Most radiobiological effects for acute exposures are actually quadratic in dose - this applies to things like mutations, chromosome aberrations, and lethal events.
The quadratic term relates (loosely) to interactions between damage from multiple ionising particles which are present in cells at the same time. When you protract exposures, more damage is repaired before subsequent ionising particles arrive, and you see a reduction in the quadratic term, to effectively a linear form at very low dose rates.
So it's not surprising that spreading the radiation out significantly reduces the yield of biological events, but actually supports a linear trend for mutation yield in low-dose and low dose-rate conditions. (Whether that tracks linearly in turn to cancer risk, on the other hand, is a topic of much more debate.)
