# Equilibrium state dynamics‐based modeling of temporal dose delivery dependencies of FLASH skin sparing

## TL;DR

This study introduces a new model to understand how FLASH radiotherapy spares normal skin tissue based on how radiation is delivered over time.

## Contribution

The novel perturbed equilibrium state dynamics (ESD) framework quantitatively models temporal dose delivery effects on FLASH skin sparing.

## Key findings

- An ESD model with three parameters accurately reproduced 52 averaged FMF values with low error.
- The model captures experimentally observed features like increased sparing with dose and saturation at high doses.
- Three ESD variants showed strong support, while others were less favored based on BIC metrics.

## Abstract

FLASH radiotherapy shows promise in sparing normal tissues while maintaining tumor control, with the magnitude of its sparing effect being strongly influenced by the temporal dose delivery (TDD) structure of ultra‐high dose rate (UHDR) irradiation. Quantitatively describing these dependencies is critical for optimizing UHDR treatments and guiding preclinical and clinical applications.

This study introduces perturbed equilibrium state dynamics (ESD) as a minimal phenomenological framework to describe TDD dependencies of FLASH normal tissue sparing.

ESD‐based modeling assumes a transient perturbation of a generic equilibrium state during irradiation, modulating the biological system's instantaneous radiosensitivity. Isoeffective dose ratios (FLASH‐modifying factors, FMF) were derived from a collection of acute skin toxicity data, consisting of both previously published and newly acquired data, from 721 mice across 60 experimental groups irradiated with pulsed electron beams and scanned proton beams spanning various TDD patterns. Eight ESD model variants were evaluated based on their ability to reproduce experimental FMF values, optimizing their three or four free parameters and assessing goodness‐of‐fit metrics.

The best‐ranked model based on the Bayesian Information Criterion (BIC) used three parameters to reproduce accurately 52 averaged FMF values (mean absolute error: 0.034, mean absolute percentage error: 4.3%). It consistently reproduces experimentally observed features of FLASH sparing in terms of FMF, including an increased sparing with dose followed by saturation at high doses. Two additional model variants received similar support by BIC, while the remaining five variants were less favored.

ESD‐based modeling offers an effective framework to describe TDD dependencies of FLASH murine skin sparing while limiting complexity and number of free parameters to avoid overfitting. Through parameter tuning, the ESD modeling aligns with key experimental data, supporting its potential as a predictive, parsimonious framework for experimental and translational research. By highlighting the potential role of a transient equilibrium perturbation modulating instantaneous radiosensitivity and quantifying respective measures, ESD‐based models may also guide the exploration and modeling of underlying mechanisms. While the present application is tissue‐specific, the approach is broadly adaptable, with clinical translation requiring refitting and validation in other contexts.