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

ORCID Icon, ORCID Icon, , ORCID Icon &
Pages 1139-1150 | Received 25 Sep 2017, Accepted 18 Sep 2019, Published online: 17 Oct 2019


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 use 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 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 2-fold. 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. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.


The authors wish to thank Mao Li (Donald Danforth Plant Science Center) and Christoph Hellmayr (Duke University) for help with the formulation of code, as well as Francesco Abate (McKinsey & Co.), Katharine Turner (Australian National University), and Jiguang Wang (Hong Kong University of Science and Technology) for helpful conversations and input on a previous version of the manuscript. The authors would also like to acknowledge The Cancer Imaging Archive (TCIA) and The Cancer Genome Atlas (TCGA) initiatives for making the imaging and the clinical data used in this study publicly available.

Disclosure Statement

The authors have declared that no competing interests exist.

Additional information


During some of this work, LC, AM, and RR were supported by the National Cancer Institute Physical Sciences–Oncology Network (NCI PS–ON) under grant no. 5 U54 CA 193313-02. AM was the PI on Pilot grant subaward no. G11124 for research on radiomics and radiogenomics. AM is also supported by the Irving Institute’s CaMPR initiative under grant no. GG011557, and would like to acknowledge the support of the New Frontiers in Research Fund–Fonds Nouvelles Frontières en Recherche (SSHRC-NFRF-FNFR Government of Canada) NFRFE-2018-00431. LC would like to acknowledge the support of grants P20GM109035 (COBRE Center for Computational Biology of Human Disease; PI Rand) and P20GM103645 (COBRE Center for Central Nervous; PI Sanes) from the NIH NIGMS, 2U10CA180794-06 from the NIH NCI and the Dana Farber Cancer Institute (PIs Gray and Gatsonis), and an Alfred P. Sloan Research Fellowship (no. FG-2019-11622). AXC would like to acknowledge support by the Columbia University Medical Scientist Training Program (MSTP). SM would like to acknowledge funding from NSF DEB-1840223, NIH R01 DK116187-01, HFSP RGP0051/2017, NSF DMS 17-13012, and NSF DMS 16-13261. This work used a high-performance computing facility partially supported by grant 2016-IDG-1013 (“HARDAC+: Reproducible HPC for Next-generation Genomics”) from the North Carolina Biotechnology Center. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of any of the funders.

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