Rep Pract Oncol Radiother, 2006; 11(2): 91-95 Review Paper Received: 2005.07.18 Accepted: 2006.01.24 Published: 2006.04.28 Isoeffect calculations based on linear quadratic equations for head and neck cancers Authors’ Contribution: A Study Design B Data Collection C Statistical Analysis D Data Interpretation E Manuscript Preparation F Literature Search G Funds Collection S. Manimaran1 ABCDEF, V. Ramasubramanian1 BCDE, K. Thayalan2 CDEF 1 2 Department of Physics, Vellore Institute of Technology – Deemed University, Vellore, India Department of Radiotherapy, Bernard Institute of Radiology & Oncology. Govt. General Hospital, Madras Medical College, Chennai, India Summary Background The linear quadratic model has led to various methods for the calculation of isoeffect relationships in radiotherapy. In this model, the tissue sensitive parameters a and b usually appear as a ratio, a/b. These parameters are used to describe the response of normal tissues to radiation insult. Different radiation induced biological end points in speciﬁc tissues and organs are associated with the characteristics of the a/b ratio. The linear quadratic model has been used clinically to address questions relating to changes in fractions in treatment schedules. Aim The process of treating cancer with ionizing radiation is complex and subject to dosimetric errors which may potentially result in early or late complications. Our objective was to correct such errors through the application of the incomplete repair linear quadratic model. Materials/Methods Repair mechanisms are affected if, owing to dosimetric error, excess dose is delivered in single or multiple fractions. Corrections for such errors were simulated, for different clinical situations, in order to avoid late ﬁbrosis in head and neck cancers. Results NSD, CRE, and TDF approach could not predict, onset of proliferation, overall treatment time, late and early complications, but linear quadratic model calculations predicts isoeffective schedules successfully with above parameters. Conclusions In head and neck cancers, a number of parameters inﬂuence the results of treatment. Isoeffect calculations show the risk factors responsible for ﬁbrosis and spinal cord damage and therefore may be used to calculate dose reductions for all remaining fractions, rather than applying shielding. Key words Full-text PDF: Word count: Tables: Figures: References: Author’s address: LQ-linear quadratic model • biological effective dose • ﬁbrosis • head and neck cancers http:/www.rpor.pl/pdf.php?MAN=8817 1761 1 — 16 S. Manimaran, Lecturer, Department of Physics, Vellore Institute of Technology, Deemed University Vellore, Tamil Nadu – 632 014, India, e-mail: [email protected] 91 Review Paper Rep Pract Oncol Radiother, 2006; 11(2): 91-95 BACKGROUND The linear quadratic model has led to various methods for the calculation of isoeffect relationships in radiotherapy. In this model, the tissue sensitive parameters a and b usually appear as a ratio, a/b. These parameters are used to describe the response of normal tissues to radiation insult. Different radiation induced biological end points in speciﬁc tissues and organs are associated with the characteristics of the a/b ratio. The linear quadratic model has been used clinically to address questions relating to changes in fractions in treatment schedules. The linear quadratic model can be used to calculate the biologically effective dose, which makes two schedules equivalent to a particular biological end point. When multiple fractions are given each day, the repair processes arising from one radiation dose may not be complete as the half time for repair is relatively long, in comparison to the time interval between fractions. Incomplete repair tends to reduce the isoeffective dose and corrections must be made for the consequent loss of tolerance. In this article, isoeffect calculations are made based on the BED concept. AIM [] E d =D 1+ a a b (2) = Extrapolated Response Dose (ERD) where S – surviving fraction, a – coefﬁcient of linear term (which determines the initial slope of the survival curve(Gy–1), b – coefﬁcient of quadratic term (which determines the shape of the shoulder of the survival curve (Gy–2), D – total dose delivered (Gy). In order to account for the loss of dose due to repopulation, Orton [3] introduced a correction to ERD, termed the BED equation as follows, BED = D [1+d/(a/b)] – K(T–To) (3) Where K is the dose required per day to counteract proliferation, T is the overall treatment time and To is the onset time for proliferation. BED is a measure of the effect [4] of a course of fractionated or continuous irradiation and has units of dose, usually expressed in grays. The aim of this present study is to calculate Biological Effective Dose in order to predict late ﬁbrosis in head and neck cancers, taking into account re-population corrections for normal cell proliferation in different clinical situations. The TE formulation is conceptually similar and has also been used in published literature [5]. In this case, we divide E by b rather than a to get. INTRODUCTION The units of Total Effect are gray2, making the results less convenient than BED. The linear quadratic model describes a wide range of fractionation schedules that are iso-effective [1]. To apply this method we must ﬁrst have a particular desired end point. The validity of the linear quadratic approach to fractionation depends principally on its ability to predict isoeffective schedules successfully [2]. There is an implicit assumption, that the isoeffect, has a direct relationship with a certain level of cell survival. Generally, the fraction of surviving cells associated with an isoeffect is unknown and it is customary to work in terms tissue effect levels, which we denote as E. Effect (E) = -logeS = D(a +bd) (1) Dividing both sides of this equation by a, we obtain 92 Total Effect = E/b=D(a/b+d) (4) But note the simple conversion of Total Effect which is the product of a/b and BED (5). MATERIALS AND METHODS When multiple fractions per day are used [6], the repair of damage caused by one radiation dose may not be complete before the next fraction is given, especially if the half time for repair T1/2 is long in relation to the time interval between fractions [7]. Incomplete repair tends to reduce the isoeffective dose and corrections must be made for the consequent loss of tolerance. This can be executed by the use of an incomplete repair model [8,9]. The amount of un-repaired damage is expressed by the function Hm which is dependent upon the number of equally spaced fractions (m). To represent the time interval between fractions, Rep Pract Oncol Radiother, 2006; 11(2): 91-95 Manimaran S et al – Isoeffect calculations… for the purpose of isoeffect calculations, an extra term is added to the basic BED formula [9]. Table 1. LQ variables. Variable For fractionated radiotherapy BED = D [1+d/(a/b)+Hm.d/(a/b)] (6) where d – dose per fraction, D – total dose. As the dose rate is reduced below the range used in external beam radiotherapy, the duration of irradiation becomes longer, and the induction of damage is counteracted by repair, leading to an increase in the isoeffective dose [10]. The BED formula for continuous irradiation incorporates the factor (g) to allow for incomplete repair [11]. (7) where D is the total dose = dose rate X time. Consider the situation when fraction size is changed without changing the overall duration of treatment time [12]. The formula required to calculate the biological effective dose is: (8) where D = total dose in n fractions of size d. Assuming the conditions for a change of fraction size in BED calculations: a) select a value for a/b for a speciﬁc tissue value, b) select the reference tolerance dose Dref, c) select a fraction size for the reference treatment (dref), d) calculate for the reference treatment: BED ref = Dref [1+ dref/(a/b)], e) for the new fraction dose, d, calculate the total dose. D = BEDref/[1+d/(a/b)] 3–3.5Gy [17] K 0.78Gy [21] To 28 days [19] (10) For the new fractional dose, d2, the, remaining total dose is given by: D2 = PE2/[1+ d2/(a/b)] (11) The same procedure can be adopted for more than two fraction sizes during treatment [11]. BED values were evaluated using equation [5] for the following values of LQ model variables [3] (Table 1). RESULTS Change of fraction size during treatment BED = D [1+ d/(a/b)] – K(T–To) α/β PE2 = BEDref-PE1 For continuous low dose rate radiotherapy BED = D [1+ D.g/(a/b)] Value (9) For the ﬁrst part of the treatment, calculate the partial BED value (PE1) from d1 and D1. The partial tolerance remaining for the second part of the treatment is: A number of clinical reports and clinical reviews have shown a signiﬁcant relationship between overall treatment time and Hendry normal tissue complication rate [13]. In order to reduce late complications, when doses in radiotherapy are changed by mistake, it is generally considered as an over dosage. In such cases. corrections must be made to alter the dose without changing the over all treatment time. The following are some example calculations, which illustrate the application of linear quadratic equations for Head and Neck Cancer. Example calculations Example 1 The planned treatment was for 70Gy in 35 fractions but, owing to dosimetric error, the ﬁrst 6 fractions were given as 4Gy/fraction, rather than 2Gy/fraction. The accumulated dose is thus 24Gy in 6 fraction (OTT – 47days). Treatment will be continued using 2Gy/fraction Question: How many fractions of 2Gy should be given in order to maintain an equal probability of late ﬁbrosis? 93 Review Paper Rep Pract Oncol Radiother, 2006; 11(2): 91-95 Assumptions; a/b = 3.5Gy Orton [3]. Solution: 1. BED = 70×(1+2/3.5) –0.76(47–28) = 95.56 Gy 2. PE1 = 24×(1+4/3.5) = 51.4Gy PE for ﬁrst 6 fr 3. PE2 = BED-PE1 = 44.16 D2 at 2Gy/fr 4. D2 = 44.16/1.57 = 28.12Gy for 2Gy/fr 28.12/2 = 14 fractions Example 2 Planned treatment was for 50Gy in 25 fractions but, owing to dosimetric error, 6 fractions were given at a dose rate of 3Gy/fr instead of 2Gy/fr Question: How many fractions of 2Gy should be given in order to maintain an equal probability of late ﬁbrosis? 1. BED = 50×(1+2/3.5) –0.76(33–28) = 74.77Gy 2. PE1 = 18×(1=3/3.5) = 33.43Gy 3. PE2 = BED-PE1 = 41.34Gy 4. PE2 = D×(1+2/3.5) = 41.34Gy 5. D2 = 41.34/1.57 =26.33Gy For 2Gy/fr 26.33/2 = 13 fractions 5. D = 112.01/1.57 = 71Gy in 2 Gy/fr. Note: Owing to the smaller volumes and different dose distributions for interstitial irradiation, the calculated BED may be too high for an external beam irradiation. It is therefore recommended to reduce the dose for all fractions. Example 4 The planned treatment is 4 fractions of 5Gy (2fr/week). After the ﬁrst fraction, by mistake, a further single dose of 12Gy was given. Question: How much dose has been given for the 3 remaining fractions? a =3Gy b for late complications [14]. 1. BED =20× 1+ 5 = 53.3Gy 3 ( ) 2. PE1 = BED =12× 1+12 = 60Gy 3 ( ) 3. PE2 BED =20× 1+ 5 = 60–53.3=6.7Gy 3 ( ) Example 3 Cancer of the oral tongue, stage T2 (3.5cm). The planned treatment is in two parts: I. External beam 50Gy in 25 fr followed by II. Interstitial implant delivering 30Gy in 3 days. Question: If the total treatment were to be given in 2Gy/fr what would be the total biologically equivalent dose for late ﬁbrosis? Assumptions: a/b=3.5Gy 4. PE2 =D2× 1+ D2 ×3 = 60–53.3=6.7Gy 3 ( 6.7 ) = D2+(D2)2/9 0.111(D2)2+D2–6.7 According to the quadratic equation = D=–b+Öb2–4ac 2a a=0.111; b=1; c=–6.7 D2=4Gy d2=3Gy T1/2 =1.0hr Spinal cord tolerance calculations: g.factor (3day) =0.04 1. PE1 =50×(1+2/3.5) – 0.76(37–28) = 71.73Gy External beam 2. PE 2 =30×(1+(30×0.04/3.5) = 40.28Gy Brachytherapy 3. BED = 71.73+40.28 = 112.01Gy 4. BED D× 1+ 2 = 112.01Gy 3.5 ( 94 ) a =2Gy b – for spinal cord damage [15] BEDreference = 50 1+ 2 = 100 2 ( ) BEDplanned = 20 1+ 5 = 70 2 ( ) Rep Pract Oncol Radiother, 2006; 11(2): 91-95 BEDplanned = 0.7 BEDreference Spinal cord tolerance below 30% 100 = Dmax =1+5 2 = 100/3.5=28.52=28Gy A maximum of 5Gy should be given in the remaining 3 fractions. DISCUSSION Deviations from the predictions of the incomplete repair LQ model have become apparent under more extreme conditions, such as reduced spinal cord tolerance in the CHART regime (3 fr/day continuous over 12 days). Most of the deviations that have so far been observed from the LQ model may have arisen from the incorrect choice of two basic parameters a/b and T1/2. The results of these calculations must only be taken as a guide to clinical practice. The linear quadratic approach to fractionation overcomes some of the deﬁciencies of the NSD and TDF concepts [16]. The validity of the equations is limited to more or less standard conditions. Deviations from the predictions of the incomplete-repair linear quadratic model have become apparent under more extreme conditions. As experience grows, applications for this method of calculation will become more evident. CONCLUSIONS Using these calculations only as a guide, the linear quadratic approach to fractionation overcomes some of the potential deﬁciencies of the TDF approach, but cannot be claimed to be universally correct. In reality it would be surprising if such simple equations satisfactorily described all the possible effects of changing dose prescriptions in radiotherapy. Neither the TDF nor the LQ based approach may be put in to clinical use directly without ﬁrst cross checking retrospective clinical data. The solutions obtained through these calculations should be considered as rough estimates only. 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