The linear-quadratic approach in clinical practice

9.1 INTRODUCTION: BIOLOGICAL EFFECT ESTIMATES ADJUSTING FOR DOSE–TIME FRACTIONATION

Since the early years of radiation therapy it has been appreciated that the biological effect of a given physical absorbed dose of ionizing radiation depends on how this dose is distributed over time. For many years, the differential response of tumours and normal tissues to changes in dose–time fractionation appeared to be the most important means of improving the therapeutic ratio. Mathematical models – often referred to as bioeffect models – were first introduced in the 1920s with the aim of quantifying the biological effect of dose-fractionation schedules on tumour control and normal-tissue side-effects. As discussed in Chapter 8, the linear-quadratic (LQ) model was introduced around 1980 and has gradually become

the model of choice for bioeffect estimation in radiotherapy. In the beginning, the use of the LQ model was conceptually linked to the target-cell hypothesis. However, there is increasing evidence that many late effects, and even some early effects, of radiation therapy are not directly related to simple killing of a defined population of target cells (see Chapter 13, and Bentzen, 2006). The most prevalent current view is that the LQ approach represents an approximate, pragmatic method for converting dose–time fractionation schedules into a biologically effective dose. The LQ model has a limited range of applicability, and extrapolations outside the range of available data should only be performed with the greatest care. Model parameters should be estimated from clinical observations and their statistical precision should be taken into account when used to estimate the biological effect of a given schedule.

While awareness has grown regarding the limitations to the LQ model and the dangers involved in using it uncritically, the application of this approach has also increased. There are several reasons for this. Intensity-modulated radiotherapy (IMRT) represents a convenient way of delivering radiation therapy with varying dose per fraction to multiple target volumes in a single session. At the same time, IMRT and conformal radiotherapy generally leads to non-uniform dose distributions in normal tissues and organs, delivering dose with a varying dose per fraction to various subvolumes – in contrast to parallel opposing field techniques that typically lead to partial organ irradiation with a dose per fraction close to the fraction size prescribed to the target volume. Finally, the realization that the fractionation sensitivities of at least some human tumours are in the same range as that typical of late normal-tis- sue effects (in contrast to what was widely assumed when the LQ model was introduced in the 1980s) has renewed the interest in the use of hypofractionation, that is, prescriptions with fraction sizes larger than 2 Gy.

9.2 QUANTITATIVE CLINICAL RADIOBIOLOGY: THE LQ FRAMEWORK

Several, mathematically equivalent, methods have been devised for performing bioeffect calculations with the LQ model as discussed in Chapter 8. The method presented in this chapter converts all phases of a treatment into the equivalent dose in 2-Gy fractions (EQD2). This has the advantage that these doses are clinically relevant and they are measured on the scale where much of the clinical experience on dose–response relationships is available. The EQD2 values from various parts of a fractionation schedule may be added directly. The EQD2 is identical to the ‘normalized total dose’ (NTD) proposed by Withers and colleagues (1983) (see also Chapter 8).

Large fractionation studies in the laboratory, mainly conducted in rodents in the 1980s, showed the ability of the LQ model to provide a close quantitative relationship between the isoeffective

doses for schedules applying varying dose per fraction. These studies also produced a number of α/βratio estimates for various normal-tissue endpoints (see Table 8.1). In parallel with these experimental studies, a number of clinical studies have produced α/βestimates for human endpoints and these are summarized in Table 9.1.

Bioeffect calculations should be used only as guidance for clinical decision-making. All of the formulae applied here have a limited field of applicability; the model assumptions may be violated in some circumstances, relevant parameters may not be known for human tissues and tumours and the uncertainty in parameter estimates, even when these are available, may give rise to considerable uncertainty in the biological effect estimates. A (self-)critical and cautious attitude is recommended and the health and safety of patients should not be compromised by reliance on the result of calculations of the type described in this chapter. We advocate the use of clinical parameter estimates whenever possible. If no clinical estimates are available we suggest using the values from experimental animal studies as a guide, but be well aware of the fact that these may not be valid for the clinical endpoints of interest. The use of ‘generic’ values, say, 3 Gy for late effects and 10 Gy for tumours, should be seen as the least evidence-based approach. There is less and less reason to believe that these values can be generalized across a wider range of human normal-tissue endpoints and tumour histologies. Therefore, calculations using these values may be seen as simply exploring the behaviour of the model – an exercise detached from the clinical reality that we should be studying. That said, a numerical estimate is often very useful when considering various therapeutic options and it is often possible to get an impression of how reliable such an estimate is, just by doing a simple calculation as illustrated in this chapter.

9.3 CHANGING THE DOSE PER FRACTION

The simplest case we will consider is when the dose per fraction is changed without change in the overall treatment time and when incomplete

repair between dose fractions is negligible. We use the Withers formula to convert a total dose D delivered with dose per fraction d into the isoeffective dose in 2-Gy fractions:

Note, that the only parameter in this formula is the α/β ratio, which is a characteristic of the endpoint of interest. Any biological dose calculation will therefore start with the identification of the tumour or normal-tissue endpoint of concern in the clinical situation. For a given fractionation schedule we may, for example, be interested in EQD2 for a squamous cell carcinoma of the lung and for lung fibrosis and we would then start by

Example 1. Converting a dose into the isoeffective dose in 2-Gy fractions

A patient with metastatic bone pain located to the 5th thoracic vertebra is considered for palliative radiotherapy using 4 5 Gy.

Problem: What is the isoeffective dose in 2-Gy fractions for spinal cord?

Solution: First, we need to choose the value of α/β. From Table 9.1 it is seen that the upper bound on α/β from human data is 3.5 Gy. Experimental animal studies (see Table 8.1) have produced estimates around 2 Gy. We choose α/β 2 Gy, insert the values for total dose, 20 Gy, and dose per fraction, 5 Gy, in equation 9.1 and get:

EQD2 20 Gy 5 Gy 2 Gy 35 Gy 2 Gy 2 Gy

Thus, 20 Gy delivered in 5-Gy fractions is biologically equivalent to 35 Gy in 2-Gy fractions for an endpoint with α/β 2.0 Gy.

selecting appropriate α/β ratios from Tables 8.1, 9.1 or 13.2.

9.4 CHANGING THE TIME-INTERVAL BETWEEN DOSE FRACTIONS

Multiple fractions per day schedules are associated with an increase in biological effect unless the interval between fractions is sufficiently long to allow full recovery between fractions. There are data to suggest that the characteristic halftime of recovery is in the order of 4–5 hours for some human late endpoints (Bentzen et al., 1999) and possibly even longer for spinal cord and brain (Dische and Saunders, 1989; Lee et al., 1999). This means that recovery will not be complete even with a 6- to 8-hour interval between fractions. In this situation it is necessary to modify the simple LQ model as described in Chapter 8. Equation 8.7 can be used under the assumption that repair is complete in the long overnight interval (i.e. between the last fraction delivered in one day and the first fraction on the following day). Even this assumption starts to be problematic with repair halftimes of 4–5 hours. Guttenberger et al. (1992) derived a formula where unrepaired damage is allowed to accumulate throughout the fractionation course. Unfortunately, this formula is not easy to tabulate as it depends not only on the repair halftime and the dose per fraction but also on the exact arrangement of dose fractions over time.

9.5 CONTINUOUS IRRADIATION

In brachytherapy, repair takes place not only after irradiation but also during the application. In this case, the apparent dose per fraction is modified by a function of the exposure time and the repair or recovery halftime, T1/2 (see Chapter 8, Section 8.8). Depending on the detailed dose rates, the isoeffective dose may depend strongly on T1/2 for a given endpoint. As these halftimes are usually not known with any useful precision from clinical data, great care should be taken when interpreting the results of isoeffect calculations for continuous irradiations.

A patient with head and neck squamous cell car-

cinoma (HNSCC) is prescribed 70 Gy in 2-Gy fractions over 7 weeks. After 50 Gy, he has an intercurrent pneumonia and cannot attend radiotherapy for 1 week. In order to finish treatment on time, it is decided to give the last 20 Gy as 2 2 Gy per day on the last five treatment days.

Problem: What is the isoeffective dose in 2-Gy fractions for subcutaneous fibrosis for a 6-hour interval and an 8-hour interval?

Solution: First, we need to choose the values

of α/β and T1/2. From Table 9.1 it is seen that α/β 1.7 Gy and from Table 9.2 we get

T1/2 4.4 hours. We now use equation 8.7:

In this case, m 2 and we use Table 8.2 to look up the values of H2. For an interfraction interval of 6 hours, we see that H2 is between 0.35

(T1/2 4.0 hours) and 0.44 (T1/2 5.0 hours). Interpolation between these values yields

H2 0.39 for T1/2 4.4 hours and therefore:

EQD2 ( t 6 hours) 50Gy

20Gy 2[1 0.39]Gy 1.7 Gy 74.2Gy

2Gy 1.7 Gy

For an 8-hour interval the value of H2 is between

0.25 (T1/2 4.0 h) and 0.33 (T1/2 5.0 hours). Interpolation between these values yields

H2 0.28 for T1/2 4.4 hours and therefore:

EQD2 ( t 8 hours) 50Gy

20Gy 2[1 0.28]Gy 1.7 Gy 73.0Gy 2Gy 1.7 Gy

Thus, about 1.2 Gy will be spared from increasing inter-fraction intervals from 6 to 8 hours.

As mentioned above, this calculation assumes that the overnight interval is sufficiently long to assure complete repair of sublethal damage. This assumption starts to break down when T1/2 is 4.4 hours. A calculation using the formula of Guttenberger et al. (1992) gives an EQD2 for the last week of twice daily treatment of 25.2 Gy (rather than 24.2 Gy as calculated in the example) for the 6-hour interval and 24.2 Gy (rather than 23.0 Gy) for the 8-hour interval.

Example 3 shows, what we would also expect intuitively, that the effect of protracting the delivery of a dose fraction is reduced when the recovery halftime is longer (i.e. less recovery will take place during the irradiation). A 5-Gy fraction delivered with acute dose rate, that is assuming no recovery at all during delivery, would correspond to an EQD2 of 7.9 Gy for α/β 3.1 Gy.

9.6 CHANGING THE OVERALL TREATMENT TIME

Very often, two fractionation schedules will differ in overall treatment time. There are good reasons to believe that overall treatment time has very little, if any, influence on late radiation effects (Bentzen and Overgaard, 1995). However, for most tumour types and for early endpoints, the biological effect of a specific dose-fractionation will decrease if overall treatment time is increased. In other words,

an extra dose will be needed to obtain the same level of effect in a longer schedule. This has traditionally been interpreted as the result of proliferation of target cells in the irradiated tissue or tumour, and many attempts have been made to include this effect in the LQ model. Experimental animal studies have shown that this is a non-linear effect as a function of time; in other words, the dose recovered per unit time will change as a function of the time since the initial trauma. At present, there is no mathematical model describing this recovery over extended intervals. Instead, the most cautious approach is to use a simple linear relationship in a fairly narrow interval around the overall time of the schedule from which it has been estimated. The magnitude of the time effect is most conveniently

quantified by Dprolif, which is the dose recovered per day due to proliferation. However, the exact mech-

anism behind this recovery is not critical for this pragmatic correction. For minor changes in overall time, say, a 4-day protraction of a schedule, the

simple estimate would then be that the EQD2 has to

be reduced by 4 Dprolif. Using this method, the isoeffective doses in 2-Gy fractions delivered over

two different times, t and T, will be related as:

Note, that if T t then EQD2,T EQD2,t. There is no simple rule specifying the maximum difference between the two times, T and t, where this linear correction is reasonable. For a 1-week difference this approximation would probably be reasonable whereas for a 3- to 4-week difference this would not

be a safe assumption. Another concern is that Dprolif may depend on the intensity of the schedule; in

other words, a very short intensive schedule may

give rise to an increased value of Dprolif. A serious warning is that little is known about the value of

Dprolif for various tumours and normal-tissue endpoints and that most of the published values only

apply towards the end of a standard 6- to 8-week schedule. For tumours, the majority of available estimates are for squamous cell carcinoma of the head and neck (Table 9.3). As mentioned above, it

appears safe to assume that Dprolif is zero for late endpoints at least for overall treatment times up to

6–8 weeks. For early reactions, a linear correction is also applicable over a limited range of treatment times, see Bentzen et al. (2001).

The Danish Head and Neck Cancer Group (DAHANCA) conducted a randomized controlled trial of 66–68 Gy in 33–34 fractions randomizing between five and six fractions per week (Overgaard et al., 2003). This trial, DAHANCA 6/7, comprised 1476 patients with HNSCC receiving definitive radiotherapy without chemotherapy.

Problem: What is the expected difference in biologically effective dose for HNSCC between the two arms of the trial?

Solution: Starting treatment on a Monday, 33 fractions delivered at five fractions per week will take 6 full weeks plus 3 additional treatment days (a total of 45 days). With six fractions

per week the overall treatment time becomes 5 full weeks plus 3 additional treatment days (a total of 38 days). From Table 9.3 we see that

Dprolif for HNSCC is about 0.7 Gy/day. There is no demonstrable difference in Dprolif among the various subsites of the head and neck region. We

insert these values in equation 9.2 and get:

EQD2,38 EQD2,45 (38 45) days

0.7 Gy/day

66 Gy 4.9 Gy 70.9 Gy

Thus, 66 Gy delivered over 38 days is biologically equivalent to 70.9 Gy in 2-Gy fractions delivered over 45 days for HNSCC. This time difference will be the same for a patient receiving 68 Gy in 34 fractions with five or six fractions per week.

If the two schedules in Example 4 had employed doses per fraction that were different from 2 Gy, we would first have calculated the equivalent EQD2 values before doing the time correction.

9.7 UNPLANNED GAPS IN TREATMENT

A problem frequently encountered in radiotherapy practice is the management of unscheduled treatment interruptions. Studies have shown that up until 1990 about a third of all patients with HNSCC experienced one or more unplanned gaps in treatment leading to a protraction of overall treatment time of more than 6 days. These interruptions were typically caused by patient-related factors (intercurrent disease, severe radiation reactions) or logistic factors (public holidays, treatment machine downtime, transport difficulties). The management of treatment gaps has been considered in some detail by a working party of the UK Royal College of Radiologists (Hendry et al., 1996). The recommendation to avoid gaps or actively modify treatment after a gap is based on the clinical evidence for a negative therapeutic effect of gaps in radiotherapy schedules. This evidence is strongest for squamous cell carcinoma (SCC) of the head and neck, non-small cell lung cancer and cancer of the uterine cervix, and it is therefore recommended that the remaining part of the treatment be modified in order to adjust for the unscheduled interruption. There is also some support for the importance of overall treatment time in SCC of the skin and vagina and in medulloblastoma. Less evidence exists for other radical treatments and there is no reason to believe that overall treatment time is a significant factor in palliative radiotherapy.

Treatment schedules may be adjusted by accelerating radiotherapy after the gap. In a planned schedule delivering one fraction per day, 5 days per week, this can be accomplished by giving more than five fractions per week, either as two fractions per day, as in Example 2 above, or preferably by treating on Saturday and/or Sunday. The idea is to deliver the planned total dose, with the prescribed dose per fraction, in as near the planned overall time as possible. If two fractions per day are delivered, these should be separated by the maximum practical interval (at least 6 hours and preferably more). Note that with long repair halftimes, incomplete repair between the dose fractions may require a dose reduction if the chance of late complications is kept fixed. Alternatively, delivering the remaining part of the treatment with hypofractionation may be considered. Whether this type of adjustment will lead to increased late sequelae or decreased tumour control depends on the exact values of α/βfor the relevant late normal-tissue endpoints and the tumour type in question.

9.8 ERRORS IN DOSE DELIVERY

Dosimetric errors in delivering the prescribed dose per fraction made early in a treatment can be corrected by modifying the dose per fraction and total dose given subsequent to discovery of the error, using the LQ model to calculate the correcting doses, which should be completed within the

A patient with colorectal cancer is planned to receive preoperative radiotherapy with five times 5 Gy from Monday to Friday (Kapiteijn et al., 2001). The first two fractions are given as planned on Monday and Tuesday, but owing to a machine breakdown, no treatment could be given on Wednesday. In order to finish as planned on Friday, delivering the isoeffective tumour dose by increasing the size of the two fractions to be given on Thursday and Friday is considered. We assume that α/β 10 Gy for colorectal cancer.

Problem: What is the required dose per fraction for the last two fractions? What is the accompanying change in the risk of rectal complications from this modified fractionation schedule?

Solution: The tumour EQD2 for the final three fractions originally planned is:

EQD2 15Gy 5Gy 10Gy 18.75Gy

2Gy 10Gy

We want to estimate the dose per fraction, x, so that delivering two fractions of this size gives an EQD2 of 18.75 Gy to the tumour:

x 10Gy EQD2 2x

2Gy 10Gy

In the following, we remember that x will have the physical units of gray and start solving this quadratic equation:

18.75 2x x 10

2 10

18.75 12 2x2 20x 2x 2 20x 225 0

The solution to a quadratic equation of the form ax bx c 0 is

(Note that only the positive root produces a physically meaningful dose.) In the present case,

we get:

x 20 202 4 2 ( 225) 2 2

20 400 1800 20 46.9 6.7 4 4

Remember the unit: gray. In other words, we would have to give fractions of 6.7 Gy on Thursday and Friday, a total of 13.4 Gy, to achieve the same tumour effect. This is of course less than the 3 5 Gy originally planned for Wednesday to Friday, and the reason for this is the larger effect per gray deriving from the increased dose per fraction in the modified schedule.

How will this affect the risk of bowel damage? From Table 9.1, we find α/βof 4 Gy. The EQD2 of the modified schedule to the bowel is now:

This value does not take the very short overall treatment time of 5 days into consideration and it is possible that such short schedules could involve an increased risk of consequential late reactions. Here, we focus only on the change in biological dose deriving from the change in dose fractionation and we note that the overall treatment time is unchanged in the two schedules compared here. The originally planned 5 Gy corresponds to an EQD2 for bowel of about 37.5 Gy. Therefore, the risk of late bowel morbidity will be increased if this modification is implemented.

Clinically, one would be concerned about increasing dose per fraction from 5 to 6.7 Gy. Biologically, it may be questioned whether the use of the LQ model is safe anyway at these large doses per fraction (see Chapter 8, Section 8.11). It should also be noted that, if we use a lower α/β, for example the 1.7 Gy estimated for fibrosis, the change in EQD2 for this endpoint is expected to be from 45 Gy to 48.5 Gy. It could be considered keeping to 5 Gy per fraction and simply accepting a 3-day protraction (i.e. finishing on Monday) or to give the fifth fraction on Saturday.

same overall time as originally prescribed. If the initial error was giving a larger dose per fraction than planned (a hypofractionated error), then the rest of the treatment should be hyperfractionated to compensate. If the initial error was giving a lower dose per fraction than planned (hyperfractionated), then the rest of the treatment should be hypofractionated to compensate.

Joiner (2004) showed how to calculate the dose per fraction used to bring the treatment back exactly to planned tolerance simultaneously for all tissues and tumour involved, following either hyperfractionated or hypofractionated errors made initially, without the need to know any α/β ratios. Defining planned treatment as p Gy per

fraction to a total dose of P Gy, suppose the initial error is e Gy per fraction given to a total of E Gy. Using the LQ model to describe all isoeffect relationships between total dose and dose per fraction for the tumour and the normal tissues, then the compensating dose per fraction of d Gy to a total dose of D Gy are given by the simple formulae:

Notably, it can be seen that the total dose for the complete treatment (error plus compensation) remains as originally prescribed.

A patient is intended to receive 35 2 Gy per fraction, with five fractions per week to an overall time of 7 weeks. By error, the treatments on each of the first 2 days are given as 4 Gy per fraction.

Problem: What is the dose per fraction that should be given in the remainder of the treatment, to exactly compensate for the initial error?

Solution: The total error dose (E) is 8 Gy; therefore from equation 9.3, the total compensating dose (D) is:

70 Gy 8 Gy 62 Gy

The dose per fraction used for the compensation, given by equation 9.4, should be:

This gives 62/1.74 35.6 fractions for the compensating 62 Gy. Since an integral number of fractions must be given, the nearest compensating treatment is 36 1.72 Gy.

In this example, there are 33 weekday treatment days left to use in the original planned overall time, so three of the 1.72 Gy doses must be given on Saturdays, or using two fractions per day on three of the Fridays, in order not to extend overall treatment time. It must be noted that equations 9.3 and 9.4 are valid only in the absence of incomplete repair; therefore compensating treatments should not be scheduled in a way that would introduce significant incomplete repair. This favours using Saturdays to deliver the additional doses, or leaving the maximum possible time between two doses given on Fridays, but at least 6 hours.

9.9 RE-IRRADIATION

A specific problem in clinical practice is the radiotherapeutic management of patients with a new primary tumour or a loco-regional recurrence in an anatomical site that necessitates re-irradiation of a previously irradiated tissue or organ.

Experimental animal data demonstrate that various

endpoints differ markedly in their capacity for long-term recovery of radiation tolerance (see Chapter 19). Quantitative clinical data are sparse and it may not be safe to assume any particular value of the recovery in a clinical setting. The most reasonable approach is probably to use the LQ model to estimate the EQD2 values for the normal-tissue endpoints of concern without any

explicit recovery assumed, and then to apply a clinical assessment of the re-irradiation tolerance based in part on the experience from animal experiments. Obviously, such an assessment would also include other clinical aspects such as the life expectancy of the patient in relation to the latent period of late damage, and the prospects for long-term benefit from the re-treatment.

9.10 FROM CHANGE IN DOSE TO CHANGE IN RESPONSE RATE

Early papers on the use of the LQ model in biological dose calculations were all concerned with estimation of isoeffective doses. In practice, two dose-fractionation schedules will most often not produce exactly the same biological effect, and even if they are isoeffective with respect to a

specific endpoint they will typically not be so with respect to other endpoints. In these situations, it is necessary to consider the steepness of the dose–response curve in order to estimate the associated change in the incidence of a clinical endpoint in going from one schedule to the other.

In the following, response rate refers to either a tumour control probability or a normal-tissue complication probability. If the response rate is R after a dose D, the change in response rate, in percentage points, after an increment in dose, D, is approximately:

where γn is the local value of the normalized dose–response gradient (see Chapter 5). The use of this formula is illustrated by the following example.

For the DAHANCA 6/7 trial, we calculated in Example 4 that the acceleration in the six fractions per week arm corresponded to an effective 4.9-Gy dose increment. Assume that the local tumour control probability (T-position alone) in the five fractions per week arm is 64 per cent (this is the actual observed control probability in this arm of the trial).

Problem: What is the expected increase in tumour control probability from the treatment acceleration?

Solution: First, we need to choose the value of γ. From Figs 5.4 and 5.5 (Chapter 5), we find that γ50 is around 1.8 for head and neck cancer with a trend towards higher values for vocal cord

tumours. At the 64 per cent level, this value is still valid (see Table 5.1). Using this value of γ allows us to calculate the expected change in response:

R4.9Gy 100% 1.8 13% 66Gy

Thus, the local tumour control probability is expected to increase by 13 per cent. The observed local tumour control rates in the trial were 64 per cent and 76 per cent in the fiveand six-fraction per week arms, respectively (Overgaard et al., 2003). This means that the tumour control rate increased by 12 per cent, in very good agreement with the expected increase calculated here.

9.11 DOUBLE TROUBLE

Dosimetric hot spots receive not only a higher total dose but also a higher dose per fraction. Rodney Withers (see, for example, Lee et al., 1995) has called this phenomenon ‘double trouble’. A hot spot could

arise as a result of internal and external inhomogeneities or because of the radiation field arrangement. A special case is in the match zone between two abutted fields where, depending on the geometrical matching technique used, a small tissue volume could be markedly overdosed or underdosed.

9.12 THE UNCERTAINTY IN BIOLOGICAL EFFECT ESTIMATES

An impression of the uncertainty in biological effect estimates may be obtained simply by vary-

ing the value of α/β or Dprolif or γ50. The idea is to insert the lower and upper 95 per cent confidence

limits of the parameter in question and use these as 95 per cent confidence limits for the biological

effect estimate. This technique is only straightforward if we are concerned with uncertainty in a single parameter, say, uncertainty in α/β. If we want to evaluate the effect of uncertainty in two

parameters, for example Dprolif and γ50, it is necessary to use more advanced methods. It is not a

good approximation simply to insert the lower and upper confidence limits for both parameters in the calculation.

Another useful approach is to identify the main clinical concern. In Examples 8 and 9, the main concern is the added risk of complications associated with the overdosage in the match zone. In Example 5, the main concern is the possible increase of late effects after hypofractionation. In these cases, it is of course cautious to assume a low value for α/β. If, on the other hand, we want to estimate the sparing from hyperfractionation in a given patient, it would be conservative (i.e. we would not be overestimating the benefit) if we assumed a relatively high value of α/β.

A final observation is that the confidence interval for α/β derived from animal studies is generally narrower than for the corresponding human endpoint. This probably reflects that the number of subjects is often larger in the experimental studies and that dose and dose per fraction is varied more systematically and over a wider range of values. As mentioned above, we would suggest that the full calculation, including the estimation of confidence limits, is done using the parameter estimates from the human data whenever possible, and that an independent calculation is done using the animal data.

9.13 SOME CURRENT ISSUES IN THE CLINICAL APPLICATION OF THE LQ MODEL

The question whether the LQ approach has been validated can only be addressed on an endpoint- by-endpoint basis. The LQ model has been applied to datasets from quite a few clinical studies, and there has in many cases been good agreement between the predicted and observed study outcome. This gives some confidence in using the model to estimate the effects of changed dose fractionation in situations where there are clinical parameter estimates. Roughly, the dose per fraction, where use of the LQ model is supported by data, ranges from about 1.0 Gy to 5 Gy or so. As mentioned in the introduction to this chapter, even then there is often a lack of appropriate parameter estimates, or the available estimates have wide 95 per cent confidence limits. Parameter estimates for clinical endpoints remain relatively scarce. Some papers derive parameter

values based on fixing one or more other parameters in the estimation. Clearly, these estimates become circular to some extent. While the clinical importance of low-dose hyper-radiosensitivity remains controversial, there is little empirical support from clinical studies for the use of the LQ model at dose per fraction of 1.0 Gy. This is an important issue as many IMRT techniques involve irradiation of large normal-tissue volumes with dose fractions in this range. At the other end of the spectrum, techniques such as extracranial stereotactic body radiotherapy, or intraoperative radiotherapy or high dose-rate brachytherapy may deliver dose fractions of 10 Gy or more. Again, there is little empirical evidence behind the use of the simple LQ model, or any of the proposed modifications of this model, in this high-dose range. In most cases, the dose fractionation issue is intertwined with the volume effect, and in practice it is difficult to separate these effects from clinical studies where patients receive roughly similar dose distributions (see also Chapter 14).

It should also be mentioned that a number of elegant formulations of the LQ model have been developed, for example for internal emitters or permanent implants. Other formulations include relative biological effectiveness (RBE) or oxygen enhancement ratio (OER) corrections, etc. The interested reader is referred to a recent book edited by Dale and Jones (2007 – see Further reading) for an overview of many of these applications and refinements of the LQ model. From the perspective of clinical radiobiology, many of these models are over-parameterized in the sense that clinical datasets typically will not have sufficient structural resolution to allow estimation of all relevant model parameters, not to mention validating the model assumptions.

Do some tumours have low α/β ratios? Parameter estimates for malignant melanoma and liposarcoma from the late 1980s suggested that this might be the case, but this hypothesis has attracted considerable interest after Brenner and Hall (1999) derived low α/βestimates for prostate cancer from a comparison between external-beam and brachytherapy outcomes. Later studies comparing external-beam only fractionation effects have provided further support for this idea (Bentzen and Ritter, 2005; Lukka et al., 2005). A

number of trials are in progress that should produce valuable data allowing a rigorous test of this hypothesis (Bentzen and Ritter, 2005; Miles and Lee, 2008). There is also emerging evidence that α/βfor subclinical breast cancer may be considerably less than the often assumed 10 Gy; see the report from the UK START Trialists Group (Bentzen et al., 2008). A cautious attitude at present is that fractionation schedules designed to exploit these presumptive low α/βvalues for some cancer types should be tested in prospective clinical trials, as they could lead to major deviations from the established practice in a particular centre.

The final note here is that the move towards increasing use of combined modality therapy in many tumour types represents a further issue regarding the applicability of standard LQ modelling. The approach often taken is to perform bioeffect calculations for the radiation therapy part in isolation. This may produce valid results as long as the chemotherapy is strictly identical in the schedules being compared. However, this approach is also challenged by data suggesting that some cytotoxic and molecular targeted drugs actually modulate the dose–time fractionation response (Bentzen et al., 2007). In this case, radiobiological parameters derived from radiationalone studies may no longer be valid.

While the LQ approach 10 years ago appeared to be rather well established, a number of new developments – mainly driven by clinical and technology advances – have made bioeffect modelling an exciting and important area of clinical and experimental radiation research again.

Key points

1.The LQ approach is the model of choice for bioeffect estimation in radiotherapy and can be used for a wide range of calculations.

2.The dose range where the LQ model is well supported by data is roughly 1–5 Gy per fraction. Extrapolations made outside this range should be done with extreme caution.

3.Clinical parameter estimates should be used in calculations whenever possible. If no clinical estimates are available, values from animal studies may be used as guidance but caution should be exercised in applying the results of such calculations to the clinical situation.

4.Estimates of uncertainty in LQ calculations should always be made, based on the uncertainty in the values of the parameters used in the calculation.

5.In combined modality therapy, it may not be valid to use parameters derived from studies using radiation alone.

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■ FURTHER READING

Dale RG, Jones B (ed.) (2007). Radiobiological modeling in radiation oncology. London: British Institute of Radiology.