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Linear Quadratic Model and Biologically Equivalent Dose for Single Fraction Treatments
Mdrcal Dosrmmy. Vol. 17, pp. 101-102 Printed
in the U.S.A.
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reserved.
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0739-0211/92 Association of Medical
1992 American
$5.00 + .oO Dosimetrists
LINEAR QUADRATIC MODEL AND BIOLOGICALLY EQUIVALENT DOSE FOR SINGLE FRACTION TREATMENTS SATISH
C.
PRASAD,
PH.D.
Radiation Physics Section, Department of Radiology, SUNY, Health Science Center, 750 East Adams Street, Syracuse, NY 13210 Abstract-The linear quadratic model has been used to calculate the biologically equivalent dose for single fraction treatments. Our calculations suggest that for late reacting tissue, such as the brain, a single fraction of 1440 cGy is equivalent to a conventional treatment of 5000 cGy in 25 fractions. Key Words: Biologically equivalent dose, Linear quadratic model. Fractionation.
where
INTRODUCIION
One of the useful purposes of biological models in radiotherapy is to be able to define biologically equivalent dose for different fractionation schedules. The nominal standard dose (NSD) and the time, dose, fractionation (TDF) tables are commonly used to compare treatments which differ in treatment time, number of fractions, and dose per fraction.’ Calculations based on the NSD and TDF formalism are, however, not valid when applied to late reacting tissues such as the brain, bone, and spinal cord. Moreover, with the advent of single fraction treatments, such as those used in radiosurgery, one needs biological models with broader applicability. The linear-quadratic (LQ) model has become popular lately.’ In the linear-quadratic (LQ) model, the biological effects of radiation are described by a ratio, a/& where (Yand p are constants for mammalian cell survival curves. The LQ model accounts for tissue type in addition to other fractionation parameters such as the dose per fraction and the total number of fractions. Two types of normal tissue are important in radiotherapy treatments. These are early reacting tissues (skin, gut, etc.) and late reacting tissues (brain, bone, etc.). The late reacting tissues are characterized by an average value of 3 Gy for the ratio a/@. In addition, the treatment time is not important for late reacting tissues.2 These simplifications make it possible to calculate biologically equivalent doses where unconventional fractionations are used. In this paper we calculate the biologically equivalent doses for single fraction treatments for late reacting tissues. METHOD
AND
MATERIAL
In the LQ model the biologically equivalent dose (BED) is defined by2: BED=nd
(1)
12= total number of fractions d = dose per fraction in Gy (~10 = is a constant depending on tissue type. Let us calculate the BED for a late reacting tissue and a conventional treatment of 5000 cGy in five weeks with a daily dose of 200 cGy. For such a treatment: n = 25, d = 2 Gy, al0 = 3 Gy
From eqn. ( 1): BED=25x2x(l+f) or, BED = 83.3 Gy A simple question that can be asked is, “What is the dose for a single treatment which will result in a BED of 83.3 Gy?” The answer to the above question can be obtained as follows. From Eqn. (1):
1 1
nd 1 + o
d
= 83.3 Gy
with n = I and (Y/P = 3 Gy. We get: $ + d - 83.3 = 0. Solving for d: d = 14.4Gy = 144OcGy.
The method can be used to calculate single fraction BED for fractionation patterns other than that considered above. RESULTS
AND
CONCLUSION
Single fraction doses that will result in biologically equivalent doses for conventional treatments
102
Medical Dosimetry
Table 1. Single fraction doses that result in biologically equivalent dose (BED) for conventional treatments consisting of 200 cGy per fraction and 5 fractions per week (results for late reacting tissue with a/P = 3 Gy) Conventional treatment
Single fraction dose
4000 cGy 4600 cGy 5000 cGy 5600 CGy 6000 cGy
1270 cGy I374 CGy 1440 CGy 1530 cGy 1590 cGy
have been calculated and summarized in Table 1. Conventional treatments in Table 1 are defined as treatments that used five fractions per week with 200 cGy per fraction. The results in Table 1 suggest that a single fraction dose of 1440 cGy is biologically equivalent to a treatment of 5000 cGy in 25 fractions. The single fraction BED can serve as a guide in treatments such as radiosurgery. The method described here can be used to calculate biologically equivalent dose for treatments where daily fraction dose is other than 200 cGy. It is worth pointing out that for late reacting tissue, such as brain, the incorporation of a treatment time factor in the LQ formalism is not important.2 Therefore, no time fac-
Volume 17, Number 2, 1992
tor has been incorporated in our calculations. Clinical data are required to support the validity of the BED for single fraction treatments reported here. Such data should be forthcoming as single fraction radiosurgery treatments become popular. The results in Table 1 should be used only as a guide in planning single fraction treatments. The effects of treatment volume and field size are important considerations that have not been taken into consideration in the present formalism. An LQ model with a volume dependence factor has to be developed before such effects can be incorporated. REFERENCES 1. Orton, C.G.; Ellis, F. A simplification in the use of the NSD
concept in practical radiotherapy. Br. J. Radiol. 46:529-537; 1973. 2. Fowler, J.F. What do we need to know to predict the effectiveness of fractionated radiotherapy schedules? In: Paliwal, B.R.; Fowler, J.F.; Herbert, D.E.; Kinsella, T.J.; Orton, C.G., editors. Prediction of responses in radiation therapy: The physical and biological basis. American Association of Physicists in Medicine Symposium Proceedings No. 7 (Part 1). New York: American Institute of Physics; 1989: l-24. 3. Flickinger, J.C.; &hell, M.C.; Larson, D.A. Estimation ofcomplications for linear accelerator radiosurgery with the integrated logistic formula. ht. J. Radiat. Oncol. Biol. Phys. 19~143-148; 1990.
in the U.S.A.
All rights
Copyright
reserved.
0
0739-0211/92 Association of Medical
1992 American
$5.00 + .oO Dosimetrists
LINEAR QUADRATIC MODEL AND BIOLOGICALLY EQUIVALENT DOSE FOR SINGLE FRACTION TREATMENTS SATISH
C.
PRASAD,
PH.D.
Radiation Physics Section, Department of Radiology, SUNY, Health Science Center, 750 East Adams Street, Syracuse, NY 13210 Abstract-The linear quadratic model has been used to calculate the biologically equivalent dose for single fraction treatments. Our calculations suggest that for late reacting tissue, such as the brain, a single fraction of 1440 cGy is equivalent to a conventional treatment of 5000 cGy in 25 fractions. Key Words: Biologically equivalent dose, Linear quadratic model. Fractionation.
where
INTRODUCIION
One of the useful purposes of biological models in radiotherapy is to be able to define biologically equivalent dose for different fractionation schedules. The nominal standard dose (NSD) and the time, dose, fractionation (TDF) tables are commonly used to compare treatments which differ in treatment time, number of fractions, and dose per fraction.’ Calculations based on the NSD and TDF formalism are, however, not valid when applied to late reacting tissues such as the brain, bone, and spinal cord. Moreover, with the advent of single fraction treatments, such as those used in radiosurgery, one needs biological models with broader applicability. The linear-quadratic (LQ) model has become popular lately.’ In the linear-quadratic (LQ) model, the biological effects of radiation are described by a ratio, a/& where (Yand p are constants for mammalian cell survival curves. The LQ model accounts for tissue type in addition to other fractionation parameters such as the dose per fraction and the total number of fractions. Two types of normal tissue are important in radiotherapy treatments. These are early reacting tissues (skin, gut, etc.) and late reacting tissues (brain, bone, etc.). The late reacting tissues are characterized by an average value of 3 Gy for the ratio a/@. In addition, the treatment time is not important for late reacting tissues.2 These simplifications make it possible to calculate biologically equivalent doses where unconventional fractionations are used. In this paper we calculate the biologically equivalent doses for single fraction treatments for late reacting tissues. METHOD
AND
MATERIAL
In the LQ model the biologically equivalent dose (BED) is defined by2: BED=nd
(1)
12= total number of fractions d = dose per fraction in Gy (~10 = is a constant depending on tissue type. Let us calculate the BED for a late reacting tissue and a conventional treatment of 5000 cGy in five weeks with a daily dose of 200 cGy. For such a treatment: n = 25, d = 2 Gy, al0 = 3 Gy
From eqn. ( 1): BED=25x2x(l+f) or, BED = 83.3 Gy A simple question that can be asked is, “What is the dose for a single treatment which will result in a BED of 83.3 Gy?” The answer to the above question can be obtained as follows. From Eqn. (1):
1 1
nd 1 + o
d
= 83.3 Gy
with n = I and (Y/P = 3 Gy. We get: $ + d - 83.3 = 0. Solving for d: d = 14.4Gy = 144OcGy.
The method can be used to calculate single fraction BED for fractionation patterns other than that considered above. RESULTS
AND
CONCLUSION
Single fraction doses that will result in biologically equivalent doses for conventional treatments
102
Medical Dosimetry
Table 1. Single fraction doses that result in biologically equivalent dose (BED) for conventional treatments consisting of 200 cGy per fraction and 5 fractions per week (results for late reacting tissue with a/P = 3 Gy) Conventional treatment
Single fraction dose
4000 cGy 4600 cGy 5000 cGy 5600 CGy 6000 cGy
1270 cGy I374 CGy 1440 CGy 1530 cGy 1590 cGy
have been calculated and summarized in Table 1. Conventional treatments in Table 1 are defined as treatments that used five fractions per week with 200 cGy per fraction. The results in Table 1 suggest that a single fraction dose of 1440 cGy is biologically equivalent to a treatment of 5000 cGy in 25 fractions. The single fraction BED can serve as a guide in treatments such as radiosurgery. The method described here can be used to calculate biologically equivalent dose for treatments where daily fraction dose is other than 200 cGy. It is worth pointing out that for late reacting tissue, such as brain, the incorporation of a treatment time factor in the LQ formalism is not important.2 Therefore, no time fac-
Volume 17, Number 2, 1992
tor has been incorporated in our calculations. Clinical data are required to support the validity of the BED for single fraction treatments reported here. Such data should be forthcoming as single fraction radiosurgery treatments become popular. The results in Table 1 should be used only as a guide in planning single fraction treatments. The effects of treatment volume and field size are important considerations that have not been taken into consideration in the present formalism. An LQ model with a volume dependence factor has to be developed before such effects can be incorporated. REFERENCES 1. Orton, C.G.; Ellis, F. A simplification in the use of the NSD
concept in practical radiotherapy. Br. J. Radiol. 46:529-537; 1973. 2. Fowler, J.F. What do we need to know to predict the effectiveness of fractionated radiotherapy schedules? In: Paliwal, B.R.; Fowler, J.F.; Herbert, D.E.; Kinsella, T.J.; Orton, C.G., editors. Prediction of responses in radiation therapy: The physical and biological basis. American Association of Physicists in Medicine Symposium Proceedings No. 7 (Part 1). New York: American Institute of Physics; 1989: l-24. 3. Flickinger, J.C.; &hell, M.C.; Larson, D.A. Estimation ofcomplications for linear accelerator radiosurgery with the integrated logistic formula. ht. J. Radiat. Oncol. Biol. Phys. 19~143-148; 1990.