title: In the linear quadratic model, the Poisson approximation and the Zaider-Minerbo formula agree on the ranking of tumor control probabilities, up to a critical cell birth rate.
abstract: To provide a rule for the agreement or disagreement of the Poisson approximation (PA) and the Zaider-Minerbo formula (ZM) on the ranking of treatment alternatives in terms of tumor control probability (TCP) in the linear quadratic model.
  
  
  
  A general criterion involving a critical cell birth rate was formally derived. For demonstration, the criterion was applied to a distinct radiobiological model of fast growing head and neck tumors and a respective range of 22 conventional and nonconventional head and neck schedules.
  
  
  
  There is a critical cell birth rate bcritbelow which PA and ZM agree on which one out of two alternative treatment schemes with single-cell survival curves S'(t) and S''(t) offers better TCP: [Formula: see text] For cell birth rates b above this critical cell birth rate, PA and ZM disagree if and only if b >bcrit> 0. In case of the exemplary head and neck schedules, out of 231 possible combinations, only 16 or 7% were found where PA and ZM disagreed. In all 231 cases the prediction of the criterion was numerically confirmed, and cell birth rates at crossovers between schedules matched the calculated critical cell birth rates.
  
  
  
  TCP estimated by PA and ZM almost never numerically coincide. Still, in many cases both formulas at least agree about which one out of two alternative fractionation schemes offers better TCP. In case of fast growing tumors featuring a high cell birth rate, however, ZM may suggest a re-evaluation of treatment options.
authors.affiliation: a Department of Radiation Oncology , University Hospitals of LMU Munich , Munich , Germany.
authors.forename: Hendrik
authors.initials: H
authors.lastname: Ballhausen
authors.affiliation: a Department of Radiation Oncology , University Hospitals of LMU Munich , Munich , Germany.
authors.forename: Claus
authors.initials: C
authors.lastname: Belka
mesh_terms: D000465:Algorithms; D049109:Cell Proliferation / Q000528:radiation effects*; D002470:Cell Survival / Q000528:radiation effects*; D003198:Computer Simulation; D004307:Dose-Response Relationship, Radiation; D006258:Head and Neck Neoplasms / Q000473:pathology / Q000503:physiopathology* / Q000532:radiotherapy*; D006801:Humans; D016014:Linear Models; D008954:Models, Biological; D016012:Poisson Distribution; D011829:Radiation Dosage; D011880:Radiotherapy Planning, Computer-Assisted / Q000379:methods
keywords: Cellular birth and death rates; head and neck schedules; linear quadratic model; radiation therapy; tumor control probability