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Biological equivalent dose assessment of the consequences of hypofractionated radiotherapy
Int. J. Radiation Oncology Biol. Phys., Vol. 47, No. 5, pp. 1379 –1384, 2000 Copyright © 2000 Elsevier Science Inc. Printed in the USA. All rights reserved 0360-3016/00/$–see front matter
PII S0360-3016(00)00571-X
BIOLOGY CONTRIBUTION
BIOLOGICAL EQUIVALENT DOSE ASSESSMENT OF THE CONSEQUENCES OF HYPOFRACTIONATED RADIOTHERAPY B. JONES, M.SC., M.D., FRCR,* R. G. DALE, PH.D.,† P. FINST, FIPEM,† S. J. KHAKSAR, M.B., B.CH. MRCP*
AND
*Cancer Therapeutics Section, Imperial College School of Medicine, Department of Clinical Oncology, Hammersmith Hospital, London, United Kingdom; and †Department of Radiation Physics and Radiobiology, Imperial College School of Medicine, Hammersmith Hospitals NHS Trust, Charing Cross Hospital, London, United Kingdom Purpose: To investigate the changes in biological effective dose (BED) that occur in high-dose regions within a target volume when radiotherapy is hypofractionated. Methods and Materials: By comparing a standard prescription of 2 Gy per fraction that is matched to give the same BED as a hypofractionated schedule at a standard intersectional prescription point, the BED increments for late-tissue effects at a higher dose region within the planning target volume (PTV) are compared. The alternative approach of BED matching between a conventional and hypofractionated schedule at the high-dose region is also considered. The results are presented as a sequence of calculations that can be understood by practicing radiation oncologists and in graphical form. Results: The BED increment at the high-dose region is marginally increased by hypofractionation, although the latter effect is relatively small: up to 5% additional BED due to hypofractionation for a 20% increase in physical dose when the prescribed fraction size is 6 –7 Gy. BED matching for late effects between a conventional and hypofractionated schedule at the high-dose region produces lower BED values throughout the remaining PTV, but at the expense of a reduced tumor control BED. Conclusion: Clinical trials that use BED isoeffect matching for late reacting tissue effects to design a hypofractioned test schedule should include comprehensive calculations of the likely BED in high-dose regions. © 2000 Elsevier Science Inc. Radiotherapy, Radiobiology, Linear– quadratic model, Clinical trials.
INTRODUCTION
2-Gy fractions, which is to be compared with an alternative schedule given in 3 Gy fractions, late normal tissue damage being required to be similar/identical in both arms. Treatment is prescribed at the intersectional point of the mid axis of the treatment beams, and there are regions of higher dose within the planning target volume owing to the irregular tissue contour and electron-density inhomogeneity as occurs for example in the treatment of the intact breast. Matching of BEDs for late effects, using an ␣/ ratio of 2 Gy appropriate for neural late-reacting tissue, is achieved at the prescription point. The alternative of a prescribed dose, matched to give the same BED for both arms of the trial at the high- dose region is also used, and the resultant tumor BEDs are compared, using ␣/ ratios of 10 Gy and 5 Gy, appropriate for squamous cell cancers and breast cancers, respectively (4). These modeling examples are not intended to simulate any particular type of radiotherapy treatment, but are designed to demonstrate a general effect.
Biological equivalent dose (BED) is an established measure for quantifying the expected biological effect of different radiation dose fractionation schedules (1–3). The BED concept is generally considered to be a good method for designing a test time-dose–fractionation schedule that is to be compared with a conventional schedule in a clinical trial. However, for the comparison to be valid, it is essential to ensure that the technical aspects, such as field sizes, field positions, etc., are identical in both arms of a trial. There are some subtle interactions of physical dose gradients and radiobiology that should be considered when trials are designed and their results analysed. To illustrate these general principles, some worked examples are presented. MATERIALS AND METHODS Consider a hypothetical clinical trial in which the control arm is a conventional schedule of 60 Gy in 30 ⫻ Reprint requests to: Dr. B. Jones, Cancer Therapeutics Section, Imperial College School of Medicine, Department of Clinical Oncology, Hammersmith Hospital, Du Cane Road, London W12 OHS, United Kingdom. E-mail: bleddyn.jones @ ic.ac.uk
Accepted for publication 7 December 1999.
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The BED is given by
冋
BED ⫽ nd 1 ⫹
d ␣/
册
(1)
where n and d are the numbers of fractions and dose per fraction, respectively. We shall consider the BED at areas where the dose is greater than that prescribed. For a dose increment of x, the BED will then be
冋
xnd 1 ⫹
册
xd . 共␣/兲
Volume 47, Number 5, 2000
Thus, if the critical tissue BED is to be maintained, hypofractionation reduces the tumor BED. This will result in a reduced tumor control probability for radical treatments, although there may be no significant change in treatment outcome where palliation is the endpoint. Now consider the effect of a higher dose region within the planning target volume (PTV) that may give rise to late-tissue complications. Assume this region has a 5% higher dose, such that both dose and dose per fraction will be multiplied by a factor of 1.05. In the control arm, the BED in the higher dose region will then be
(2)
冉
30 ⫻ 2 ⫻ 1.05 ⫻ 1 ⫹
The control arm BED is
冉 冊
2 ⫽ 120 Gy 2 2
60 1 ⫹
(3)
where the ␣/ ratio (or fractionation sensitivity) for late effects in the critical normal tissue is 2 Gy. To find the number (n) of fractions at 3 Gy per fraction to provide the same BED,
冉 冊
n⫻3 1⫹
3 ⫽ 120 Gy 2 2
(4)
leading to n ⫽ 16 and so the total dose would equal 48 Gy. RESULTS The tumor BED values (proliferation ignored), assuming ␣/ ratios of 10 Gy or 5 Gy, based on the prescription point BED would be as follows. For the control arm:
冉
冊
(5)
冉 冊
(6)
60 1 ⫹
2 ⫽ 72 Gy 10 10
or 60 1 ⫹
2 ⫽ 84 Gy 5. 5
For the hypofractionated arm:
冉
冊
(7)
冉 冊
(8)
48 1 ⫹
3 ⫽ 62.4 Gy 10 10
or 48 1 ⫹
3 ⫽ 76.8 Gy 5. 5
冊
2 ⫻ 1.05 ⫽ 129.15 Gy 2 2
(9)
which represents a 7.6% increase in BED2 relative to a plan in which no high dose occurs. In the hypofractionated arm, the BED in the higher dose region is
冉
16 ⫻ 3 ⫻ 1.05 1 ⫹
冊
3 ⫻ 1.05 ⫽ 129.78 Gy 2. 2
(10)
This represents an 8.2% increase in BED2 in the high-dose region compared with an increase of 7.6% for the conventional treatment in the same region. Such a small increase (of 0.6% in BED) may not be sufficiently significant to cause a detectable increase in normal tissue toxicity in the example given, but there is undoubtedly a marginal enhancement of the BED. Thus, it can be seen that combination of a higher physical dose and a larger fraction size leads to a proportionately greater increase in BED: a variant of the well-known “double trouble” effect, which can be problematic in many treatment situations (5, 6). The previous studies of the “double trouble” effect did not extend the argument to include hypofractionation: the modeling was based on prescribed doses of 2 Gy per fraction. The effect described above, due to hypofractionation, will be greater: (1) if dose per fraction is further increased in the test arm of a trial , and (2) if a higher percentage physical dose occurs in normal tissues within the PTV. When clinically significant, the effect could be termed “treble trouble,” because the “double trouble” effect at high-dose regions treated to 2 Gy per fraction is compounded by hypofractionation. The BED enhancements that result may be described by incremental ratios of BED equations, which are derived as follows. For the control arm (n fractions of dose d), the BED at the region of high dose, obtained by multiplication of the prescribed dose and dose per fraction by a factor x (such as 1.05 in the above example for a 5% increase), is divided by the BED at the prescription point to give:
Biological equivalent dose assessment of the consequences of hypofractionated radiotherapy
冉 冉
冊 冉 冊 冉
冊 冊
xd xd x 1⫹ ␣/ / BED RATIO 1 ⫽ ⫽ . d d nd 1 ⫹ 1⫹ ␣/ ␣/ xnd 1 ⫹
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(11)
For a nonconventional hypofractionated schedule (m fractions of dose h), the appropriate ratio will be
冉 冉
冊 冉 冊 冉
冊 冊
xh xh x 1⫹ ␣/ ␣/ ⫽ . BED RATIO 2 ⫽ h h mh 1 ⫹ 1⫹ ␣/ ␣/ xmh 1 ⫹
(12)
The ratio that describes the additional fractional increment in BED (termed ⌬BED) that occurs as a direct result of hypofractionation will be ⌬BED ⫽
BEDRATIO2 . BEDRATIO1
(13)
Therefore,
冉 冉 冊冉 冉 冊冉
hx d 1⫹ 共␣/兲 ␣/ h dx 1⫹ 1⫹ ␣/ ␣/
1⫹
⌬BED ⫽
冊
冊
冉 冊冉 冊 冉 冊冉 冊
␣ ␣ ⫹ hx ⫹d   ⫽ . ␣ ␣ ⫹h ⫹ dx  
(14)
Equation 3 is independent of the number of fractions in each arm of the trial, but depends on the dose per fraction in each case. This equation provides the additional increment in BED due to hypofractionation within the high-dose region between the two schedules as the treatment inhomogeneity (x) increases. In the example already given above, ⌬BED is 129.8/129.2 ⫽ 1.005. Such a value is very small (and clinically insignificant) but it must be recalled that this result expresses the further BED increment that occurs in addition to the higher BED that occurs at high-dose regions during conventionally fractionated therapy, i.e., an increment in BED beyond the “double trouble” effect. More general results are given in Fig. 1, where x and dose per fraction of the test/hypofractionated schedule are varied. The ⌬BED ratio increases further, reaching a value of 5% for a 25% dose per fraction increment due to dose inhomogeneity but this degree of change is only seen when the prescribed hypofractionated schedule dose per fraction is increased to be as large as 6 –7.5 Gy per fraction. The effect can also be illustrated in terms of the calculated equivalent total dose at 2 Gy per fraction that would be delivered to the higher dose region. This form of presenta-
Fig. 1. The relationship between the enhancement of biological equivalent dose (BED) at a high-dose region and the prescribed dose per fraction for variable values of x, the physical dose increment factor. When x ⫽ 1.05, there is a dose of 105% relative to the prescription dose, etc.
tion of high dose is far easier for clinicians to comprehend than a BED increment and involves no further assumptions (which would be required for the calculation of normal tissue complication probabilities). In Fig. 2, it can be seen that, for a prescribed 2 Gy per fraction, there is a significant stepwise increment in the equivalent total dose (calculated at 2 Gy per fraction) due to the increasing “hot spot” or “double trouble” effect: from a baseline of 50 Gy in 25 fractions to 54 Gy for a 5% additional “hot spot” dose, to 58 Gy for a 10% additional “hot spot” dose, etc. Then, as the prescribed dose per fraction increases beyond 2 Gy, there is an additional gradual increase in the equivalent total dose. For example, at the 20% high-dose region (an x factor of 1.20), a change in the prescribed dose per fraction from 2 Gy to an extreme value of 7 Gy increases the equivalent dose at the “hot spot” from approximately 66 Gy to 70 Gy
Fig. 2. Plot of estimated total dose standardized to 2 Gy per fraction at high-dose regions for a prescribed dose of 50 Gy in 25 fractions, for variable values of x, the physical dose increment factor at the hot spot regions.
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(in 2-Gy fractions). Such BED increments are larger where the higher dose is greater. These secondary increments again appear small, but these may be clinically significant in large treatment volumes or in situations where the trial endpoint (e.g., neurological complications) may increase steeply with total dose and dose per fraction. Thus, a trial that uses BED calculations to match a conventional dose-fractionation schedule at the prescription point may be prejudicial against a hypofractionated treatment, despite the apparent isoeffect. The alternative is to calculate the isoeffect at the high-dose region of interest that is relevant to normal tissue complications. The penalty for doing this would be that a smaller tumor BED would be delivered in the hypofractionated arm at the prescription point and this may adversely effect the tumor control. For example, if a significant volume of normal tissue has a 7% high dose within the PTV in the original trial control arm referred to above (60 Gy in 30 fractions) and the BED allowed is to be 120 Gy2 in this region, and that it is decided to specify a lower dose at the standard prescription point, the modified dose prescription will be changed as follows. Dose per fraction for the conventional arm will be given by the solution of
冉
冊
d ⫻ 1.07 30 ⫻ d ⫻ 1.07 1 ⫹ ⫽ 120 2
(15)
Dose per fraction (h) for hypofractionated arm will be given by the solution of
冊
(16)
if 16 fractions are given, and h ⫽ 2.80 Gy per fraction. These doses will result in BED‘s in normal tissues that receive the same dose as at the prescription point of :
冉
1.87 ⫻ 30 1 ⫹
冊
1.87 ⫽ 108.6 Gy 2 2
(17)
for the control arm and
冉
2.8 ⫻ 16 1 ⫹
冊
2.8 ⫽ 107.5 Gy 2 2
冉
30 ⫻ 1.87 1 ⫹
冊
1.87 ⫽ 66.6Gy 10. 10
(19)
This represents a 7.5% reduction compared with a BED of 72 Gy10 when the dose prescription is 2 Gy per fraction. (b) for an ␣/ of 5 Gy
冉
30 ⫻ 1.87 1 ⫹
冊
1.87 ⫽ 77.1Gy 5. 5
(20)
This represents an 8.2% reduction compared with a BED of 84 Gy5 when the dose prescription is 2 Gy per fraction. 2. Hypofractionated arm tumor BED (a) for an ␣/ of 10 Gy
冉
冊
2.8 ⫽ 57.3Gy 10. 10
(21)
This represents an 8.2% reduction compared with a BED of 62.4 Gy10 when the dose per fraction is 3 Gy (b) for an ␣/ of 5 Gy
d ⫽ 1.87 Gy per fraction.
冉
ated arm of the trial than in the conventional arm: a finding that is not intuituve. There are, however, further implications for tumor BEDs. In each case these will be: 1. Control arm tumor BED (a) for an ␣/ of 10 Gy
16 ⫻ 2.8 1 ⫹
or
h ⫻ 1.07 ⫽ 120 16 ⫻ h ⫻ 1.07 1 ⫹ 2
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(18)
for the hypofractionated arm. Thus, the probability of normal damage within the whole PTV will now be reduced rather more in the hypofraction-
冉
16 ⫻ 2.8 1 ⫹
冊
2.8 ⫽ 69.9 Gy 5. 5
(22)
This represents a 9% reduction compared with a BED of 76.8 Gy5 when the dose per fraction is 3 Gy. Thus, to guard against excess normal tissue damage by prescribing to the area of higher dosage, a reduction of tumor control may result. The reduction in tumor BED may not necessarily be significant in situations where: (a) relatively low tumor BED values may be sufficient to provide good outcomes, such as in postoperative radiotherapy, because the number of tumor clonogens may be relatively small (7), or where partial tumor shrinkage or clonogen sterilization is required prior to definitive surgery; (b) there is excellent normal tissue sparing due to focal forms of radiotherapy, e.g., conformal radiotherapy, protons and high-dose-rate brachytherapy (3); then, a higher tumor BED may be safely given; (c) highly radiosensitive tumors are treated; and (d) the aim of treatment is palliative, because there is usually a less-clear dose–response relationship for outcomes such as hemostasis or pain relief. A further example of the above problem would be if the conventional treatment continues to be prescribed at the standard intersectional point to 60 Gy in 30 fractions. The 7% additional dose at the hot spot would then result in a BED of
Biological equivalent dose assessment of the consequences of hypofractionated radiotherapy
Table 1. Matching for normal tissue isoeffect at intersectional prescription point BED Matching Location
Conventional fractionation
Hypofractionation
Intersectional prescription point “Hot spot” region
BED1 BED2 (⬎BED1)
BED1 BED3 (⬎BED2)
冉
30 ⫻ 2 ⫻ 1.07 ⫻ 1 ⫹
冊
2 ⫻ 1.07 ⫽ 132.9 Gy 2. 2
(23)
The hypofractionated arm could be matched to this BED (if it is thought that this BED will be well tolerated). Consequently, the hypofractionated dose per fraction at the “hot spot” would be given by the solution for d in
冉 冊
16 ⫻ d 1 ⫹
d ⫽ 132.9. 2
(24)
From which d ⫽ 3.2 Gy At the original prescription point, the hypofractionated dose per fraction would then be d/1.07 ⫽ 2.99 Gy. Now, the tumor BED would be (a) for ␣/ ⫽ 10 Gy
冉
16 ⫻ 2.99 ⫻ 1 ⫹
冊
2.99 ⫽ 62.1 Gy 10. 10
(25)
This represents a 13.8% reduction when compared with the tumor BED obtained with the conventional fractionation (72 Gy10) (b) for ␣/ ⫽ 5 Gy
冉
16 ⫻ 2.99 ⫻ 1 ⫹
冊
2.99 ⫽ 76.4 Gy 10 5
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DISCUSSION Wide field treatments can be associated with significant dose inhomogeneity. For example, Neal et al. (8) quote areas of up to 28% dose increments in radiotherapy of the intact breast. The modeling examples presented in this article are dependent on the validity of the linear– quadratic model, but have shown the potential enhancement of “double trouble” consequences of high-dose regions. The results presented have used a normal tissue ␣/ ratio of 2 Gy and lower enhancement values will occur by use of a 3-Gy value, as is used for the majority of isoeffect calculations. The “double trouble” effect is capable of working in two directions, namely the effect is deleterious when the treatment is prescribed conventionally, but is protective to normal tissues if the dose is prescribed at the “hot spot” region. The directionality is further compounded by the dissociation in BED values between normal tissue and tumor if the isoeffective dose is determined by a normal tissue isoeffect, so that reduced tumor control is predicted with hypofractionation regardless of the prescription point used. Ideally, clinical trial design for the assessment of hypofractionated schedules should take serious note of these issues. The expected maximum normal tissue BED and the tumor BEDs should be separately calculated for each arm of a trial. A single value of BED at one point (e.g., the prescription point) will not adequately describe an entire treatment when there is significant inhomogeneity of dose within the PTV. It is recommended that BED should be calculated for the prescribed dose and at any normal tissue regions that contain a significantly higher dose within the PTV. For tumor control, the BED for any significant region that receives a lower dose should also be considered. In future, the use of integrated BED values taken across specific regions of interest (9) and dose–volume histograms converted to include BED values (10) with three-dimensional radiation planning should provide more scope for the application of these concepts. This paper has concentrated upon the effects of hypofractionation, but hyperfractionation and treatment acceleration will be considered in a further report.
(26)
representing a 9.0% reduction when compared with the tumor BED obtained with the conventional fractionation (84 Gy5). A qualitative summary of the general findings are given in Tables 1 and 2.
Table 2. Matching for normal tissue effect at “hot spot” region BED Matching Location
Conventional fractionation
Hypofractionation
Intersectional prescription point “Hot spot” region
BED2 (⬍BED1) BED1
BED3 (⬍BED2) BED1
SUMMARY High-dose regions within a PTV, when examined in terms of biological equivalent dose, can provide the potential for “double trouble” effects that manifest as normal tissue damage. Such effects are further exacerbated by use of hypofractionation. The additional BED in critical normal tissue due to hypofractionation is relatively modest but is capable of causing further toxicity. This problem can be overcome by matching the BED values for different schedules at the high-dose region, in which case the hypofractionated schedule is predicted to produce a lower toxicity than conventional fractionation. Tumor control is potentially lower in each case and particularly in the hypofractionated arm. Considerable care is therefore necessary in the treatment planning and dose prescription of hypofractionated radiotherapy.
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REFERENCES 1. Dale RG. The application of the linear quadratic theory to fractionated and protracted radiotherapy. Br J Radiol 1985; 58:515–528. 2. Fowler JF The linear quadratic formula and progress in fractionated radiotherapy. Br J Radiol 1989;62:679 – 694. 3. Dale RG, Jones B. The clinical radiobiology of brachytherapy. Br J Radiol 1998;71:465– 483. 4. Jones B, Dale RG. Mathematical models of tumour and normal tissue response. Acta Oncol 1999;38:883– 893. 5. Scalliet P, Cosset J-M, Wambersie A. Application of the LQ model to the interpretation of absorbed dose distribution in the daily practice of radiotherapy. Radiother Oncol 1991;22:180 – 189. 6. Lee SP, Leu MY, Smithers JB, et al. Biologically effective dose
7. 8. 9. 10.
distribution based on the linear– quadratic model and its clinical relevance. Int J Radiat Oncol Biol Phys 1995;33:375–389. Bates TD. A prospective clinical trial of post operative radiotherapy delivered in three fractions per week versus two fractions per week. Clin Radiol 1975;26:297–304. Neal AJ, Mayles WPM, Yarnold JR. Invited review: Tangential breast irradiation—Rationale and methods for improving dosimetry. Br J Radiol 1994;67:1149 –1154. Dale RG, Coles IP, Deehan C, et al. The calculation of integrated biological response in brachyhterapy. Int J Radiat Oncol Biol Phys 1997;38:633– 642. Wheldon TE, Deehan C, Wheldon EG, et al. The linear quadratic transformation of dose volume histograms in fractionated radiotherapy. Radiother Oncol 1998;46:285–295.
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BIOLOGY CONTRIBUTION
BIOLOGICAL EQUIVALENT DOSE ASSESSMENT OF THE CONSEQUENCES OF HYPOFRACTIONATED RADIOTHERAPY B. JONES, M.SC., M.D., FRCR,* R. G. DALE, PH.D.,† P. FINST, FIPEM,† S. J. KHAKSAR, M.B., B.CH. MRCP*
AND
*Cancer Therapeutics Section, Imperial College School of Medicine, Department of Clinical Oncology, Hammersmith Hospital, London, United Kingdom; and †Department of Radiation Physics and Radiobiology, Imperial College School of Medicine, Hammersmith Hospitals NHS Trust, Charing Cross Hospital, London, United Kingdom Purpose: To investigate the changes in biological effective dose (BED) that occur in high-dose regions within a target volume when radiotherapy is hypofractionated. Methods and Materials: By comparing a standard prescription of 2 Gy per fraction that is matched to give the same BED as a hypofractionated schedule at a standard intersectional prescription point, the BED increments for late-tissue effects at a higher dose region within the planning target volume (PTV) are compared. The alternative approach of BED matching between a conventional and hypofractionated schedule at the high-dose region is also considered. The results are presented as a sequence of calculations that can be understood by practicing radiation oncologists and in graphical form. Results: The BED increment at the high-dose region is marginally increased by hypofractionation, although the latter effect is relatively small: up to 5% additional BED due to hypofractionation for a 20% increase in physical dose when the prescribed fraction size is 6 –7 Gy. BED matching for late effects between a conventional and hypofractionated schedule at the high-dose region produces lower BED values throughout the remaining PTV, but at the expense of a reduced tumor control BED. Conclusion: Clinical trials that use BED isoeffect matching for late reacting tissue effects to design a hypofractioned test schedule should include comprehensive calculations of the likely BED in high-dose regions. © 2000 Elsevier Science Inc. Radiotherapy, Radiobiology, Linear– quadratic model, Clinical trials.
INTRODUCTION
2-Gy fractions, which is to be compared with an alternative schedule given in 3 Gy fractions, late normal tissue damage being required to be similar/identical in both arms. Treatment is prescribed at the intersectional point of the mid axis of the treatment beams, and there are regions of higher dose within the planning target volume owing to the irregular tissue contour and electron-density inhomogeneity as occurs for example in the treatment of the intact breast. Matching of BEDs for late effects, using an ␣/ ratio of 2 Gy appropriate for neural late-reacting tissue, is achieved at the prescription point. The alternative of a prescribed dose, matched to give the same BED for both arms of the trial at the high- dose region is also used, and the resultant tumor BEDs are compared, using ␣/ ratios of 10 Gy and 5 Gy, appropriate for squamous cell cancers and breast cancers, respectively (4). These modeling examples are not intended to simulate any particular type of radiotherapy treatment, but are designed to demonstrate a general effect.
Biological equivalent dose (BED) is an established measure for quantifying the expected biological effect of different radiation dose fractionation schedules (1–3). The BED concept is generally considered to be a good method for designing a test time-dose–fractionation schedule that is to be compared with a conventional schedule in a clinical trial. However, for the comparison to be valid, it is essential to ensure that the technical aspects, such as field sizes, field positions, etc., are identical in both arms of a trial. There are some subtle interactions of physical dose gradients and radiobiology that should be considered when trials are designed and their results analysed. To illustrate these general principles, some worked examples are presented. MATERIALS AND METHODS Consider a hypothetical clinical trial in which the control arm is a conventional schedule of 60 Gy in 30 ⫻ Reprint requests to: Dr. B. Jones, Cancer Therapeutics Section, Imperial College School of Medicine, Department of Clinical Oncology, Hammersmith Hospital, Du Cane Road, London W12 OHS, United Kingdom. E-mail: bleddyn.jones @ ic.ac.uk
Accepted for publication 7 December 1999.
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The BED is given by
冋
BED ⫽ nd 1 ⫹
d ␣/
册
(1)
where n and d are the numbers of fractions and dose per fraction, respectively. We shall consider the BED at areas where the dose is greater than that prescribed. For a dose increment of x, the BED will then be
冋
xnd 1 ⫹
册
xd . 共␣/兲
Volume 47, Number 5, 2000
Thus, if the critical tissue BED is to be maintained, hypofractionation reduces the tumor BED. This will result in a reduced tumor control probability for radical treatments, although there may be no significant change in treatment outcome where palliation is the endpoint. Now consider the effect of a higher dose region within the planning target volume (PTV) that may give rise to late-tissue complications. Assume this region has a 5% higher dose, such that both dose and dose per fraction will be multiplied by a factor of 1.05. In the control arm, the BED in the higher dose region will then be
(2)
冉
30 ⫻ 2 ⫻ 1.05 ⫻ 1 ⫹
The control arm BED is
冉 冊
2 ⫽ 120 Gy 2 2
60 1 ⫹
(3)
where the ␣/ ratio (or fractionation sensitivity) for late effects in the critical normal tissue is 2 Gy. To find the number (n) of fractions at 3 Gy per fraction to provide the same BED,
冉 冊
n⫻3 1⫹
3 ⫽ 120 Gy 2 2
(4)
leading to n ⫽ 16 and so the total dose would equal 48 Gy. RESULTS The tumor BED values (proliferation ignored), assuming ␣/ ratios of 10 Gy or 5 Gy, based on the prescription point BED would be as follows. For the control arm:
冉
冊
(5)
冉 冊
(6)
60 1 ⫹
2 ⫽ 72 Gy 10 10
or 60 1 ⫹
2 ⫽ 84 Gy 5. 5
For the hypofractionated arm:
冉
冊
(7)
冉 冊
(8)
48 1 ⫹
3 ⫽ 62.4 Gy 10 10
or 48 1 ⫹
3 ⫽ 76.8 Gy 5. 5
冊
2 ⫻ 1.05 ⫽ 129.15 Gy 2 2
(9)
which represents a 7.6% increase in BED2 relative to a plan in which no high dose occurs. In the hypofractionated arm, the BED in the higher dose region is
冉
16 ⫻ 3 ⫻ 1.05 1 ⫹
冊
3 ⫻ 1.05 ⫽ 129.78 Gy 2. 2
(10)
This represents an 8.2% increase in BED2 in the high-dose region compared with an increase of 7.6% for the conventional treatment in the same region. Such a small increase (of 0.6% in BED) may not be sufficiently significant to cause a detectable increase in normal tissue toxicity in the example given, but there is undoubtedly a marginal enhancement of the BED. Thus, it can be seen that combination of a higher physical dose and a larger fraction size leads to a proportionately greater increase in BED: a variant of the well-known “double trouble” effect, which can be problematic in many treatment situations (5, 6). The previous studies of the “double trouble” effect did not extend the argument to include hypofractionation: the modeling was based on prescribed doses of 2 Gy per fraction. The effect described above, due to hypofractionation, will be greater: (1) if dose per fraction is further increased in the test arm of a trial , and (2) if a higher percentage physical dose occurs in normal tissues within the PTV. When clinically significant, the effect could be termed “treble trouble,” because the “double trouble” effect at high-dose regions treated to 2 Gy per fraction is compounded by hypofractionation. The BED enhancements that result may be described by incremental ratios of BED equations, which are derived as follows. For the control arm (n fractions of dose d), the BED at the region of high dose, obtained by multiplication of the prescribed dose and dose per fraction by a factor x (such as 1.05 in the above example for a 5% increase), is divided by the BED at the prescription point to give:
Biological equivalent dose assessment of the consequences of hypofractionated radiotherapy
冉 冉
冊 冉 冊 冉
冊 冊
xd xd x 1⫹ ␣/ / BED RATIO 1 ⫽ ⫽ . d d nd 1 ⫹ 1⫹ ␣/ ␣/ xnd 1 ⫹
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(11)
For a nonconventional hypofractionated schedule (m fractions of dose h), the appropriate ratio will be
冉 冉
冊 冉 冊 冉
冊 冊
xh xh x 1⫹ ␣/ ␣/ ⫽ . BED RATIO 2 ⫽ h h mh 1 ⫹ 1⫹ ␣/ ␣/ xmh 1 ⫹
(12)
The ratio that describes the additional fractional increment in BED (termed ⌬BED) that occurs as a direct result of hypofractionation will be ⌬BED ⫽
BEDRATIO2 . BEDRATIO1
(13)
Therefore,
冉 冉 冊冉 冉 冊冉
hx d 1⫹ 共␣/兲 ␣/ h dx 1⫹ 1⫹ ␣/ ␣/
1⫹
⌬BED ⫽
冊
冊
冉 冊冉 冊 冉 冊冉 冊
␣ ␣ ⫹ hx ⫹d   ⫽ . ␣ ␣ ⫹h ⫹ dx  
(14)
Equation 3 is independent of the number of fractions in each arm of the trial, but depends on the dose per fraction in each case. This equation provides the additional increment in BED due to hypofractionation within the high-dose region between the two schedules as the treatment inhomogeneity (x) increases. In the example already given above, ⌬BED is 129.8/129.2 ⫽ 1.005. Such a value is very small (and clinically insignificant) but it must be recalled that this result expresses the further BED increment that occurs in addition to the higher BED that occurs at high-dose regions during conventionally fractionated therapy, i.e., an increment in BED beyond the “double trouble” effect. More general results are given in Fig. 1, where x and dose per fraction of the test/hypofractionated schedule are varied. The ⌬BED ratio increases further, reaching a value of 5% for a 25% dose per fraction increment due to dose inhomogeneity but this degree of change is only seen when the prescribed hypofractionated schedule dose per fraction is increased to be as large as 6 –7.5 Gy per fraction. The effect can also be illustrated in terms of the calculated equivalent total dose at 2 Gy per fraction that would be delivered to the higher dose region. This form of presenta-
Fig. 1. The relationship between the enhancement of biological equivalent dose (BED) at a high-dose region and the prescribed dose per fraction for variable values of x, the physical dose increment factor. When x ⫽ 1.05, there is a dose of 105% relative to the prescription dose, etc.
tion of high dose is far easier for clinicians to comprehend than a BED increment and involves no further assumptions (which would be required for the calculation of normal tissue complication probabilities). In Fig. 2, it can be seen that, for a prescribed 2 Gy per fraction, there is a significant stepwise increment in the equivalent total dose (calculated at 2 Gy per fraction) due to the increasing “hot spot” or “double trouble” effect: from a baseline of 50 Gy in 25 fractions to 54 Gy for a 5% additional “hot spot” dose, to 58 Gy for a 10% additional “hot spot” dose, etc. Then, as the prescribed dose per fraction increases beyond 2 Gy, there is an additional gradual increase in the equivalent total dose. For example, at the 20% high-dose region (an x factor of 1.20), a change in the prescribed dose per fraction from 2 Gy to an extreme value of 7 Gy increases the equivalent dose at the “hot spot” from approximately 66 Gy to 70 Gy
Fig. 2. Plot of estimated total dose standardized to 2 Gy per fraction at high-dose regions for a prescribed dose of 50 Gy in 25 fractions, for variable values of x, the physical dose increment factor at the hot spot regions.
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(in 2-Gy fractions). Such BED increments are larger where the higher dose is greater. These secondary increments again appear small, but these may be clinically significant in large treatment volumes or in situations where the trial endpoint (e.g., neurological complications) may increase steeply with total dose and dose per fraction. Thus, a trial that uses BED calculations to match a conventional dose-fractionation schedule at the prescription point may be prejudicial against a hypofractionated treatment, despite the apparent isoeffect. The alternative is to calculate the isoeffect at the high-dose region of interest that is relevant to normal tissue complications. The penalty for doing this would be that a smaller tumor BED would be delivered in the hypofractionated arm at the prescription point and this may adversely effect the tumor control. For example, if a significant volume of normal tissue has a 7% high dose within the PTV in the original trial control arm referred to above (60 Gy in 30 fractions) and the BED allowed is to be 120 Gy2 in this region, and that it is decided to specify a lower dose at the standard prescription point, the modified dose prescription will be changed as follows. Dose per fraction for the conventional arm will be given by the solution of
冉
冊
d ⫻ 1.07 30 ⫻ d ⫻ 1.07 1 ⫹ ⫽ 120 2
(15)
Dose per fraction (h) for hypofractionated arm will be given by the solution of
冊
(16)
if 16 fractions are given, and h ⫽ 2.80 Gy per fraction. These doses will result in BED‘s in normal tissues that receive the same dose as at the prescription point of :
冉
1.87 ⫻ 30 1 ⫹
冊
1.87 ⫽ 108.6 Gy 2 2
(17)
for the control arm and
冉
2.8 ⫻ 16 1 ⫹
冊
2.8 ⫽ 107.5 Gy 2 2
冉
30 ⫻ 1.87 1 ⫹
冊
1.87 ⫽ 66.6Gy 10. 10
(19)
This represents a 7.5% reduction compared with a BED of 72 Gy10 when the dose prescription is 2 Gy per fraction. (b) for an ␣/ of 5 Gy
冉
30 ⫻ 1.87 1 ⫹
冊
1.87 ⫽ 77.1Gy 5. 5
(20)
This represents an 8.2% reduction compared with a BED of 84 Gy5 when the dose prescription is 2 Gy per fraction. 2. Hypofractionated arm tumor BED (a) for an ␣/ of 10 Gy
冉
冊
2.8 ⫽ 57.3Gy 10. 10
(21)
This represents an 8.2% reduction compared with a BED of 62.4 Gy10 when the dose per fraction is 3 Gy (b) for an ␣/ of 5 Gy
d ⫽ 1.87 Gy per fraction.
冉
ated arm of the trial than in the conventional arm: a finding that is not intuituve. There are, however, further implications for tumor BEDs. In each case these will be: 1. Control arm tumor BED (a) for an ␣/ of 10 Gy
16 ⫻ 2.8 1 ⫹
or
h ⫻ 1.07 ⫽ 120 16 ⫻ h ⫻ 1.07 1 ⫹ 2
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for the hypofractionated arm. Thus, the probability of normal damage within the whole PTV will now be reduced rather more in the hypofraction-
冉
16 ⫻ 2.8 1 ⫹
冊
2.8 ⫽ 69.9 Gy 5. 5
(22)
This represents a 9% reduction compared with a BED of 76.8 Gy5 when the dose per fraction is 3 Gy. Thus, to guard against excess normal tissue damage by prescribing to the area of higher dosage, a reduction of tumor control may result. The reduction in tumor BED may not necessarily be significant in situations where: (a) relatively low tumor BED values may be sufficient to provide good outcomes, such as in postoperative radiotherapy, because the number of tumor clonogens may be relatively small (7), or where partial tumor shrinkage or clonogen sterilization is required prior to definitive surgery; (b) there is excellent normal tissue sparing due to focal forms of radiotherapy, e.g., conformal radiotherapy, protons and high-dose-rate brachytherapy (3); then, a higher tumor BED may be safely given; (c) highly radiosensitive tumors are treated; and (d) the aim of treatment is palliative, because there is usually a less-clear dose–response relationship for outcomes such as hemostasis or pain relief. A further example of the above problem would be if the conventional treatment continues to be prescribed at the standard intersectional point to 60 Gy in 30 fractions. The 7% additional dose at the hot spot would then result in a BED of
Biological equivalent dose assessment of the consequences of hypofractionated radiotherapy
Table 1. Matching for normal tissue isoeffect at intersectional prescription point BED Matching Location
Conventional fractionation
Hypofractionation
Intersectional prescription point “Hot spot” region
BED1 BED2 (⬎BED1)
BED1 BED3 (⬎BED2)
冉
30 ⫻ 2 ⫻ 1.07 ⫻ 1 ⫹
冊
2 ⫻ 1.07 ⫽ 132.9 Gy 2. 2
(23)
The hypofractionated arm could be matched to this BED (if it is thought that this BED will be well tolerated). Consequently, the hypofractionated dose per fraction at the “hot spot” would be given by the solution for d in
冉 冊
16 ⫻ d 1 ⫹
d ⫽ 132.9. 2
(24)
From which d ⫽ 3.2 Gy At the original prescription point, the hypofractionated dose per fraction would then be d/1.07 ⫽ 2.99 Gy. Now, the tumor BED would be (a) for ␣/ ⫽ 10 Gy
冉
16 ⫻ 2.99 ⫻ 1 ⫹
冊
2.99 ⫽ 62.1 Gy 10. 10
(25)
This represents a 13.8% reduction when compared with the tumor BED obtained with the conventional fractionation (72 Gy10) (b) for ␣/ ⫽ 5 Gy
冉
16 ⫻ 2.99 ⫻ 1 ⫹
冊
2.99 ⫽ 76.4 Gy 10 5
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DISCUSSION Wide field treatments can be associated with significant dose inhomogeneity. For example, Neal et al. (8) quote areas of up to 28% dose increments in radiotherapy of the intact breast. The modeling examples presented in this article are dependent on the validity of the linear– quadratic model, but have shown the potential enhancement of “double trouble” consequences of high-dose regions. The results presented have used a normal tissue ␣/ ratio of 2 Gy and lower enhancement values will occur by use of a 3-Gy value, as is used for the majority of isoeffect calculations. The “double trouble” effect is capable of working in two directions, namely the effect is deleterious when the treatment is prescribed conventionally, but is protective to normal tissues if the dose is prescribed at the “hot spot” region. The directionality is further compounded by the dissociation in BED values between normal tissue and tumor if the isoeffective dose is determined by a normal tissue isoeffect, so that reduced tumor control is predicted with hypofractionation regardless of the prescription point used. Ideally, clinical trial design for the assessment of hypofractionated schedules should take serious note of these issues. The expected maximum normal tissue BED and the tumor BEDs should be separately calculated for each arm of a trial. A single value of BED at one point (e.g., the prescription point) will not adequately describe an entire treatment when there is significant inhomogeneity of dose within the PTV. It is recommended that BED should be calculated for the prescribed dose and at any normal tissue regions that contain a significantly higher dose within the PTV. For tumor control, the BED for any significant region that receives a lower dose should also be considered. In future, the use of integrated BED values taken across specific regions of interest (9) and dose–volume histograms converted to include BED values (10) with three-dimensional radiation planning should provide more scope for the application of these concepts. This paper has concentrated upon the effects of hypofractionation, but hyperfractionation and treatment acceleration will be considered in a further report.
(26)
representing a 9.0% reduction when compared with the tumor BED obtained with the conventional fractionation (84 Gy5). A qualitative summary of the general findings are given in Tables 1 and 2.
Table 2. Matching for normal tissue effect at “hot spot” region BED Matching Location
Conventional fractionation
Hypofractionation
Intersectional prescription point “Hot spot” region
BED2 (⬍BED1) BED1
BED3 (⬍BED2) BED1
SUMMARY High-dose regions within a PTV, when examined in terms of biological equivalent dose, can provide the potential for “double trouble” effects that manifest as normal tissue damage. Such effects are further exacerbated by use of hypofractionation. The additional BED in critical normal tissue due to hypofractionation is relatively modest but is capable of causing further toxicity. This problem can be overcome by matching the BED values for different schedules at the high-dose region, in which case the hypofractionated schedule is predicted to produce a lower toxicity than conventional fractionation. Tumor control is potentially lower in each case and particularly in the hypofractionated arm. Considerable care is therefore necessary in the treatment planning and dose prescription of hypofractionated radiotherapy.
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REFERENCES 1. Dale RG. The application of the linear quadratic theory to fractionated and protracted radiotherapy. Br J Radiol 1985; 58:515–528. 2. Fowler JF The linear quadratic formula and progress in fractionated radiotherapy. Br J Radiol 1989;62:679 – 694. 3. Dale RG, Jones B. The clinical radiobiology of brachytherapy. Br J Radiol 1998;71:465– 483. 4. Jones B, Dale RG. Mathematical models of tumour and normal tissue response. Acta Oncol 1999;38:883– 893. 5. Scalliet P, Cosset J-M, Wambersie A. Application of the LQ model to the interpretation of absorbed dose distribution in the daily practice of radiotherapy. Radiother Oncol 1991;22:180 – 189. 6. Lee SP, Leu MY, Smithers JB, et al. Biologically effective dose
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distribution based on the linear– quadratic model and its clinical relevance. Int J Radiat Oncol Biol Phys 1995;33:375–389. Bates TD. A prospective clinical trial of post operative radiotherapy delivered in three fractions per week versus two fractions per week. Clin Radiol 1975;26:297–304. Neal AJ, Mayles WPM, Yarnold JR. Invited review: Tangential breast irradiation—Rationale and methods for improving dosimetry. Br J Radiol 1994;67:1149 –1154. Dale RG, Coles IP, Deehan C, et al. The calculation of integrated biological response in brachyhterapy. Int J Radiat Oncol Biol Phys 1997;38:633– 642. Wheldon TE, Deehan C, Wheldon EG, et al. The linear quadratic transformation of dose volume histograms in fractionated radiotherapy. Radiother Oncol 1998;46:285–295.