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The influence of dose heterogeneity on tumour control probability in fractionated radiation therapy

Theoretical modelling of tumour control probability (TCP) with respect to non-uniformity in the dose to the tumour, alternate fractionation schemes and tumour kinetics is a very useful tool for assessment of the influence of changes in dosimetric or radiobiological factors on the outcome of the treatment. Various attempts have been made to also include effects from non-uniform dose to the tumour volume, but the problem has not been fully solved and many factors were totally neglected or not accurately taken into account. This paper presents derivations of analytical expressions of TCP for macroscopic inter-cell dose variations and for random inter-fractional variations in average tumour dose, based on binomial statistics for the TCP and the well-known linear quadratic model for the cell survival. Numerical calculations have been performed to validate the analytical expressions. An analysis of the influence of the deterministic and stochastic heterogeneity in dose delivery on the TCP was performed. The precision requirements in dose delivery are discussed briefly with the support of the presented results. The main finding of this paper is that it is primarily the shape of the cell survival curve that governs how the response is affected by macroscopic dose variations. The analytical expressions for TCP accounting for heterogeneity in dose can quite well describe the TCP for varying dose from cell to cell and random dose in each fraction. An increased TCP is seen when a large number of fractions are used and the variations in dose to the cells are rather high for tissues with low α/β.