Equivalent constant dose rate as a unifying dose rate concept for FLASH across all beam time structures

The FLASH effect has been investigated experimentally using beam modalities with different time structures such as pulsed electron beams and pencil beam scanning (PBS) proton fields. The different beam time structures make direct comparisons between experiments difficult. A biologically founded dose rate concept that is applicable across all beam time structures is highly needed.

For any dose delivery with a time-varying dose rate, we define the equivalent constant dose rate ( $\dot{D}$ _eq,const) as the dose rate that would give the same FLASH sparing effect if the same dose had been delivered with a constant dose rate. This study develops a mathematical framework for $\dot{D}$ _eq,const and uses this to investigate $\dot{D}$ _eq,const for a range of beam time structures.

Calculation of $\dot{D}$ _eq,const requires a model of the FLASH sparing effect. Here, $\dot{D}$ _eq,const was investigated using a phenomenological oxygen enhancement ratio (OER) weighted dose model that has previously been used for successful quantification of the acute skin toxicity variations observed in mice across various proton PBS irradiations. Analytical equations were developed to calculate $\dot{D}$ _eq,const for any piece-wise constant dose rate irradiation and used to determine $\dot{D}$ _eq,const for several pulsed beams and PBS fields. $\dot{D}$ _eq,const was characterized systematically for pulsed beams and PBS fields, and the ability of $\dot{D}$ _eq,const to describe acute skin toxicity occurrence in previous mice experiments was investigated.

For pulsed beams with mean dose rate $\dot{D}$ _mean, n pulses, pulse doses up to 5 Gy and pulse repetition times up to 100 ms, $\dot{D}$ _eq,const was approximately given by n/(n+1)⋅ $\dot{D}$ _mean. $\dot{D}$ _eq,const for a 40 Gy PBS transmission proton field with $\dot{D}$ _mean = 60 Gy/s previously used for mouse irradiations was 155 Gy/s without repainting, 64 Gy/s with four repaintings, and 7.7 Gy/s when splitting the field into two deliveries separated by 2 minutes. $\dot{D}$ _eq,const was 98 Gy/s for spread-out Bragg peak irradiation without repainting. The occurrence of acute skin toxicity in mice after ∼40 Gy irradiations with various beam time structures, including pulsed electron beams and proton PBS, followed the same trend as function of $\dot{D}$ _eq,const, demonstrating its radiobiological relevance. $\dot{D}$ _eq,const only depended slightly on the phenomenological OER-model parameters.

The generally applicable biologically founded dose rate concept $\dot{D}$ _eq,const allows direct comparison of FLASH dose rates across all beam time structures.