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

Abstract

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.

A Statistical Pipeline for Identifying Physical Features that Differentiate Classes of 3D Shapes

“…c = 1) is ineffective at capturing enough variation to identify class-specific regions (Figs. 2(d)-2(f)), which supports the intuition that seeing more of a shape leads to an improved ability to understand its complete structure [1–3]. Our empirical results show that more power can be achieved by summarizing the shapes with filtrations taken over multiple directions.…”

“…In the first step of the SINATRA pipeline, we use a tool from integral geometry and differential topology called the Euler characteristic (EC) transform [1–4]. For a mesh , the Euler characteristic is an accessible topological invariants derived from: where denote the number of vertices (corners), edges, and faces of the mesh, respectively.…”

“…More generally, the sub-image analysis problem can be framed as a regression-based task: given a collection of shapes, find the properties that explain the greatest variation in some response variable (continuous or binary). One example is identifying the structures of glioblastoma tumors that best indicate signs of potential relapse and other clinical outcomes [1]. From a statistical perspective, the sub-image selection problem is directly related to the variable selection problem — given high-dimensional covariates and a univariate outcome, we want to infer which variables are most relevant in explaining or predicting variation in the observed response.…”

“…The transformation should lose a minimum amount of geometric information and apply to a wide range of shape and imaging datasets. In this paper, we use a tool from integral geometry and differential topology called the Euler characteristic (EC) transform [1–4], which maps shapes into vectors without requiring pre-specified landmark points or pairwise correspondences. This property is central to our innovations.…”

“…Detailed mathematical analyses have also been provided [3]. A nonlinear regression framework which uses the EC transform to predict outcomes of disease free survival in glioblastoma [1] is most relevant to this paper. This works shows that the EC transform reduces the problem of regression with shape covariates into a problem in functional data analysis (FDA), and that nonlinear regression models are more accurate than linear models when predicting complex phenotypes and traits.…”