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Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans and Bayesian Inference

Jana Lipkova, Panagiotis Angelikopoulos, Stephen Wu, Esther Alberts, Benedikt Wiestler, Christian Diehl, Christine Preibisch, Thomas Pyka, Stephanie Combs, Panagiotis Hadjidoukas, Koen Van Leemput, Petros Koumoutsakos, John S. Lowengrub, Bjoern Menze

arXiv:1807.00499v3cs.CE

TL;DR

Glioblastoma cells extend beyond scan-visible lesions, limiting population-based radiotherapy planning. The paper combines multimodal MRI and FET-PET with a Bayesian tumor model to infer patient-specific cell density and uncertainty from single-time-point scans. The resulting plans spare more healthy tissue while maintaining comparable tumor coverage and identify high-density regions for dose escalation.

  • Problem

    Glioblastoma infiltrates beyond visible lesions, while existing radiotherapy plans rely mainly on population studies and do not adequately account for patient-specific tumor distributions.

  • Method

    A Bayesian framework combines a patient-specific tumor growth model with multimodal MRI and FET-PET observations to infer tumor cell density while propagating modeling and imaging uncertainties.

  • Results

    The personalized plans spare more healthy tissue while maintaining tumor coverage comparable to standard radiotherapy, and inferred high-density regions coincide with areas supporting dose-escalation design.

  • Takeaways & Limitations

    Single-time-point multimodal scans and mathematical modeling provide patient-specific tumor estimates that can assist personalized radiotherapy design.

Abstract

from arXiv · show

Glioblastoma is a highly invasive brain tumor, whose cells infiltrate surrounding normal brain tissue beyond the lesion outlines visible in the current medical scans. These infiltrative cells are treated mainly by radiotherapy. Existing radiotherapy plans for brain tumors derive from population studies and scarcely account for patient-specific conditions. Here we provide a Bayesian machine learning framework for the rational design of improved, personalized radiotherapy plans using mathematical modeling and patient multimodal medical scans. Our method, for the first time, integrates complementary information from high resolution MRI scans and highly specific FET-PET metabolic maps to infer tumor cell density in glioblastoma patients. The Bayesian framework quantifies imaging and modeling uncertainties and predicts patient-specific tumor cell density with confidence intervals. The proposed methodology relies only on data acquired at a single time point and thus is applicable to standard clinical settings. An initial clinical population study shows that the radiotherapy plans generated from the inferred tumor cell infiltration maps spare more healthy tissue thereby reducing radiation toxicity while yielding comparable accuracy with standard radiotherapy protocols. Moreover, the inferred regions of high tumor cell densities coincide with the tumor radioresistant areas, providing guidance for personalized dose-escalation. The proposed integration of multimodal scans and mathematical modeling provides a robust, non-invasive tool to assist personalized radiotherapy design.

I. INTRODUCTION

Glioblastoma infiltrates beyond scan-visible lesions, making uniform, population-based radiotherapy margins poorly suited to patient-specific disease patterns. The proposed framework combines a patient-specific tumor growth model, multimodal imaging, and Bayesian inference to estimate tumor density and guide personalized treatment plans.

  • I. INTRODUCTION: Glioblastoma infiltrates surrounding healthy-appearing tissue, while standard radiotherapy uniformly expands visible tumors using population-derived margins despite uncertain residual-cell distributions.The extent of the clinical target volume margin varies across radiotherapy guidelines, and infiltration is anisotropic.
  • A. Tumor growth model: The framework combines a Fisher-Kolmogorov tumor growth model with a stochastic imaging model linking simulated cell density to MRI and FET-PET observations.The growth equation models proliferation and infiltration in a patient-specific brain anatomy reconstructed from MRI scans.
  • C. Parameters estimation and uncertainty propagation: Bayesian inference estimates patient-specific parameters while accounting for modeling and measurement uncertainties, then propagates them into robust tumor-density predictions.The framework produces maximum-a-posteriori, mean, and standard-deviation estimates for tumor cell density.
  • B. Imaging model: MRI segmentations provide morphological tumor observations, while FET-PET and other modality observations are related to the modeled tumor cell density at corresponding voxels.The framework uses complementary observations from multimodal scans within a shared patient-specific anatomy.
  • IV. Personalized Radiotherapy: The personalized plan spares more healthy tissue while maintaining tumor coverage comparable to the standard protocol and supports dose-escalation design using inferred high-density regions.Observed recurrences are used to compare treatment plans, and high-density regions are marked for escalation planning.

B. Multimodal imaging model

The imaging model combines MRI segmentations with FET-PET observations to relate multimodal scans to modeled tumor cell density while accounting for patient-specific uncertainty.

  • B. Multimodal imaging model: MRI provides binary morphological segmentations, while FET-PET supplies a normalized metabolic signal related to tumor cell density.The model treats observations from T1Gd, FLAIR, and FET-PET scans as modality-specific data at corresponding voxel locations.
  • B. Multimodal imaging model: The probability of observing each MRI segmentation is modeled with a Bernoulli distribution whose visibility depends on tumor cell density and an unknown threshold.The threshold c represents the cell density below which tumor cells are not visible in MRI.
  • B. Multimodal imaging model: The FET-PET observation model assumes proportionality between normalized signal and tumor cell density, with patient-specific proportionality and noise parameters.Prediction error is modeled as normally distributed, and the PET scan is registered to the higher-resolution MRI scans.
  • B. Multimodal imaging model: The imaging model accounts for PET resolution by using voxels separated by 4 mm when evaluating the modeled PET signal.The PET scan has 4 mm resolution, whereas the MRI scans have 1 mm resolution.

C. Parameters estimation and uncertainty propagation

Bayesian calibration updates prior parameter distributions using multimodal scan data, then propagates posterior uncertainty to obtain tumor-density predictions and MAP estimates.

  • C. Parameters estimation and uncertainty propagation: Bayesian calibration combines prior information with scan data through the posterior distribution P(θ|D, M), using a likelihood for the observed modalities.The likelihood treats observations from different medical scans as independent because they capture different physiological processes.
  • C. Parameters estimation and uncertainty propagation: Transitional Markov Chain Monte Carlo samples the posterior through intermediate probability distributions and supports efficient exploration with parallel chains.The algorithm uses automatically computed intermediate exponents and a parallel implementation.
  • C. Parameters estimation and uncertainty propagation: Posterior parameter uncertainty is propagated through the tumor and imaging models to produce robust predictions of tumor cell density.The resulting predictions can be summarized using posterior means and variances.
  • C. Parameters estimation and uncertainty propagation: The maximum a posteriori estimate selects the most probable tumor cell density conditioned on the observed data and model.It is defined as uMAP = argmaxθ P(u|D, M).
  • C. Parameters estimation and uncertainty propagation: In the synthetic test case, multimodal calibration closely agrees with ground truth, whereas MRI-only calibration cannot correctly recover the tumor cell-density profile.The comparison is shown for the synthetic ground-truth density and corresponding image observations.

III. RESULTS

The study evaluates the Bayesian framework first on synthetic data and then on clinical data for patient-specific tumor-density inference and personalized radiotherapy planning.

  • III. RESULTS: The evaluation uses synthetic data to test inference sensitivity and multimodal image information, followed by clinical data for tumor-density inference and radiotherapy design.Tumor recurrence patterns are used to compare personalized and standard radiotherapy plans.

A. Sensitivity study

Synthetic-data calibration shows that single-time-point observations cannot uniquely identify growth-speed parameters, while multimodal data still support accurate tumor-density inference.

  • Synthetic setup: The synthetic observations were generated by thresholding ground-truth densities for T1Gd and FLAIR segmentations and adding normalized Gaussian noise to the FET-PET signal.A sensitivity study indicated that 6000 samples were adequate for the model.
  • Identifiability: Single-time-point tumor observations cannot uniquely identify the time-dependent parameters Dw, ρ, and T because compensating parameter combinations generate similar densities.The posterior captures plausible parameter distributions rather than forcing a single estimate.
  • Synthetic calibration: The calibration table reports MAP, mean, and standard deviation for growth and tumor-location parameters relative to the synthetic ground truth.The reported units include mm2/day for Dw, 1/day for ρ, days for T, and millimeters for tumor-center coordinates.
  • Synthetic calibration: Multimodal calibration produced MAP and mean tumor-density estimates almost indistinguishable from the ground-truth tumor despite large parametric uncertainties.Low posterior standard deviations indicate that the multimodal observations were sufficient for tumor-density inference from a single time point.
  • Multimodal information: MRI primarily constrains tumor morphology, whereas FET-PET constrains cell-density profiles in regions of high tumor infiltration.Using MRI alone produced central density deviations and higher uncertainty relative to the ground truth.

B. Patient study

A retrospective study of eight glioblastoma patients used preoperative multimodal scans for Bayesian inference, yielding patient-specific infiltration predictions with low uncertainty.

  • Study design: The retrospective clinical study included eight patients with glioblastoma who received surgery followed by combined radiotherapy and chemotherapy.Preoperative scans from patients P1-P8 were used for Bayesian inference after no visible tumor remained post-treatment.
  • Patient predictions: The model accurately predicted collateral-hemisphere infiltration for P5, P7, and P8 while restricting predictions to one hemisphere for P1-P4.These patient-specific spatial patterns were evaluated against recurrence patterns.
  • Patient predictions: For visually similar preoperative tumors in P1 and P2, the model predicted more infiltrative behavior for P1, consistent with its recurrence pattern.Low posterior standard deviations reflected high confidence in the predictions.

C. Personalized radiotherapy design

Patient-specific tumor-density predictions are used to design clinical target-volume margins and identify high-cellularity regions for potential dose escalation.

  • Personalized design: Personalized radiotherapy can use MAP or mean-plus-standard-deviation tumor-density scenarios to design CTV margins and identify potentially radioresistant regions.Because MAP and mean estimates were similar and standard deviations were small, this study used MAP estimates.
  • Evaluation: CTV efficiency is defined as the relative volume of recurrent tumor contained within the clinical target volume.Recurrence patterns provide the basis for evaluating whether target-volume designs cover recurrent disease.

1) Dose distribution:

The personalized MAP-based plan reduced irradiated healthy tissue while maintaining efficiency comparable to the standard RTOG protocol and supported dose-escalation targeting.

  • Dose distribution: Figure 5 combines preoperative MRI and FET-PET, inferred MAP and uncertainty maps, recurrence scans, CTV margins, and proposed dose-escalation outlines for P1-P8.The figure contrasts standard RTOG margins with MAP-based personalized margins and escalation regions.
  • Dose distribution: The CTV MAP plan used a smaller irradiation volume while achieving comparable efficiency to the CTV RTOG plan.Figure 6 compares overall irradiated CTV volume with corresponding efficiency for the two plans.
  • CTV margins: The personalized CTV was obtained by thresholding MAP tumor-cell density at u = 0.1% to match the efficiency of the RTOG CTV.The standard RTOG margin used a 2 cm expansion around the visible tumor.
  • Clinical comparison: Personalized plans spared more healthy tissue while maintaining standard-protocol efficiency, although both plans had reduced efficiency for P7-P8.The reduced efficiency for P7-P8 was mainly around the ventricles and may reflect preoperative-to-recurrence anatomical misalignment.
  • Dose escalation: High-cellularity regions identified from FET-PET enhancement or MAP estimates were compared as candidate targets for dose escalation.The escalation analysis evaluates how well each plan targets T1Gd-enhanced recurrent tumor regions.

2) Dose-escalation:

The framework uses inferred tumor cell-density maps to propose personalized dose-escalation plans, while emphasizing that optimal designs require further study.

  • 2) Dose-escalation:: High-cellularity regions coincided with tumor-recurrence areas better than FET-PET enhancement alone, supporting personalized dose-escalation design.The authors present this as a preliminary result and identify optimal dose-escalation design as requiring more extensive studies.
  • 2) Dose-escalation:: Two candidate plans used thresholded MAP tumor density: a single-level escalation at u = 30% and a cascaded four-level plan at u = [0.1, 25, 50, 75]%.
  • 2) Dose-escalation:: Patient-specific tumor estimates extend radiotherapy planning beyond visible lesion outlines and can identify regions for dose-escalation.
  • 2) Dose-escalation:: The software and data used in the work are publicly released to facilitate clinical translation and future improvements.

A. Prior PDF

The Bayesian calibration assigns uniform prior distributions to patient-specific tumor and imaging parameters within specified biologically motivated ranges.

  • A. Prior PDF: The unknown parameters θ = {θu, θFT1Gd, θFFLAIR, θFFET} are assumed independent with uniform prior distribution.
  • A. Prior PDF: Prior ranges include Dw ∈[0.013, 3.8] mm2/day, ρ ∈[0.0027, 0.19] /day, and T ∈[30, Tmax + 365] day.Tmax is defined from the distance the minimum-speed propagation wave must travel across the visible tumor.
  • A. Prior PDF: The initial tumor location prior is a cube centered at the tumor center of mass with side length 2rmax.
  • A. Prior PDF: Imaging-model priors use α ∈[0.05, 0.1], σ ∈[0.015, 0.2], and b ∈[0.5, 1.02].The lower bound for b reflects that T1Gd and FET-PET scans usually do not enhance low-grade gliomas.

C. Effect of the brain anatomy on the tumor cell density

Brain anatomy and anisotropic growth produce spatially variable tumor cell densities across visible segmentation borders, making patient-specific imaging parameters important.

  • C. Effect of the brain anatomy on the tumor cell density: Tumor cell densities varied significantly along T1Gd and FLAIR segmentation borders and among patients because of anatomical restrictions and anisotropic growth.
  • C. Effect of the brain anatomy on the tumor cell density: At borders far from ventricles and skull, inferred tumor densities varied less and were more consistent among patients.
  • C. Effect of the brain anatomy on the tumor cell density: The findings imply that imaging parameters should be patient-specific, particularly for complex anatomies such as the brain.
  • C. Effect of the brain anatomy on the tumor cell density: MRI-based registration and tissue segmentation reconstruct patient brain anatomy for the imaging and tumor-growth analyses.
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