Realistic closed-form TCP model including cell sensitivity dependence

Objective: To develop a mechanistic extension of the Poisonnian linear quadratic (LQ) tumor control probability (TCP) formulation by incorporating tumor volume and cell sensitivity inter-patient variations which can be applied to a cohort of patients. Approach: A novel closed-form expression for TCP was derived from first principles, incorporating inter-individual variations in tumor volume and cell sensitivity within the LQ model of tumor control. Furthermore, an exponential time dependence of local control (LC) in terms of TCP was introduced. The proposed model was fitted to 22 datasets of early-stage non-small cell lung cancer (NSCLC), encompassing various dose regimes, tumor volumes, treatment duration and outcome values over different follow-up periods. A log-likelihood algorithm was employed for the fitting process. Main results: The fit of the population TCP model, which adopts tumor volume and cell radiosensitivities uniformly distributed, resulted in a cell sensitivity value of α ¯ U = 0.37 [ 0.13 − 0.47 ] Gy$^{−1}$, its corresponding bandwidth Δ α = 0.37 [ 0.04 − 0.42 ] Gy$^{−1}$, β = 0. 015 [ 0.009 − 0.039 ] Gy$^{−2}$, the characteristic time at which LC reaches TCP, t 1 / 2 = 19.6 [7.3-90.8] months, and the cell population doubling time T $_{d}$ = 2.0 [0.2 4.9] days. The parameters α ¯ U , Δ α and β were found to be significant (p < 0.05), while t 1 / 2 and T $_{d}$ proved non-statistically significant for the model under Wald test. This model describes data from 1675 lesions and offers a better fit compared to alternative approaches incorporating Gaussian or log-normal radiosensitivity distributions. Significance: A closed form of TCP population model was derived by including cell sensitivity and tumor size heterogeneities. A relation between TCP and LC was established by modeling LC as an exponential function of follow-up time. The derived TCP population model facilitates direct application to clinical datasets and was tested against NSCLC clinical data. Individual TCP can be estimated from the radiobiological parameters of the population.