British Journal of Radiology (2006) 79, 254-257
© 2006 British Institute of Radiology
doi: 10.1259/bjr/49977661
Calculation of high-LET radiotherapy dose required for compensation of overall treatment time extensions
B Jones, MD, FRCR, MedFIPEM
1
A Carabe-Fernandez, MSc, MPhys
2 and
R G Dale, PhD, FinstP, FIPEM
2
1 Birmingham Cancer Centre, Queen Elizabeth University Hospital, Birmingham B15 2TH, 2 Department of Radiotherapy Physics & Radiobiology, Charing Cross Hospital, London W6 8RF, UK
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Abstract
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A method is presented that allows biological effective dose (BED) equations to be used to calculate compensatory doses for treatment time extensions when high-LET (linear energy transfer) radiotherapy schedules are used. The principles involved are the same as those for low-LET radiations, but incorporate two relative biological effectiveness (RBE) factors, RBEmax and RBEmin, which represent the RBE at very low and very high fraction doses, respectively, with the actual RBE changing between these extremes. The method has the advantage that low-LET 
/
ratios and low-LET daily dose-equivalent repopulation factors are used in the calculations. The daily dose repopulation equivalents and increments in dose per fraction in the case of high LET radiotherapy are smaller than those for low LET.
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Introduction
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The loss of tumour control following an extension in treatment duration can, in principle, be overcome by increasing the total dose after the extension. This can be achieved – as well as our assumptions allow – by calculating the dose per day (dc) that should compensate for the additional tumour cell repopulation. The magnitude of dc can be estimated from a logical extension of the linear-quadratic (LQ) model of radiation effect through consideration of the biologically effective dose (BED) concept [1–4]. Dale et al [5] have stressed that the compensatory dose per day is normally expressed in BED Gy units, which are those that pertain for a specific tumour / ratio and from this the actual physical dose (dc) can be separately calculated. There are some additional subtleties in the case of high-LET radiotherapy because additional RBE (relative biological effectiveness) correction factors need to be included. The general mathematical approach is that originally followed by Bewley [6] for high LET radiations, but is now adapted to the LQ model and the BED concept in particular, such that the use of the Cobalt equivalent Gy concept is avoided.
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Methods
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RBE is conventionally defined as the ratio of the dose per fraction of the low LET radiation (dL) to the dose per fraction of the high LET radiation (dH), so that it is usually greater than 1, as in:
As shown in Appendix A, RBE is maximized at low dose to a value given as RBEmax, where:
where H and L are the respective radiosensitivities for high and low LET radiations. Similarly, from Appendix A, at very high dose RBE approaches an asymptotic minimum value (RBEmin), where:
Consequently, where RBEmin is significantly greater than unity and in contradiction with the conventional assumption first put forward in the Theory of Dual Radiation Action by Kellerer and Rossi [7], it follows that H>L. Also, the high-LET parameter can be expressed in terms of the low-LET value and the RBEmin, at high dose per fraction, i.e.:
These Equations (2–4) are used to form BED equations for high LET radiations as shown in Appendix B.
Next we consider how to compensate for a practical problem using the standard BED equations for low LET radiations and also the newly derived equation for the high LET BED (BEDH):
where N is the number of fractions, d the dose per fraction and (L/L) is the low LET / ratio. The repopulation equivalent equations are also given in Appendix B: essentially, the repopulation terms in the high LET equations are the same as for low LET.
Worked example
A schedule of 45 Gy in 25 fractions using megavoltage X-rays is to be followed by a highly localized "boost" of 6 Gy in 2 fractions of 3 Gy each, using a high-LET radiation for which RBEmin = 1.3 and RBEmax = 8; these values are assumed to apply for both cancer and normal tissues. There is a delay of 7 days in the provision of the boost, due to patient illness. The tumour type is assumed to have a daily repopulation equivalent of 0.6 Gy per day after a lag interval of 25 days during megavoltage X-ray treatment. The normal tissue BED is assumed to be governed by / = 2 Gy and the tumour / = 10 Gy.
The intended BED to normal tissue from X-rays = 45 x (1+1.8/2) = 85.5 Gy2.
The intended BED to any normal tissue that receives the added high-LET boost of 2 fractions of 3 Gy = 6 x (8+1.32 x 3/2) = 63.2 Gy2, so that the total intended maximum BED to same volume of normal tissue = 85.5+63.2 = 148.7 Gy2.
The intended BED to tumour by X-rays, BEDL = 45 x (1+1.8/10) = 53.1 Gy10, plus the intended BED to tumour by high LET, BEDH = 6 x (8+1.32 3/10) = 51.04 Gy10, so that the total tumour BED is 104.14 Gy10 before allowing for repopulation.
The additional 7 days of repopulation must be allowed for because of the treatment interruption in providing the boost, which is equivalent to 0.6 x 7 = 4.2 Gy10.
The boost must therefore accommodate the original high-LET BED plus 4.2 Gy, i.e. 51.04+4.2 = 55.24 Gy10.
As this is to be given in two fractions, then: 2 x d x (8+1.32d/10) = 55.24, and the solution for d is 3.23 Gy/fraction instead of the originally prescribed 3 Gy per fraction, prescribed before the treatment gap had occurred.
The normal tissue BED will then be: 2 x 3.23 x (8+1.32 x 3.23/2) = 69.31 Gy2.
Thus the total (low- plus high-LET) normal tissue maximum BED will have increased by 69.31–63.2 = 6.11 Gy2, an increase of 4.1% on an already high BED in the localized boost volume, in order to maintain the same tumour BED. This could cause enhanced tissue side effects.
In practice a compromise solution such as a dose per fraction of 3.15 Gy instead of 3.23 Gy might be used. This would lead to 67.17 Gy2 maximum high-LET BED to the normal tissues and 53.75 Gy10 to the tumour.
A plot of the calculated compensatory dose per day, given as a single fraction, for high-LET radiations is shown in Figure 1
. It can be seen that the / ratio has little influence on the value of the compensatory dose, with marginal differences only if / is very low and when K is large. This result is dominated by the RBEmax value, since small fractional doses are operational – e.g. for a K value of 1 Gy per day the equivalent high LET dose per day is 0.12 Gy per day which is close to the ratio 1/RBEmax = 1/8 ( = 1.25). It needs to be stressed that these calculated doses, which are single fraction equivalents, must not be used as additional doses added to the prescribed dose per fraction; to be correctly incorporated the method used in the worked example should be followed to adjust the dose per fraction.
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Discussion
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As in the case of low-LET radiations, extension of treatment time can cause significant dilemmas to the radiation oncologist. Gaps occurring later in the treatment are particularly difficult to compensate for since an extension to the prescribed treatment time becomes unavoidable [4]. Compensations for overall treatment time extensions typically involve a compromise between delivery of a higher BED to normal tissues and a reduced BED to tumour, or the same BED to tumour and an even higher BED to normal tissues. The incorporation of RBE within BED equations has enabled the calculation of estimated compensatory doses for high LET radiations.
The use of a fixed RBE weighting factors is an alternative approach. Although approved in international definitions, this method will under estimate RBE at low doses per fraction and over estimate RBE at high doses per fraction. In contrast, the use of RBEmax and RBEmin overrides this potential problem. The RBEmax dominates the effective RBE at low dose per fraction, whereas RBEmin dominates the RBE at large doses per fraction. We have used these two RBE parameters because of the increasing use of large fraction sizes in high LET radiotherapy [8].
The actual high LET daily dose correction factors are much lower than the K value doses, as are the increments in dose per fraction required when compared with those required in low LET radiotherapy.
These aspects should be additionally considered in situations where protraction of relatively high LET radiotherapy occurs, e.g. radioiodine seed implants and in the case of hadrontherapy, for instance using ion beam or neutron radiotherapy schedules. For high-energy proton beams with significant spread out Bragg peaks, the correction factors will be much smaller, since the average RBE for protons is only around 1.1, although the refinement of a variable RBE with dose per fraction might lead to better results [9, 10]. In the case of ions beams, treatments are hypofractionated [8, 11], varying from 1 fraction to 20 fractions with treatment times of up to 1 month, but these times – apart from the single fraction case – could be extended for a variety of reasons such as patient illness, synchrotron breakdown etc.
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Acknowledgments and declarations
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Author AC-F is a Cyclotron Trust Research Fellow. All authors are members of the EPSRC-funded medical applications of ion beams network. BJ and RGD are Hon. Professors at the University of Caen as part of the ASCLEPIOS project, part of the French National Hadrontherapy Project.
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Appendix A
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To obtain RBE terms
RBEs are derived by intercomparing the single doses (low- and high-LET) required to obtain a given iso-effect. Since only single doses are involved there is no need to consider the repopulation effect, i.e. the iso-effect equation is simply:
At near-zero doses the quadratic dose terms become negligible and:
Leading to:
Similarly, at exceedingly high doses, the quadratic terms dominate, i.e.:
and:
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Appendix B
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To obtain high LET BED equations
The standard linear quadratic equation for the low LET surviving fraction (SF) with compensation for cellular repopulation is:
where L and L are the low LET radiosensitivity parameters, NL is the number of low LET fractions of dose dL, T is the overall treatment time in days, TDEL is the delay time in days before which repopulation becomes significant and is optional according to tumour type and TEFF is the effective cellular doubling time in days.
For high LET radiations the surviving fraction expression is changed to be:
where the subscripts are changed to H in order to refer to high LET radiations. Next, taking the natural logarithm of each side and multiply by –1 to, respectively, obtain from Equations (B1) and (B2) the log cell kills designated by EL and EH:
Assume that Equations (B3) and (B4) refer to the same biological effect.
To obtain the BED, divide each equation by L, the low LET radiosensitivity parameter, then, for Low LET, the BED is:
and where
is the daily dose equivalent for repopulation in units of BED Gy per day.
The High LET BED is also obtained by dividing by the low LET L parameter to give
Replacement of the high LET radiosensitivity parameters in Equation (B6) with RBEmax and RBEmin – as given in Equations (2–4) in the main text – results in:
For calculations involving overall treatment time variations and compensation of unintended treatment interruptions, the same low-LET daily BED dose equivalent values (K) are used in both cases.
The first component of the right hand side of Equation (B7) represents the BED which must be delivered to offset the effect of TH days-worth of repopulation, as quantified by the second (subtractive) factor. Equating these two components we obtain:
Once the time-point (TDEL) is passed, the BED-equivalent of repopulation for each additional day (for which TH–TDEL = 1) is found by setting NH = 1 in Equation (B8). On rearrangement, this leads to:
In Equation (B9) the repopulation term K is expressed as an equivalent BED dose per day. The actual (physical) dose per day (dc) – given as a single fraction – required to compensate for repopulation is the solution for dH in Equation (B9), i.e.:
Received for publication July 6, 2005.
Revision received August 15, 2005.
Accepted for publication September 5, 2005.
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References
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Abstract
Introduction
