The application of the linear-quadratic dose-effect equation to fractionated and protracted radiotherapy

The linear-quadratic (LQ) dose-effect formalism is currently providing new perspectives into the ways in which alterations in the dose per fraction in conventional radiotherapy may be used to bring about improved results with respect to early or late normal tissue reactions. In this paper, using a model initially developed by Roesch, the LQ equations are explored further in terms of dose-rate rather than dose. By the incorporation of one other parameter, mu, which relates to the rate of repair of sub-lethal radiation damage, a more general formalism is obtained. In particular, equations are derived which can be used to examine the relative effectiveness of different treatment regimes, including those involving decaying sources. Such equations are of wider applicability than other LQ derivations which relate only to dose-response relationships. The extended equations, which are fully consistent with the existing LQ method, are also shown to lead directly to other independently established, relationships for protracted irradiation. The nature of the link between high and low dose-rate treatments is discussed, and some worked examples provide indications of how the new equations may be used to assess further the potential clinical benefits of low dose-rate treatments and permanent implants.