Quantifying the position and steepness of radiation dose-response curves

Radiation dose-response curves are of fundamental importance both in practical radiotherapy and as the basis of more theoretical considerations concerning the potential benefit to be gained from modified dose-fractionation schedules or of the effects of dosimetric and biological variability. The steepness of the dose-response curve is a key parameter and quantitative measures of steepness derived from clinical data are strongly needed. Unfortunately, there are many ambiguities associated with quantifying the steepness of radiation dose-response curves and these are identified and discussed in the present paper. The following problems are reviewed. (1) In the literature, various descriptors of 'steepness' are reported. We focus on the normalized dose-response gradient, gamma, and the dose-response slope, theta. The mathematical properties and the relationship between these are discussed. (2) Steepness estimates depend on the mathematical model used to describe the dose response relationship. Three standard formulations are considered: the Poisson, the logistic and the probit dose-response model. The magnitude of the model dependence is influenced by the range of the empirical dose-response data available, and is most pronounced for data concentrated around very low or very high response levels. (3) Reparametrizations of the standard models in terms of position and steepness are given, and it is pointed out that some previously published formulas are only approximations. (4) The method of analysis can influence the steepness estimate. An analysis of a specific data set shows that the use of the least-squares method rather than the preferred maximum likelihood method may influence both the steepness estimate and its confidence interval. (5) Dose-response data generated with a fixed number of fractions rather than a fixed dose per fraction will produce steeper dose-response curves. The approximation involved in describing such a set of dose-response data by a position and a single steepness parameter is discussed. (6) The importance of specifying the statistical uncertainty of the steepness estimate is stressed. All of these problems are illustrated by a practical example, in which dose-response data from the literature are re-analysed.