Abstract
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.
Highlights
With the advent of next-generation high-throughput sequencing (Roychowdhury et al, 2011; Reuter et al, 2015), improved medical imaging (Wang, 2016; Tahmassebi et al, 2018; Aiello et al, 2019), and an increased focus on personalized medicine (Dilsizian and Siegel, 2014; Gu and Taylor, 2014; Alyass et al, 2015; Suwinski et al, 2019), more data is being collected than ever before
Though we focus on persistent homology here, it is worth noting that there have been many notable successes of the application of other Topological data analysis (TDA) methods, such as the Mapper algorithm (Singh et al, 2007)
Unlike many other methods which focus on the analysis of zero dimensional homology groups (DeWoskin et al, 2010; Nicolau et al, 2011), performing analyses which are topologically equivalent to clustering, this study focused their efforts on identifying loops of one dimensional homology groups which persist over a large range of values of the proximity parameter, hypothesizing that connections around holes imply nontrivial interactions among genes and biological functions which could have implications for tumorigenesis
Summary
With the advent of next-generation high-throughput sequencing (Roychowdhury et al, 2011; Reuter et al, 2015), improved medical imaging (Wang, 2016; Tahmassebi et al, 2018; Aiello et al, 2019), and an increased focus on personalized medicine (Dilsizian and Siegel, 2014; Gu and Taylor, 2014; Alyass et al, 2015; Suwinski et al, 2019), more data is being collected than ever before. The data points could represent customers’ preferences or patient gene expression In this case if a product or drug were targeted to the average person, the target would be the center of the circle and would miss the data set entirely. While this is a simple made-up example, it illustrates the importance of understanding the shape of data. Mapper was recently used to extract information from high-throughput microarray data and define a new subtype of breast cancer, c-MYB+, characterized by high c-MYB expression and low levels of innate inflammatory genes, with corresponding patients exhibiting 100% survival and no metastasis (Nicolau et al, 2007). Before delving into the applications of persistent homology in cancer, we introduce some of the key mathematical underpinnings needed to understand these results
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