The reduction of tumour control with increasing overall time: Mathematical considerations

The rate of loss of tumour control (dP/dt) with extension of treatment time is analysed to assess the relative contributions of radiobiological parameters (radiosensitivity, clonogen doubling time, clonogen numbers and fractionation schedule) on such loss. Linear quadratic modelling and Poisson statistics are used to study individual tumour responses. A heterogeneous tumour population is constructed by the use of random sampling techniques to allow for variations in intrinsic radiosensitivity and clonogen doubling times. Average tumour control probability is calculated for this population for two different fractionation schedules (60 Gy in 30 fractions and 50 Gy in 15 fractions), each given over 15-60 days. The magnitude of dP/dt will depend upon the tumour cure probability (P): the loss of control will be most significant for tumours which have a cure of 37% when the Poisson survival model is used. The analysis suggests that compensation for short unscheduled treatment gaps (e.g. by increasing the total dose or rescheduling with use of weekend treatment sessions) may only be required for difficult tumours (i.e. radioresistant and/or with short clonogen doubling times). Where pre-treatment clonogen numbers are relatively low as in small volume tumours or after surgical debulking, the model predicts that correction for short treatment gaps is probably not required if the average effective clonogen doubling times are longer than 5 days. Different dose-time-fractionation schedules, even though producing similar overall cure rates in clinical practice, may actually be achieving cures in different subpopulations within a population of tumours, since the value of dP/dt in each individual tumour will depend upon the set of radiobiological parameters given above. For a hypothetical randomly selected heterogeneous tumour population the predicted rates of loss of tumour control produced by an extension in treatment time are 0.9 and 1.1% per day, respectively, for the above fractionation schedules. These values are close to those reported in the clinical literature for the first 2 weeks of treatment prolongation (1-2% per day for squamous cell carcinomas). The Poisson method, when combined with random sampling techniques, consequently provides realistic data. Modelling of this clinical problem provides an insight into how tumour sub-populations, each characterized by its own set of radiobiological parameters, can influence the overall rate of loss of tumour control in a heterogeneous population. Random sampling techniques should be considered as necessary precursors for the assessment of the choices of dose/fractionation in future clinical trials particularly when more precise data regarding the radiobiological parameters and their statistical variations become available.