Investigating spatial fractionation and radiation induced bystander effects: a mathematical modelling approach

Can dose convolution modelling explain bath and shower effects in rat spinal cord?

Objective. ‘Bath and shower’ effects were first seen in proton irradiations of rat spinal cord, where a low dose ‘bath’ reduced the smaller field ‘shower’ dose needed for limb paralysis giving the appearance of sensitisation of the cord or disproportionate response. This was difficult to reconcile with existing tissue complication models. The purpose of this investigation is to explore a different approach using a dose convolution algorithm to model the 50% isoeffect endpoint. Approach. Bath and shower dose distributions were convolved with Gaussian functions with widths specified by the σ parameter. The hypothesis was that the maximum value from the convolved distributions was constant for isoeffect across the modelled scenarios. A simpler field length dependent relative biological effectiveness (FLRBE) approach was also used for a subset of the data which gave results independent of σ. Main results. The maximum values from the convolved distributions were constant within ±17% across the bath and shower experiments for σ = 3.5 mm, whereas the maximum dose varied by a factor of four. The FLRBE results were also within ±14% confirming the validity of the dose convolution approach. Significance. A simple approach using dose convolution modelling of the 50% isotoxicity gave compelling consistency with the full range of bath and shower results, while the FLRBE approach confirmed the results for the symmetric field data. Convolution modelling and the effect of time interval were consistent with a signalling factor diffusion mechanism such as the ‘bystander effect’. The results suggest biological effectiveness is reduced for very small field sizes, requiring a higher isoeffect dose. By implication, the bath dose does not sensitise the cord to the shower dose; when biological effectiveness is accounted for, a small increase in the bath dose requires a significantly larger reduction in shower dose for isoeffect.