Modeling Dose and Local Control in Radiotherapy

Abstract

We discuss models for predicting local control (prevention of tumor recurrence) after therapeutic radiation in cancer patients. The probability of control is first formulated from theoretical precepts. Biophysical principles dictate that the three factors in therapy that most universally influence outcome are total dose, number of sessions in which the dose is administered, and total time under treatment. We show that these principles also suggest the scale, or link function, on which local control probability for a tumor of given size is a linear function of these predictors. The probabilities are given clinical relevance by assigning a mixing distribution to tumor size; effective size, the number of actively dividing cells in a tumor, is an unmeasurable but of course quite influential quantity. We show in this case that a gamma distribution on tumor size induces linearity on a subset of the class of links first proposed by Burr. Next, we discuss methods of modeling control by a finite follow-up time. We demonstrate a new result, that minor assumptions on the effects of size on recurrence allow models developed for permanent control to be applied directly to recurrence by a finite time. We also describe adjustments for accommodating losses to follow-up before that time. Finally, we develop inference on the mixing distribution of tumor size along with results of the effect of misspecifying the distribution. We illustrate the methods with a new analysis of a radiotherapy study. Though developed for a specific type of failure data, many of the results also apply to any time-dependent binary outcome.

Key Words:

• Binary regression
• Linear-quadratic formula
• Link tests
• Logistic regression
• Gamma distribution
• Radiotherapy