Minimum dose rate estimation for pulsed FLASH radiotherapy: A dimensional analysis

Abstract

Purpose/objectives: To provide an order of magnitude estimate of the minimum dose rate ( $R_{\min }$ ) required by pulsed ultra-high dose rate radiotherapy (FLASH RT) using dimensional analysis.

Materials/methods: In this study, we postulate that radiation-induced transient hypoxia inside normal tissue cells during FLASH RT results in better normal tissue sparing over conventional dose rate radiotherapy. We divide the process of cell irradiation by an ultra-short radiation pulse into three sequential phases: (a) The radiation pulse interacts with the normal tissue cells and produces radiation-induced species. (b) The radiation-induced species react with oxygen molecules and reduce the cell environmental oxygen concentration ( $\left(O_2\right)$ ). (c) Oxygen molecules, from nearest capillaries, diffuse slowly back into the resulted low $\left(O_2\right)$ regions. By balancing the radiation-induced oxygen depletion in phase II and diffusion-resulted $\left(O_2\right)$ replenishment in phase III, we can estimate the maximum allowed pulse repetition interval to produce a pulse-to-pulse superimposed $\left(O_2\right)$ reduction against the baseline $\left(O_2\right)$ . If we impose a threshold in radiosensitivity reduction to achieve clinically observable radiotherapy oxygen effect and combine the processes mentioned above, we could estimate the $R_{\min }$ required for pulsed FLASH RT through dimensional analysis.

Results: The estimated $R_{\min }$ required for pulsed FLASH RT is proportional to the product of the oxygen diffusion coefficient and $\left(O_2\right)$ inside the cell, and inversely proportional to the product of the square of the oxygen diffusion distance and the drop of intracellular $\left(O_2\right)$ per unit radiation dose. Under typical conditions, our estimation matches the order of magnitude with the dose rates observed in the recent FLASH RT experiments.

Conclusions: The $R_{\min }$ introduced in this paper can be useful when designing a FLASH RT system. Additionally, our analysis of the chemical and physical processes may provide some insights into the FLASH RT mechanism.

Keywords: FLASH RT; diffusion process; dose rate effect; hypoxia; oxygen effect.