Mathematical derivation of optimal uniform treatment schedules for the fractionated irradiation of human tumors

Abstract

Wheldon and Kirk [22] developed a mathematical model which permitted estimation of optimal treatment schedules for the radiotherapy of exponentially growing tumors. The present work is a direct development of their analysis. The mathematical model developed here permits the determination of optimal treatment schedules for radiotherapy of tumors whose associated growth curves belong to a large class of curves which include the exponential, the Gompertzian, and the Verhulst growth curves as special cases. It is shown that various mathematical approximations employed by Wheldon and Kirk [22] may be deleted, thus strengthening both the mathematical foundation of their model and the validity of subsequent results. In particular the present analysis allows dicussion of schedules for the radiotherapy of tumor cells which undergo exponential survival. Such a discussion was precluded by the analysis of Wheldon and Kirk [22]. Calculations based on the present analysis employing available data for tumors possessing either exponential or Gompertzian growth curves indicate that optimal schedules may achieve a significantly better tumor-cell kill than a conventional schedule, without connective-tissue tolerance being exceeded. Once data are available for tumors possessing other growth curves belonging to the large class of curves discussed in this paper, appropriate optimal schedules could be determined.