Calculation of integrated biological response in brachytherapy.

Purpose: To present analytical methods for calculating or estimating the integrated biological response in brachytherapy applications, and which allow for the presence of dose gradients.

Methods And Materials: The approach uses linear-quadratic (LQ) formulations to identify an equivalent biologically effective dose (BEDeq) which, if applied to a specified tissue volume, would produce the same biological effect as that achieved by a given brachytherapy application. For simple geometrical cases, BED multiplying factors have been derived which allow the equivalent BED for tumors to be estimated from a single BED value calculated at a dose reference point. For more complex brachytherapy applications a voxel-by-voxel determination of the equivalent BED will be more accurate. Equations are derived which when incorporated into brachytherapy software would facilitate such a process.

Results: At both high and low dose rates, the BEDs calculated at the dose reference point are shown to be lower than the true values by an amount which depends primarily on the magnitude of the prescribed dose; the BED multiplying factors are higher for smaller prescribed doses. The multiplying factors are less dependent on the assumed radiobiological parameters. In most clinical applications involving multiple sources, particularly those in multiplanar arrays, the multiplying factors are likely to be smaller than those derived here for single sources. The overall suggestion is that the radiobiological consequences of dose gradients in well-designed brachytherapy treatments, although important, may be less significant than is sometimes supposed. The modeling exercise also demonstrates that the integrated biological effect associated with fractionated high-dose-rate (FHDR) brachytherapy will usually be different from that for an "equivalent" continuous low-dose-rate (CLDR) regime. For practical FHDR regimes involving relatively small numbers of fractions, the integrated biological effect to tissues close to the treatment sources will be higher with HDR than for LDR. Conversely, the integrated biological effect on structures more distant from the sources will be less with HDR. This provides quantitative confirmation of an idea proposed elsewhere, and suggests the existence of a potentially useful biological advantage for HDR brachytherapy delivered in relatively small fraction numbers and which is not apparent when considering radiobiological effect only at discrete reference points.

Conclusion: The estimation and direct calculation of integrated biological response in brachytherapy are both relatively straightforward. Although the tabular data presented here result from considering only simple geometrical cases, and may thus overestimate the consequences of dose gradients in multiplanar clinical applications, the methods described may open the way to the development of more realistic radiobiological software, and to more systematic approaches for correlating physical dose and biological effect in brachytherapy.