Fingerprints of cancer by persistent homology

Abstract

We have carried out a topological data analysis of gene expressions for different databases based on the Fermat distance between the z scores of different tissue samples. There is a critical value of the filtration parameter at which all clusters collapse in a single one. This critical value for healthy samples is gapless and smaller than that for cancerous ones. After collapse in a single cluster, topological holes persist for larger filtration parameter values in cancerous samples. Barcodes, persistence diagrams and Betti numbers as functions of the filtration parameter are different for different types of cancer and constitute fingerprints thereof.

𝓗1 persistent features of the resting-state connectome in healthy subjects

“…In physics, topological voids (𝓗 2 structures) were found in the study of the baryon acoustic oscillation related to the galaxy distribution ( Kono, Takeuchi, Cooray, Nishizawa, & Murakami, 2020 ). Finally, in medicine, Carpio, Bonilla, Mathews, and Tannenbaum (2019) use the topological descriptors for the two first dimensions, that is, 𝓗 0 and 𝓗 1 . They found that different cancer cells have distinct topological values at these dimensions, indicating the descriptors’ usefulness as biomarkers.…”

“…Figure 5 shows the existence of an interesting number of features that are not spurious. Therefore, extending the analysis of loops to a percentage of the most persistent could provide a new perspective for the analysis, because with more cycles, it is possible to (a) enrich the R-fMRI topological description of each subject, which can be used in other developments like classification ( Bhaskar et al, 2021 ; Carpio et al, 2019 ), and (b) identify the nodes that are involved in more than one persistent loop because they could be relevant in the study of brain high-order processes. Another concern is the use of the proposed approach for comparison between population and/or specific brain regions.…”

Clustering methods and Bayesian inference for the analysis of the evolution of immune disorders

“…Nowadays, the increasing availability of medical data related to all sorts of illnesses is fostering the development of machine learning techniques [2] for medical diagnosis and treatment. While neural networks are often used for image based diagnosis [25,16], supervised and unsupervised clustering techniques [22] are now widely employed to investigate the role of genes in sickness [17,24,4,20] and to study the response to therapies [15,6], as well as for assisted clinical diagnosis using information from digital devices [18,21]. Developing tools to assess the reliability of such automatic procedures and to choose the best method for different situations and clinical environments has become essential [18].…”

Intrinsic persistent homology via density-based metric learning

“…Topological Data Analysis (TDA) arises naturally in several fields like surface reconstruction [23], quasi periodicity classification in video data and other time series [53,54,58], protein formation [31,60], identification of tumors [8], detection of stock market crashes [33], algebraic computational geometry [18], and many others. Understanding topological features from data is also crucial in the experimental study of chaotic dynamical systems since the main strategy to validate or reject a physical model in the presence of chaotic behavior is to determine if the data and the model itself share the same topology [5,32,47,48,50].…”