The linear-quadratic model is an appropriate methodology for determining isoeffective doses at large doses per fraction

Abstract

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, the LQ model has the following useful properties for predicting isoeffect doses: (1) it is a mechanistic, biologically based model; (2) it has sufficiently few parameters to be practical; (3) most other mechanistic models of cell killing predict the same fractionation dependencies as does the LQ model; (4) it has well-documented predictive properties for fractionation/dose-rate effects in the laboratory; and (5) 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 the LQ model has been applied in the clinic.

Fig. 1

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 are correctly restituted, but a few undergo binary misrepair. As shown in 1B, binary misrepair can result in a dicentric chromosome aberration, which generally destroys the clonogenic viability of the cell. In about half the binary misrepair events, the two DSB shown in Fig 1A lead to a translocation, shown in 1C; translocations involve large scale rearrangements, and can cause potentially precarcinogenic alterations in cellular phenotype, but most do not impair cellular survival.

Fig. 2

Survival of x-irradiated CHO cells, determined by flow cytometry population counting, five days after treatment (22). The curve is the corresponding LQ model ft.

Fig. 3

Goodness of fit of LQ model to measured cell survival data, as a function of the dose range which was fitted (23). The quantity plotted is χ² per degree of freedom, hence smaller values represent better fits to the LQ model. For example the left-most point represents a good fit of the LQ model to cell-survival data in the dose range from 0 Gy to 4 Gy, and the right-most point represents a less good fit of the LQ model to cell-survival data in the dose range from 0 Gy to 16 Gy.

Fig. 4

Isoeffect data for late response from three (□○Δ) different regions of the rat spinal cord (25), for acute skin reactions (◆) in mice (26), and for early (●) and late (⊕) murine intestinal damage (27). The data are plotted in a “reciprocal-dose F_e” form (26) such that, if they follow a linear-quadratic relationship, the points fall on a straight line.

Fig. 5

For a single acute dose fraction, shown is the percent relative difference [100(S_SR-S_LQ)/S_SR] between survival calculated exactly using the SR model and survival calculated using the corresponding LQ approximation. Calculations reported in Ref (32), based on the parameter set from Kiefer and Löbrich (45).