221 R A D I O N 01073 E V A L U A T I O N O F B I O L O G I C A L E F F E C T IN R A D I O T H E R APY B I O L O G I C A L E Q U I V A L E N T D O S E V E R S U S SURVIVING FRACTION CRITERIA
To the Editors, According to the LQ model, the biological effect is measured by E = - lnSF, the logarithm of the surviving fraction. When applying this model to the calculation o f equivalent fractionation schedules for the same tissue, it is indifferent to take either E or E/a as the relevant magnitude, because the same value o f a is assumed. However, some authors, for instance Scalliet [ 1 ], when converting physical dose to biological equivalent dose, use the concept of normalized dose by applying the LQ model to different tissues, using the "standard" fraction size o f 2 Gy. This, as will be shown, leads to results which do not represent the relative biological effect o f the irradiation treatment, for different tissues, to which correspond different values of the physical dose and biological parameters. By applying the LQ model, it is possible to make a quantitative analysis, which will show the degree o f distortion introduced by the current method of normalization and its dependency on various parameters. If we take E = - l n S F , as a measure of biological effect, considering two tissues 1 and 2, treated with n fractions, and having biological parameters, respectively al, (a/b)l and a2, (a/b)2, we may write:
E1 : nql d t [ l +
qldt/(a/b) 1] a l
al (a/b)2 [(a/b)l + d] k l / k 2 : a2 (a/b)l [(a/b)2 + d] The last expression shows that if al = a2 and (a/b)l = (a/b)2; then k l = k2, for whatever value of d. In general, however, because tissues 1 and 2 have difl?rent values for these parameters, k2 is different from kl, and the degree of distortion introduced by the normalization procedure depends not only on the relative values of these parameters but also on the value of d, the fraction dose size taken as "standard". In principle, there is no particular reason for considering the value of d = 2 Gy, as the standard value, except, of course, that it is a very c o m m o n value of tumor dose fraction size in current practice. Other values for d could in principle be used, as for instance d = 0, which would correspond to taking E/a as equivalence criterion. To illustrate this distortion effect let us take, as a numerical example, the case presented by Scalliet et al. [1 ]:
For tissue 1 (spinal cord) we have: ql = 0.75 (75 }o ); (a/b)l = 2 Gy; al = 0.093 Gy ( - 1 ) . For tissue 2 (skin): q2 = 0.5 (505o); (a/b)2 = 10 Gy; a2 = 0.033 Gy (-1). We have taken for al and a2 the tentative values presented in Table 3.2 of the book by Thames and Hendry, "Fractionation in Radiotherapy" [2]. Introducing these values in expressions 1 and 2 and making d = dt = 2 Gy, we obtain:
E 2 = nq2 art [ 1 + q2dt/(a/b) 2] a2 kl = 6.7 and k2 = 1.43 where q l. 100 °Jo and q2.100 %, are the percentage physical dose values pertaining to tissues 1 and 2 dt is the tumor fraction size. From the above expression we may calculate:
ql al (a/b)2 [(a/b)l + qldt] k l = E l ~ E 2 = q2 a2 (a/b)l [(a/b)2 + q2dt]
(1)
If we now follow the current normalization procedure, using the equivalent dose criterion and taking d as the "standard" dose fraction, we have:
For tissue 1: nq l dt [ 1 + (q 1dt/(a/b) 1)] = D leq [ 1 + (d/(a/b) 1 ] For tissue 2." nq2 dt [ 1 + (q2dt/(a/b)2)] = O2eq [ 1 + (d/(a/b)2].
In terms o f percentages these values represent 15 and 70~o, respectively. According to the above results, there is almost a 5-fold distortion factor when using the current normalization procedure, instead of the surviving fraction criterion. Obviously not too much significance should be placed on these figures, because of the well known unreliability of the al and a2 values. We think, however that this unreliability should not serve as an excuse for disregarding the distortion effects which will most surely be introduced by adopting the current normalization procedure. Sincerely,
From these expression we have: F. Pulido Valente (received January 1992, accepted July 1992)
ql [(a/b)2 + d] [(a/b)l + qldt] k2 = Dleq/D2eq = q2 [(a/b)l + d] [(a/b)2 + q2dt]
(2)
From Eqns. (1) and (2) it results:
Av. das Tulipas It 10, 2esq. Mmiraflores - Algds 1495 Lisboa, Portugal
References 1 Scalliet, P. et al. Application of the LQ model to the interpretation of absorbed dose distribution in the daily practice of radiotherapy. Radiother. Oncol., 22: 180-189, 1991.
2 Thames, H. D. and Hendry, J. H. & Francis (1987).
Fractionation in Radiotherapy. Taylor
To the Editors, According to the LQ model, the biological effect is measured by E = - lnSF, the logarithm of the surviving fraction. When applying this model to the calculation o f equivalent fractionation schedules for the same tissue, it is indifferent to take either E or E/a as the relevant magnitude, because the same value o f a is assumed. However, some authors, for instance Scalliet [ 1 ], when converting physical dose to biological equivalent dose, use the concept of normalized dose by applying the LQ model to different tissues, using the "standard" fraction size o f 2 Gy. This, as will be shown, leads to results which do not represent the relative biological effect o f the irradiation treatment, for different tissues, to which correspond different values of the physical dose and biological parameters. By applying the LQ model, it is possible to make a quantitative analysis, which will show the degree o f distortion introduced by the current method of normalization and its dependency on various parameters. If we take E = - l n S F , as a measure of biological effect, considering two tissues 1 and 2, treated with n fractions, and having biological parameters, respectively al, (a/b)l and a2, (a/b)2, we may write:
E1 : nql d t [ l +
qldt/(a/b) 1] a l
al (a/b)2 [(a/b)l + d] k l / k 2 : a2 (a/b)l [(a/b)2 + d] The last expression shows that if al = a2 and (a/b)l = (a/b)2; then k l = k2, for whatever value of d. In general, however, because tissues 1 and 2 have difl?rent values for these parameters, k2 is different from kl, and the degree of distortion introduced by the normalization procedure depends not only on the relative values of these parameters but also on the value of d, the fraction dose size taken as "standard". In principle, there is no particular reason for considering the value of d = 2 Gy, as the standard value, except, of course, that it is a very c o m m o n value of tumor dose fraction size in current practice. Other values for d could in principle be used, as for instance d = 0, which would correspond to taking E/a as equivalence criterion. To illustrate this distortion effect let us take, as a numerical example, the case presented by Scalliet et al. [1 ]:
For tissue 1 (spinal cord) we have: ql = 0.75 (75 }o ); (a/b)l = 2 Gy; al = 0.093 Gy ( - 1 ) . For tissue 2 (skin): q2 = 0.5 (505o); (a/b)2 = 10 Gy; a2 = 0.033 Gy (-1). We have taken for al and a2 the tentative values presented in Table 3.2 of the book by Thames and Hendry, "Fractionation in Radiotherapy" [2]. Introducing these values in expressions 1 and 2 and making d = dt = 2 Gy, we obtain:
E 2 = nq2 art [ 1 + q2dt/(a/b) 2] a2 kl = 6.7 and k2 = 1.43 where q l. 100 °Jo and q2.100 %, are the percentage physical dose values pertaining to tissues 1 and 2 dt is the tumor fraction size. From the above expression we may calculate:
ql al (a/b)2 [(a/b)l + qldt] k l = E l ~ E 2 = q2 a2 (a/b)l [(a/b)2 + q2dt]
(1)
If we now follow the current normalization procedure, using the equivalent dose criterion and taking d as the "standard" dose fraction, we have:
For tissue 1: nq l dt [ 1 + (q 1dt/(a/b) 1)] = D leq [ 1 + (d/(a/b) 1 ] For tissue 2." nq2 dt [ 1 + (q2dt/(a/b)2)] = O2eq [ 1 + (d/(a/b)2].
In terms o f percentages these values represent 15 and 70~o, respectively. According to the above results, there is almost a 5-fold distortion factor when using the current normalization procedure, instead of the surviving fraction criterion. Obviously not too much significance should be placed on these figures, because of the well known unreliability of the al and a2 values. We think, however that this unreliability should not serve as an excuse for disregarding the distortion effects which will most surely be introduced by adopting the current normalization procedure. Sincerely,
From these expression we have: F. Pulido Valente (received January 1992, accepted July 1992)
ql [(a/b)2 + d] [(a/b)l + qldt] k2 = Dleq/D2eq = q2 [(a/b)l + d] [(a/b)2 + q2dt]
(2)
From Eqns. (1) and (2) it results:
Av. das Tulipas It 10, 2esq. Mmiraflores - Algds 1495 Lisboa, Portugal
References 1 Scalliet, P. et al. Application of the LQ model to the interpretation of absorbed dose distribution in the daily practice of radiotherapy. Radiother. Oncol., 22: 180-189, 1991.
2 Thames, H. D. and Hendry, J. H. & Francis (1987).
Fractionation in Radiotherapy. Taylor
