Applications of computational homology to the analysis of treatment response in breast cancer patients

DeWoskin, Daniel; Climent, Joan; Cruz-White, I.; Vázquez, Mariel; Park, C.; Arsuaga, Javier

doi:10.1016/j.topol.2009.04.036

Cited by 33 publications

(34 citation statements)

References 9 publications

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“…The overall spherical shape of the complex is due to the fact that expression values of consecutive probes along chromosomes are mostly independent from each other since they do not necessarily belong to the same pathways. This shape is in contrast with the overall ellipsoidal shape of the CGH data due to the correlation in the copy number values of consecutive probes (see [13] for an example).…”

Section: Our Topological Analysis Distinguishes Most But Not All Breamentioning

confidence: 79%

“…We here argue that computational homology provides such methods. In [13] we proposed a new algorithm that associates a set of point clouds (and corresponding simplicial complexes and topological invariants) to copy number measurements along the genome by means of a sliding window algorithm. Here we have extended our work by introducing Takens' theorem as a plausible mathematical framework for the mapping of the data to a point cloud.…”

Section: Resultsmentioning

confidence: 99%

“…The method proposed in [13] was aimed at identifying CNAs, such as amplifications and deletions of the genome, from array CGH. These data consists of fluorescence measurements obtained from probes, ordered according to their physical location, along the genome.…”

Section: A Topological Approach To Study Array Datamentioning

confidence: 99%

“…In [13] we introduced a new computational homology method to analyze array comparative genomic hybridization (CGH) with the goal of identifying recurrent copy number aberrations (CNAs) in cancer. This method uses a sliding window algorithm to associate a set of point clouds to each array CGH.…”

Section: Introductionmentioning

confidence: 99%

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Topological analysis of gene expression arrays identifies high risk molecular subtypes in breast cancer

Arsuaga

Baas

DeWoskin

et al. 2012

AAECC

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Genomic technologies measure thousands of molecular signals with the goal of understanding complex biological processes. In cancer these molecular signals have been used to characterize disease subtypes, signaling pathways and to identify subsets of patients with specific prognosis. However molecular signals for any disease type are so vast and complex that novel mathematical approaches are required for further analyses. Persistent and computational homology provide a new method for these analyses. In our previous work we presented a new homology-based supervised classification method to identify copy number aberrations from comparative genomic hybridization arrays. In this work we first propose a theoretical framework for our classification method and second we extend our analysis to gene expression data. We analyze a published breast cancer data set and find that that our method can distinguish most, but not all, different breast cancer subtypes. This result suggests 123 4 J. Arsuaga et al. that specific relationships between genes, captured by our algorithm, help distinguish between breast cancer subtypes. We propose that topological methods can be used for the classification and clustering of gene expression profiles.

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Section: Our Topological Analysis Distinguishes Most But Not All Breamentioning

confidence: 79%

Section: Resultsmentioning

confidence: 99%

Section: A Topological Approach To Study Array Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Topological analysis of gene expression arrays identifies high risk molecular subtypes in breast cancer

Arsuaga

Baas

DeWoskin

et al. 2012

AAECC

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show abstract

“…Persistent homology is a recent branch of applied algebraic topology that has applications in data analysis [9], image processing and recognition [1,13], and more [5,16]. It is also a subject of active research, from both the computational [17] and theoretical [10,11] points of view.…”

Section: Introductionmentioning

confidence: 99%

Weighted persistent homology

Ren¹,

Wu²,

Wu³

2018

Rocky Mountain J. Math.

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In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. We also prove a theorem that allows us to calculate the mod p 2 weighted persistent homology given some information on the mod p weighted persistent homology.2010 Mathematics Subject Classification. Primary 55N35, 55T99; Secondary 55U20, 55U10.

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An introduction to persistent homology for time series

Равишанкер

Chen

2021

WIREs Computational Stats

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Topological data analysis (TDA) uses information from topological structures in complex data for statistical analysis and learning. This paper discusses persistent homology, a part of computational (algorithmic) topology that converts data into simplicial complexes and elicits information about the persistence of homology classes in the data. It computes and outputs the birth and death of such topologies via a persistence diagram. Data inputs for persistent homology are usually represented as point clouds or as functions, while the outputs depend on the nature of the analysis and commonly consist of either a persistence diagram, or persistence landscapes. This paper gives an introductory level tutorial on computing these summaries for time series using R, followed by an overview on using these approaches for time series classification and clustering. This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data Applications of Computational Statistics > Computational Mathematics

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Applications of computational homology to the analysis of treatment response in breast cancer patients

Cited by 33 publications

References 9 publications

Topological analysis of gene expression arrays identifies high risk molecular subtypes in breast cancer

Topological analysis of gene expression arrays identifies high risk molecular subtypes in breast cancer

Weighted persistent homology

An introduction to persistent homology for time series

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