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Tables of equivalent dose in 2 gy fractions: A simple application of the linear quadratic formula
Int. J. Radiation
Oncology
Pergamon
Biol.
Phys., Vol. 31. No. 2, pp. 371-378, 1995 Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0360-3016/95 $9.50 + .OO
0360-3016(94)E0126-5
l
Oncology Intelligence
TABLES
OF EQUIVALENT DOSE IN 2 GY FRACTIONS: A SIMPLE OF THE LINEAR QUADRATIC FORMULA MICHAEL
BARTON,
APPLICATION
F.R.A.C.R.
Division of Radiation Oncology, Westmead Hospital, Darcy St., Westmead, NSW 2 145, Australia Purpose: To provide a simple method of showing the equivalent dose in 2 Gy fractions for various treatment regimens using the linear quadratic (LQ) formula. Methods and Materials: Tables of equivalent dose for common total doses and fraction sizes have been calculated using the LQ formula for representative ~$3 ratios (1, 3, 10). Corrections have not been made for repopulation. Examples are given. Conclusion: The tables offer a simnle method of eauating one fractionation regimen with another, recognizing the limitation of this method. Linear quadratic
formula, Radiotherapy.
INTRODUCT
ION
Biologically
The linear quadratic (LQ) formula ( 1,4, 12) has replaced empirical systems such as nominal standard dose (NSD) and related formulas such as cumulative radiation effect (CRE) and time, dose and fractionation (TDF) because it is based on a mechanistic model of radiation effect on deoxyribonucleic acid (DNA) (3) and it better accounts for the differences between tumor and normal tissue reactions. Extensive evaluation with clinical, in vivo and in vitro data has confirmed its usefulness (9). Despite this, some radiation oncologists may find that the formula is difficult to use. This may be because the original equations describe a “biologically equivalent dose.” To be more understandable to the practicing clinician it is easier to think of equivalent doses as if they were given in a standard fraction size of 2 Gy. Using the LQ formula it is possible to equate doses of any fractionation regimen to an equivalent total dose in 2 Gy fractions. This paper provides simple tables to convert most fractionation schemes to an equivalent in 2 Gy fractions (Tables l-6). Theoretical discussion is kept to a minimum. The specific values of a:/3 have been chosen from the literature where there is good agreement (14). It must be remembered that, like the results of all models, these tables should be regarded only as a guide which may assist clinical decision making. METHODS
effective dose (BED) = and + pnd*, (Eq. 1)
where n is the number of fractions, d is the dose per fraction, (Yand p are coefficients of the effect of dose. One may divide throughout by p to get the following equation: BED = aJp.nd + nd2 = nd(crJP + d). P
0%. 2)
If two regimens (n,, cl,) and (n2, d2) have the same biological effect then the two corresponding expressions for BED//3 can be equated as follows: ntdt(4P
+ dJ = wb(dP
+ dd
0%. 3)
Therefore:
0%. 4) To find an equivalent total dose n2d2 in 2 Gy fractions, 2 is substituted for d2.
AND MATERIALS
This formula has been used to produce the accompanying tables (Tables 1, 2, 3, 4, 5 and 6). An ol:p of 1 has been assumed for spinal cord and neurological tissue, an (Y:/? of 3 for other late reacting tissues, and an (~$3 of 10 for
The theoretical basis for these calculations is well described by Yaes et al. (15). The classical LQ equation is as follows: Accepted for publication 24 February 1994. 371
1. J. Radiation Oncology 0 Biology 0 Physics
312
Table 1. Dose biologically
equivalent
Volume 3 I, Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
OC/@= 1
(Gy)
of fractions
1.O
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40
1 1 2 3 3 4 5 5 6 I 1 8 9 9 10 11 11 12 13 13 14 15 15 16 17 17 18 19 19 20 21 21 22 23 23 24 25 25 26 27
1 3 4 5 6 8 9 10 11 13 14 15 16 18 19 20 21 23 24 25 26 28 29 30 31 33 34 35 36 38 39 40 41 43 44 45 46 48 49 50
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
3 6 9 12 15 18 20 23 26 29 32 35 38 41 44 41 50 53 55 58 61 64 61 70 73 76 19 82
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 11 16 21 26 32 31 42 47 53 58 63 68 14 79 84
I 13 20 21 33 40 47 53 60 67 73 80
8 17 25 33 41 50 58 66 74 83
10 20 30 40 50 60 IO 80
14 28 42 56 IO 84
19 37 56 75 93
24 48 12 96
30 60 90
31 73 110
A simple application of the linear quadratic formula l M. BARTON
Table 2. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
373
CY/@= 3
(Gy)
of fractions
1.O
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
I 2 2 3 4 5 6 6 7 8 9 10 10 11 12 13 14 14 15 16 17 18 18 19 20 21 22 22 23 24 25 26 26 27 28 29 30 30 31 32
1 3 4 5 7 8 9 II 12 14 15 16 18 19 20 22 23 24 26 27 28 30 31 32 34 35 36 38 39 41 42 43 45 46 47 49 50 51 53 54
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
3 6 8 11 14 17 19 22 25 28 30 33 36 39 41 44 47 50 52 55 58 61 63 66 69 72 74 77 80 83
4 7 11 14 18 22 25 29 32 36 40 43 47 50 54 58 61 65 68 72 76 79 83
5 9 14 18 23 27 32 36 41 46 50 55 59 64 68 73 77 82
6 11 17 22 28 34 39 45 50 56 62 67 73 78 84
7 14 20 27 34 41 47 54 61 68 74 81
8 16 24 32 40 48 56 64 72 80
I1 22 32 43 54 65 76 86
14 28 42 56 70 84 98
18 35 53 70 88
22 43 65 86
26 52 78 104
314
I. J. Radiation Oncology 0 Biology 0 Physics
Table 3. Dose biologically
equivalent
Volume 31. Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
C-I/@= 10
(Gy)
of fractions
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1 2 3 4 5 6 6 7 8 9 10 11 12 13 14 15 16 17 17 18 19 20 21 22 23 24 25 26 27 28 28 29 30 31 32 33 34 35 36 37
1 3 4 6 7 9 10 12 13 14 16 17 19 20 22 23 24 26 27 29 30 32 33 35 36 37 39 40 42 43 45 46 47 49 50 52 53 55 56 58
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76
3 5 8 10 13 16 18 21 23 26 29 31 34 36 39 42 44 47 49 52 55 51 60 63 65 68 70 73 76 78 81
3 7 10 13 16 20 23 26 29 33 36 39 42 46 49 52 55 59 62 65 68 72 75 78 81
4 8 12 16 20 24 28 32 35 39 43 47 51 55 59 63 67 71 75 79 83
5 9 14 19 23 28 33 37 42 41 51 56 61 65 70 15 79 84
5 11 16 22 27 33 38 44 49 54 60 65 71 76 82
6 13 19 25 31 38 44 50 56 63 69 75 81
8 16 24 32 40 48 56 64 72 80
10 20 30 40 50 60 69 79 89
12 24 36 48 60 72 84
14 29 43 57 71 86
17 33 50 61 83
A simple application of the linear quadratic formula 0 M. BARTON
Table 4. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.0
1.5
2.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
1 3 4 5 7 8 9 11 12 13 15 16 17 19 20 21 23 24 25 27 28 29 31 32 33 35 36 37 39 40 41 43 44 45 47 48 49 51 52 53
2 3 5 7 8 10 12 13 15 17 18 20 22 23 25 27 28 30 32 33 35 37 38 40 42 43 45 47 48 50 52 53 55 57 58 60 62 63 65 67
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
375
2.5
3.0
3.5
4.0
5 7 9 12 14 16 19 21 23 26 28 30 33 35 37 40 42 44 47 49 51 54 56 58 61 63 65 68 70 72 75 77 79 82
5 8 11 13 16 19 21 24 27 29 32 35 37 40 43 45 48 51 53 56 59 61 64 67 69 72 75 77 80
6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
7 10 13 17 20 23 27 30 33 37 40 43 47 50 53 57 60 63 67 70 73 77 80
a/P = 1
(Gy) 4.5
5.0
6.0
11 15 18 22 26 29 33 37 40 44 48 51 55 59 62 66 70 73 77 81
12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
14 19 23 28 33 37 42 47 51 56 61 65 70 75 79 84
7.0
8.0
21 27 32 37 43 48 53 59 64 69 75 80
24 30 36 42 48 54 60 66 72 78 84
9.0
10.0
33 40 47 53 60 67 73 80
37 44 51 59 66 73 81
376
1. J. Radiation Oncology 0 Biology 0 Physics
Table 5. Dose biologically
equivalent
Volume 3 I, Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.0
1.5
2.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2 3 5 6 8 10 11 13 14 16 18 19 21 22 24 26 21 29 30 32 34 35 37 38 40 42 43 45 46 48 50 51 53 54 56 58 59 61 62 64
2 4 5 7 9 11 13 14 16 18 20 22 23 25 27 29 31 32 34 36 38 40 41 43 45 47 49 50 52 54 56 58 59 61 63 65 67 68 70 72
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2.5
3.0
3.5
4.0
4 7 9 11 13 15 18 20 22 24 26 29 31 33 35 37 40 42 44 46 48 51 53 55 57 59 62 64 66 68 70 73 75 77 79 81
5 7 10 12 14 17 19 22 24 26 29 31 34 36 38 41 43 46 48 50 53 55 58 60 62 65 67 70 72 74 77 79 82
5 8 10 13 16 18 21 23 26 29 31 34 36 39 42 44 47 49 52 55 57 60 62 65 68 70 73 75 78 81
6 8 11 14 17 20 22 25 28 31 34 36 39 42 45 48 50 53 56 59 62 64 67 70 73 76 78 81
u/B = 3
(Gy) 4.5
5.0
6.0
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
10 13 16 19 22 26 29 32 35 38 42 45 48 51 54 58 61 64 67 70 74 77 80
11 14 18 22 25 29 32 36 40 43 47 50 54 58 61 65 68 72 76 79 83
7.0
8.0
16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
18 22 26 31 35 40 44 48 53 57 62 66 70 75 79 84’
9.0
10.0
24 29 34 38 43 48 53 58 62 67 72 77 82
26 31 36 42 47 52 57 62 68 73 78 83
A simple application of the linear quadratic formula 0 M. BARTON
Table 6. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.O
1.5
2.0
2.5
3.0
3.5
4.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2 4 6 7 9 11 13 15 17 18 20 22 24 26 28 29 31 33 35 37 39 40 42 44 46 48 50 51 53 55 57 59 61 62 64 66 68 70 72 73
2 4 6 8 IO 12 13 15 17 19 21 23 25 27 29 31 33 35 36 38 40 42 44 46 48 50 52 54 56 58 59 61 63 65 67 69 71 73 75 77
2 4 6 8 IO 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
4 6 8 10 13 15 17 19 21 23 25 27 29 31 33 35 38 40 42 44 46 48 50 52 54 56 58 60 63 65 67 69 71 73 75 77 79 81
4 7 9 11 13 15 17 20 22 24 26 28 30 33 35 37 39 41 43 46 48 50 52 54 56 59 61 63 65 67 69 72 74 76 78 80
5 7 9 11 14 16 18 20 23 25 27 29 32 34 36 38 41 43 45 47 50 52 54 56 59 61 63 65 68 70 72 74 77 79 81
5 7 9 12 14 16 19 21 23 26 28 30 33 35 37 40 42 44 47 49 51 54 56 58 61 63 65 68 70 72 75 77 79 82
377
cy//I = IO
(Gy) 4.5
5.0
6.0
7.0
8.0
9.0
10.0
7 IO 12 15 17 19 22 24 27 29 31 34 36 39 41 44 46 48 51 53 56 58 60 63 65 68 70 73 75 77 80
8 IO 13 15 18 20 23 25 28 30 33 35 38 40 43 45 48 50 53 55 58 60 63 65 68 70 73 75 78 80
8 11 13 16 19 21 24 27 29 32 35 37 40 43 45 48 51 53 56 59 61 64 67 69 72 75 77 80
11 14 17 20 23 26 28 31 34 37 40 43 45 48 51 54 57 60 62 65 68 71 74 77 79 82
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
16 19 22 25 29 32 35 38 41 44 48 51 54 57 60 63 67 70 73 76 79 82
17 20 23 27 30 33 37 40 43 47 50 53 57 60 63 67 70 73 77 80
378
1.J. Radiation Oncology0 Biology0 Physics
tumors and acute reacting tissues (7, 11, 14). Doses converted to 2 Gy fractions may be added without further correction. No account has been taken of the effect of treatment time as this is still unclear. Experimental evidence is consistent with some recovery in late reacting tissues (8), however this has not been quantified for clinical application. A conservative approach is to assume no repair between courses of treatment by simply adding the biologically equivalent doses in 2 Gy fractions for late effects (a$ = 1 or 3). The reported corrections for squamous cancer of the head and neck (2,6, 13) and cervix (5) can be made directly to 2 Gy equivalent doses. Further correction factors are available in the literature ( 10, 13, 14). When multiple fractions per day are given, the effect of interfraction interval may be seen with late effects. Clinical evidence (9) shows that repair is not complete by 4.5 h but may be by 6 h. Hence late reactions may be enhanced above the calculated values if the interfraction interval is 5 4.5 h. Examples 1. A 75 year-old man with prostate cancer presents with thoracic spinal cord compression. He has had previous radiation therapy to the same area to a dose of 30 Gy in ten fractions. What is the equivalent total dose in 2 Gy fractions? a@ = 1 is used for neural tissue.
Volume 3 1, Number 2, 1995
(4 From Table 1 select row 10 from the “number of fractions” column and look across to the 3 Gy/fraction column. The total equivalent dose in 2 Gy fractions is given at the intersection, i.e., 40 Gy. Therefore, if spinal cord tolerance is, say, 50 Gy in 2 Gy fractions a further 10 Gy in 2 Gy fractions, or its equivalent, could be given. (b) Alternatively Table 4 can be used by referring to the “30” row in the “total dose column” and the 3 Gy column. Again the equivalent total dose is
40Gy. 2. A new accelerated fractionation treatment protocol is proposed for head and neck cancers. It is desired that 1.5 Gy should be given twice a day with a minimum of 6 h interfraction interval to allow for repair and that the total dose should give equivalent late effects to 50 Gy in 2 Gy fractions. a$ = 3 is used for late effects. (a) In Table 2 (a$ = 3) one refers to the 1.5 Gy column and looks for the value 50. From the “number of fractions” column it can be seen that 37 fractions of 1.5 Gy, that is, 56 Gy, is equivalent to 50 Gy in 2 Gy fractions for late effects. (b) Using Table 4 ((~$3 = 3), again refer to the 1.5 Gy column and find the value 50. From the “total dose” column the value 56 Gy is obtained.
REFERENCES I. Barendsen, G. W. Dose fractionation, dose rate and iso2.
3. 4. 5. 6.
7.
effect relationships for normal tissue responses. Int. J. Radiat. Oncol. Biol. Phys. 81981-1997; 1982. Barton, M. B.; Keane, T. J.; Gadalla, T.; Maki, E. The effect of treatment time and treatment interruption on tumour control following radical radiotherapy of laryngeal cancer. Radiother. Oncol. 23(3):137-143; 1992. Brenner, D. J.; Hall, E. J. The origins and basis of the linearquadratic model. Int. J. Radiat. Oncol. Biol. Phys. 23(l): 252; 1992. Douglas, B. G.; Fowler, J. F. Fractionation schedules and a quadratic dose-effect relationship. Br. J. Radiol. 48:502504; 1975. Fyles, A.; Keane, T.; Barton, M.; Simm, J. The effect of treatment duration in the local control of cervix cancer. Radiother Oncol. 25:273-279; 1992. Maciejewski, B.; Preuss-Bayer, G.; Trott, K. R. The influence of the number of fractions and of overall treatment time on the local control and late complication rate in squamous cell carcinoma of the larynx. Int. J. Radiat. Oncol. Biol. Phys. 9:321-328; 1983. Maciejewski, B.; Withers, H. R.; Taylor, J. M. G.; Hliniak, A. Dose fractionation and regeneration in radiotherapy for cancer of the oral cavity and oropharynx: tumour dose-response and repopulation. Int. J. Radiat. Oncol. Biol. Phys. 16:831-843; 1989.
8. Mason, K. A.; Withers, H. R.; Chiang, C. S. Late effects of radiation on the lumbar spinal cord of guinea pigs: retreatment tolerance. Int. J. Radiat. Oncol. Biol. Phys. 26:643648; 1993. 9. Thames, H. D.; Hendry, J. H. Fractionation in radiotherapy. London: Taylor & Francis; 1987. 10. Trott, K. R.; Kummermehr, J. The time factor and repopulation in tumors and normal tissues. Sem. Radiat. Oncol. 3(2):115-125; 1993. 11. Turesson, I.; Notter, G. The influence of fraction size on the late normal tissue reaction. I. Comparison of the effects of dailv and once-a-week fractionation on normal skin. Int. J. Radiat. Oncol. Biol. Phys. 10:593-598; 1984. 12. Withers, H. R.; Thames, H. D.; Peters, L. J. A new iso-
effectcurve for change in dose per fraction. Radiother. Oncol. 1:187-191; 1983. 13. Withers, H. R.; Taylor, J. M. G.; Maciejewski, B. The hazard of accelerated tumour clongen repopulation during radiotherapy. Acta Oncol. 27:98-l 13; 1988. 14. Withers, H. R. Some changes in concepts of dose fractionation over 20 years. Front. Radiat. Ther. Oncol. 22: l-l 3; 1988. 15. Yaes, R. J.; Patel, P.; Maruyama, Y. On using the linearquadratic model in daily clinical practice. Int. J. Radiat. Oncol. Biol. Phys. 20(6): 1353- 1362; 199 1.
Oncology
Pergamon
Biol.
Phys., Vol. 31. No. 2, pp. 371-378, 1995 Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved 0360-3016/95 $9.50 + .OO
0360-3016(94)E0126-5
l
Oncology Intelligence
TABLES
OF EQUIVALENT DOSE IN 2 GY FRACTIONS: A SIMPLE OF THE LINEAR QUADRATIC FORMULA MICHAEL
BARTON,
APPLICATION
F.R.A.C.R.
Division of Radiation Oncology, Westmead Hospital, Darcy St., Westmead, NSW 2 145, Australia Purpose: To provide a simple method of showing the equivalent dose in 2 Gy fractions for various treatment regimens using the linear quadratic (LQ) formula. Methods and Materials: Tables of equivalent dose for common total doses and fraction sizes have been calculated using the LQ formula for representative ~$3 ratios (1, 3, 10). Corrections have not been made for repopulation. Examples are given. Conclusion: The tables offer a simnle method of eauating one fractionation regimen with another, recognizing the limitation of this method. Linear quadratic
formula, Radiotherapy.
INTRODUCT
ION
Biologically
The linear quadratic (LQ) formula ( 1,4, 12) has replaced empirical systems such as nominal standard dose (NSD) and related formulas such as cumulative radiation effect (CRE) and time, dose and fractionation (TDF) because it is based on a mechanistic model of radiation effect on deoxyribonucleic acid (DNA) (3) and it better accounts for the differences between tumor and normal tissue reactions. Extensive evaluation with clinical, in vivo and in vitro data has confirmed its usefulness (9). Despite this, some radiation oncologists may find that the formula is difficult to use. This may be because the original equations describe a “biologically equivalent dose.” To be more understandable to the practicing clinician it is easier to think of equivalent doses as if they were given in a standard fraction size of 2 Gy. Using the LQ formula it is possible to equate doses of any fractionation regimen to an equivalent total dose in 2 Gy fractions. This paper provides simple tables to convert most fractionation schemes to an equivalent in 2 Gy fractions (Tables l-6). Theoretical discussion is kept to a minimum. The specific values of a:/3 have been chosen from the literature where there is good agreement (14). It must be remembered that, like the results of all models, these tables should be regarded only as a guide which may assist clinical decision making. METHODS
effective dose (BED) = and + pnd*, (Eq. 1)
where n is the number of fractions, d is the dose per fraction, (Yand p are coefficients of the effect of dose. One may divide throughout by p to get the following equation: BED = aJp.nd + nd2 = nd(crJP + d). P
0%. 2)
If two regimens (n,, cl,) and (n2, d2) have the same biological effect then the two corresponding expressions for BED//3 can be equated as follows: ntdt(4P
+ dJ = wb(dP
+ dd
0%. 3)
Therefore:
0%. 4) To find an equivalent total dose n2d2 in 2 Gy fractions, 2 is substituted for d2.
AND MATERIALS
This formula has been used to produce the accompanying tables (Tables 1, 2, 3, 4, 5 and 6). An ol:p of 1 has been assumed for spinal cord and neurological tissue, an (Y:/? of 3 for other late reacting tissues, and an (~$3 of 10 for
The theoretical basis for these calculations is well described by Yaes et al. (15). The classical LQ equation is as follows: Accepted for publication 24 February 1994. 371
1. J. Radiation Oncology 0 Biology 0 Physics
312
Table 1. Dose biologically
equivalent
Volume 3 I, Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
OC/@= 1
(Gy)
of fractions
1.O
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 21 28 29 30 31 32 33 34 35 36 31 38 39 40
1 1 2 3 3 4 5 5 6 I 1 8 9 9 10 11 11 12 13 13 14 15 15 16 17 17 18 19 19 20 21 21 22 23 23 24 25 25 26 27
1 3 4 5 6 8 9 10 11 13 14 15 16 18 19 20 21 23 24 25 26 28 29 30 31 33 34 35 36 38 39 40 41 43 44 45 46 48 49 50
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
3 6 9 12 15 18 20 23 26 29 32 35 38 41 44 41 50 53 55 58 61 64 61 70 73 76 19 82
4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 11 16 21 26 32 31 42 47 53 58 63 68 14 79 84
I 13 20 21 33 40 47 53 60 67 73 80
8 17 25 33 41 50 58 66 74 83
10 20 30 40 50 60 IO 80
14 28 42 56 IO 84
19 37 56 75 93
24 48 12 96
30 60 90
31 73 110
A simple application of the linear quadratic formula l M. BARTON
Table 2. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
373
CY/@= 3
(Gy)
of fractions
1.O
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
I 2 2 3 4 5 6 6 7 8 9 10 10 11 12 13 14 14 15 16 17 18 18 19 20 21 22 22 23 24 25 26 26 27 28 29 30 30 31 32
1 3 4 5 7 8 9 II 12 14 15 16 18 19 20 22 23 24 26 27 28 30 31 32 34 35 36 38 39 41 42 43 45 46 47 49 50 51 53 54
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
3 6 8 11 14 17 19 22 25 28 30 33 36 39 41 44 47 50 52 55 58 61 63 66 69 72 74 77 80 83
4 7 11 14 18 22 25 29 32 36 40 43 47 50 54 58 61 65 68 72 76 79 83
5 9 14 18 23 27 32 36 41 46 50 55 59 64 68 73 77 82
6 11 17 22 28 34 39 45 50 56 62 67 73 78 84
7 14 20 27 34 41 47 54 61 68 74 81
8 16 24 32 40 48 56 64 72 80
I1 22 32 43 54 65 76 86
14 28 42 56 70 84 98
18 35 53 70 88
22 43 65 86
26 52 78 104
314
I. J. Radiation Oncology 0 Biology 0 Physics
Table 3. Dose biologically
equivalent
Volume 31. Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Number
C-I/@= 10
(Gy)
of fractions
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6.0
7.0
8.0
9.0
10.0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
1 2 3 4 5 6 6 7 8 9 10 11 12 13 14 15 16 17 17 18 19 20 21 22 23 24 25 26 27 28 28 29 30 31 32 33 34 35 36 37
1 3 4 6 7 9 10 12 13 14 16 17 19 20 22 23 24 26 27 29 30 32 33 35 36 37 39 40 42 43 45 46 47 49 50 52 53 55 56 58
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76
3 5 8 10 13 16 18 21 23 26 29 31 34 36 39 42 44 47 49 52 55 51 60 63 65 68 70 73 76 78 81
3 7 10 13 16 20 23 26 29 33 36 39 42 46 49 52 55 59 62 65 68 72 75 78 81
4 8 12 16 20 24 28 32 35 39 43 47 51 55 59 63 67 71 75 79 83
5 9 14 19 23 28 33 37 42 41 51 56 61 65 70 15 79 84
5 11 16 22 27 33 38 44 49 54 60 65 71 76 82
6 13 19 25 31 38 44 50 56 63 69 75 81
8 16 24 32 40 48 56 64 72 80
10 20 30 40 50 60 69 79 89
12 24 36 48 60 72 84
14 29 43 57 71 86
17 33 50 61 83
A simple application of the linear quadratic formula 0 M. BARTON
Table 4. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.0
1.5
2.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
1 3 4 5 7 8 9 11 12 13 15 16 17 19 20 21 23 24 25 27 28 29 31 32 33 35 36 37 39 40 41 43 44 45 47 48 49 51 52 53
2 3 5 7 8 10 12 13 15 17 18 20 22 23 25 27 28 30 32 33 35 37 38 40 42 43 45 47 48 50 52 53 55 57 58 60 62 63 65 67
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
375
2.5
3.0
3.5
4.0
5 7 9 12 14 16 19 21 23 26 28 30 33 35 37 40 42 44 47 49 51 54 56 58 61 63 65 68 70 72 75 77 79 82
5 8 11 13 16 19 21 24 27 29 32 35 37 40 43 45 48 51 53 56 59 61 64 67 69 72 75 77 80
6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
7 10 13 17 20 23 27 30 33 37 40 43 47 50 53 57 60 63 67 70 73 77 80
a/P = 1
(Gy) 4.5
5.0
6.0
11 15 18 22 26 29 33 37 40 44 48 51 55 59 62 66 70 73 77 81
12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
14 19 23 28 33 37 42 47 51 56 61 65 70 75 79 84
7.0
8.0
21 27 32 37 43 48 53 59 64 69 75 80
24 30 36 42 48 54 60 66 72 78 84
9.0
10.0
33 40 47 53 60 67 73 80
37 44 51 59 66 73 81
376
1. J. Radiation Oncology 0 Biology 0 Physics
Table 5. Dose biologically
equivalent
Volume 3 I, Number 2, 1995
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.0
1.5
2.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2 3 5 6 8 10 11 13 14 16 18 19 21 22 24 26 21 29 30 32 34 35 37 38 40 42 43 45 46 48 50 51 53 54 56 58 59 61 62 64
2 4 5 7 9 11 13 14 16 18 20 22 23 25 27 29 31 32 34 36 38 40 41 43 45 47 49 50 52 54 56 58 59 61 63 65 67 68 70 72
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2.5
3.0
3.5
4.0
4 7 9 11 13 15 18 20 22 24 26 29 31 33 35 37 40 42 44 46 48 51 53 55 57 59 62 64 66 68 70 73 75 77 79 81
5 7 10 12 14 17 19 22 24 26 29 31 34 36 38 41 43 46 48 50 53 55 58 60 62 65 67 70 72 74 77 79 82
5 8 10 13 16 18 21 23 26 29 31 34 36 39 42 44 47 49 52 55 57 60 62 65 68 70 73 75 78 81
6 8 11 14 17 20 22 25 28 31 34 36 39 42 45 48 50 53 56 59 62 64 67 70 73 76 78 81
u/B = 3
(Gy) 4.5
5.0
6.0
9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
10 13 16 19 22 26 29 32 35 38 42 45 48 51 54 58 61 64 67 70 74 77 80
11 14 18 22 25 29 32 36 40 43 47 50 54 58 61 65 68 72 76 79 83
7.0
8.0
16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
18 22 26 31 35 40 44 48 53 57 62 66 70 75 79 84’
9.0
10.0
24 29 34 38 43 48 53 58 62 67 72 77 82
26 31 36 42 47 52 57 62 68 73 78 83
A simple application of the linear quadratic formula 0 M. BARTON
Table 6. Dose biologically
equivalent
to 2 Gy fractions; (no repopulation) Dose/fraction
Total dose (Gy)
1.O
1.5
2.0
2.5
3.0
3.5
4.0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
2 4 6 7 9 11 13 15 17 18 20 22 24 26 28 29 31 33 35 37 39 40 42 44 46 48 50 51 53 55 57 59 61 62 64 66 68 70 72 73
2 4 6 8 IO 12 13 15 17 19 21 23 25 27 29 31 33 35 36 38 40 42 44 46 48 50 52 54 56 58 59 61 63 65 67 69 71 73 75 77
2 4 6 8 IO 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80
4 6 8 10 13 15 17 19 21 23 25 27 29 31 33 35 38 40 42 44 46 48 50 52 54 56 58 60 63 65 67 69 71 73 75 77 79 81
4 7 9 11 13 15 17 20 22 24 26 28 30 33 35 37 39 41 43 46 48 50 52 54 56 59 61 63 65 67 69 72 74 76 78 80
5 7 9 11 14 16 18 20 23 25 27 29 32 34 36 38 41 43 45 47 50 52 54 56 59 61 63 65 68 70 72 74 77 79 81
5 7 9 12 14 16 19 21 23 26 28 30 33 35 37 40 42 44 47 49 51 54 56 58 61 63 65 68 70 72 75 77 79 82
377
cy//I = IO
(Gy) 4.5
5.0
6.0
7.0
8.0
9.0
10.0
7 IO 12 15 17 19 22 24 27 29 31 34 36 39 41 44 46 48 51 53 56 58 60 63 65 68 70 73 75 77 80
8 IO 13 15 18 20 23 25 28 30 33 35 38 40 43 45 48 50 53 55 58 60 63 65 68 70 73 75 78 80
8 11 13 16 19 21 24 27 29 32 35 37 40 43 45 48 51 53 56 59 61 64 67 69 72 75 77 80
11 14 17 20 23 26 28 31 34 37 40 43 45 48 51 54 57 60 62 65 68 71 74 77 79 82
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81
16 19 22 25 29 32 35 38 41 44 48 51 54 57 60 63 67 70 73 76 79 82
17 20 23 27 30 33 37 40 43 47 50 53 57 60 63 67 70 73 77 80
378
1.J. Radiation Oncology0 Biology0 Physics
tumors and acute reacting tissues (7, 11, 14). Doses converted to 2 Gy fractions may be added without further correction. No account has been taken of the effect of treatment time as this is still unclear. Experimental evidence is consistent with some recovery in late reacting tissues (8), however this has not been quantified for clinical application. A conservative approach is to assume no repair between courses of treatment by simply adding the biologically equivalent doses in 2 Gy fractions for late effects (a$ = 1 or 3). The reported corrections for squamous cancer of the head and neck (2,6, 13) and cervix (5) can be made directly to 2 Gy equivalent doses. Further correction factors are available in the literature ( 10, 13, 14). When multiple fractions per day are given, the effect of interfraction interval may be seen with late effects. Clinical evidence (9) shows that repair is not complete by 4.5 h but may be by 6 h. Hence late reactions may be enhanced above the calculated values if the interfraction interval is 5 4.5 h. Examples 1. A 75 year-old man with prostate cancer presents with thoracic spinal cord compression. He has had previous radiation therapy to the same area to a dose of 30 Gy in ten fractions. What is the equivalent total dose in 2 Gy fractions? a@ = 1 is used for neural tissue.
Volume 3 1, Number 2, 1995
(4 From Table 1 select row 10 from the “number of fractions” column and look across to the 3 Gy/fraction column. The total equivalent dose in 2 Gy fractions is given at the intersection, i.e., 40 Gy. Therefore, if spinal cord tolerance is, say, 50 Gy in 2 Gy fractions a further 10 Gy in 2 Gy fractions, or its equivalent, could be given. (b) Alternatively Table 4 can be used by referring to the “30” row in the “total dose column” and the 3 Gy column. Again the equivalent total dose is
40Gy. 2. A new accelerated fractionation treatment protocol is proposed for head and neck cancers. It is desired that 1.5 Gy should be given twice a day with a minimum of 6 h interfraction interval to allow for repair and that the total dose should give equivalent late effects to 50 Gy in 2 Gy fractions. a$ = 3 is used for late effects. (a) In Table 2 (a$ = 3) one refers to the 1.5 Gy column and looks for the value 50. From the “number of fractions” column it can be seen that 37 fractions of 1.5 Gy, that is, 56 Gy, is equivalent to 50 Gy in 2 Gy fractions for late effects. (b) Using Table 4 ((~$3 = 3), again refer to the 1.5 Gy column and find the value 50. From the “total dose” column the value 56 Gy is obtained.
REFERENCES I. Barendsen, G. W. Dose fractionation, dose rate and iso2.
3. 4. 5. 6.
7.
effect relationships for normal tissue responses. Int. J. Radiat. Oncol. Biol. Phys. 81981-1997; 1982. Barton, M. B.; Keane, T. J.; Gadalla, T.; Maki, E. The effect of treatment time and treatment interruption on tumour control following radical radiotherapy of laryngeal cancer. Radiother. Oncol. 23(3):137-143; 1992. Brenner, D. J.; Hall, E. J. The origins and basis of the linearquadratic model. Int. J. Radiat. Oncol. Biol. Phys. 23(l): 252; 1992. Douglas, B. G.; Fowler, J. F. Fractionation schedules and a quadratic dose-effect relationship. Br. J. Radiol. 48:502504; 1975. Fyles, A.; Keane, T.; Barton, M.; Simm, J. The effect of treatment duration in the local control of cervix cancer. Radiother Oncol. 25:273-279; 1992. Maciejewski, B.; Preuss-Bayer, G.; Trott, K. R. The influence of the number of fractions and of overall treatment time on the local control and late complication rate in squamous cell carcinoma of the larynx. Int. J. Radiat. Oncol. Biol. Phys. 9:321-328; 1983. Maciejewski, B.; Withers, H. R.; Taylor, J. M. G.; Hliniak, A. Dose fractionation and regeneration in radiotherapy for cancer of the oral cavity and oropharynx: tumour dose-response and repopulation. Int. J. Radiat. Oncol. Biol. Phys. 16:831-843; 1989.
8. Mason, K. A.; Withers, H. R.; Chiang, C. S. Late effects of radiation on the lumbar spinal cord of guinea pigs: retreatment tolerance. Int. J. Radiat. Oncol. Biol. Phys. 26:643648; 1993. 9. Thames, H. D.; Hendry, J. H. Fractionation in radiotherapy. London: Taylor & Francis; 1987. 10. Trott, K. R.; Kummermehr, J. The time factor and repopulation in tumors and normal tissues. Sem. Radiat. Oncol. 3(2):115-125; 1993. 11. Turesson, I.; Notter, G. The influence of fraction size on the late normal tissue reaction. I. Comparison of the effects of dailv and once-a-week fractionation on normal skin. Int. J. Radiat. Oncol. Biol. Phys. 10:593-598; 1984. 12. Withers, H. R.; Thames, H. D.; Peters, L. J. A new iso-
effectcurve for change in dose per fraction. Radiother. Oncol. 1:187-191; 1983. 13. Withers, H. R.; Taylor, J. M. G.; Maciejewski, B. The hazard of accelerated tumour clongen repopulation during radiotherapy. Acta Oncol. 27:98-l 13; 1988. 14. Withers, H. R. Some changes in concepts of dose fractionation over 20 years. Front. Radiat. Ther. Oncol. 22: l-l 3; 1988. 15. Yaes, R. J.; Patel, P.; Maruyama, Y. On using the linearquadratic model in daily clinical practice. Int. J. Radiat. Oncol. Biol. Phys. 20(6): 1353- 1362; 199 1.