Determination of the optimum dose per fraction in fractionated radiotherapy when there is delayed onset of tumour repopulation during treatment. 1C I ARMPILIA ...
The British Journal of Radiology, 77 (2004), 765–767 DOI: 10.1259/bjr/47388747
E
2004 The British Institute of Radiology
Determination of the optimum dose per fraction in fractionated radiotherapy when there is delayed onset of tumour repopulation during treatment 1
C I ARMPILIA, MSc, 1R G DALE, PhD, FIPEM and 2B JONES, MD, FRCR
1
Radiation Physics and Radiobiology, Hammersmith Hospitals NHS Trust/Imperial College Faculty of Medicine, Charing Cross Hospital, London W6 8RF and 2Clinical Oncology, Imperial College Faculty of Medicine, Hammersmith Hospital, London W12 0HS, UK
Introduction The choice of fractionation in radiotherapy involves a compromise between a number of competing radiobiological factors. The severity of late normal tissue effects may be highly dependent on fraction size, thus requiring the dose per fraction to be relatively small. This in turn leads to more fractions being used in order to maintain the tumour effect, resulting in a longer overall treatment time. In contrast, tumours possessing short clonogen doubling times ideally require shorter treatment times for improved tumour cell kill in order to reduce the impact of tumour cell repopulation. A mathematical model has been previously developed to investigate ways of optimizing the theoretical treatment outcome by estimating the optimum dose per fraction required for a given set of radiobiological parameters [1, 2]. The earlier model assumed that tumour repopulation proceeds at a constant rate right from the initiation of treatment, i.e. that there is no delay time before the repopulation process begins. However, especially for head and neck tumours, there is evidence for the existence of a lag time before the onset of compensatory re-growth [3, 4]. This report describes a modification to the earlier model and which provides a non-analytical, re-iterative, method for deriving the optimum dose per fraction which produces the maximum biologically effective dose (BED) to the tumour in the presence of a delayed start to repopulation.
Brief description of model Within the linear–quadratic (LQ) cell survival model, BED is defined by: d ð1Þ BED~nd 1z (a=b) where n and d are, respectively, the number and size of the dose fractions, and a and b are the respective linear and quadratic radiosensitivity coefficients of the irradiated Received 12 January 2004 and in revised form 12 May 2004, accepted 26 May 2004.
tissue. The ratio (a/b) is an inverse measure of the fractionation sensitivity of the tissue in question. BED is a measure of the biological dose delivered to a tumour or organ and is the theoretical total dose that would be required to produce a particular isoeffect using an infinitely large number of infinitesimally small dose fractions. BED is a measure of effect in units of Gyx, where the suffix x indicates the value of a/b assumed in the calculation. Equation (1) makes no allowance for tissue repopulation during treatment. The incorporation of the repopulation effect is achieved through the use of a subtractive repopulation factor which takes account of the treatment time and the repopulation rate. The expression for BED is then modified as: d {K T{Tdelay ð2Þ BED~nd 1z (a=b) where T is the overall time of treatment (the elapsed time between the first and last fraction) and Tdelay is the time after the start of treatment at which proliferation begins. K (in units of Gy day21) is the biological dose per day required to compensate for ongoing tumour cell repopulation, once this has started. K is the daily BED that must be delivered simply to stop the tumour growing further during treatment. At present there are little data relating to K and Tdelay factors for each tumour. For head and neck tumours recommended values of K and Tdelay are 0.9 Gy day21 and 28 days, respectively [4, 5]. Both a/b and K are tissue-specific parameters and appropriate values must be selected for each when calculating tumour BEDs. For most late-responding normal tissues the proliferation rate is usually so small that K , 0 and may be neglected, i.e. for such tissues the BED is calculated using Equation (1). To simplify what follows, Equations (1) and (2) may be re-written in two forms, the BED suffixes, respectively, relating to tumour and late-responding tissues: d {K T {Tdelay ð3Þ BEDtum ~nd 1z ctum
Current address for C I Armpilia, Cosmote, 14122 Athens, Greece.
d BEDlate ~nd 1z clate
Current address for Dr B Jones, Clinical Oncology, Queen Elizabeth University Hospital, Birmingham B15 2TH, UK.
where ctum and clate are the respective a/b ratios for tumour or normal tissue late effects, to be used as appropriate.
Address correspondence to Dr R G Dale.
The British Journal of Radiology, September 2004
ð4Þ
765
C I Armpilia, R G Dale and B Jones
Equation (4) may be re-arranged as: n~
BEDlate d ð1zd=clate Þ
ð5Þ
and the relationship between T and n can be approximately expressed as: T ~fn{1
ð6Þ
where f is the mean interfraction interval. The numerical value of f to be used is the ratio of seven (the number of week days) divided by the number of fractions per week, e.g. for five fractions per week, f57/5 [1]. Equation (3) may then be rewritten then as: BEDlate BEDtum ~ ð1zd=ctum Þ{ 1zd=clate BEDlate ð7Þ K f {1{Tdelay d ð1zd=clate Þ Putting Tdelay50 in Equation (7) and differentiating BEDtum with respect to d, the optimum dose per fraction (dopt) for the special case where repopulation proceeds at a constant rate throughout treatment will be the solution for d of: ð1{clate =ctum Þd 2 {2fKd{clate fK~0
ð8Þ
N
For head and neck tumours the value is around 0.9 Gy day21 [4, 6]. A tolerance normal tissue value of (BED)late5100 Gy3 was assumed. (This corresponds to a treatment delivering 60 Gy in 30 fractions.)
Results and discussion When no repopulation delay time is considered, the optimum dose per fraction increases near-linearly with the K factor (Figure 1). This means that, to compensate for a higher repopulation rate, as exists particularly in head and neck tumours and others of squamous cell histology, it is necessary to increase the dose per fraction and reduce fraction number in order to maintain the normal tissue isoeffect. However, with a delay time of 28 days, the optimum dose per fraction reaches a plateau region (2.6 Gy for f57/5 and 2.0 Gy for f57/7) when K exceeds 0.5 Gy day21. The reason for the existence of the plateau is that the optimization process is selecting a treatment time which is less than Tdelay, i.e. treatment is completed before repopulation begins. For this region the tumour BED maintains the value of 67.6 Gy10 (Figure 2). In Figure 2 the BED decreases with increasing K factor, as expected, for Tdelay50. The plateau region occurs when considering a delay period before the onset of significant repopulation and corresponds to
i.e. dopt ~
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2fKz2 f 2 K 2 zð1{clate =ctum Þclate fK ð1{clate =ctum Þ
ð9Þ
It is seen from Equation (9) that the optimum dose per fraction (dopt) is independent of the normal tissue BED (BEDlate) but is dependent on the normal tissue a/b ratio (clate). The derived dopt value can be used in Equation (5) to determine the number of fractions required to achieve the required normal tissue isoeffect. For any particular fractionation pattern (determined by f) the associated overall treatment time is then found from Equation (6). Equation (9) represents the condition examined in detail in earlier papers dealing with this issue [1, 2]. Unfortunately, however, when Tdelay is non-zero, there is no comparable analytical expression from which the optimum dose per fraction may be determined directly and a re-iterative computer method must be employed. Such a program has been written and, for a range of parameter values, samples Equation (7) in order to determine the dose per fraction which maximizes the tumour BED. Two cases in particular are discussed here; one assuming constant proliferation throughout the entire treatment (i.e. with Tdelay50), and the other assuming proliferation delay of 28 days, as might be appropriate to head and neck tumours [4]. The calculations involved the following parameters and assumptions:
N N N N N 766
An a/b ratio of ctum510 Gy for tumour was used. The normal tissue a/b ratio (clate) was assumed to be 3 Gy. Fractionation parameters f of 7/5 and of 7/7 were used for five and seven fractions per week, respectively. It was assumed that complete repair of sublethal radiation injury occurs between fractions in all cases. The K factor was varied from 0 to 1.5 Gy10 day21.
Figure 1. The relationship between optimum dose per fraction and K factor (biological dose per day required to compensate for ongoing tumour cell repopulation). The cases for which the time after the start of treatment at which proliferation begins (Tdelay)50 may be derived via Equation (8). Cases for which Tdelay is non-zero muct be derived using the re-iterative method discussed in this article.
Figure 2. The relationship between tumour biologically effective dose (BED) and K factor (biological dose per day required to compensate for ongoing tumour cell repopulation). The British Journal of Radiology, September 2004
Optimum dose per fraction when there is delayed repopulation
the plateau region for the optimum dose per fraction. Once again, the plateau region occurs because treatment is completed before the initiation of accelerated repopulation. As expected, therefore, for tumours possessing a high K factor (i.e. repopulate rapidly), treatments completed within the delay period are significantly better. When the K factor is small and approaches zero the maximum achievable BED approaches 100 Gy10. In the case of seven fractions per week (f57/7) Figures 1 and 2 show that the value of optimum dose per fraction is 2.05 Gy, with an associated tumour BED of 71.6 Gy10 for K¢0.6 Gy day21. For K50.9 Gy day21 and with f57/5, the optimum dose per fraction for Tdelay50 and Tdelay528 days are 4.7 Gy and 2.6 Gy, respectively. The corresponding tumour BED values are 47.7 Gy and 67.6 Gy, respectively. This result shows how significant the delayed onset of repopulation can be on achievable tumour cell kill. As stated earlier, it is still not clear how repopulation kinetics change with time during treatment. In the reiterative model presented here we have assumed that no repopulation occurs during the lag period, i.e. that the onset of a fixed rate of repopulation is essentially a ‘‘step function’’. The dose required for a given tumour effect would therefore change, with increasing treatment time, according to the solid line shown in Figure 3. In reality, however, it is more likely that the rate of repopulation increases continuously, with the mathematical analysis requiring a more complicated model involving cell loss factors [6, 7]. The dotted line in Figure 3 is the schematic representation of how dose for a given tumour effect would increase with increasing treatment time in the latter scenario. Investigation showed no difference at all in the derived optimum dose per fraction when considering an initial phase of slow fixed repopulation of 0.1 Gy (as opposed to the case where there is no repopulation at all) during the lag period of 28 days. Also, the corresponding tumour BED values were reduced by around only 0.1 Gy. Future developments of this work will consider how the alternative continuous repopulation model, outlined above, may be incorporated into the method for deriving an optimum dose per fraction.
Conclusion This study confirms that, where individual tumours vary in terms of repopulation rates and delay times before such repopulation begins, there is a range of doses per fraction which would give the maximum tumour cell kill while respecting a given level of normal tissue tolerance. For a heterogeneous group of tumours, embracing a wide range of repopulation characterstics, the model proposed here can be used to identify a single fractional dose which is optimal for that whole group and this will form the basis of future studies. Although further experimental work and examples from
The British Journal of Radiology, September 2004
Figure 3. Schematic showing how the dose required for a given tumour cell kill increases with prolongation of treatment time. The rise in the curves is a direct result of the repopulation that occurs during treatment. The ‘‘dog-leg’’ pattern of change is assumed for the purpose of the calculations in this article. The more plausible ‘‘continuous’’ pattern of change is shown for comparison. Tdelay, the time after the start of treatment at which proliferation begins.
clinical practice are required to identify a larger parameter dataset, the concepts presented here could shed interesting new light in the debate over how to produce improved results in radiotherapy. Also, and as has been discussed elsewhere, the identification of a radiobiologically-optimized dose per fraction for a given set of radiobiological constraints has implications for exploring cost-optimization in radiotherapy [8].
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