  Brain tumor growth is unique to each glioma patient and extends beyond what
is visible in imaging scans, infiltrating surrounding brain tissue.
Understanding these hidden patient-specific progressions is essential for
effective therapies. Current treatment plans for brain tumors, such as
radiotherapy, typically involve delineating a uniform margin around the visible
tumor on pre-treatment scans to target this invisible tumor growth. This "one
size fits all" approach is derived from population studies and often fails to
account for the nuances of individual patient conditions. We present the
GliODIL framework, which infers the full spatial distribution of tumor cell
concentration from available multi-modal imaging, leveraging a
Fisher-Kolmogorov type physics model to describe tumor growth. This is achieved
through the newly introduced method of Optimizing the Discrete Loss (ODIL),
where both data and physics-based constraints are softly assimilated into the
solution. Our test dataset comprises 152 glioblastoma patients with
pre-treatment imaging and post-treatment follow-ups for tumor recurrence
monitoring. By blending data-driven techniques with physics-based constraints,
GliODIL enhances recurrence prediction in radiotherapy planning, challenging
traditional uniform margins and strict adherence to the Fisher-Kolmogorov
partial differential equation (PDE) model, which is adapted for complex cases.
