A survival model for fractionated radiotherapy with an application to prostate cancer

This paper explores the applicability of a mechanistic survival model, based on the distribution of clonogens surviving a course of fractionated radiation therapy, to clinical data on patients with prostate cancer. The study was carried out using data on 1,100 patients with clinically localized prostate cancer who were treated with three-dimensional conformal radiation therapy. The patients were stratified by radiation dose (group 1: <67.5 Gy; group 2: 67.5-72.5 Gy; group 3: 72.5-77.5 Gy; group 4: 77.5-87.5 Gy) and prognosis category (favourable, intermediate and unfavourable as defined by pre-treatment PSA and Gleason score). A relapse was recorded when tumour recurrence was diagnosed or when three successive prostate specific antigen (PSA) elevations were observed from a post-treatment nadir PSA level. PSA relapse-free survival was used as the primary end point. The model, which is based on an iterated Yule process, is specified in terms of three parameters: the mean number of tumour clonogens that survive the treatment, the mean of the progression time of post-treatment tumour development and its standard deviation. The model parameters were estimated by the maximum likelihood method. The fact that the proposed model provides an excellent description both of the survivor function and of the hazard rate is prima facie evidence of the validity of the model because closeness of the two survivor functions (empirical and model-based) does not generally imply closeness of the corresponding hazard rates. The estimated cure probabilities for the favourable group are 0.80, 0.74 and 0.87 (for dose groups 1-3, respectively); for the intermediate group: 0.25, 0.51, 0.58 and 0.78 (for dose groups 1-4, respectively) and for the unfavourable group: 0.0, 0.27, 0.33 and 0.64 (for dose groups 1-4, respectively). The distribution of progression time to tumour relapse was found to be independent of prognosis group but dependent on dose. As the dose increases the mean progression time decreases (41, 28.5, 26.2 and 14.7 months for dose groups 1-4, respectively). This analysis confirms that, in terms of cure rate, dose escalation has a significant positive effect only in the intermediate and unfavourable groups. It was found that progression time is inversely proportional to dose, which means that patients recurring in higher dose groups have shorter recurrence times, yet these groups have better survival, particularly long-term. The explanation for this seemingly illogical observation lies in the fact that less aggressive tumours, potentially recurring after a long period of time, are cured by higher doses and do not contribute to the recurrence pattern. As a result, patients in higher dose groups are less likely to recur; however, if they do, they tend to recur earlier. The estimated hazard rates for prostate cancer pass through a clear-cut maximum, thus revealing a time period with especially high values of instantaneous cancer-specific risk; the estimates appear to be nonproportional across dose strata.