On the inclusion of proliferation in tumour control probability calculations for inhomogeneously irradiated tumours

In previous modelling of tumour control probability (TCP) for inhomogeneously irradiated tumours we used an expression that did not include a proliferation correction term, which should be lambda(T - Tk). We did not use that term in the specific examples in our previous work to model slowly growing tumours in order to avoid unnecessary mathematical complexity. We have now considered how to do so in more detail, and there are some variations, such as schedules that depart from a number of equal fractions over the entire course of treatment, if one wishes to compensate for proliferation in the remaining fractions by increasing the dose per fraction after the kick-off time has passed in order to achieve the same TCP when proliferation is neglected.