SU‐FF‐T‐74: An Objective Function for TCP/NTCP Curve Fitting

Purpose: To propose an objective function for TCP/NTCP curve fitting. Method and Materials: One of the widely used objective functions by physicists, when fitting theoretical models to experimental data, is the χ 2 one, where: χ 2  =   Σ i( ( Y theoretical i ‐ Y experimental i) /   σ experimental i ) 2which is the maximum likelihood function in case of Gaussian random variables. In the case of a binary outcome, as is the nature of clinical or experimental animal radiation outcome data, the random variable has a binomial distribution and the maximum likelihood function for ungrouped data becomes: L = Σ responders ln ( P theor ( par , DVH) )   + Σ non ‐ responders ln ( 1 ‐ ( P theor ( par , DVH ) ) ) , where P theor stands for TCP theor or NTCP theor . The maximization of L is often used by different authors for the estimation of the TCP/NTCP model parameters. Results: However, sometimes χ 2 is used to fit TCP/NTCP functions for different purposes like theoretical comparison between different models. In this case the application of χ 2 becomes inaccurate and inadequate especially when the function values are close to the ends of the interval in which they are defined, namely 0 or 1. This is why we propose t he application of the double logarithmic transformation, presuming that the random variable is not normally but log‐log‐normally distributed. Conclusion: Thus the objective function in case of model comparison would become: χ 2  =   Σ i( ( − ln ( − ln ( P theoretical i) + ln ( − ln ( P ‘ experimental ’ i) ) / σ log _ ‘ experimental ’ i ) 2where σ log _ ‘ experimental ’ i  =   −   σ ‘ experimental ’ i  /   ( P ‘ experimental ’ i   ln ( P ‘ experimental ’ i) ) . Here P theoretical istands for the values of P predicted by one of the models that are being compared and P ‘ experimental ’ istands for the values of P predicted by the other model.