Predicting the radiation control probability of heterogeneous tumour ensembles: data analysis and parameter estimation using a closed-form expression

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Abstract
A closed-form formula describing the tumour control probability (tcp) of a heterogeneous collection of tumours has been obtained by analytically averaging the homogeneous double-exponential tcp formula over inter-tumour distributions of clonogen radiosensitivity, density and repopulation rate, tumour volume and dose. The formula can be straightforwardly and relatively quickly fitted to clinical data, yielding radiobiological parameter values for use in tcp modelling. The formula was fitted to published tcp data which catalogued tumour control records grouped by dose and tumour volume, and treatment duration. Fitted parameter values, confidence intervals and goodness-of-fit statistics were determined. The sets of parameter values obtained are unique only to within a scaling factor. The formula provides non-rejectable fits to data which grouped tcp by dose and volume when radiosensitivity parameters take values close to laboratory estimates, the fitted volume dependence parameter, however, taking rather high values. Good fits are obtainable with the intuitively reasonable volume parameter value of one, but with radiosensitivity values around one-third of their laboratory estimates. Non-rejectable fits to data which grouped tcp by dose and treatment duration may be obtained with radiosensitivity and repopulation rate parameters lying close to laboratory estimates.

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