QUANTIFYING THE POSITION AND STEEPNESS OF RADIATION DOSE-RESPONSE CURVES

Radiation dose-response curves are of fundamental importance both in p ractical radiotherapy and as the basis of more theoretical considerati ons concerning the potential benefit to be gained from modified dose-f ractionation schedules or of the effects of dosimetric and biological variability. The steepness of the dose-response curve is a key paramet er and quantitative measures of steepness derived from clinical data a re strongly needed. Unfortunately, there are many ambiguities associat ed with quantifying the steepness of radiation dose-response curves an d these are identified and discussed in the present paper. The followi ng problems are reviewed. (1) In the literature, various descriptors o f 'steepness' are reported. We focus on the normalized dose-response g radient, gamma, and the dose-response slope, theta. The mathematical p roperties and the relationship between these are discussed. (2) Steepn ess estimates depend on the mathematical model used to describe the do se-response relationship. Three standard formulations are considered: the Poisson, the logistic and the probit dose-response model. The magn itude of the model dependence is influenced by the range of the empiri cal dose-response data available, and is most pronounced for data conc entrated around very low or very high response levels. (3) Reparametri zations of the standard models in terms of position and steepness are given, and it is pointed out that some previously published formulas a re only approximations. (4) The method of analysis can influence the s teepness estimate. An analysis of a specific data set shows that the u se of the least-squares method rather than the preferred maximum likel ihood method may influence both the steepness estimate and its confide nce interval. (5) Dose-response data generated with a fixed number of fractions rather than a fixed dose per fraction will produce steeper d ose-response curves. The approximation involved in describing such a s et of dose-response data by a position and a single steepness paramete r is discussed. (6) The importance of specifying the statistical uncer tainty of the steepness estimate is stressed. All of these problems ar e illustrated by a practical example, in which dose-response data from the literature are re-analysed.