Personalized Predictions of Glioblastoma Infiltration: Mathematical Models, Physics-Informed Neural Networks and Multimodal Scans

TL;DR

This work presents a physics-informed neural network framework to infer patient-specific GBM growth parameters from a single MRI snapshot by coupling a Fisher–KPP reaction–diffusion model with a diffuse-domain approach to complex brain geometry. A non-dimensional scaling strategy yields tractable, near-unity parameters, enabling a two-stage training where a characteristic PDE solution guides pre-training before patient-specific fine-tuning. Validation on synthetic data and 24 patient cases shows that the method can predict tumor infiltration patterns and support personalized radiotherapy planning, with SEG data often sufficing and FET offering incremental gains. The approach provides a practical, data-efficient pathway for mechanistic, image-guided tumor prognosis and treatment design, while acknowledging limitations of the simplified GBM dynamics and the need for uncertainty quantification.

Abstract

Predicting the infiltration of Glioblastoma (GBM) from medical MRI scans is crucial for understanding tumor growth dynamics and designing personalized radiotherapy treatment plans.Mathematical models of GBM growth can complement the data in the prediction of spatial distributions of tumor cells. However, this requires estimating patient-specific parameters of the model from clinical data, which is a challenging inverse problem due to limited temporal data and the limited time between imaging and diagnosis. This work proposes a method that uses Physics-Informed Neural Networks (PINNs) to estimate patient-specific parameters of a reaction-diffusion PDE model of GBM growth from a single 3D structural MRI snapshot. PINNs embed both the data and the PDE into a loss function, thus integrating theory and data. Key innovations include the identification and estimation of characteristic non-dimensional parameters, a pre-training step that utilizes the non-dimensional parameters and a fine-tuning step to determine the patient specific parameters. Additionally, the diffuse domain method is employed to handle the complex brain geometry within the PINN framework. Our method is validated both on synthetic and patient datasets, and shows promise for real-time parametric inference in the clinical setting for personalized GBM treatment.

Figures (22)

• Figure 1: Overview of the parameter estimation framework. I) Medical Images show preoperative patient T1Gd and FLAIR scans. II) Mathematical Model includes a Partial Differential Equation (PDE) for the tumor cell density $u$, and an imaging model that relates medical images with $u$ by thresholding (segmentation). The unknown parameters that need to be estimated are tabulated. III) Physics-informed Neural Network (PINN) is used to solve the PDE and estimate parameters from data. See Fig. \ref{['f:workflow']} below for a detailed workflow. IV) Prediction and Validation: The estimated parameters can be used in the PDE model to predict the 3D tumor cell density (right), and can be validated via DICE scores (center). Left: The personalized clinical target volume (CTV, blue) based on model predictions is compared with Radiation Therapy Oncology Group (RTOG) CTV (green) to evaluate radiation volume and coverage of tumor recurrence (efficiency).
• Figure 2: Workflow for patient-specific parameter estimation via PINN. I. Preprocessing: Register the brain atlas to the T1Gd patient scan. Solve the Cahn-Hilliard equation to acquire the geometry. Compute centers and radii of T1Gd and FLAIR segmentations ($R^{\rm T1Gd}_{\rm seg}$ and $R^{\rm FLAIR}_{\rm seg}$). II. Grid Search: In a spherically symmetric geometry, use a grid search algorithm to identify patient-specific characteristic values of $\bar{D}/\bar{\rho}$ and $L$ such that the radii derived from the PDE solution are close to $R^{\rm T1Gd}_{\rm seg}$ and $R^{\rm FLAIR}_{\rm seg}$. III. Pre-training: Solve the PDE using a Finite Difference Method (FDM) with the patient-specific characteristic parameters (and $\mu_\mathcal{D}=\mu_\mathcal{R}=1$) in the patient geometry to obtain a characteristic solution $\bar{u}^{FDM}$. Train the PINN to solve the PDE using $\bar{u}^{FDM}$ as data. IV. Fine-Tuning: Use the segmentation data to fine-tune the PINN and learn the patient specific parameters $\mu_\mathcal{D}$ and $\mu_\mathcal{R}$. These estimated parameters are used for tumor cell density predictions.
• Figure 3: Validation using synthetic data. Row 1: (a)-(d). ground truth (GT) data; Row 2: (e)-(h). Training data-- GT with correlated noise; Row 3: (i)-(l). Parameter estimation using noisy, synthetic FET+SEG data and predictions by evaluating the PINN solution, $u^{PINN}$; Row 4: (m)-(p). Parameter estimation using noisy FET+SEG data and predictions using the FDM solution of the PDE, $u^{FDM}$, with inferred parameters; Row 5: (q)-(t). Parameter estimation using only the noisy SEG data and predictions using $u^{FDM}$. (t): Comparison of contours at 1%, 25%, 50%, and 75% between $u^{FDM}$ (solid) and GT $u$ (dotted). Column 1: 3D isosurfaces of tumor cell density at 1% and $y^{\rm FLAIR}$; Column 2: Cell densities; Column 3: Synthetic segmentations mimicking T1Gd and FLAIR data; Column 4: Synthetic FET-PET distributions. Further, (k), (o) and (s) show comparisons of predicted segmentations (lines--dashed, solid) with GT segmentations (filled). Comparing rows 3 and 4, $u^{PINN}$ is less accurate than $u^{FDM}$ as a solution of the PDE with the inferred parameters because $u^{PINN}$ is more strongly influenced by the noisy data. Notably, the estimated cell densities from the $u^{FDM}$ using FET+SEG and SEG in rows 4 and 5, respectively, are very similar and yield accurate approximations of the GT data.
• Figure 4: Predicted tumor morphologies at times different from that of training data. Morphologies are shown at the normalized times $\bar{t}$ = 0.6, 0.8, 1.0, 1.2 and are obtained using $u^{FDM}$ with parameters inferred from the synthetic data in Fig. \ref{['f:syndetail']}. Only SEG data is used for inference and the training data corresponds to the normalized time $\bar{t}=1$. Row 1: 3D isosurfaces of $u^{FDM}$ at 1% and 30% Row 2: Comparision of predicted contours (solid; $1\%,~25\%,~50\%$ and $75\%$) with GT (dotted).
• Figure 5: Predicted patient-specific tumor morphologies, cell densities and FET-PET distributions from Patient 5 (see text). Row 1: Medical images: (a) T1Gd; (b) FLAIR; (c) Post-processed FET-PET (see text); (d) Original FET-PET data. Row 2: $u^{FDM}$ predictions using FET+SEG data. Row 3: $u^{FDM}$ predictions using SEG data only. (e) and (i): Predicted cell density distributions from $u^{FDM}$ together with boundaries of the segmentations (curves) from the MRI images. (f) and (j): Comparisons of the contours of $u^{FDM}$ at the inferred segmentation thresholds (curves) with MRI segmentations (filled). Margins for $\rm CTV^{\rm RTOG}$ (green) and $\rm CTV^{\rm P}$ (blue); (g) predicted FET-PET distribution (see text). (h) and (l): 3D visualization of 1% and $u_c^{\rm FLAIR}$ isosurfaces. (k) Comparison of 1%, $u_c^{\rm FLAIR}$, and $u_c^{\rm T1Gd}$ contours of $u^{FDM}$ (solid) and $u^{PINN}$ (dashed).