title: A realistic closed-form radiobiological model of clinical tumor-control data incorporating intertumor heterogeneity.
abstract: OBJECTIVE
  To investigate the role of intertumor heterogeneity in clinical tumor control datasets and the relationship to in vitro measurements of tumor biopsy samples. Specifically, to develop a modified linear-quadratic (LQ) model incorporating such heterogeneity that it is practical to fit to clinical tumor-control datasets.
  
  
  METHODS
  We developed a modified version of the linear-quadratic (LQ) model for tumor control, incorporating a (lagged) time factor to allow for tumor cell repopulation. We explicitly took into account the interpatient heterogeneity in clonogen number, radiosensitivity, and repopulation rate. Using this model, we could generate realistic TCP curves using parameter estimates consistent with those reported from in vitro studies, subject to the inclusion of a radiosensitivity (or dose)-modifying factor. We then demonstrated that the model was dominated by the heterogeneity in alpha (tumor radiosensitivity) and derived an approximate simplified model incorporating this heterogeneity. This simplified model is expressible in a compact closed form, which it is practical to fit to clinical datasets. Using two previously analysed datasets, we fit the model using direct maximum-likelihood techniques and obtained parameter estimates that were, again, consistent with the experimental data on the radiosensitivity of primary human tumor cells. This heterogeneity model includes the same number of adjustable parameters as the standard LQ model.
  
  
  RESULTS
  The modified model provides parameter estimates that can easily be reconciled with the in vitro measurements. The simplified (approximate) form of the heterogeneity model is a compact, closed-form probit function that can readily be fitted to clinical series by conventional maximum-likelihood methodology. This heterogeneity model provides a slightly better fit to the datasets than the conventional LQ model, with the same numbers of fitted parameters. The parameter estimates of the clinically important time factors and lag periods are very similar to those obtained from the conventional LQ model, but with slightly narrower confidence intervals, reflecting the better fit to the clinical data.
  
  
  CONCLUSIONS
  We have demonstrated, as have others, the importance of intertumor heterogeneity in the response of patient populations to radiotherapy. With the possible inclusion of a radiosensitivity-modifying factor (in vitro/in vivo) of around 1.7, the in vivo data can be made consistent with the in vitro SF2 and Tpot data. Fitting two previously analyzed multicenter datasets indicated that previous analyses based on conventional LQ models gave results for clinically important time factors and lags periods that were not significantly biased by the failure to include intertumor heterogeneity, with slightly narrower confidence intervals, reflecting the better fit to the clinical data. The simple closed-form model we have developed allows direct estimation of the heterogeneity in radiosensitivity within clinical series, and should prove useful in the analysis of other clinical series.
authors.affiliation: CRC Biomathematics and Computing Unit, Paterson Institute for Cancer Research, Christie Hospital NHS Trust, Manchester, UK.
authors.forename: S A
authors.initials: SA
authors.lastname: Roberts
authors.forename: J H
authors.initials: JH
authors.lastname: Hendry
mesh_terms: D004307:Dose-Response Relationship, Radiation; D006258:Head and Neck Neoplasms / Q000532:radiotherapy*; D006801:Humans; D007822:Laryngeal Neoplasms / Q000532:radiotherapy; D016013:Likelihood Functions; D016014:Linear Models; D008954:Models, Biological; D010610:Pharyngeal Neoplasms / Q000532:radiotherapy; D011853:Radiobiology / Q000379:methods*