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
), 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
), 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.