Applications of Topological Data Analysis in Oncology

The emergence of the information age in the last few decades brought with it an explosion of biomedical data. But with great power comes great responsibility: there is now a pressing need for new data analysis algorithms to be developed to make sense of the data and transform this information into knowledge which can be directly translated into the clinic. Topological data analysis (TDA) provides a promising path forward: using tools from the mathematical field of algebraic topology, TDA provides a framework to extract insights into the often high-dimensional, incomplete, and noisy nature of biomedical data. Nowhere is this more evident than in the field of oncology, where patient-specific data is routinely presented to clinicians in a variety of forms, from imaging to single cell genomic sequencing. In this review, we focus on applications involving persistent homology, one of the main tools of TDA. We describe some recent successes of TDA in oncology, specifically in predicting treatment responses and prognosis, tumor segmentation and computer-aided diagnosis, disease classification, and cellular architecture determination. We also provide suggestions on avenues for future research including utilizing TDA to analyze cancer time-series data such as gene expression changes during pathogenesis, investigation of the relation between angiogenic vessel structure and treatment efficacy from imaging data, and experimental confirmation that geometric and topological connectivity implies functional connectivity in the context of cancer.

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Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

Bulletin of Mathematical Biology ◽

10.1007/s11538-020-00847-3 ◽

2021 ◽

Vol 83 (3) ◽

Author(s):

Maria-Veronica Ciocanel ◽

Riley Juenemann ◽

Adriana T. Dawes ◽

Scott A. McKinley

Keyword(s):

Time Series ◽

Data Analysis ◽

Actin Filaments ◽

Time Series Data ◽

Polymer Networks ◽

Topological Data Analysis ◽

Series Data ◽

Topological Features ◽

Topological Data

AbstractIn developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

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Using Topological Data Analysis to Process Time-series Data: A Persistent Homology Way

Journal of Physics Conference Series ◽

10.1088/1742-6596/1550/3/032082 ◽

2020 ◽

Vol 1550 ◽

pp. 032082

Author(s):

Gang Ma

Keyword(s):

Time Series ◽

Data Analysis ◽

Time Series Data ◽

Persistent Homology ◽

Topological Data Analysis ◽

Series Data ◽

Process Time ◽

Topological Data

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Topological data analysis for true step detection in periodic piecewise constant signals

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2018.0027 ◽

2018 ◽

Vol 474 (2218) ◽

pp. 20180027 ◽

Cited By ~ 4

Author(s):

Firas A. Khasawneh ◽

Elizabeth Munch

Keyword(s):

Data Analysis ◽

Persistent Homology ◽

Topological Data Analysis ◽

Future Research ◽

Accurate Identification ◽

Piecewise Constant ◽

Higher Dimensional ◽

Powerful Approach ◽

Hadamard Transforms ◽

Topological Data

This paper introduces a simple yet powerful approach based on topological data analysis for detecting true steps in a periodic, piecewise constant (PWC) signal. The signal is a two-state square wave with randomly varying in-between-pulse spacing, subject to spurious steps at the rising or falling edges which we call digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables accurate and automatic calculations of the spindle speed without any choice of parameters. The results are compared with the frequency and sequency methods of the Fourier and Walsh–Hadamard transforms, respectively. Both our approach and the Fourier analysis yield comparable results for pulses with regular spacing and digital ringing while the latter causes large errors using the Walsh–Hadamard method. Further, the described approach significantly outperforms the frequency/sequency analyses when the spacing between the peaks is varied. We discuss generalizing the approach to higher dimensional PWC signals, although using this extension remains an interesting question for future research.

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Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan

Frontiers in Physics ◽

10.3389/fphy.2021.572216 ◽

2021 ◽

Vol 9 ◽

Author(s):

Peter Tsung-Wen Yen ◽

Siew Ann Cheong

Keyword(s):

Data Analysis ◽

Stock Markets ◽

Persistent Homology ◽

Betti Numbers ◽

Stock Index ◽

Topological Data Analysis ◽

Series Data ◽

Topological Features ◽

Market Crashes ◽

Topological Data

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

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Persistence Landscape based Topological Data Analysis for Personalized Arrhythmia Classification

10.20944/preprints201908.0320.v1 ◽

2019 ◽

Author(s):

Yan Yan ◽

Kamen Ivanov ◽

Jian Cen ◽

Qiu-Hua Liu ◽

Lei Wang

Keyword(s):

Data Analysis ◽

High Performance ◽

Topological Analysis ◽

Point Clouds ◽

Topological Data Analysis ◽

Reconstruction Technique ◽

Biomedical Data ◽

Training Set ◽

Set Size ◽

Topological Data

Data can be illustrated in shapes, and the shapes could provide insight for data modeling and information extraction. Topological data analysis provides an alternative insight in biomedical data analysis and knowledge discovery with the algebra topology tools. In present work, we study the application of topological data analysis for personalized electrocardiographic signal classification toward arrhythmia analysis. Using phase space reconstruction technique, the signal samples are converted into point clouds for topological analysis facility. With topological techniques the persistence landscapes from the point clouds are extracted as features to perform the arrhythmia classification task. We find that the proposed method is robust to the training set size, with only a training set size of 20% percents, the normal heartbeat class are 100% recognized, ventricular beats for 97.13%, supra-ventricular beats for 94.27% and fusion beats for 94.27% within the corresponding experiments. The property of keeping high performance when using smaller training sample proves that the proposed method is especially applicable to personalized analysis. With the present study, we show that the topological data analysis technique could be a useful tool in biomedical signal analysis, and provide powerful ability in personalized analysis.

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