Series model volume effects in a population of non-identical patients: how low is low?

Working with several mechanisms of critical local tissue damage, formulae are analytically derived that describe normal tissue complication probabilities (ntcps) for series-type radiotherapy complications arising in heterogeneous patient populations. Using the formulae, values are calculated for deltaD50(10)-the increase in dose leading to a 50% series-type complication rate (D50) when irradiated organ volume is reduced tenfold. From the structure of the ntcp formulae derived, it follows that dose-levels leading to clinically relevant serious complication rates (less than 5%) will change less with irradiated volume than will D50. Calculated values of deltaD50(10) for the heterogeneous series model are low-generally less than 6 Gy; such values are much lower than those calculated for the non-heterogeneous series model (27-37 Gy). These results suggest that if the dose-limiting toxicity of a radiotherapy treatment is a series-type complication with a local damage mechanism similar to any of those studied in this work, then even very substantial improvements in technique-leading to large reductions in highly dosed normal tissue volumes-would be unlikely to allow a useful degree of escalation of the dose delivered to the tumour, unless highly dosed normal tissue volumes can be reduced below the length-scale of a functional subunit.