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- Abstract
- Mechanistic background to the Linear-Quadratic model
- The LQ formalism
- Does the LQ model predict the variation of effect with dose in the 2 Gy to 18 Gy dose per fraction range?
- Is the LQ model correct?
- Conclusions
- References

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Semin Radiat Oncol. Author manuscript; available in PMC 2009 October 1.

Published in final edited form as:

PMCID: PMC2750078

NIHMSID: NIHMS69828

David J. Brenner, Ph.D., D.Sc.

David J. Brenner, From the Center for Radiological Research, Columbia University Medical Center, 630 West 168th Street, New York, NY.

The publisher's final edited version of this article is available at Semin Radiat Oncol

See other articles in PMC that cite the published article.

The tool most commonly used for quantitative predictions of dose / fractionation dependencies in radiotherapy is the mechanistically-based linear-quadratic (LQ) model. The LQ formalism is now almost universally used for calculating radiotherapeutic isoeffect doses for different fractionation/protraction schemes. In summary, LQ has the following useful properties for predicting isoeffect doses: First, it is a mechanistic, biologically-based model; second, it has sufficiently few parameters to be practical; third, most other mechanistic models of cell killing predict the same fractionation dependencies as does LQ; fourth, it has well documented predictive properties for fractionation/dose-rate effects in the laboratory; fifth, it is reasonably well validated, experimentally and theoretically, up to about 10 Gy / fraction, and would be reasonable for use up to about 18 Gy per fraction. To date, there is no evidence of problems when LQ has been applied in the clinic.

Let us start from the premise that we need *some* model for calculating iso-effect doses when alternate fractionation schemes are considered. In addition, apart from increasing interest in alternative fractionation/protraction schemes, it is essential that we know how to compensate appropriately for missed radiotherapy treatments.

The tool most commonly used for quantitative predictions of dose / fractionation dependencies is the linear-quadratic (LQ) formalism (1–5). In radiotherapeutic applications, the LQ formalism is now almost universally used for calculating isoeffect doses for different fractionation/protraction schemes.

In contrast to earlier methodologies, such as CRE, NSD and TDF (6, 7), which were essentially empirical descriptions of past clinical data, the LQ formalism has become the preferred tool largely because it has a somewhat more biological basis, with tumor control and normal tissue complications specifically attributed to cell killing. By contrast, descriptive empirical models can go disastrously wrong if used outside the dose/fractionation range from which they were derived - as when NSD was applied to large doses per fraction (8, 9).

It is clear that radiotherapeutic response, both for tumor control and for complications, is dominated by cell killing (2, 10, 11) - and LQ is a mechanistic model of cell killing. Underlying the application of LQ to fractionation/protraction effects is pairwise misrepair of primary lesions such as double-strand breaks (DSB) or base damage (hereon in, we shall refer to DSB as the primary lesion, but base damage sites may well also be relevant here (12)). As schematized in Fig. 1, cell killing occurs via chromosome aberrations such as dicentric aberrations (13), formed when pairs of nearby DSB wrongly rejoin to one another (14). Protracting the exposure time potentially allows the first DSB to be repaired before the second is produced, and the LQ approach quantifies this effect (5). Nowadays, this binary DSB misrepair model is the most usual way to motivate the standard LQ approach, but different biological rationales for the same mathematical formalism have also been given, as we will discuss.

Examples of binary misrepair: 1A shows two chromosomes; each has one double strand break (DSB), shown as a gap. Centromeres, which are needed for proper transmission of chromosomes to daughter cells at mitosis, are shown as black constrictions. Most DSB **...**

It is important to stress here that the standard LQ formalism, as applied to time-dose relationships, is *not* merely a truncated power series in dose. Its key feature here is a specific mechanistically-based functional form for the protraction factor, usually designated by *G*, which takes into account dose protraction or fractionation. Expressions for special cases of the time factor, *G*, were derived by Lea and Catcheside (4, 15); a general form was subsequently derived (16), and has since been rederived from several different points of view (9). We refer to this general form of the time factor, given explicitly in Eq. 4 below, as the generalized Lea-Catcheside time factor, *G*.

The LQ model, in its most usual current version, describes cell killing in terms of the following mechanisms:

- Radiation produces DNA double strand breaks (DSBs) proportionate to the dose.
- These DSBs can be repaired, with first-order rate constant λ (=
*ln*2/*T*_{½}, where*T*_{½}is the repair half time). In practice, there may be more than one class of DSB that may be repaired with different rate constants; the LQ formalism can be simply extended to take this into account. - In competition with DSB repair, binary misrepair of pairs of DSBs produced from
*different*radiation tracks (i.e., different photons) can produce lethal lesions (often identified as predominantly dicentric chromosomal aberrations), the yield being proportional to the*square*of the dose (see the quadratic term in Eq. 1 and Eq. 3). The two independent radiation tracks can occur at different times during the overall regimen, allowing repair of the first DSB to take place before it can undergo pairwise misrepair with the second; it is this phenomenon which is the heart of the fractionation/protraction dependence in the LQ formalism. - In addition,
*single*radiation tracks can produce lethal lesions, possibly by a variety of mechanisms, the yield being linearly proportional to the dose.

Overall, in the LQ formalism, the yield (*Y*) of lethal lesions, and the corresponding survival (*S*) equation are

$$Y\phantom{\rule{thinmathspace}{0ex}}\propto \phantom{\rule{thinmathspace}{0ex}}\mathrm{\alpha}D\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}G\mathrm{\beta}{D}^{2}.$$

(1)

(note that the biologically effective dose, BED, is defined as *Y*/α(1)). Then, assuming the lethal lesions are Poisson distributed from cell to cell, the surviving fraction will be

$$\mathrm{S}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{exp}}(-Y),$$

(2)

and thus

$$S\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{exp}}[-\phantom{\rule{thinmathspace}{0ex}}(\mathrm{\alpha}D\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thinmathspace}{0ex}}G\mathrm{\beta}{D}^{2})].$$

(3)

In Eq. 1 and Eq. 3, *G* is the generalized Lea-Catcheside time factor, which accounts quantitatively for fractionation/protraction; it is important to note that *G* acts only on the quadratic component, as described in Point 3 above. The generalized time factor has the form (16)

$$G=(2/{D}^{2}){\displaystyle \underset{-\infty}{\overset{\infty}{\int}}\stackrel{.}{D}(t)dt{\displaystyle \underset{-\infty}{\overset{t}{\int}}{e}^{-\lambda (t-t\prime )}\stackrel{.}{D}(t\prime )dt\prime .}}$$

(4)

Here *(t)* describes the variation in dose rate over the entire course of the radiotherapy, and λ is a characteristic damage repair rate. Generically, the term after the second integral sign refers to the first of a pair of DSBs required to produce a lethal lesion - the exponential term describing the reduction in numbers of such DSB through repair; similarly, the term after the first integral sign refers to the second DSB, which can interact with DSBs, produced earlier, that still remain after repair.

The time factor, *G*, can be calculated for any fractionation/protraction scheme, and systematically accounts for the effects of protracting the dose delivery in any way. *G* can take values from zero to one, with *G*=1 for a single acute dose, leading to the simplest single-fraction LQ formalism:

$$S=\mathit{\text{exp}}[-(\mathrm{\alpha}D+\mathrm{\beta}{D}^{2})].$$

(5)

The interpretation of *G* <1 is a reduction in cell killing due to repair which occurs during protracted radiotherapy. Two special cases, which illustrate the main features of the general expression for *G* are: 1) for irradiation with *n* short fractions, each separated by a time *T* (17):

$$G=[2\mathrm{\theta}/(1-\mathrm{\theta})][n-(1-{\mathrm{\theta}}^{n})/(1-\mathrm{\theta})]$$

(6)

where θ = *exp*(−λ*T*). When the time between the short fractions is very long, then

$$G=1/n,$$

(7)

and the most simple LQ model for well-separated fractionated exposures become

$$S=\mathit{\text{exp}}[-D(\mathrm{\alpha}+\mathrm{\beta}D/n)].$$

(8)

Different fractionation/protraction schemes have different time factors, *G*, any of which can be calculated (3, 18) from Eq. (4).

The goal here is to be able to make equi-effect regimen extrapolations from standard regimes, typically with 1.8 to 2 Gy per fraction, to hypofractionated regimes. Possible hypofractionated doses per fraction are, for example, up to about 7 Gy for prostate (19), and up to about 15 Gy for non-small cell lung cancer (20). For fractionated stereotactic radiotherapy, we are interested in extrapolating in the opposite direction, from single-fraction doses as high as 18 Gy, to hypofractionated exposures with lower doses per fraction (21).

So the general question is whether the linear quadratic formalism described in the previous section describes radiotherapeutically-relevant dose-responses in the dose-per-fraction range from, say, 2 Gy to 18 Gy. For many situations, such as prostate-cancer hypofractionation, the dose-per-fraction range of interest is probably about 2 to 7 Gy.

We shall first discuss some *in vitro* and *in vivo* experimental data to address the question, and then discuss some alternate mechanistic models to the LQ equations, and their potential impact, relative to LQ-based predictions.

It is not easy to generate precise, accurate measurements of cell survival at high radiation doses. One recent approach to this problem is to use DNA flow cytometry for counting cell numbers, rather than colonies. This approach can produce very high statistics data, and establish precise survival curves. Figure 2 shows such survival data for irradiated CHO cells (22), where it can be seen that the standard errors can be made very small with this approach. The single-fraction LQ-model fit (Eq. 5) to these data clearly fits very well, indicating that it can predict the pattern of dose responses over the key 2 to 7 Gy dose range.

Similar results have recently been reported by Garcia *et al* (23), using a more standard colony assay to measure cell survival. In this case, surviving fractions in the dose range from zero to 16 Gy were measured. In order to estimate dose regions where the single-fraction LQ model (Eq. 5) did / did not fit the data, the authors fitted the data from 0 Gy to 4 Gy, and then from 0 Gy to progressively larger doses, each time assessing the goodness of fit of the model to the data. Typical results are shown in Fig 3, where it can be seen that the quality of fit to the LQ model does not decline significantly until doses above 15 Gy are included.

There are a fairly wide range of quantitative in-vivo endpoints for which it is possible to test concordance with the LQ predictions in the 2 to 20 Gy range (24). Fig 4, for example, shows some iso-effect results from Van der Kogel (25) for late-responding damage to the rat spinal cord, from Douglas and Fowler (26) for acute damage in mouse skin, and from Peck and Gibbs (27) for early and late damage to the murine small intestine. The form of the plot, the so called reciprocal-dose F_{e} plot (24, 26), is such that, if the LQ formalism applies, the data would fall on a straight line. Although the reciprocal dose plot approach is not an optimal methodology for parameter estimation (28, 29), it does provide a visual indication of how well *in-vivo* data agrees with the LQ model, in the dose range of interest. All the quantitative *in-vivo* endpoints in Fig 4 are consistent with the LQ model, over a wide range of doses per fraction, including those of interest in hypofractionation. Extensive F_{e} plot analyses by Barendsen (24), using 12 normal tissue response endpoints, reached the same conclusion. While more sophisticated methods are available for assessing agreement with the LQ model (30), given the inherent uncertainties in the data, it is clear that all these data, including those for single fractions of ~20 Gy, are consistent with the LQ model.

We have shown that the LQ model is reasonably predictive of dose-response relations, both *in vitro* and *in vivo*, in the dose per fraction range of 2 to 15 Gy. Of course it goes without saying that no mechanistic model describing dose-time patterns can be fully complete or correct, so we discuss here some of the main mechanistic uncertainties associated with the LQ model.

Again it is argued that cell killing is the dominant process mediating radiotherapeutic response (2, 10, 11), both for early and late effects, including vascular effects. But not all cell killing is mediated through chromosome aberrations produced by pairwise misrepair. However, other cell killing mechanisms - such as apoptosis and induction of small mutations, are dose-rate independent (31). LQ incorporates both dose-rate-independent mechanisms such as these (in the linear term), and dose-rate-dependent mechanisms (in the quadratic term). Assuming LQ correctly models the mechanisms involved in the dose-rate dependent term, then with appropriate parameters (α/β actually is the ratio of the dose-rate-independent to the dose-rate-dependent term), LQ should be adequate for predicting fractionation/protraction effects.

It has been known for some time that, in addition to the LQ model, various other binary misrepair models lead to the same generalized Lea-Catcheside time factor, *G*, in an appropriate approximation, and thus predict virtually the same time-dose relations as does the LQ approach. This result was demonstrated by several authors (17, 32–35) for the RMR (Repair-Misrepair) model (36), LPL (Lethal-Potentially Lethal) model (33), and more general binary misrepair models. In light of the conceptual similarities between the LQ model and other binary misrepair models, the fact that the corresponding formalisms make virtually equivalent predictions for dose-time relationships is not, in retrospect, particularly surprising.

This Poisson assumption, embodied in Eq. 2, is used in *all* models which calculate the fraction of inactivated cells from the average yield of lethal lesions. It is certainly the case that, based on microdosimetric considerations, this distribution is not exactly Poisson (37). For densely-ionizing radiations such as heavy ions or neutrons, this point is well taken and important. But for sparsely-ionizing radiation such as x or γ rays, below ~50 Gy per fraction, deviations from a Poisson distribution are quite minor (5, 38).

Various high-dose saturable repair (SR) models have been considered (39–43), all having in common the notion that per-lesion repair rate decreases as the dose - and the production of initial damage – increases. This might arise, for example, if repair enzymes can be overloaded. Such a change in repair rates with increasing dose might in principle result in different dose-response relations at high doses, compared with the LQ approach where repair rates are dose independent.

In fact there is not, as yet, strong evidence that saturable repair mechanisms are important at the doses and dose rates of relevance to radiotherapy (44). However, since SR models appear, *prima facie*, to be mechanistically different from binary misrepair models, there is the possibility that they could make significantly different predictions of fractionation/protraction effects, casting doubt on the validity of LQ-based iso-effect dose calculations. In fact, the formalisms describing most saturable repair (SR) models also lead, in an appropriate approximation, to the same time-dose relations as does the LQ formalism (32). This comes about because these SR formalisms reduce to the specific form of the generalized Lea-Catcheside time factor, *G*, which describes protraction effects in the LQ approach.

Figure 5 gives numerical estimates comparing the predictions of a typical SR formalism (39, 45), with the corresponding LQ predictions, at doses of relevance to radiotherapy (32). Specifically, in a comparison of any two practical fractionated external beam radiotherapeutic regimens, the predicted iso-effect doses are very similar using either the LQ or the SR formalism. For example, for a single-fraction dose of 5 Gy, using equivalent parameters, the LQ and the SR model would give cell survival predictions which differed by only about 0.18%; thus for a ten-fraction 5 Gy per fraction prostate hypofractionation protocol, the LQ and the SR model would differ in cell survival predictions by only about 1.8%.

Whilst Fig 5 refers to the practical equivalence of a particular SR formalism to LQ, other SR models also show this equivalence. For example, for the SR model described by Sontag (43), which is conceptually similar though described by a slightly different formalism, a corresponding theorem on equivalence to LQ can be proved (32). More generally, it can be shown that formalisms describing a very broad class of radiobiological reaction rate models, whether based on binary misrepair or saturable repair, all lead to the same generalized Lea-Catcheside time factor, *G*, for dose protraction (32).

Typing “linear-quadratic” and “radiotherapy” into PubMed results in over 600 hits, so LQ is very widely used. Even with this extensive use, there is to date no evidence that use of LQ has resulting in significant underdosing or overdosing, for alternate fractionation schemes.

It is important to distinguish here between validity of the LQ model, and the appropriate parameters to use in the LQ approach – these are different issues. For example, the suggestion that the α/β ratio for prostate cancer is anomalously low (46, 47), has resulted in several large randomized trials, all designed using LQ, comparing conventional fractionation to hypofractionation (48). Such trials are explicitly based on mechanistic considerations quantified with the LQ model, and would today be inconceivable without the conceptual framework of an established mechanistic model.

What is the dose-per-fraction range for which the LQ model should be used? It has been argued here, based both on experimental and theoretical considerations, that LQ is a reliable mechanistically-plausible model for designing protocols in the dose-per-fraction range from 2 to 10 Gy. Above 10 Gy, the model would be expected to become progressively less accurate but, based on animal data, still acceptable for the design of clinical trials based on doses per fraction of 15 to 18 Gy.

That the LQ model is useful over such a wide dose range is related to the observation that almost all mechanistic models of cell killing predict essentially the same dependencies for fractionation as does LQ, so the use of LQ is not as model dependent as one might expect. Of course the basic LQ model does not tell the whole story. Fractionation/protraction effects are controlled (49) by the 4 R’s (repair, redistribution, reoxygenation, repopulation). At the cost of extra parameters, the remaining 3 R’s can be modeled using LQ (50, 51), though it is clear that fractionation effects are dominated by repair.

In summary, LQ has the following useful properties for predicting isoeffect doses:

- It is a mechanistic, biologically-based model;
- It has sufficiently few parameters to be practical;
- Most other mechanistic models of cell killing predict the same fractionation dependencies as does LQ;
- It has well documented predictive properties for fractionation/dose-rate effects in the laboratory;
- It is reasonably well validated, experimentally and theoretically, up to about 10 Gy / fraction, and would be reasonable for use up to about 18 Gy per fraction;
- To date, there is no evidence of problems when LQ has been applied in the clinic.

Alfred North Whitehead commented that "*There is no more common error than to assume that, because prolonged and accurate mathematical calculations have been made, the application of the result to some fact of nature is absolutely certain*". This is certainly true for the LQ model and all other mechanistically based models used to design alternate fractionation protocols. Adding clinical judgment to the results of radiobiological modeling is a must.

Work supported by NIH grants U19 AI-067773 and P41 RR-11623.

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