title: A closed-form description of tumour control with fractionated radiotherapy and repopulation.
abstract: OBJECTIVE
  To derive a closed form expression of tumour control probability (TCP) following the geometric stochastic approach of Tucker and Taylor.
  
  
  METHODS
  A model was constructed based upon a Galton-Watson branching process with cell killing represented by a Bernoulli random variable, and repopulation represented by a Yule Fury process. A closed-form expression of the probability-generating function was derived, which yielded an explicit expression for the mean number of surviving clonogens and the TCP.
  
  
  RESULTS
  The mean number of surviving cells, after i clonogens have been treated with n fractions of irradiation, was [equation: see text], where s is the surviving fraction, lambda is the rate of cell division, and delta t is the interfraction time interval. The tumour control probability was [equation: see text].
  
  
  CONCLUSIONS
  Tucker and Taylor provided improvements upon the conventional Poisson model for TCP, mainly through numerical simulation. Here a model based upon their geometric stochastic approach has been derived in closed form. The resultant equations provide a simpler alternative to numerical simulation allowing the effects of fractionated radiotherapy on a replicating population of tumour cells to be more easily predicted.
authors.affiliation: Department of Radiation Oncology, The Ottawa Regional Cancer Center, Ontario, Canada.
authors.forename: W S
authors.initials: WS
authors.lastname: Kendal
mesh_terms: D002455:Cell Division / Q000528:radiation effects; D002470:Cell Survival / Q000528:radiation effects; D008432:Mathematical Computing; D008954:Models, Biological; D009369:Neoplasms / Q000473:pathology* / Q000532:radiotherapy*