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Estimating glioblastoma migration-to-proliferation ratio from single time-point MRI data via radial density distribution

Rafael BravoiD, David A. HormuthiD, Ernesto A. B. F. LimaiD, Bahar TercaniD, Boris H. AguilariD, Thomas E. YankeeloviD

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Abstract

Glioblastoma (GBM) tumors with a high migration-to-proliferation ratio (D/k) are associated with treatment resistance, motivating the estimation of D/k from routine pre-treatment magnetic resonance imaging (MRI) data and the investigation of mechanisms underlying its heterogeneity across patients. Unlike previous approaches based on nested spherical approximations of enhancing and non-enhancing tumor volumes, we incorporate the spatial arrangement of the enhancing and non-enhancing tumor regions by deriving a 1D tumor density distribution from their segmentations and fitting a 1D reaction-diffusion model to estimate D/k. We achieved an error of under 5% when comparing our method to synthetic density distributions and found excellent correlation with synthetic tumor segmentations (Pearson correlation coefficient (PCC) = 0.996). We then evaluated associations between patient-specific D/k values and their clinical, transcriptomic, and radiomic features. When applied to 102 pre-treatment MRI scans from the Brain Tumor Segmentation (BraTS) 2017 dataset, elevated D/k significantly correlated with reduced overall survival (log-rank p = 0.01), older age (p = 0.04), and a trend toward lower Karnofsky Performance Status (p = 0.059). In a subset of 91 patients with available microarray data, mesenchymal-like phenotypes consistently exhibited elevated D/k across three established molecular subtype classifications (p = 0.013, p = 0.055, and p = 0.127). Elastic Net regularization followed by overrepresentation analysis revealed an enrichment of glycosylation-related pathways among genes associated with D/k variability (p = 0.0123 and p = 0.0368). Furthermore, Multi-Omic Factor Analysis identified a radiomic factor correlated with D/k (r = 0.57, p = 4x10-9), with the highest-weighted features corresponding to smoothness of the non-enhancing tumor region on T2 and FLAIR.

In-depth summary

Claude Opus 5.5 · xhigh effort · 7 Nov 2026 · from the full text of version 1 (31 Oct 2026) · CC BY 4.0

A reaction-diffusion model fitted to shell-averaged tumor density from one pre-treatment MRI estimates glioblastoma migration/proliferation ratio (D/k); in BraTS 2017, higher D/k went with shorter overall survival.

Current D/k methods fit a reaction-diffusion model to enhancing and non-enhancing volumes treated as nested spheres, so a spiculated tumor and a near-spherical one with similar voxel counts cannot be told apart, although irregular shape carries a worse prognosis. Here a radial density profile around the tumor center is fitted instead, so an irregular shape adds to the inferred migration, and the estimate is then linked to survival, molecular subtypes and gene expression.

Key results

  1. SuggestedOn synthetic tumors grown with the same reaction-diffusion equation, the fitting recovered known D/k closely from line profiles. Noise degraded low-D/k estimates more than high-D/k ones, and segmentation-based inputs gave offset but tightly correlated estimates.

    Line profiles: CCC 0.9979 (k fixed) vs 0.9978 (D fixed); median error -2.01% (IQR -2.66% to -1.48%) at dx = 1 mm. Noise σ = 0.05: mean error 20.1% (SD 20.5%) at D/k 0.1 mm vs -1.2% (2.7%) at 10 mm. Simulated segmentations: CCC = 0.813, PCC = 0.996. · Validating D/k fitting using simulated density…; Validating the D/k pipeline using simulated…; D/k pipeline confirmed with synthetic density…

  2. SuggestedAcross BraTS 2017 patients, those above an optimized D/k cut point had worse overall survival. D/k also correlated with age and showed a trend toward lower Karnofsky Performance Score (KPS).

    Cut point 2.081 mm from a log-rank sweep over the 10th to 90th percentiles; high D/k n = 54 vs low n = 47, log-rank p = 9.89 × 10⁻³ (not adjusted for cut-point selection). Spearman: age p = 0.04, KPS p = 0.06. Similar trends with the Swanson D/ρ method. · Clinical Data and Survival Analysis; Correlation with clinical variables and…; D/k is correlated with age and overall survival

  3. SuggestedMesenchymal-subtype tumors tended to have higher D/k than all other subtypes combined, but the difference was significant in only one of three subtype schemes.

    Mann-Whitney U test, mesenchymal vs the rest (neural subtype excluded): microarray subtypes p = 0.0547 (n = 91); Transcriptome subtype p = 0.127 (n = 81); Random Forest Sturm methylation clusters p = 0.0133 (n = 22). · Correlation with clinical variables and…; Mesenchymal tumors tend to exhibit increased D/k; Figure legends

  4. SuggestedElastic Net regression on microarray expression predicted held-out D/k moderately well, and the genes it selected were enriched for N-linked glycosylation pathways.

    LOOCV predicted vs actual D/k: Pearson r = 0.42, p = 3.3 × 10^-5 (91 patients). The full-data model kept 98 genes. Bonferroni-adjusted ORA: Reactome Asparagine N-linked Glycosylation p = 0.0123, Synthesis of UDP N Acetyl Glucosamine p = 0.0368; no MSigDB Hallmark set was significant. · Elastic Net analysis; Glycosylation Pathway associations identified by…; Glycosylation pathways associated with increased…

  5. SuggestedIn a joint MOFA of microarray and radiomic data, one factor tracked D/k and was dominated by texture features of the non-enhancing region on T2 and FLAIR. Coarser, more uniform texture went with higher D/k.

    MOFA gave 20 factors that were mostly modality-specific. Factor 6 vs D/k: r = 0.57, p = 4 × 10⁻⁹. Its highest-weighted features were all T2 or FLAIR radiomic features of the non-enhancing region (from 704 radiomic features per patient). · BraTS 2017 dataset: imaging, radiomic, and…; Multi-omics factor analysis finds radiomic…; Radiomic texture of the non-enhancing region may…

Methods, data and reproducibility

Segmentations became density maps (90% enhancing, 45% non-enhancing, 0% normal-appearing), averaged over 1 mm spherical shells around the enhancing-region center of mass, and fitted with a 1D finite-volume spherical reaction-diffusion model (grid search, then Levenberg-Marquardt, with k fixed). D/k was checked on simulated tumors, then related to survival, subtypes, gene expression and MOFA factors.

  • BraTS 2017 pre-treatment glioblastoma MRI (pre/post-contrast T1, T2, FLAIR) with manual segmentations, plus clinical data (age, KPS, overall survival) · human (glioblastoma patients) · n = 102 patients (survival missing for 1)
  • BraTS 2017 radiomic features (intensity, shape, texture) from pre-treatment MRI and segmentations · human · n = 704 features per patient · 10.7937/K9/TCIA.2017.KLXWJJ1Q
  • TCGA glioblastoma mRNA microarray, RMA-normalized, downloaded from FireBrowse · human · n = 91 patients
  • Synthetic 3D tumors from forward-Euler solves of the reaction-diffusion model (199 x 199 x 199 mm domain), with resolution, noise, segmentation and boundary-proximity variants · n = 10 D/k values; 20 noise replicates per D/k and noise level
  • Published subtype labels: microarray subtypes (202 TCGA glioma samples), RNA-seq subtypes (667 samples), DNA methylation random-forest clusters (324 samples) · human · n = 91, 81 and 22 BraTS patients, respectively
Methods & tools
Python, numba, Levenberg-Marquardt optimization, lifelines, survminer (surv_cutpoint), Elastic Net regression, MOFA, GSVA, Reactome, MSigDB Hallmark gene sets, FireBrowse
Assay conditions & compute
Simulations ran on a MacBook Air with a 10-core Apple M4 chip and 16 GB RAM; run times are not reported.
Reproducibility gaps
The original MRI, clinical and molecular data are not redistributed; they must be obtained from BraTS (registration), TCIA and FireBrowse. The repository includes patient-level D/k for 102 cases and coefficients for 98 genes. No software versions are stated.
Limitations (6)
  • Each segmented region gets a fixed density (90%, 45%, 0%), which ignores necrosis and the true density gradient, and normal-appearing brain is assumed tumor-free despite infiltration there. ADC maps, which could estimate density, were not available in BraTS 2017. · stated by the authors
  • The model omits anatomy that affects migration (grey vs white matter rates, brain boundaries, tissue compression); adding it would raise model complexity and likely require more imaging time points. · stated by the authors
  • Expression comes from microarrays of one biopsy site per tumor. That site may not represent the whole imaged tumor, microarrays profile less than RNA-seq, and transcript data cannot identify which glycoproteins are modified. · stated by the authors
  • The survival p-value is not corrected for the data-driven choice of cut point, and the association between D/k and survival may not be independent of age. · stated by the authors
  • The synthetic validation generates tumors with the same isotropic reaction-diffusion equation that is fitted, so it does not test model misspecification; on simulated segmentations the estimates were offset (CCC = 0.813). · noted in this summary
  • All patient-level associations come from one cohort (102 patients, 91 with microarray) with no external validation, and key effects are modest (mesenchymal comparison p = 0.0547; LOOCV prediction r = 0.42). · noted in this summary

Read this if you model glioma growth from imaging, or want a single-scan invasiveness measure with open Python code to relate to survival, subtypes or expression data in your own cohort.

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