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Classification of Histopathology Slides with Persistent Homology Convolutions

Shrunal Pothagoni, Benjamin Schweinhart

TL;DR

A novel method is presented that generates local persistent homology-based data using a modified version of the convolution operator called Persistent Homology Convolutions and indicates that persistent homology convolutions extract meaningful geometric information from the histopathology slides.

Abstract

Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as histopathology, topology is an important descriptor that can be used to distinguish between disease-indicating tissue by analyzing the shape characteristics of cells. Current literature suggests that reintroducing topological information using persistent homology can improve medical diagnostics; however, previous methods utilize global topological summaries which do not contain information about the locality of topological features. To address this gap, we present a novel method that generates local persistent homology-based data using a modified version of the convolution operator called Persistent Homology Convolutions. This method captures information about the locality and translation invariance of topological features. We perform a comparative study using various representations of histopathology slides and find that models trained with persistent homology convolutions outperform conventionally trained models and are less sensitive to hyperparameters. These results indicate that persistent homology convolutions extract meaningful geometric information from the histopathology slides.

Classification of Histopathology Slides with Persistent Homology Convolutions

TL;DR

A novel method is presented that generates local persistent homology-based data using a modified version of the convolution operator called Persistent Homology Convolutions and indicates that persistent homology convolutions extract meaningful geometric information from the histopathology slides.

Abstract

Convolutional neural networks (CNNs) are a standard tool for computer vision tasks such as image classification. However, typical model architectures may result in the loss of topological information. In specific domains such as histopathology, topology is an important descriptor that can be used to distinguish between disease-indicating tissue by analyzing the shape characteristics of cells. Current literature suggests that reintroducing topological information using persistent homology can improve medical diagnostics; however, previous methods utilize global topological summaries which do not contain information about the locality of topological features. To address this gap, we present a novel method that generates local persistent homology-based data using a modified version of the convolution operator called Persistent Homology Convolutions. This method captures information about the locality and translation invariance of topological features. We perform a comparative study using various representations of histopathology slides and find that models trained with persistent homology convolutions outperform conventionally trained models and are less sensitive to hyperparameters. These results indicate that persistent homology convolutions extract meaningful geometric information from the histopathology slides.

Paper Structure

This paper contains 19 sections, 3 theorems, 9 equations, 6 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Persistent Homology of Histopathology Slides
  4. The Thickening Filtration and the Alpha Complex
  5. The Extended Lower Star Filtration
  6. Ordinary Persistent Homology
  7. Extended Persistent Homology
  8. Vectorization of Persistence Data
  9. Persistent Homology Convolutions
  10. Experiments
  11. Experimental Setup
  12. Histopathology Dataset
  13. PHC Data Generation
  14. Intrinsic Dimensionality of Data
  15. Model, Training, and Implementation Details
  16. ...and 4 more sections

Key Result

Proposition 1

The Persistent Homology Diagram of the thickening filtration of $X\subset \left[ 0,1 \right]^2$ is a multiset whose elements consist of intervals of the form $(b,d)$. For each interval, the initial coordinate $b$ represents the value of $\epsilon$ at which a feature is born and the second coordinate

Figures (6)

  • Figure 1: An example of differing tissue structure of between Non-Tumor (left), Necrotic Tumor (center), and Viable Tumor (right) samples in the Osteosarcoma image dataset from Leavey2019-jsarunachalam2019viable.
  • Figure 2: Images showing the cellular structure of nonwoody stems in a squash (left, Cucurbita sp) and in a castor oil plant (right, Ricinus communis). The tissues contain similar numbers of smaller and larger cells in noticeably different arrangements. Images were taken from BCCBioscienceImageLibrary.
  • Figure 3: Top left: a subimage of a necrotic tumor slide. Bottom left: the threshold set $X$ from the lighter regions around the cells. Right: a simplified representation of $X$; $\mathrm{TPH}_p(X)$ is approximated by taking the alpha complex of $X.$ A visualization of the topological features that emerge is shown for the thickening filtration on the sample (left) and the topologically equivalent alpha complex (right). Type 1 features emerge when connected components merge as $\epsilon$ increases. Type 2 features are born when holes form and die in the union of balls centered at the finite point sample.
  • Figure 4: Top left: histopathology slide of necrotic tumor cells. Top right: the graph of $f$ in the vicinity of a cell with multi-nucleation. $f$ possesses many minima, saddle point, and maxima; pairs of neighboring critical points correspond to different types of persistent homology intervals. Each colored box on the density plot has a simplified representation of the geometric feature below that appears in the filtration. There is a single Type 1 feature (magenta box) corresponding to the absolute minimum of $f.$ Type 2 features (black box) pair local minima with neighboring saddle points (or maxima on the boundary); they represent cellular or subcellular structures darker than neighboring pixels. The illustrated feature quantifies multi-nucleation. Type 3 features (blue box) pair local maxima with neighboring saddle points; these pairs capture measurements related to structures lighter than their neighbors. Here, a void between cells gives rise to a Type 3 feature.
  • Figure 5: Top: density plot of a necrotic histopathology slide. Bottom: visual representation of features generated by quotienting the superlevel sets. There are two instances in which loops are generated in $\mathrm{XPH}_1(\tilde{f})$ as shown in the blue box. This occurs when the maxima on the boundary (shown by (a)) or the relative maxima within the surface (shown by (b)) are identified in the quotient. Lastly, voids are created if there exists a cavity in the surface, allowing for the surrounding ridge to collapse to a point, creating a sphere as shown in the magenta box. These voids correspond with features in $\mathrm{XPH}_2(\tilde{f})$.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Definition 1