A Topological Data Analysis Framework for Quantifying Necrosis in Glioblastomas

TL;DR

This work tackles the problem of quantifying necrosis morphology in Glioblastoma beyond simple necrotic fraction by introducing an interior function, a TDA-based descriptor that leverages persistence landscapes and the Persistence Histogram Transform (PHT). It defines subcomplex lacunarity indices $\eta$, $\tau$, and derived measures $\sigma$ and $\rho$, and constructs a primary index diagram to capture how a 2D tumor image fills its surroundings. Through analysis of 93 GBM patients (1065 images), the authors perform a four-cluster delineation driven by lacunarity polarity and disorder, using 2D cubical complexes and HPC-accelerated persistent homology computations. The framework provides topology-informed metrics and a diagnostic diagram for characterizing necrosis patterns, with potential implications for prognosis and tumor stratification, while acknowledging computational costs and 2D limitations that motivate future 3D extensions.

Abstract

In this paper, we introduce a shape descriptor that we call "interior function". This is a Topological Data Analysis (TDA) based descriptor that refines previous descriptors for image analysis. Using this concept, we define subcomplex lacunarity, a new index that quantifies geometric characteristics of necrosis in tumors such as conglomeration. Building on this framework, we propose a set of indices to analyze necrotic morphology and construct a diagram that captures the distinct structural and geometric properties of necrotic regions in tumors. We present an application of this framework in the study of MRIs of Glioblastomas (GB). Using cluster analysis, we identify four distinct subtypes of Glioblastomas that reflect geometric properties of necrotic regions.

Figures (16)

• Figure 1: The image in the upper left corner represents the core image of an MRI scan of a glioblastoma, while the bottom left image shows its enhanced version. On the right, a visualization depicts the graphs of the corresponding interior functions.
• Figure 2: The image presents a table displaying various thresholded MRIs of glioblastomas alongside their corresponding primary and secondary index values. Each table entry contains three pairs of images: the upper row shows core versions, while the lower row displays their corresponding enhanced versions. Above each image pair, the patient tag and image number are indicated. The index values for each pair are displayed at the bottom of the images and are arranged in ascending order from left to right.
• Figure 3: Primary index diagram of thresholded glioblastoma MRIs presented in Example 3.1. The different index groups are distinguished by the shape of the markers in the plot, while the legend indicates the corresponding image tags.
• Figure 4: Primary index diagram pertaining to all the data in the dataset. The diagram shows that the data points are contained in $[0,1]^2$, and the data points are heavily situated around the lower and rightmost areas of the diagram.
• Figure 5: Image depicting Silhouette plots and their respecting cluster plots for number of clusters $n=3,4,5,6$ using K-means.

Theorems & Definitions (13)

• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3: BubenikSTDA, p. 83
• Definition 2.4: BubenikSTDA, p.84
• Definition 2.5
• Proposition 2.6
• proof
• Example 2.7
• Definition 3.1