Mathematical Modeling in Gel Design and Applications

In this work an innovative approach was developed to address a significant challenge in the field of radiation dosimetry: the accurate measurement of spatial dose distributions using Fricke gel dosimeters. Hydrogels are widely used in radiation dosimetry due to their ability to simulate the tissue-equivalent properties of human tissue, making them ideal for measuring and mapping radiation dose distributions. Among the various gel dosimeters, Fricke gels exploit the radiation-induced oxidation of ferrous ions to ferric ions and are particularly notable due to their sensitivity. The concentration of ferric ions can be measured using various techniques, including magnetic resonance imaging (MRI) or spectrophotometry. While Fricke gels offer several advantages, a significant hurdle to their widespread application is the diffusion of ferric ions within the gel matrix. This phenomenon leads to a blurring of the dose distribution over time, compromising the accuracy of dose measurements. To mitigate the issue of ferric ion diffusion, researchers have explored various strategies such as the incorporation of additives or modification of the gel composition to either reduce the mobility of ferric ions or stabilize the gel matrix. The computational method proposed leverages the power of artificial intelligence, particularly deep learning, to mitigate the effects of ferric ion diffusion that can compromise measurement precision. By employing Physics Informed Neural Networks (PINNs), the method introduces a novel way to apply physical laws directly within the learning process, optimizing the network to adhere to the principles governing ion diffusion. This is particularly advantageous for solving the partial differential equations that describe the diffusion process in 2D and 3D. By inputting the spatial distribution of ferric ions at a given time, along with boundary conditions and the diffusion coefficient, the model can backtrack to accurately reconstruct the original ion distribution. This capability is crucial for enhancing the fidelity of 3D spatial dose measurements, ensuring that the data reflect the true dose distribution without the artifacts introduced by ion migration. Here, multidimensional models able to handle 2D and 3D data were developed and tested against dose distributions numerically evolved in time from 20 to 100 h. The results in terms of various metrics show a significant agreement in both 2D and 3D dose distributions. In particular, the mean square error of the prediction spans the range $1\times {10}^{-6}$–$1\times {10}^{-4}$, while the gamma analysis results in a 90–100% passing rate with 3%/2 mm, depending on the elapsed time, the type of distribution modeled and the dimensionality. This method could expand the applicability of Fricke gel dosimeters to a wider range of measurement tasks, from simple planar dose assessments to intricate volumetric analyses. The proposed technique holds great promise for overcoming the limitations imposed by ion diffusion in Fricke gel dosimeters. Full article