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Predicting Clinical Outcomes in Glioblastoma: An Application of Topological and Functional Data Analysis

Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, Raúl Rabadán

arXiv:1611.06818v6stat.AP

TL;DR

Quantifying GBM tumor geometry for statistical analysis is challenging, while collecting molecular data is invasive and costly despite the availability of MRI. The paper develops SECT to summarize tumor shape for regression and finds that SECT explains more DFS variance than other covariates across a wide variety of models.

  • Problem

    Quantifying MRI tumor geometry for statistical analysis is challenging, while collecting GBM molecular data is invasive and costly despite readily available MRIs.

    Source: §1 · p. 1

  • Method

    The paper develops SECT, a smooth-curve summary of MRI tumor shape with an inner product structure, for use as a covariate in functional regression models.

    Source: §1 · p. 2, §3 · p. 7

  • Results

    SECT explained more DFS variance than all other covariates across a wide variety of models and kernel functions.

    Source: §5 · p. 12

  • Takeaways & Limitations

    MRI-derived tumor shape has a clinically relevant connection to variation in GBM survival-based outcomes.

    Source: §1 · p. 2

    Coarse height-function filtrations can miss very local shape undulations, and texture-based features may not be captured by SECT.

    Source: §2.3 · p. 7

Abstract

from arXiv · show

Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining molecular data make other readily-available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to utilize information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic---the smooth Euler characteristic transform (SECT)---that quantifies magnetic resonance images (MRIs) of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus twofold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics.

1 Introduction

GBM’s aggressive course and the invasive, costly collection of molecular data motivate using readily available MRIs. The paper introduces SECT to bring tumor shape into statistical models and examine its relevance to GBM survival outcomes.

1 Introduction

  • GBM is rapidly progressive, with a post-diagnosis survival period of 12–15 months and limited therapies.The introduction also notes spatial diffusivity and molecular heterogeneity as characteristics of GBM.Source: §1 · p. 1
  • Molecular information requires an invasive procedure and costs time and resources, whereas GBM MRIs are readily available.The paper motivates computational use of MRI data as a potential source for cancer research and future treatment strategies.Source: §1 · p. 2
  • SECT summarizes GBM MRI shape as smooth curves, allowing tumor shape to enter regression frameworks using functional data analysis.The paper’s two aims are to quantify tumor images for statistical models and explore imaging information in GBM clinical studies.Source: §1 · p. 2
  • The study combines publicly available TCIA MRIs with matched TCGA genomic and clinical data to examine shape and survival outcomes.Its predictive analysis reports a clinically relevant connection between brain-tumor shape and variation in survival-based outcomes.Source: §1 · p. 2
  • The case study compares gene expression, morphometric and volumetric tumor features, and proposed shape summaries using Gaussian process regression with MCMC inference.The paper describes comparisons across regressions generated by different covariance functions.Source: §1 · p. 2

2 Quantifying Tumor Images Using Topology

This section develops a topological shape summary designed for regression, contrasting it with landmark and persistence-based approaches. It introduces homology as a way to characterize tumor structure and track shape information across scales.

2 Quantifying Tumor Images Using Topology

  • The proposed summary maps tumor shapes into a Hilbert space with an inner product, making shape covariates usable in general regression models.The section identifies injectivity and a well-defined inner product as key properties of this mapping.Source: §2 · p. 2

2.1 Background on Summary Statistics for Shape Data

  • Landmark methods can incur correspondence, alignment, and mapping errors on highly dissimilar shapes, while PHT avoids landmarks but yields summaries unsuitable for general functional models.SECT builds on PHT while constructing a topological summary that can integrate shape information into regression-based methods.Source: §2.1 · pp. 2–3

2.2 Homology and Persistence

  • Homology describes holes in shapes, which can represent MRI-visible necrosis or the segregated masses of multifocal tumors; tracking their occurrences may indicate disease severity.The text notes that greater necrosis or multifocality has been suggested to accompany more aggressive GBM.Source: §2.2 · p. 3
  • Homology degree distinguishes connected components, loops, and voids, corresponding in the GBM application to tumor masses and lesions or necrosis.The interpretation of degrees 1 and 2 as necrosis depends on whether the analysis uses two-dimensional MRI slices or the three-dimensional tumor.Source: §2.2 · p. 3
  • Persistent homology tracks how homology changes across a filtration of simplicial complexes, encoding shape at successive scales.Simplicial complexes discretize shapes into vertices, edges, faces, and higher-dimensional elements; discretization supports efficient computation and application to finite data.Source: §2.2 · pp. 3–4, Persistent Homology · p. 4

Persistent Homology.

Persistent homology summarizes shape through features tracked across directional filtrations, while the SECT turns Euler-characteristic curves into smooth functions. Its Hilbert-space structure supports broader statistical modeling, with injectivity for two- and three-dimensional shapes subject to using enough directions.

Persistent Homology.

  • Persistence intervals represent topological features by their birth and death along a filtration, and the PHT collects persistence diagrams across homology degrees and shape directions.The PHT is injective for two- and three-dimensional shapes.Source: Persistent Homology · p. 4
  • In persistent homology, each interval’s endpoints mark when a topological feature is born and dies, and its length indicates how long the feature persists.Persistence diagrams plot birth time against death time.Source: Persistent Homology · p. 4
  • Considering all directions in the PHT accommodates variation in MRI head positioning between patients and scans, which can cause image-registration issues.The SECT retains this all-directions summary strategy.Source: Persistent Homology · p. 4

2.3 Smooth Euler Characteristic Transform

  • The SECT smooths a centered Euler-characteristic curve to replace the ECT’s piecewise-constant, integer-valued functions with continuous, piecewise-linear curves in L2.The resulting SEC has compact support; the SECT collects these directional curves.Source: §2.3 · p. 6
  • SECT summaries are curves in Hilbert space, enabling quantitative comparisons with functional and nonparametric methods beyond the limited nonparametric adaptations available for PHT.The PHT instead produces persistence diagrams, whose geometry does not provide the same inner-product structure.Source: Persistent Homology · p. 5, §2.3 · p. 7
  • The SECT is injective for two- and three-dimensional shapes, provided enough directions are used; a single directional EC curve is not injective.The paper reports reasonable robustness of its GBM prediction results to the final numerical parameter choices.Source: §2.3 · p. 7
  • Coarse filtrations with too few sublevel sets can miss local shape undulations, potentially reducing prediction performance when small focal lesions matter to survival.The paper notes this as a possible information loss for texture-based features.Source: §2.3 · p. 7

3 Functional Regression Models with Tumor Shape Information as Covariates

The section treats SECT tumor-shape summaries as functional covariates in regression. Gaussian process regression provides a flexible model, with posterior predictions for patients whose tumor summaries are observed.

3 Functional Regression Models with Tumor Shape Information as Covariates

  • SECT represents tumor shape as curves in a Hilbert space, enabling functional data analysis and Gaussian-process regression on MRI-derived shape information.The framework models continuous functions as Hilbert-space elements using stochastic processes.Source: §3 · p. 7, §3.1 · p. 7

3.1 Gaussian Process Regression

  • Because a linear relationship between outcomes and functional tumor summaries may be too restrictive, the authors use Gaussian process regression to accommodate possible nonlinear interactions.The GP model uses a positive-definite covariance function and its induced reproducing kernel Hilbert space.Source: §3.1 · p. 8

3.2 Posterior Predictive Inference

  • For a new tumor, its SECT summary is used to derive a posterior predictive distribution for the unobserved patient's clinical response.The joint distribution links observed responses to function values at test images, with covariance entries representing relationships between training and test tumors.Source: §3.2 · pp. 8–9

4 Predicting Clinical Outcomes in Glioblastoma

The study compares SECT with molecular, morphometric, and geometric predictors of disease-free and overall survival in 48 patients with GBM. SECT performs best overall, while the authors discuss why topological summaries may capture information missed by narrower tumor measurements.

4.1 Genomic and Radiomic Data

  • The analysis used diagnostic post-contrast axial T1 MRIs and matched genomic and clinical data from 48 TCIA patients, selected to avoid treatment-related imaging or genomic confounding.The cohort selection also favored a common MRI modality and the axial representation with the most observations.Source: §4.1 · p. 9
  • From segmented slices, the authors derived 212 morphometric predictors, five geometric measures, and SECT summaries using 100 sublevel sets across 72 directions.Aggregated SECT curves produced 7200-dimensional vectors per patient; summaries were aggregated across slices as a proxy for 3D tumors.Source: §4.1 · p. 10

4.2 Prediction Results

  • They compared GP predictions of DFS and OS using 1000 random 80/20 train-test splits and linear, Gaussian, and Cauchy kernels, with kernel bandwidths selected by cross-validation.Prediction accuracy was assessed with R2 and the frequency each data type had the greatest R2.Source: §4.2 · pp. 10–11
  • Under Cauchy-kernel GP regression, SECT achieved the greatest R2 for DFS (0.237) and OS (0.158), and was optimal for 40.5% and 36.5% of splits, respectively.The authors report that SECT was the most accurate survival predictor, particularly for DFS.Source: §4.2 · p. 11
  • Topological summaries provide information about tumor size and texture, whereas volumetric and morphometric measures capture more limited aspects; all tumor characteristics generally predicted DFS better than OS.The authors note that DFS can provide a cleaner signal because recurrence events occur earlier than death from disease.Source: §4.2 · p. 11

4.3 MRI Specific Consequences on Results

  • Because the brain is fibrous, metric-based shape quantification may require difficult choices of metric or geodesic, while topology avoids these requirements and permits comparisons across tumor sizes.The authors hypothesize that this flexibility may contribute to SECT's predictive performance.Source: §4.3 · p. 11

4.4 More Biological Implications

  • The authors suggest tumor topology may correlate with molecular heterogeneity and recurrence mechanisms, including multifocal cases where recurrence occurs in only one hemisphere.They argue that SECT may offer insight into biological phenomena beyond simple geometric proximity.Source: §4.4 · pp. 11–12

5 Discussion

The discussion presents SECT as a functional, inner-product-compatible representation of GBM MRI shape and reports its survival-prediction performance relative to molecular and shape covariates. It also identifies sample size and unresolved biological and methodological questions as future-work boundaries.

5 Discussion

  • SECT maps tumor shapes into an inner-product space suited to standard functional and nonlinear regression models.The authors use this representation with functional Gaussian process regression to predict GBM patient survival.Source: §5 · p. 12
  • Across varied kernel models, SECT explained more DFS variance than the other covariates and outperformed them for both DFS and OS with Gaussian and Cauchy kernels.Source: §5 · p. 12
  • The study focused on survival prediction, and its sample size was too small for effective neural-network training; spatial feature selection and partial shape reconstruction remain open directions.The authors also propose investigating links between tumor shape, treatment, pathways, and GBM etiologies.Source: §5 · p. 12

Author Contributions Statement

The authors state that the study was jointly conceived and written, with method development, algorithm development, implementation, and analysis divided among named contributors.

Author Contributions Statement

  • LC and AM developed the methods; LC, AM, and AXC developed algorithms; LC implemented the software and performed analyses.LC, AM, SM, and RR conceived the study, and all authors wrote and revised the manuscript.Source: Author Contributions Statement · p. 13

Appendix

This appendix section gives formal mathematical details on persistent homology and topological data analysis for shapes and images.

Appendix

  • The section develops formal mathematical details for persistent homology and topological data analysis.It directs readers seeking a complete discussion of TDA and applied topology to the literature.Source: Appendix · p. 14
  • The mathematical discussion concerns topological data analysis for shapes and images.Source: Appendix · p. 14

Homology, Simplicial Complexes, and Persistence

This section develops simplicial homology concepts used to study holes and structure in shapes, defining the complexes, chains, and homology groups that support the paper’s topological analysis.

Homology, Simplicial Complexes, and Persistence

Persistent Homology

Persistent homology tracks how homology changes across a nested filtration of spaces, capturing topological structure across multiple scales.

Persistent Homology

  • A filtration is a nested family of simplicial complexes whose homology can change as the parameter increases.The section illustrates height-based and Vietoris–Rips radius-based filtrations.Source: Persistent Homology · p. 15
  • Persistent homology records the homology of each filtered space and the maps relating spaces at different parameter values.Source: Persistent Homology · p. 15
  • Persistent homology studies topological structure as a proximity rule is progressively relaxed across multiple scales.Source: Persistent Homology · p. 15

Barcodes and Persistence Diagrams

Barcodes and persistence diagrams summarize the lifetimes of topological features across a filtration; such summaries can reduce dimensionality in high-dimensional statistical problems.

Barcodes and Persistence Diagrams

  • Barcodes encode topological feature lifetimes, with birth marking appearance and death marking disappearance or merger.When features merge, the convention retains the one that appeared first.Source: Barcodes and Persistence Diagrams · p. 15
  • The illustrated barcode links filtration stages to H0 components, H1 cycles, and H2 cycles, showing their formation, merging, and filling.Source: Barcodes and Persistence Diagrams · pp. 15–16
  • A persistence diagram represents each barcode interval by its birth and death times as a point in a scatterplot.The diagonal has infinite multiplicity and supports well-defined metrics on diagrams or barcodes.Source: Barcodes and Persistence Diagrams · p. 16
  • Topological summary statistics can reduce dimensionality in high-dimensional statistical problems where predictors greatly outnumber observations.Source: Barcodes and Persistence Diagrams · p. 16

Figures and Tables

The figures illustrate how filtrations track topological features, how EC curves are smoothed into SEC curves, and how tumor SECT curves are assembled for analysis. The accompanying table caption describes the survival-prediction comparisons, while supplementary figures show tumor examples and topology concepts.

Figures and Tables

  • Figure 1 illustrates a height filtration in which vertices, edges, and faces enter successive nested complexes according to their maximal height in direction ν.The figure’s inclusions show the evolving simplicial complex.Source: Figure 1 · p. 23
  • Figure 2 contrasts the stepwise EC curve for a hand contour with a smooth SEC curve formed by centering and integrating the EC curve.With no holes in the hand, EC tracks the connected components appearing during filtration.Source: Figure 2 · p. 24
  • Figure 3 shows a segmented tumor and example SECT curves; 72 direction curves, each with 100 filtration steps, form a 7,200-dimensional patient covariate.The displayed directions are ν = 1, 36, and 72.Source: Figure 3 · p. 24
  • Table 1’s caption specifies DFS and OS comparisons across gene expression, morphometry, geometry, and SECT using R2 and Optimal%, based on 1,000 random 80–20 splits.It also describes three covariance functions and reports standard errors and cross-validated bandwidths; the caption alone does not establish which method performs best.Source: Table 1 · p. 25

Supplementary Figures

  • Supplementary figures show a tumor-and-ventricle mesh, examples of necrotic and multifocal tumors from different patients, and original MRI versus MITKats-segmented image data.The segmented examples are attributed to MITKats [48], and the MRI data are identified as coming from TCIA and TCGA.Source: Figure S1 · p. 26, Figure S2 · p. 27, Figure S4 · p. 28
  • Supplementary illustrations show that a fixed-direction EC curve can match for different shapes, while the torus and annulus examples depict cycles and persistent-homology barcodes.The torus has Betti numbers β0 = 1, β1 = 2, and β2 = 1; the annulus barcode includes a persistent H0 component and a comparatively short H2 bar.Source: Figure S3 · p. 28, Figure S5 · p. 29, Figure S6 · p. 30

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