32 1-390 (0-33)
6-51 1-428 (0-27)
6-72 1-457 (0-33)
6-98 1-553 (0-24)
7-33 1-594 (0-33)
7-66 1-669 (0-37)
814 1-701 (0-37)
Radiation
6x4
8x4
8x6
10x5
10x8
15x6
15x10
20x5
20x10
20x15
HVT 10 mm Cu Q 50 cm SSD M A
5-22 1-392 (0-29)
5-39 1-399 (0-28)
5-61 1-462 (0-26)
5-64 1-440 (0-27)
5-97 1-513 (0-25)
5-97 1-495 (0-25)
6-35 1-601 (0-19)
5-97 1-450 (0-33)
6-54 1-600 (0-24)
6-86 1-675 (0-28)
HVT 1 -5 mm Cu Q 50 cm SSD M A HVT2 0mm Cu Q 50 cm SSD M A HVT 2-5 mm Cu Q 50 cm SSD M A
5-53 1-388 (0-20)
5-70 1-397 (0-22)
5-94 1-462 (0-14)
5-95 1-448 (0-17)
6-31 1-521 (0-16)
6-33 1-496 (0-18)
6-72 1-618 (0-13)
6-31 1-460 (0-21)
6-88 1-637 (0-18)
7-18 1-732 (0-23)
5-68 1-379 (0-23)
5-84 1-393 (0-23)
6 09 1-462 (0-16)
612 1-442 (0-19)
6-43 1-540 (0-18)
6-49 1-494 (0-22)
6-87 1-625 (0-18)
6-51 1-446 (0-23)
7-12 1-621 (0-21)
7-48 1-703 (0-24)
5-82 1-373 (0-23)
5-97 1-391 (0-22)
6-22 1-457 (0-22)
6-25 1-440 (0-23)
6-56 1-537 (016)
6-63 1-493 (0-22)
7-02 1-622 (0-16)
6-64 1-444 (0-26)
7-27 1-620 (0-20)
7-65 1-697 (0-23)
HVT 3 0 mm Cu Q 50 cm SSD M A
5-96 1-361 (0-25)
6 09 1-383 (0-21)
6-37 1-442 (0-26)
6-40 1-426 (0-24)
6-72 1-521 (0-16)
6-77 1-486 (0-26)
7-18 1-612 (0-13)
6-77 1-436 (0-34)
7-38 1-616 (0-24)
7-73 1-700 (0-28)
X-rays
Rectangular
fields
Closed applicators
A denotes the root mean square value of the differences between the computed and published data.
Tissue-air ratios The equation . (3) has been used to represent the variation of the tissueair ratio (TAR) with depth of overlying tissue d beyond the dose maximum. Q and M are constants
for a given field size (at the centre of rotation) and S is the peak scatter factor. Values of Q and M for 60Co radiation which correspond to tissue-air ratios measured at the British Columbia Cancer Institute are given in Table VI. Values for 60Co and other radiations derived from
742
SEPTEMBER 1975
Empirical equations for the representation of depth-dose data for computerized treatment planning TABLE IV VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l — (1 — EXP ( — D/Q)
)M
FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11 137
Radiation
Cs, 60Co y-rays, 2 MV X-rays
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
Q M
7-88 1-124 (0-20)
8-09 1-135 (0-19)
8-28 1145 (0-18)
8-47 1-150 (0-22)
8-66 1152 (0-24)
8-96 1162 (0-26)
9-25 1159 (0-26)
9-59 1-157 (0-32)
9-99 1-153 (0-35)
137 Cs y rays 40 cm SSD
Q M
8-31 1199 (0-14)
8-53 1-216 (0-19)
8-70 1-231 (0-15)
8-89 1-240 (0-20)
910 1-244 (0-20)
9-50 1-249 (0-20)
9-75 1-257 (0-22)
1010 1-264 (0-20)
10-54 1-258 (0-23)
137
Cs y rays 50 cm SSD
Q M
8-59 1-239 (0-17)
8-77 1-266 (0-21)
8-96 1-287 (0-21)
915 1-302 (0-23)
9-32 1-316 (0-26)
9-66 1-329 (0-29)
10-00 1-333 (0-30)
10-39 1-336 (0-30)
10-96 1-331 (0-31)
137
Cs y rays Infinite SSD*
Q M
10-60 1-507 (0-26)
10-69 1-567 (0-22)
10-77 1-631 (0-28)
10-92 1-673 (0-26)
11 06 1-719 (0-31)
11-37 1-787 (0-36)
11-66 1-839 (0-40)
1215 1-892 (0-43)
12-90 1-945 (0-44)
2 MV X rays 100 cm SSD
Q M
11-06 1-172 (0-26)
1117 1-211 (0-25)
11-30 1-245 (0-22)
11-46 1-274 (0-22)
11-61 1-302 (0-25)
11-95 1-339 (0-32)
12-29 1-360 (0-32)
12-75 1-376 (0-34)
13-44 1-371 (0-27)
2 MV X rays Infinite SSD*
Q M
13-17 1-240 (0-30)
13-12 1-303 (0-29)
13-19 1-352 (0-32)
13-25 1-403 (0-28)
13-26 1-459 (0-28)
13-48 1-533 (0-30)
13-76 1-594 (0-33)
14-30 1-641 (0-33)
15-18 1-668 (0-23)
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15X15
20x20
10-90 1-068 (0-14)
11-00 1099 (0-17)
1113 1-124 (0-19)
11-29 1-142 (0-21)
11-52 1146 (0-22)
11-94 1-153 (0-22)
12-46 1-140 (0-23)
13-24 1119 (0-25)
14-32 1-097 (0-16)
11-27 1-105 (0-23)
11-44 1-129 (0-22)
11-60 1-152 (0-20)
11-80 1164 (0-22)
12-00 1176 (0-21)
12-46 1178 (0-25)
12-98 1169 (0-28)
13-71 1155 (0-28)
14-72 1139 (0-15)
12-22 1-155 (0-29)
12-36 1-184 (0-27)
12-56 1-200 (0-27)
12-76 1-212 (0-25)
13-24 1-212 (0-27)
13-68 1-221 (0-25)
14-31 1-217 (0-19)
15-34 1194 (0-20)
137
Cs y rays 30cmSSD
A
A
A A
Radiation 60
Co y rays 50 cm SSD
Q M
60
Co y rays 60 cm SSD
Q M
60
Co y rays 80 cm SSD
Q M
A
12 09 1-122 (0-25)
60
Co y rays 100 cm SSD
Q M A
12-63 1137 (0-20)
12-77 1-171 (0-26)
12-92 1-201 (0-27)
13-11 1-219 (0-28)
13-33 1-232 (0-30)
13-74 1-249 (0-35)
14-20 1-251 (0-35)
14-85 1-248 (0-29)
15-73 1-233 (0-23)
60
Q M
15-12 1-229 (0-28)
15-18 1-280 (0-28)
15-23 1-330 (0-27)
15-35 1-367 (0-26)
15-50 1-401 (0-27)
15-92 1-440 (0-22)
16-44 1-463 (0-21)
17-38 1-462 (0-17)
18-64 1-472 (0-13)
12-07 1-204 (0-22)
12-34 1-215 (0-15)
12-62 1-224 (0-12)
12-80 1-230 (0-10)
13-39 1-239 (0-10)
13-84 1-242 (0-12)
14-47 1-242 (0-15)
15-31 1-232 (0-17)
A A
Co y rays Infinite SSD*
A «°Co y rays 80 cm SSD tB.C.C.I. data
Q M A
*Values from B.J.R. tables of tissue-air ratios, normalised to 1 -000 at reference depth. fMeasured values obtained at the B.C. Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit. A denotes the root mean square value of the differences between the computed and published data.
743
VOL.
48, No. 573 R. O. Kornelsen and M. E. J. Young TABLE V VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l —(1 —EXP (—D/Q) ) M FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11
Megavoltage X-rays Radiation
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
4 MV X rays 80 cm SSD
Q M
13-18 1-147 (0-52)
13-34 1-174 (0-51)
13-50 1194 (0-48)
13-68 1-215 (0-39)
13-91 1-221 (0-37)
14-31 1-228 (0-32)
14-73 1-224 (0-34)
15-19 1-219 (0-38)
15-81 1-221 (0-42)
4 MV X rays 90 cm SSD
Q M
13-55 1-154 (0-51)
13-69 1-185 (0-47)
13-83 1-212 (0-42)
14-03 1-229 (0-40)
14-26 1-235 (0-37)
14-71 1-237 (0-34)
15-12 1-237 (0-35)
15-65 1-231 (0-39)
16-28 1-235 (0-42)
4 MV X rays 100 cm SSD
Q M
13-85 1161 (0-49)
14-00 1-193 (0-46)
14-18 1-215 (0-45)
14-37 1-233 (0-42)
14-61 1-242 (0-39)
15-04 1-249 (0-35)
15-44 1-252 (0-35)
15-99 1-248 (0-35)
16-64 1-250 (0-43)
4 MV X rays Infinite SSD
Q M
17-92 1-236 (0-58)
18-09 1-282 (0-58)
18-26 1-319 (0-58)
18-48 1-353 (0-56)
18-82 1-371 (0-52)
19-45 1-391 (0-49)
20-04 1-405 (0-49)
20-93 1-406 (0-49)
2201 1-423 (0-58)
A
A A A
Megavoltage X rays Radiation 6 MV X rays 100 cm SSD
Q M
A
3x3
4x4
6x6
8x8
10x10
12x12
15x15
20x20
25x25
15-42 1104 (0-35)
15-65 1126 (0-40)
16-35 1143 (0-32)
16-83 1167 (0-29)
17-27 1-179 (0-25)
17-75 1-178 (0-20)
18-39 1-182 (0-16)
19-28 1-183 (0-17)
19-78 1191 (0-14)
6 MV X rays Infinite SSD
Q M
20-57 1-167 (0-40)
21-03 1193 (0-42)
21-90 1-245 (0-39)
22-89 1-270 (0-41)
23-41 1-304 (0-35)
24-07 1-323 (0-33)
24-95 1-348 (0-20)
26-41 1-366 (0-15)
27-28 1-390 (0-13)
8 MV X rays 100 cm SSD
Q M
16-71 1167 (0-40)
17-06 1161 (0-40)
17-65 1165 (0-30)
18-32 1:160 (0-24)
18-93 1-159 (0-29)
19-49 1-154 (0-19)
20 06 1-157 (0-18)
20-66 1-155 (0-19)
21-17 1-147 (0-35)
8 MV X rays Infinite SSD
Q M
22-65 1-270 (0-50)
23-21 1-272 (0-45)
24-23 1-282 (0-38)
25-16 1-297 (0-27)
26-23 1-303 (0-23)
27-15 1-311 (0-20)
28-26 1-316 (0-16)
29-65 1-312 (0-22)
30-62 1-309 (0-31)
A
A A
TABLE VI TISSUE AIR RATIOS FOR
60
CO RADIATION
Values of the constants Q and M in the equation T = S [1 - ( l - e x p ( - ( y - 0 - 5 ) / Q ) ) M ] Field (cm) 5x5 7x7 10x10 12x12 15x15
Q*
M*
15-98 16-55 17-46 18-00 18-85
1-216 1-270 1-326 1-347 1-365
A (0-23) (0-18) (0-16) (0-15) (0-18)
Q+
M+
15-99 16-56 17-43 18-00 18-85
1-215 1-271 1-326 1-347 1-366
Q* and M* are the values which best fit the individual depth doses measured on an A.E.C.L. Theratron F Unit. Q"1" and M + are calculated from equations (4) and (5). A denotes the root mean square value of the differences between measured dose values and those computed using Q* and M* in equation (3). 744
SEPTEMBER
1975
Empirical equations for the representation of depth-dose data for computerized treatment planning
data in The British Journal of Radiology, Supplement 11, are given in Tables IV and V in the sections labelled "infinite SSD". At the British Columbia Cancer Institute we use tissue-air ratios, not only for the evaluation of dose when we use constant source-axis techniques, but also in determining corrections for inhomogeneities, such as lung (Batho, 1964; Young, 1967). Therefore, we need to be able to evaluate TAR for any field size. To do this, Q, M and S are expressed as functions of the equivalent radius (R). For our own data (measured on an AECL Theratron F machine) Q = 14-55+0-5144R . . . (4) M = l + 0-3845(l-exp(-0-2378Ri- 2 ))(5) S = l+0-1068 (1-exp (-0-1084 R)) . (6) PART II
Off-axis doses
We have used two different approaches in determining the dose at points other than those on the central axis:
(1) a single function of the off-axis distance (called the off-axis ratio) is used to relate the dose at all points at a given depth to the axial dose at that depth; (2) a more detailed treatment, using different equations in the main beam, in the penumbra and outside the beam. Off-axis ratios The off-axis ratio (OAR) (also called the offcentre ratio, the profile function, or the transverse distribution) is a function of the off-axis distance and parameters such as the depth, field-size and source distance, and may be quite complex in form. See, for example, the summaries given by Van de Geijn (1972) and by Weinkam, Kolde and Sterling (1973). A graph of off-axis ratio plotted against off-axis distance is reminiscent of the Fermi-Dirac distribution function of solid state physics. For 60Co radiation fields of medium size, defined by main
TABLE VII COMPARISON OF MEASURED* AND COMPUTED OFF-AXIS RATIOS
10 x 10 field
6 x 6 field Measured Dose y ~6 cm
y = 15 cm
y ~ 24 cm
Measured
Calc.
8
Calc.
8
OAR
OAR
(mm)
OAR
(mm)
— 0-2 0-6 0-6 0-6 0-4 0-2 01 01 0-2 0-2 0-2 —
1-00 0-850 0-768 0-692 0-582 0-500 0-398 0-309 0-250 0-200 0151 0-077 0062
— 0-2 01 0-6 0-2 0-5 0-3 0-4 0-2 0-2 0-2 2-5 —
—.
SEPTEMBER 1975
Empirical equations for the representation of depth dose data for computerized treatment planning By R. O. Kornelsen, Ph.D., and M. E. J. Young, M.Sc. British Columbia Cancer Institute, Vancouver, B.C., Canada {Received October 1974)
value by less than 1 per cent of the maximum dose, or by an amount corresponding to a displacement of less than 1 mm. This paper describes empirical equations which satisfy these criteria and have been used at the British Columbia Cancer Institute. In Part I we consider the representation of central axis data. The equation suggested was originally developed to fit cobalt 60 depth dose data but by fitting data from The British Journal of Radiology, Supplement 11, we have shown that it can also be used at both lower and higher energies. In Part II the representation of dose at points other than those lying on the axis is considered.
ABSTRACT
Equations of the form P=100 ( l - ( l - e x p ( - d / Q ) ) M ) and TAR = S ( 1 - ( 1 - e x p (-d/Q)) M ) have been used to represent the variation of central axis percentage depth dose P or tissue-air ratio (TAR) with depth d below the dose maximum. These equations were originally developed for the representation of cobalt 60 depth dose data but have also been fitted to the central axis depth dose data published in the British Journal of Radiology Supplement 11, for radiations ranging in energy from 1 -5 mm Cu HVT to 8 MV. Values of the constants Q and Mfor standard field sizes are presented together with an estimate of the goodness of fit in each case. Two different approaches have been used in determining the dose at points other than those on the central axis. In the simpler method, used for rotation techniques, the off-axis ratio (OAR) is calculated from the equation. OAR = &i + (l -ki) [1/(1 +exp (k2 (x-0-5 w))) ] where x is the off-axis distance, w the field width at the depth and ki and kz are constants. In the more accurate method, used for fixed field techniques, different equations are used within the main beam, within the geometrical penumbra and outside the beam.
In the preparation of radiation treatment plans by computer, it is necessary to provide information from which depth doses can be obtained. The method will depend upon whether the treatment is to be delivered by standard fields, which have been pre-measured in a phantom, or by fields which are irregular in shape, or unique in some other way. The present paper is concerned only with the representation of pre-measured standard fields. For such fields, dose data may be stored as a point-by-point array but this requires considerable storage if acceptable accuracy is to be maintained. If the data can be fitted by analytical equations, then relatively little storage is required and the calculation of dose may be made both faster and cheaper. In this context, we have arbitrarily defined "acceptable accuracy" to mean the following: (1) on the central axis, a computed dose value differs from the measured or tabulated value by less than 0-5 per cent of the maximum dose. (2) elsewhere in the field (whether within the main beam, within the penumbra or outside the beam) a computed value differs from the measured *Based on a paper read at the Canadian Association of Radiologists' Meeting at Vancouver, Canada, on April 23, 1974.
PART I
Central axis data There is a marked similarity in shape between a graph of percentage depth dose against depth and a graph of the fraction of surviving cells against X- or y-ray dose. One of the equations that has been used successfully to represent the survival curve is surviving fraction=n/no = l —(1 —e D/ Do ) m where D denotes dose and Do and m are constants. By analogy, it might be expected that percentage depth doses (P) could be represented by an equation of the form P/100 = l - ( l - e - d / Q ) M . . (1) where d denotes depth beyond the peak of the dose curve and M and Q are constants. The form of the equation ensures that after an initial shoulder, the curve will gradually become exponential, the extent of the shoulder being determined by M and the gradient of the exponential portion by Q. The fact that only the two constants M and Q have to be determined empirically is an advantage compared with equations giving comparable accuracy which have been used by others (Glover (1966), Sterling, Perry and Weinkam (1967), Van de Geijn (1970), Melski (1970), Patomaki (1971), Weinkam, Kolde and Sterling (1973)). The disadvantage is that the equation cannot be used to describe doses in the build-up region for high energy radiation. For 60Co y radiation, we have fitted the equation . . (2)
739
VOL.
48, No. 573 R. O. Kornelsen and M. E. J. Young
(where y denotes depth below the surface in cm and Q is expressed in units of cm"1) to central axis data for more than 160 field sizes on different treatment machines. Table I illustrates how well our own experimental data are represented by the computed values for three typical fields (a fairly small square, a fairly large square and a medium-sized elongated field). We have calculated the root mean square value of the difference between computed and
measured values for all field sizes (denoted by A in Table I and hereafter) and A has never exceeded 0-25 and is usually less than 0 - 20. Table II shows that the goodness of fit is relatively independent of field size and shape. To test whether the same form of equation could be used at other energies, we have fitted data from The British Journal of Radiology, Supplement 11, for X rays from 1 mm to 4 mm copper half-value
TABLE I COMPARISON OF MEASURED* CENTRAL AXIS PERCENTAGE DEPTH DOSES FOR 6 0 C O y RADIATION WITH VALUES COMPUTED USING THE EQUATION p / 1 0 0 = l —(1 —EXP ( — ( Y — 0 - 5 ) / Q ) ) M
Field size (cm) 15x15
6x6 Depth (cm)
Meas.
Calc.
Diff. -0-1 -0-2 -0-2 -0-2 -0-2
5x20
Calc.
Diff.
100-0 98-2 94-0 89-5 85-0 80-4
1000 98-5 94.4 89-8 85-2 80-6
+ 0-3 + 0-4 + 0-3 + 0-2 + 0-2
100-0 98-1 93-3 88-2 82-9 77-8
100-0 98-2 93-4 88-2 82-9 77-7
761 71-7 67-5 63-5 59-7
0-0 0-0 -0-2 -0-1 -0-2
72-8 68-1 63-4 59-1 55-0
72-7 67-9 63-4 59-1 55-0
-0-1 -0-2 00 00 00 0-0 00 00
Meas.
Calc.
Meas.
Diff.
0-5 1-0 2-0 3-0 40 5-0
100 0 98-1 93-0 87-5 81-9 76-5
100 0 98-0 92-8 87-3 81-7 76-3
60 7-0 8-0 9-0 100
71-2 66-3 61-4 57-1 52-8
71-1 66-2 61-5 571 53-0
+ 0-2
76-1 71-7 67-7 63-6 59-9
11-0 120 130 140 15-0
49 0 45-4 42-0 38-9 36-0
49-2 45-5 42-2 39-1 361
+ 0-2 + 01 + 0-2 + 0-2 + 0-1
56-2 52-7 49-4 46-4 43-5
560 52-6 49-3 46-3 43-3
-0-2 -0-1 -01 -0-1 -0-2
51-2 47-6 44-2 41 0 38-0
51-2 47-6 44-2 411 38-2
160 170 18-0 19-0 20-0
33-3 30-8 28-5 26-4 24-4
33-4 30-9 28-7 26-4 24-4
4-0-1 0-0 0-0
40-7 38-1 35-6 33-3 31-2
40-6 38-0 35-6 33-3 21-3
-0-1 -01 0-0 00 0-0
35-2 32-8 30-4 28-2 26-2
35-4 32-9 30-5 28-3 26-2
+ 01 + 0-1 + 01 + 0-1
21-0 22-0 23-0 24-0 25-0
22-6 20-9 19-3 17-9 16-5
22-6 20-8 19-3 17-8 16-4
00 -0-1 0-0 -01 -01
291 27-2 25-4 23-8 22-2
29-2 27-3 25-5 23-9 22-3
+ 0-1 + 0-1 + 0-1 + 01 + 01
24-3 22-5 20-9 19-4 18-0
24-3 22-5 20-9 19-4 17-9
0-0 00 00 00 -0-1
26-0 27-0 28-0 29-0 30-0
15-3 14-2 131 12-2 11-3
15-2 140 12-9 11-9 11-0
-0-1 -0-2 -0-2 -0-3 -0-3
20-8 19-4 18-1 16-9 15-8
20-8 19-5 18-2 171 16-0
+ 0-1 + 01 + 0-2
16-7 15-5 14-4 13-4 12-4
16-6 15-4 14-3 13-2 12-2
-01 -0-1 -01 -0-2 -0-2
Q = 12-:54 M = 1-:>15 A = 0-1 5
-0-1 -0-1
+ 01 00
+ 0-1 + 0-2
Q =14-47 M = 1-242 A = 0-15
0-0
+ 0-2
+ 01 + 01 00 00 -01
+ 01 + 0-2
0-0
Q = 1 2 *14 M = 1-:132 A = 0-1 0
*Measured values obtained at the British Columbia Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit with 80 cm SSD. Adenotes the root mean square value of the differences between measured and computed values.
740
SEPTEMBER 1975
Empirical equations for the representation of depth-dose data for computerized treatment planning TABLE II The effect of field size and shape on the constantsf Q and M and the root mean square difference A between measured* and computed values Rectangular fields
Square fields Field size
Q
M
A
Field size
Q
M
A
5x5 6x6 7x7 8x8 10x10 12x12 15x15 20x20
12-07 12-34 12-62 12-80 13-39 13-84 14-47 15-31
1-204 1-215 1-224 1-230 1-239 1-242 1-242 1-232
0-22 0-15 012 0-10 0-10 0-12 0-15 0-17
6x4 6x5 6x6 6x8 6x10 6x12 6x15 6x20
1203 12-19 12-34 12-56 12-75 12-85 13-00 1314
1-198 1-208 1-215 1-224 1-229 1-234 1-235 1-236
0-23 0-18 0-15 0-11 0-11 009 010 011
fThe constants refer to the equation P/100 = l - ( 1 -exp(—(y —0-5)/Q) ) M *Measured values obtained at the British Columbia Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit with 80 cm SSD.
5
5
10 EQUIVALENT
IS DIAMETER
10 EQUIVALENT
15 DIAMETER
(cm)
(cm)
FIG. 1. Variation of the empirical constant M with the field size 60 for Co radiation, 80 cm SSD. British Columbia Cancer Institute data.
FIG. 2. Variation of the empirical constant Q with field size for 60 Co radiation, 80 cm SSD. British Columbia Cancer Institute data.
thickness, for 137Cs and 60Co y rays and for megavoltage X rays up to 8 MV.* Values of the constants Q and M and the root mean square value of the difference, A> between computed and tabulated values for the Supplement 11 data are given in Tables III, IV and V. Although the published data are not fitted quite so well as our own 60Co data, the agreement (shown by the value of A) in Tables III and IV is certainly acceptable. For X rays of higher energy (Table V), the equation represents the data quite well for field sizes of 6 X 6 cm2 and larger at both 6 MV and 8 MV, but at 4 MV the fit is not so good. For the sake of completeness, we have included the values of the constants for 4 MV radiation
but when A is of the order of 0-5, individual values of the dose calculated from the equation may differ from the tabulated values by as much as 1 per cent. The major discrepancies appear to occur at dose points in the vicinity of the peak. It should be pointed out, however, that real differences of the order of 1 per cent or more appear to exist between depth dose data measured at different centres. Figures 1 and 2, respectively, show the constants M and Q for the British Columbia Cancer Institute 60 Co fixed field data plotted against equivalent diameter. It is clear that M and Q could be expressed as functions of the equivalent diameter and evaluated when required. We have not considered it useful to do this when depth doses for constant sourceskin distance techniques are being computed, but for the corresponding constants which appear in the equation for tissue-air ratios, we do take this extra step.
*The data were fitted using a Fletcher-Powell minimization routine to solve the least squares curve-fitting problem. We are indebted to Mr. Robert Harrison, M.Sc, of the Biophysics Department of this Institute for this computer program. 741
VOL. 48, No. 573
R. O. Kornelsen and M. E. J. Young TABLE III VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l —(1 — EXP ( —D/Q) ) M FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11
X-rays
Square fields
Closed applicators
Radiation
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
HVT 1 0 mm Cu Q 50 cm SSD M A
5-02 1-381 (0-26)
5-23 1-413 (0-29)
5-46 1-436 (0-30)
5-64 1-470 (0-26)
5-82 1-499 (0-23)
6 09 1-567 (0-20)
6-34 1-619 (0-21)
6-64 1-681 (0-26)
7-12 1-691 (0-37)
HVT 1-5 mm Cu Q 50 cm SSD M A HVT 2 0 mm Cu Q 50 cm SSD M A
5-38 1-353 (0-37)
5-60 1-390 (0-24)
5-79 1-432 (0-22)
5-98 1-469 (0-19)
614 1-506 (0-17)
6-47 1-567 (0-13)
6-72 1-631 (0-16)
7-02 1-703 (0-18)
7-38 1-783 (0-32)
5-46 1-361 (0-31)
5-70 1-398 (0-22)
5-90 1-440 (0-17)
610 1-474 (0-18)
6-28 1-514 (0-17)
6-58 1-591 (0-15)
6-86 1-642 (0-18)
7-19 1-703 (0-23)
7-87 1-732 (0-31)
HVT 2-5 mm Cu Q 50 cm SSD M A HVT 3 0 mm Cu Q 50 cm SSD M A
5-59 1-356 (0-32)
5-82 1-394 (0-25)
603 1-434 (0-19)
6-25 1-464 (0-22)
6-43 1-506 (0-20)
6-75 1-581 (0-22)
6-99 1-646 (0-20)
7-35 1-700 (0-22)
8 00 1-715 (0-31)
5-71 1-353 (0-32)
5-97 1-380 (0-31)
6-20 1-411 (0-28)
6-41 1-445 (0-28)
6-60 1-481 (0-25)
6-90 1-570 (0-20)
7-18 1-629 (0-19)
7-48 1-699 (0-22)
8-07 1-714 (0-34)
HVT4 0mmCu Q 50 cm SSD M A
5-85 1-355 (0-22)
6-07 1-378 (0-22)
6-32 1-390 (0-33)
6-51 1-428 (0-27)
6-72 1-457 (0-33)
6-98 1-553 (0-24)
7-33 1-594 (0-33)
7-66 1-669 (0-37)
814 1-701 (0-37)
Radiation
6x4
8x4
8x6
10x5
10x8
15x6
15x10
20x5
20x10
20x15
HVT 10 mm Cu Q 50 cm SSD M A
5-22 1-392 (0-29)
5-39 1-399 (0-28)
5-61 1-462 (0-26)
5-64 1-440 (0-27)
5-97 1-513 (0-25)
5-97 1-495 (0-25)
6-35 1-601 (0-19)
5-97 1-450 (0-33)
6-54 1-600 (0-24)
6-86 1-675 (0-28)
HVT 1 -5 mm Cu Q 50 cm SSD M A HVT2 0mm Cu Q 50 cm SSD M A HVT 2-5 mm Cu Q 50 cm SSD M A
5-53 1-388 (0-20)
5-70 1-397 (0-22)
5-94 1-462 (0-14)
5-95 1-448 (0-17)
6-31 1-521 (0-16)
6-33 1-496 (0-18)
6-72 1-618 (0-13)
6-31 1-460 (0-21)
6-88 1-637 (0-18)
7-18 1-732 (0-23)
5-68 1-379 (0-23)
5-84 1-393 (0-23)
6 09 1-462 (0-16)
612 1-442 (0-19)
6-43 1-540 (0-18)
6-49 1-494 (0-22)
6-87 1-625 (0-18)
6-51 1-446 (0-23)
7-12 1-621 (0-21)
7-48 1-703 (0-24)
5-82 1-373 (0-23)
5-97 1-391 (0-22)
6-22 1-457 (0-22)
6-25 1-440 (0-23)
6-56 1-537 (016)
6-63 1-493 (0-22)
7-02 1-622 (0-16)
6-64 1-444 (0-26)
7-27 1-620 (0-20)
7-65 1-697 (0-23)
HVT 3 0 mm Cu Q 50 cm SSD M A
5-96 1-361 (0-25)
6 09 1-383 (0-21)
6-37 1-442 (0-26)
6-40 1-426 (0-24)
6-72 1-521 (0-16)
6-77 1-486 (0-26)
7-18 1-612 (0-13)
6-77 1-436 (0-34)
7-38 1-616 (0-24)
7-73 1-700 (0-28)
X-rays
Rectangular
fields
Closed applicators
A denotes the root mean square value of the differences between the computed and published data.
Tissue-air ratios The equation . (3) has been used to represent the variation of the tissueair ratio (TAR) with depth of overlying tissue d beyond the dose maximum. Q and M are constants
for a given field size (at the centre of rotation) and S is the peak scatter factor. Values of Q and M for 60Co radiation which correspond to tissue-air ratios measured at the British Columbia Cancer Institute are given in Table VI. Values for 60Co and other radiations derived from
742
SEPTEMBER 1975
Empirical equations for the representation of depth-dose data for computerized treatment planning TABLE IV VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l — (1 — EXP ( — D/Q)
)M
FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11 137
Radiation
Cs, 60Co y-rays, 2 MV X-rays
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
Q M
7-88 1-124 (0-20)
8-09 1-135 (0-19)
8-28 1145 (0-18)
8-47 1-150 (0-22)
8-66 1152 (0-24)
8-96 1162 (0-26)
9-25 1159 (0-26)
9-59 1-157 (0-32)
9-99 1-153 (0-35)
137 Cs y rays 40 cm SSD
Q M
8-31 1199 (0-14)
8-53 1-216 (0-19)
8-70 1-231 (0-15)
8-89 1-240 (0-20)
910 1-244 (0-20)
9-50 1-249 (0-20)
9-75 1-257 (0-22)
1010 1-264 (0-20)
10-54 1-258 (0-23)
137
Cs y rays 50 cm SSD
Q M
8-59 1-239 (0-17)
8-77 1-266 (0-21)
8-96 1-287 (0-21)
915 1-302 (0-23)
9-32 1-316 (0-26)
9-66 1-329 (0-29)
10-00 1-333 (0-30)
10-39 1-336 (0-30)
10-96 1-331 (0-31)
137
Cs y rays Infinite SSD*
Q M
10-60 1-507 (0-26)
10-69 1-567 (0-22)
10-77 1-631 (0-28)
10-92 1-673 (0-26)
11 06 1-719 (0-31)
11-37 1-787 (0-36)
11-66 1-839 (0-40)
1215 1-892 (0-43)
12-90 1-945 (0-44)
2 MV X rays 100 cm SSD
Q M
11-06 1-172 (0-26)
1117 1-211 (0-25)
11-30 1-245 (0-22)
11-46 1-274 (0-22)
11-61 1-302 (0-25)
11-95 1-339 (0-32)
12-29 1-360 (0-32)
12-75 1-376 (0-34)
13-44 1-371 (0-27)
2 MV X rays Infinite SSD*
Q M
13-17 1-240 (0-30)
13-12 1-303 (0-29)
13-19 1-352 (0-32)
13-25 1-403 (0-28)
13-26 1-459 (0-28)
13-48 1-533 (0-30)
13-76 1-594 (0-33)
14-30 1-641 (0-33)
15-18 1-668 (0-23)
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15X15
20x20
10-90 1-068 (0-14)
11-00 1099 (0-17)
1113 1-124 (0-19)
11-29 1-142 (0-21)
11-52 1146 (0-22)
11-94 1-153 (0-22)
12-46 1-140 (0-23)
13-24 1119 (0-25)
14-32 1-097 (0-16)
11-27 1-105 (0-23)
11-44 1-129 (0-22)
11-60 1-152 (0-20)
11-80 1164 (0-22)
12-00 1176 (0-21)
12-46 1178 (0-25)
12-98 1169 (0-28)
13-71 1155 (0-28)
14-72 1139 (0-15)
12-22 1-155 (0-29)
12-36 1-184 (0-27)
12-56 1-200 (0-27)
12-76 1-212 (0-25)
13-24 1-212 (0-27)
13-68 1-221 (0-25)
14-31 1-217 (0-19)
15-34 1194 (0-20)
137
Cs y rays 30cmSSD
A
A
A A
Radiation 60
Co y rays 50 cm SSD
Q M
60
Co y rays 60 cm SSD
Q M
60
Co y rays 80 cm SSD
Q M
A
12 09 1-122 (0-25)
60
Co y rays 100 cm SSD
Q M A
12-63 1137 (0-20)
12-77 1-171 (0-26)
12-92 1-201 (0-27)
13-11 1-219 (0-28)
13-33 1-232 (0-30)
13-74 1-249 (0-35)
14-20 1-251 (0-35)
14-85 1-248 (0-29)
15-73 1-233 (0-23)
60
Q M
15-12 1-229 (0-28)
15-18 1-280 (0-28)
15-23 1-330 (0-27)
15-35 1-367 (0-26)
15-50 1-401 (0-27)
15-92 1-440 (0-22)
16-44 1-463 (0-21)
17-38 1-462 (0-17)
18-64 1-472 (0-13)
12-07 1-204 (0-22)
12-34 1-215 (0-15)
12-62 1-224 (0-12)
12-80 1-230 (0-10)
13-39 1-239 (0-10)
13-84 1-242 (0-12)
14-47 1-242 (0-15)
15-31 1-232 (0-17)
A A
Co y rays Infinite SSD*
A «°Co y rays 80 cm SSD tB.C.C.I. data
Q M A
*Values from B.J.R. tables of tissue-air ratios, normalised to 1 -000 at reference depth. fMeasured values obtained at the B.C. Cancer Institute using an A.E.C.L. Eldorado 8 Therapy Unit. A denotes the root mean square value of the differences between the computed and published data.
743
VOL.
48, No. 573 R. O. Kornelsen and M. E. J. Young TABLE V VALUES OF THE CONSTANTS Q AND M IN THE EQUATION P/100 = l —(1 —EXP (—D/Q) ) M FITTED TO DATA GIVEN IN B . J . R . SUPPLEMENT 11
Megavoltage X-rays Radiation
4x4
5x5
6x6
7x7
8x8
10x10
12x12
15x15
20x20
4 MV X rays 80 cm SSD
Q M
13-18 1-147 (0-52)
13-34 1-174 (0-51)
13-50 1194 (0-48)
13-68 1-215 (0-39)
13-91 1-221 (0-37)
14-31 1-228 (0-32)
14-73 1-224 (0-34)
15-19 1-219 (0-38)
15-81 1-221 (0-42)
4 MV X rays 90 cm SSD
Q M
13-55 1-154 (0-51)
13-69 1-185 (0-47)
13-83 1-212 (0-42)
14-03 1-229 (0-40)
14-26 1-235 (0-37)
14-71 1-237 (0-34)
15-12 1-237 (0-35)
15-65 1-231 (0-39)
16-28 1-235 (0-42)
4 MV X rays 100 cm SSD
Q M
13-85 1161 (0-49)
14-00 1-193 (0-46)
14-18 1-215 (0-45)
14-37 1-233 (0-42)
14-61 1-242 (0-39)
15-04 1-249 (0-35)
15-44 1-252 (0-35)
15-99 1-248 (0-35)
16-64 1-250 (0-43)
4 MV X rays Infinite SSD
Q M
17-92 1-236 (0-58)
18-09 1-282 (0-58)
18-26 1-319 (0-58)
18-48 1-353 (0-56)
18-82 1-371 (0-52)
19-45 1-391 (0-49)
20-04 1-405 (0-49)
20-93 1-406 (0-49)
2201 1-423 (0-58)
A
A A A
Megavoltage X rays Radiation 6 MV X rays 100 cm SSD
Q M
A
3x3
4x4
6x6
8x8
10x10
12x12
15x15
20x20
25x25
15-42 1104 (0-35)
15-65 1126 (0-40)
16-35 1143 (0-32)
16-83 1167 (0-29)
17-27 1-179 (0-25)
17-75 1-178 (0-20)
18-39 1-182 (0-16)
19-28 1-183 (0-17)
19-78 1191 (0-14)
6 MV X rays Infinite SSD
Q M
20-57 1-167 (0-40)
21-03 1193 (0-42)
21-90 1-245 (0-39)
22-89 1-270 (0-41)
23-41 1-304 (0-35)
24-07 1-323 (0-33)
24-95 1-348 (0-20)
26-41 1-366 (0-15)
27-28 1-390 (0-13)
8 MV X rays 100 cm SSD
Q M
16-71 1167 (0-40)
17-06 1161 (0-40)
17-65 1165 (0-30)
18-32 1:160 (0-24)
18-93 1-159 (0-29)
19-49 1-154 (0-19)
20 06 1-157 (0-18)
20-66 1-155 (0-19)
21-17 1-147 (0-35)
8 MV X rays Infinite SSD
Q M
22-65 1-270 (0-50)
23-21 1-272 (0-45)
24-23 1-282 (0-38)
25-16 1-297 (0-27)
26-23 1-303 (0-23)
27-15 1-311 (0-20)
28-26 1-316 (0-16)
29-65 1-312 (0-22)
30-62 1-309 (0-31)
A
A A
TABLE VI TISSUE AIR RATIOS FOR
60
CO RADIATION
Values of the constants Q and M in the equation T = S [1 - ( l - e x p ( - ( y - 0 - 5 ) / Q ) ) M ] Field (cm) 5x5 7x7 10x10 12x12 15x15
Q*
M*
15-98 16-55 17-46 18-00 18-85
1-216 1-270 1-326 1-347 1-365
A (0-23) (0-18) (0-16) (0-15) (0-18)
Q+
M+
15-99 16-56 17-43 18-00 18-85
1-215 1-271 1-326 1-347 1-366
Q* and M* are the values which best fit the individual depth doses measured on an A.E.C.L. Theratron F Unit. Q"1" and M + are calculated from equations (4) and (5). A denotes the root mean square value of the differences between measured dose values and those computed using Q* and M* in equation (3). 744
SEPTEMBER
1975
Empirical equations for the representation of depth-dose data for computerized treatment planning
data in The British Journal of Radiology, Supplement 11, are given in Tables IV and V in the sections labelled "infinite SSD". At the British Columbia Cancer Institute we use tissue-air ratios, not only for the evaluation of dose when we use constant source-axis techniques, but also in determining corrections for inhomogeneities, such as lung (Batho, 1964; Young, 1967). Therefore, we need to be able to evaluate TAR for any field size. To do this, Q, M and S are expressed as functions of the equivalent radius (R). For our own data (measured on an AECL Theratron F machine) Q = 14-55+0-5144R . . . (4) M = l + 0-3845(l-exp(-0-2378Ri- 2 ))(5) S = l+0-1068 (1-exp (-0-1084 R)) . (6) PART II
Off-axis doses
We have used two different approaches in determining the dose at points other than those on the central axis:
(1) a single function of the off-axis distance (called the off-axis ratio) is used to relate the dose at all points at a given depth to the axial dose at that depth; (2) a more detailed treatment, using different equations in the main beam, in the penumbra and outside the beam. Off-axis ratios The off-axis ratio (OAR) (also called the offcentre ratio, the profile function, or the transverse distribution) is a function of the off-axis distance and parameters such as the depth, field-size and source distance, and may be quite complex in form. See, for example, the summaries given by Van de Geijn (1972) and by Weinkam, Kolde and Sterling (1973). A graph of off-axis ratio plotted against off-axis distance is reminiscent of the Fermi-Dirac distribution function of solid state physics. For 60Co radiation fields of medium size, defined by main
TABLE VII COMPARISON OF MEASURED* AND COMPUTED OFF-AXIS RATIOS
10 x 10 field
6 x 6 field Measured Dose y ~6 cm
y = 15 cm
y ~ 24 cm
Measured
Calc.
8
Calc.
8
OAR
OAR
(mm)
OAR
(mm)
— 0-2 0-6 0-6 0-6 0-4 0-2 01 01 0-2 0-2 0-2 —
1-00 0-850 0-768 0-692 0-582 0-500 0-398 0-309 0-250 0-200 0151 0-077 0062
— 0-2 01 0-6 0-2 0-5 0-3 0-4 0-2 0-2 0-2 2-5 —
—.
