Personalized Spatiotemporal Radiotherapy for GBM: A PDE-Constrained Optimization Study.

We develop a personalized optimization framework for designing spatiotemporal radiotherapy strategies for Glioblastoma Multiforme (GBM). The tumor dynamics are described by a reaction diffusion model on patient-specific brain geometries, and the treatment objective is formulated as a PDE-constrained optimization problem in which the admissible targeting region is allowed to vary on each day of therapy. The resulting optimization requires repeated solutions of the state and adjoint equations, and we implement an adjoint-driven gradient algorithm with line search and projection to enforce clinical dose constraints. To improve robustness, the algorithm is first validated on a simplified surrogate model before being applied to full patient data. The proposed framework generates patient-specific targeting volumes informed by clinical and radiological parameters, producing a dynamic treatment region that adapts to tumor evolution rather than relying on standard fixed or population-based margins. Our results demonstrate that incorporating daily adaptability into the optimization process can lead to substantially different and potentially more effective treatment targets. This study highlights the potential of PDE-constrained optimization as a mathematical tool for personalized radiotherapy planning.