Individualizing Glioma Radiotherapy Planning by Optimization of Data and Physics-Informed Discrete Loss

TL;DR

The paper presents GliODIL, a data–physics hybrid framework that infers the full spatial distribution of glioma cell density by coupling a Fisher-Kolmogorov reaction–diffusion model with multi-modal imaging through Optimizing a Discrete Loss. By discretizing the forward PDE on a multi-grid, enforcing PDE and initial-condition constraints while fitting MRI and FET-PET signals, GliODIL generates patient-specific tumor maps that inform radiotherapy planning. In a study of 152 GBM patients (58 with pre-treatment FET-PET), GliODIL achieves superior recurrence coverage compared with the Standard Plan and PDE-only baselines, demonstrating robust improvements in targeting complex tumor shapes while maintaining clinically standard treatment volumes. The framework balances data fidelity and physics priors, supports efficient computation, and opens avenues for time-series imaging and uncertainty quantification to advance personalized radiotherapy.

Abstract

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, 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 model, which is adapted for complex cases.

Figures (8)

• Figure 1: GliODIL Overview.a Multi-modal patient data comprising MR and PET imaging. Tissue extraction using atlas registration lipkova2019personalized and automatic brain tumor segmentation kofler2020brats are performed to define the tumor boundaries and microenvironments. Automatically segmented tumor regions include three components:(i) edema, characterized by tissue swelling due to fluid accumulation; (ii) enhancing core, indicative of active tumor growth and characterized by vascular leakage, and (iii) necrotic core, showing tissue death due to hypoxia or nutrient deprivation. Corresponding FET-PET scans provide metabolic insight, further aiding in accurate tumor delineation. b Prior knowledge about the tumor growth process and the imaging signatures of tumor cells. The physics of tumor growth is described by a partial differential equation (PDE), while the relationship between tumor cells and the available imaging data is modeled through the Imaging Model. c Spatio-temporal progression of a tumor within patient anatomy. Calculation of PDE's residual $L_{\text{PDE}}$ and single focal initial condition $L_{\text{IC}}$. Unknown fields are stored on a 4-dimensional multi-resolution grid. Optimization utilizes automatic differentiation of each gridpoint and is guided by the loss function. Since we model the progression based on a single time-point input data, the growth parameters are being resolved up to a timescale. Calculation of discrepancy between patient's tumor characteristics $L_{\text{DATA}}$ and proposed by GliODIL tumor cell-distribution at the final time-point. d GliODIL outputs. The framework successfully infers the complete distribution of tumor cells, facilitating the development of a radiotherapy plan. This plan effectively covers areas of tumor recurrence identified in post-operative data, while maintaining the total radiotherapy volume in line with standard clinical guidelines.
• Figure 2: Calibration of PDE Loss Weight in Synthetic Dataset Experiments for Tumor Segmentation. a Illustration of GliODIL's synthetic results for localized multi-focal and single focal tumors. In these simulations, all loss weights remain constant except for the $L_{\text{PDE}}$ weight, $\lambda_{\text{PDE}}$. A value of $\lambda_{\text{PDE}} = 1$ was selected to strike a balance in the model between fit to the data and the tumor growth equations. Dice scores were calculated at 10% tumor cell concentration, which is outside of the segmented region (edema at least at 20%). b The first and second rows represent a segmentation and a synthetic FET-PET that serve as input to GliODIL. Comparison: a forward run with ground truth parameters, tumor cells distribution inferred by GliODIL, and an initial guess (I. guess) that serves as a starting point of the optimization. c Comparison of exact and inferred parameters. d Input for the multi-focal experiment. e Tumor cell distribution of GliODIL with progressively relaxed $\lambda_{\text{PDE}}$. Root Mean Square Error (RMSE) is calculated for low-cell tumor concentration (at 10%), meaning tumor cell distribution outside of visible tumor on MRI.
• Figure 3: Tumor Cell Concentration Inference in Real Patient Data: Comparative Analysis of Predictive Models. a, b Comparison of tumor cell density predictions from various models with corresponding data inputs. Threshold segmentation values for Dice scores for each model are determined through a grid search, since the LMI model does not infer them. c Average data-fit scores for each model.
• Figure 4: Illustration of Radiotherapy Planning: Uniform Distance Margin (Standard Plan) vs GliODIL vs. $\text{PDE}$ Solution.a) Model inputs including the FET-PET metabolic map and tumor segmentation data. b) Illustration of the distance from the tumor core segmentation and its 1.5 cm isoline, adjusted for brain boundaries. This serves to define the Standard Plan and compare tumor cell distributions for both GliODIL and $\text{PDE}_{\text{GliODIL}}$, ensuring equal total 3D volumes across plans. Also shown is the absolute difference in distribution between GliODIL and $\text{PDE}_{\text{GliODIL}}$, exceeding 20%. c) Visualization of radiotherapy plans including the Standard Plan, GliODIL, and $\text{PDE}_{\text{GliODIL}}$.
• Figure 5: Recurrence Coverage Analysis of Edema, Enhancing Core, and Necrotic Core in Real Patient Radiotherapy Plansa,b Recurrence coverage of selected volume radiotherapy plans. All radiotherapy plans have the same total volume. Output tumor cell distribution thresholds found through a grid search to match the Standard Plan volumes. c,d Average Recurrence Coverage and direct patient-by-patient comparisons to the Standard Plan.