Tackling the problem of diffusion in Fricke gel dosimeters through a Physics-informed neural network algorithm

The use of radiation in medical applications demands accurate dose measurements, particularly with the advent of high-dose rates and mini-beam fields. In this context, Fricke Gels (FGs) are gaining interest for their tissue-equivalent properties. FGs rely on Fe $ ^{2+}$ to Fe3+ oxidation upon radiation exposure, that can be quantified by MRI or Optical Absorption (OA) spectroscopy, enabling access to 3D Dose Distributions (DD). Ion diffusion bring to blurring effects in the recorded DD. Furthermore adding chelating agents, such as methylthymol blue (MTB), induces auto-oxidation processes. Both physical phenomena strongly limit FG's clinical utility. Although the processes can be described by modified Diffusion Equation, reconstructing the backward time DD leads to a challenging inverse problem. A promising solution is given by the Physics-Informed Neural Networks (PINNs), which integrate physical laws with machine learning to solve partial differential equations. In this study, we trained PINNs to predict pre-diffusion DD in 1D MTB FGs, exploiting diffused data up to 8 hours post-irradiation. Predictions were compared with OA data from irradiated MTB-GTA gel. Results in terms of Mean Squared Error (MSE) ( $ 1\times 10^{-6} \text{--} 1\times 10^{-5}OD^2$) indicate the PINNs potential in overcoming FG limitations.