Proton Therapy: Analytical Formulation of Three-Dimensional Geometric Tumor Regression in Pediatric Medulloblastoma by Magnetic Resonance Imaging

Abstract

Despite the major importance of medical imaging in pediatric neuro-oncology, there is still a limited connection between clinical image interpretation and rigorous mathematical formalism. Pediatric medulloblastoma is usually described through radiological, surgical, and pathological findings, while its geometric evolution is rarely translated into a quantitative and reproducible framework. This study addresses that gap by proposing an innovative mathematical approach based on real clinical imaging data from the literature. Starting from classical tumor-growth theory, the work develops a regression-oriented model based on differential equations to describe the three-dimensional evolution of the lesion. The tumor is approximated by simplified analytical geometries, allowing its dimensions to be incorporated into a tractable volumetric framework. Numerical implementation in Python was then used to generate three-dimensional representations consistent with the radiological states reported in the clinical case. The results show that qualitative imaging findings can be converted into measurable mathematical objects, improving the interpretation of tumor burden and residual disease. The study also demonstrates that mathematical language can move beyond abstraction and become a practical tool for quantitative medical imaging. In this sense, the work contributes to an underexplored field by linking medical physics, applied mathematics, and pediatric oncology through a clinically meaningful modeling strategy.