Persistent Homology for the Quantitative Evaluation of Architectural Features in Prostate Cancer Histology

Dory: Computation of persistence diagrams up to dimension two for Vietoris-Rips filtrations of large data sets

Persistent homology (PH) is an approach to topological data analysis (TDA) that computes multi-scale topologically invariant properties of high-dimensional data that are robust to noise. While PH has revealed useful patterns across various applications, computational requirements have limited applications to small data sets of a few thousand points. We present Dory, an efficient and scalable algorithm that can compute the persistent homology of sparse Vietoris-Rips complexes on larger data sets, up to and including dimension two and over the field Z 2 . As an application, we compute the PH of the human genome at high resolution as revealed by a genome-wide Hi-C data set containing approximately three million points. Extant algorithms were unable to process it, whereas Dory processed it within five minutes, using less than five GB of memory. Results show that the topology of the human genome changes significantly upon treatment with auxin, a molecule that degrades cohesin, corroborating the hypothesis that cohesin plays a crucial role in loop formation in DNA.

Advancing Precision Medicine: Algebraic Topology and Differential Geometry in Radiology and Computational Pathology

Precision medicine aims to provide personalized care based on individual patient characteristics, rather than guideline-directed therapies for groups of diseases or patient demographics. Images-both radiology- and pathology-derived-are a major source of information on presence, type, and status of disease. Exploring the mathematical relationship of pixels in medical imaging ("radiomics") and cellular-scale structures in digital pathology slides ("pathomics") offers powerful tools for extracting both qualitative and, increasingly, quantitative data. These analytical approaches, however, may be significantly enhanced by applying additional methods arising from fields of mathematics such as differential geometry and algebraic topology that remain underexplored in this context. Geometry's strength lies in its ability to provide precise local measurements, such as curvature, that can be crucial for identifying abnormalities at multiple spatial levels. These measurements can augment the quantitative features extracted in conventional radiomics, leading to more nuanced diagnostics. By contrast, topology serves as a robust shape descriptor, capturing essential features such as connected components and holes. The field of topological data analysis was initially founded to explore the shape of data, with functional network connectivity in the brain being a prominent example. Increasingly, its tools are now being used to explore organizational patterns of physical structures in medical images and digitized pathology slides. By leveraging tools from both differential geometry and algebraic topology, researchers and clinicians may be able to obtain a more comprehensive, multi-layered understanding of medical images and contribute to precision medicine's armamentarium.

Persistent homology reveals strong phylogenetic signal in 3D protein structures

Changes that occur in proteins over time provide a phylogenetic signal that can be used to decipher their evolutionary history and the relationships between organisms. Sequence comparison is the most common way to access this phylogenetic signal, while those based on 3D structure comparisons are still in their infancy. In this study, we propose an effective approach based on Persistent Homology Theory (PH) to extract the phylogenetic information contained in protein structures. PH provides efficient and robust algorithms for extracting and comparing geometric features from noisy datasets at different spatial resolutions. PH has a growing number of applications in the life sciences, including the study of proteins (e.g. classification, folding). However, it has never been used to study the phylogenetic signal they may contain. Here, using 518 protein families, representing 22,940 protein sequences and structures, from 10 major taxonomic groups, we show that distances calculated with PH from protein structures correlate strongly with phylogenetic distances calculated from protein sequences, at both small and large evolutionary scales. We test several methods for calculating PH distances and propose some refinements to improve their relevance for addressing evolutionary questions. This work opens up new perspectives in evolutionary biology by proposing an efficient way to access the phylogenetic signal contained in protein structures, as well as future developments of topological analysis in the life sciences.

CCF-GNN: A Unified Model Aggregating Appearance, Microenvironment, and Topology for Pathology Image Classification

Pathology images contain rich information of cell appearance, microenvironment, and topology features for cancer analysis and diagnosis. Among such features, topology becomes increasingly important in analysis for cancer immunotherapy. By analyzing geometric and hierarchically structured cell distribution topology, oncologists can identify densely-packed and cancer-relevant cell communities (CCs) for making decisions. Compared to commonly-used pixel-level Convolution Neural Network (CNN) features and cell-instance-level Graph Neural Network (GNN) features, CC topology features are at a higher level of granularity and geometry. However, topological features have not been well exploited by recent deep learning (DL) methods for pathology image classification due to lack of effective topological descriptors for cell distribution and gathering patterns. In this paper, inspired by clinical practice, we analyze and classify pathology images by comprehensively learning cell appearance, microenvironment, and topology in a fine-to-coarse manner. To describe and exploit topology, we design Cell Community Forest (CCF), a novel graph that represents the hierarchical formulation process of big-sparse CCs from small-dense CCs. Using CCF as a new geometric topological descriptor of tumor cells in pathology images, we propose CCF-GNN, a GNN model that successively aggregates heterogeneous features (e.g., appearance, microenvironment) from cell-instance-level, cell-community-level, into image-level for pathology image classification. Extensive cross-validation experiments show that our method significantly outperforms alternative methods on H&E-stained and immunofluorescence images for disease grading tasks with multiple cancer types. Our proposed CCF-GNN establishes a new topological data analysis (TDA) based method, which facilitates integrating multi-level heterogeneous features of point clouds (e.g., for cells) into a unified DL framework.

A topological classifier to characterize brain states: When shape matters more than variance

Despite the remarkable accuracies attained by machine learning classifiers to separate complex datasets in a supervised fashion, most of their operation falls short to provide an informed intuition about the structure of data, and, what is more important, about the phenomena being characterized by the given datasets. By contrast, topological data analysis (TDA) is devoted to study the shape of data clouds by means of persistence descriptors and provides a quantitative characterization of specific topological features of the dataset under scrutiny. Here we introduce a novel TDA-based classifier that works on the principle of assessing quantifiable changes on topological metrics caused by the addition of new input to a subset of data. We used this classifier with a high-dimensional electro-encephalographic (EEG) dataset recorded from eleven participants during a previous decision-making experiment in which three motivational states were induced through a manipulation of social pressure. We calculated silhouettes from persistence diagrams associated with each motivated state with a ready-made band-pass filtered version of these signals, and classified unlabeled signals according to their impact on each reference silhouette. Our results show that in addition to providing accuracies within the range of those of a nearest neighbour classifier, the TDA classifier provides formal intuition of the structure of the dataset as well as an estimate of its intrinsic dimension. Towards this end, we incorporated variance-based dimensionality reduction methods to our dataset and found that in most cases the accuracy of our TDA classifier remains essentially invariant beyond a certain dimension.

The shape of cancer relapse: Topological data analysis predicts recurrence in paediatric acute lymphoblastic leukaemia

Although children and adolescents with acute lymphoblastic leukaemia (ALL) have high survival rates, approximately 15-20% of patients relapse. Risk of relapse is routinely estimated at diagnosis by biological factors, including flow cytometry data. This high-dimensional data is typically manually assessed by projecting it onto a subset of biomarkers. Cell density and "empty spaces" in 2D projections of the data, i.e. regions devoid of cells, are then used for qualitative assessment. Here, we use topological data analysis (TDA), which quantifies shapes, including empty spaces, in data, to analyse pre-treatment ALL datasets with known patient outcomes. We combine these fully unsupervised analyses with Machine Learning (ML) to identify significant shape characteristics and demonstrate that they accurately predict risk of relapse, particularly for patients previously classified as 'low risk'. We independently confirm the predictive power of CD10, CD20, CD38, and CD45 as biomarkers for ALL diagnosis. Based on our analyses, we propose three increasingly detailed prognostic pipelines for analysing flow cytometry data from ALL patients depending on technical and technological availability: 1. Visual inspection of specific biological features in biparametric projections of the data; 2. Computation of quantitative topological descriptors of such projections; 3. A combined analysis, using TDA and ML, in the four-parameter space defined by CD10, CD20, CD38 and CD45. Our analyses readily extend to other haematological malignancies.

The Stria Vascularis in Mice and Humans Is an Early Site of Age-Related Cochlear Degeneration, Macrophage Dysfunction, and Inflammation

Age-related hearing loss, or presbyacusis, is a common degenerative disorder affecting communication and quality of life for millions of older adults. Multiple pathophysiologic manifestations, along with many cellular and molecular alterations, have been linked to presbyacusis; however, the initial events and causal factors have not been clearly established. Comparisons of the transcriptome in the lateral wall (LW) with other cochlear regions in a mouse model (of both sexes) of "normal" age-related hearing loss revealed that early pathophysiological alterations in the stria vascularis (SV) are associated with increased macrophage activation and a molecular signature indicative of inflammaging, a common form of immune dysfunction. Structure-function correlation analyses in mice across the lifespan showed that the age-dependent increase in macrophage activation in the stria vascularis is associated with a decline in auditory sensitivity. High-resolution imaging analysis of macrophage activation in middle-aged and aged mouse and human cochleas, along with transcriptomic analysis of age-dependent changes in mouse cochlear macrophage gene expression, support the hypothesis that aberrant macrophage activity is an important contributor to age-dependent strial dysfunction, cochlear pathology, and hearing loss. Thus, this study highlights the SV as a primary site of age-related cochlear degeneration and aberrant macrophage activity and dysregulation of the immune system as early indicators of age-related cochlear pathology and hearing loss. Importantly, novel new imaging methods described here now provide a means to analyze human temporal bones in a way that had not previously been feasible and thereby represent a significant new tool for otopathological evaluation.SIGNIFICANCE STATEMENT Age-related hearing loss is a common neurodegenerative disorder affecting communication and quality of life. Current interventions (primarily hearing aids and cochlear implants) offer imperfect and often unsuccessful therapeutic outcomes. Identification of early pathology and causal factors is crucial for the development of new treatments and early diagnostic tests. Here, we find that the SV, a nonsensory component of the cochlea, is an early site of structural and functional pathology in mice and humans that is characterized by aberrant immune cell activity. We also establish a new technique for evaluating cochleas from human temporal bones, an important but understudied area of research because of a lack of well-preserved human specimens and difficult tissue preparation and processing approaches.

Euler characteristic curves and profiles: a stable shape invariant for big data problems

Tools of topological data analysis provide stable summaries encapsulating the shape of the considered data. Persistent homology, the most standard and well-studied data summary, suffers a number of limitations; its computations are hard to distribute, and it is hard to generalize to multifiltrations and is computationally prohibitive for big datasets. In this article, we study the concept of Euler characteristics curves for 1-parameter filtrations and Euler characteristic profiles for multiparameter filtrations. While being a weaker invariant in one dimension, we show that Euler characteristic-based approaches do not possess some handicaps of persistent homology; we show efficient algorithms to compute them in a distributed way, their generalization to multifiltrations, and practical applicability for big data problems. In addition, we show that the Euler curves and profiles enjoy a certain type of stability, which makes them robust tools for data analysis. Lastly, to show their practical applicability, multiple use cases are considered.

Machine learning techniques on homological persistence features for prostate cancer diagnosis

The rapid evolution of image processing equipment and techniques ensures the development of novel picture analysis methodologies. One of the most powerful yet computationally possible algebraic techniques for measuring the topological characteristics of functions is persistent homology. It's an algebraic invariant that can capture topological details at different spatial resolutions. Persistent homology investigates the topological features of a space using a set of sampled points, such as pixels. It can track the appearance and disappearance of topological features caused by changes in the nested space created by an operation known as filtration, in which a parameter scale, in our case the intensity of pixels, is increased to detect changes in the studied space over a range of varying scales. In addition, at the level of machine learning there were many studies and articles witnessing recently the combination between homological persistence and machine learning algorithms. On another level, prostate cancer is diagnosed referring to a scoring criterion describing the severity of the cancer called Gleason score. The classical Gleason system defines five histological growth patterns (grades). In our study we propose to study the Gleason score on some glands issued from a new optical microscopy technique called SLIM. This new optical microscopy technique that combines two classic ideas in light imaging: Zernike’s phase contrast microscopy and Gabor’s holography. Persistent homology features are computed on these images. We suggested machine learning methods to classify these images into the corresponding Gleason score. Machine learning techniques applied on homological persistence features was very effective in the detection of the right Gleason score of the prostate cancer in these kinds of images and showed an accuracy of above 95%.

Spatial Transcriptomic Analysis Reveals Associations between Genes and Cellular Topology in Breast and Prostate Cancers

BACKGROUND

Cancer is the leading cause of death worldwide with breast and prostate cancer the most common among women and men, respectively. Gene expression and image features are independently prognostic of patient survival; but until the advent of spatial transcriptomics (ST), it was not possible to determine how gene expression of cells was tied to their spatial relationships (i.e., topology).

METHODS

We identify topology-associated genes (TAGs) that correlate with 700 image topological features (ITFs) in breast and prostate cancer ST samples. Genes and image topological features are independently clustered and correlated with each other. Themes among genes correlated with ITFs are investigated by functional enrichment analysis.

RESULTS

Overall, topology-associated genes (TAG) corresponding to extracellular matrix (ECM) and Collagen Type I Trimer gene ontology terms are common to both prostate and breast cancer. In breast cancer specifically, we identify the ZAG-PIP Complex as a TAG. In prostate cancer, we identify distinct TAGs that are enriched for GI dysmotility and the IgA immunoglobulin complex. We identified TAGs in every ST slide regardless of cancer type.

CONCLUSIONS

These TAGs are enriched for ontology terms, illustrating the biological relevance to our image topology features and their potential utility in diagnostic and prognostic models.

Topological data analysis in biomedicine: A review

Significant technological advances made in recent years have shepherded a dramatic increase in utilization of digital technologies for biomedicine- everything from the widespread use of electronic health records to improved medical imaging capabilities and the rising ubiquity of genomic sequencing contribute to a "digitization" of biomedical research and clinical care. With this shift toward computerized tools comes a dramatic increase in the amount of available data, and current tools for data analysis capable of extracting meaningful knowledge from this wealth of information have yet to catch up. This article seeks to provide an overview of emerging mathematical methods with the potential to improve the abilities of clinicians and researchers to analyze biomedical data, but may be hindered from doing so by a lack of conceptual accessibility and awareness in the life sciences research community. In particular, we focus on topological data analysis (TDA), a set of methods grounded in the mathematical field of algebraic topology that seeks to describe and harness features related to the "shape" of data. We aim to make such techniques more approachable to non-mathematicians by providing a conceptual discussion of their theoretical foundations followed by a survey of their published applications to scientific research. Finally, we discuss the limitations of these methods and suggest potential avenues for future work integrating mathematical tools into clinical care and biomedical informatics.

0-Dimensional Persistent Homology Analysis Implementation in Resource-Scarce Embedded Systems

Persistent Homology (PH) analysis is a powerful tool for understanding many relevant topological features from a given dataset. PH allows finding clusters, noise, and relevant connections in the dataset. Therefore, it can provide a better view of the problem and a way of perceiving if a given dataset is equal to another, if a given sample is relevant, and how the samples occupy the feature space. However, PH involves reducing the problem to its simplicial complex space, which is computationally expensive and implementing PH in such Resource-Scarce Embedded Systems (RSES) is an essential add-on for them. However, due to its complexity, implementing PH in such tiny devices is considerably complicated due to the lack of memory and processing power. The following paper shows the implementation of 0-Dimensional Persistent Homology Analysis in a set of well-known RSES, using a technique that reduces the memory footprint and processing power needs of the 0-Dimensional PH algorithm. The results are positive and show that RSES can be equipped with this real-time data analysis tool.

The seen and the unseen: Molecular classification and image based-analysis of gastrointestinal cancers

Gastrointestinal cancers account for 22.5% of cancer related deaths worldwide and represent circa 20% of all cancers. In the last decades, we have witnessed a shift from histology-based to molecular-based classifications using genomic, epigenomic, and transcriptomic data. The molecular based classification revealed new prognostic markers and may aid the therapy selection. Because of the high-costs to perform a molecular classification, in recent years immunohistochemistry-based surrogate classification were developed which permit the stratification of patients, and in parallel multiple groups developed hematoxylin and eosin whole slide image analysis for sub-classifying these entities. Hence, we are witnessing a return to an image-based classification with the purpose to infer hidden information from routine histology images that would permit to detect the patients that respond to specific therapies and would be able to predict their outcome. In this review paper, we will discuss the current histological, molecular, and immunohistochemical classifications of the most common gastrointestinal cancers, gastric adenocarcinoma, and colorectal adenocarcinoma, and will present key aspects for developing a new artificial intelligence aided image-based classification of these malignancies.

A Domain-Oblivious Approach for Learning Concise Representations of Filtered Topological Spaces for Clustering

Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this paper, we propose a persistence diagram hashing framework that learns a binary code representation of persistence diagrams, which allows for fast computation of distances. This framework is built upon a generative adversarial network (GAN) with a diagram distance loss function to steer the learning process. Instead of using standard representations, we hash diagrams into binary codes, which have natural advantages in large-scale tasks. The training of this model is domain-oblivious in that it can be computed purely from synthetic, randomly created diagrams. As a consequence, our proposed method is directly applicable to various datasets without the need for retraining the model. These binary codes, when compared using fast Hamming distance, better maintain topological similarity properties between datasets than other vectorized representations. To evaluate this method, we apply our framework to the problem of diagram clustering and we compare the quality and performance of our approach to the state-of-the-art. In addition, we show the scalability of our approach on a dataset with 10k persistence diagrams, which is not possible with current techniques. Moreover, our experimental results demonstrate that our method is significantly faster with the potential of less memory usage, while retaining comparable or better quality comparisons.

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Quantifying and comparing complex spatial biological datasets is crucial for medical applications and remains an active area of research. As datasets become more heterogeneous and complicated, so must the methods that are used to understand them. Multiparameter topology is built upon the assumption that the shape of data depends on multiple parameters, such as scale, outliers, or other parameters (e.g., cell density and oxygen levels in the case of tumors). A key difficulty encountered in multiparameter persistent homology (MPH) is interpreting and comparing data. The present work uses statistical MPH landscapes to overcome this difficulty and quantifies differences in synthetic data of immune cell infiltration as well as clinical tumor histology data of T cells, macrophages, and hypoxia.

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data—often with outliers, artifacts, and mislabeled points—such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

Persistent homology of tumor CT scans is associated with survival in lung cancer

PURPOSE

Radiomics, the objective study of nonvisual features in clinical imaging, has been useful in informing decisions in clinical oncology. However, radiomics currently lacks the ability to characterize the overall topological structure of the data. This niche can be filled by persistent homology, a form of topological data analysis that analyzes high-level structure. We hypothesized that persistent homology features quantified using cubical complexes could be extracted from lung tumor scans and related to survival.

METHODS

We obtained segmented computed tomography (CT) lung scans (n = 565) from the NSCLC-Radiomics and NSCLC-Radiogenomics datasets in The Cancer Imaging Archive. These scans are three-dimensional images whose pixel intensity corresponds to a number of Hounsfield units. Cubical complexes are a topological image analysis method that effectively analyzes the number of topological features in an image as the image is thresholded at different intensities. We calculated a novel output called a feature curve by plotting the number of zero-dimensional (0D) topological features counted from the cubical complex filtration against each Hounsfield value. This curve's first moment of distribution was utilized as a summary statistic to show association with survival in a Cox proportional hazards model. We hypothesized that persistent homology features quantified using cubical complexes could be extracted from lung tumor scans and related to survival.

RESULTS

After controlling for tumor image size, age, and stage, the first moment of the 0D topological feature curve was associated with poorer survival (HR = 1.118; 95% CI = 1.026-1.218; p = 0.01). The patients in our study with the lowest first moment scores had significantly better survival (1238 days; 95% CI = 936-1599) compared to the patients with the highest first moment scores (429 days; 95% CI = 326-601; p = 0.0015).

CONCLUSIONS

We have shown that persistent homology can generate useful clinical correlates from tumor CT scans. Our 0D topological feature curve statistic predicts survival in lung cancer patients. This novel statistic may be used in tandem with standard radiomics variables to better inform clinical oncology decisions.

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry

Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

Development and validation of a nomogram based on multiparametric magnetic resonance imaging and elastography-derived data for the stratification of patients with prostate cancer

Background

This study sought to develop and validate a nomogram combining the elastographic Q-analysis score (EQS), the Prostate Imaging Reporting and Data System (PI-RADS) score, and clinical parameters for the stratification of patients with prostate cancer (PCa).

Methods

A retrospective study was conducted of 375 patients with 375 lesions who underwent volume-navigation transrectal ultrasound (TRUS) and multiparametric magnetic resonance imaging (MP-MRI)-fusion targeted biopsies between April 2017 and January 2020. Based on a multivariate logistic regression model, a nomogram was created to assess any PCa and high-risk PCa [Gleason score (GS) ≥4+3] using data from patients diagnosed between April 2017 and June 2019 (n=271), and was validated in patients diagnosed after July 2019 (n=104). The nomogram's performance was evaluated based on its discrimination, calibration, and clinical usefulness.

Results

The areas under the curve (AUCs) of the nomogram for predicting any PCa and high-risk PCa were 0.949 [95% confidence interval (CI), 0.921 to 0.978] and 0.936 (95% CI, 0.906 to 0.965), respectively, in the training cohort, and 0.946 (95% CI, 0.894 to 0.997) and 0.971 (95% CI, 0.9331 to 1), respectively, in the validation cohort. The nomogram was well calibrated, and no significant difference was found between the predicted and observed probabilities. A decision curve analysis (DCA) for the nomogram with and without the EQS showed that the threshold probability of for any PCa was <67%.

Conclusions

The nomogram that combined elastography-derived and MP-MRI data was more clinically useful than the model based on PI-RADS and clinical parameters alone. Our nomogram could aid urologists to make decisions and avoid unnecessary biopsies.

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.

Homological scaffold via minimal homology bases

The homological scaffold leverages persistent homology to construct a topologically sound summary of a weighted network. However, its crucial dependency on the choice of representative cycles hinders the ability to trace back global features onto individual network components, unless one provides a principled way to make such a choice. In this paper, we apply recent advances in the computation of minimal homology bases to introduce a quasi-canonical version of the scaffold, called minimal, and employ it to analyze data both real and in silico. At the same time, we verify that, statistically, the standard scaffold is a good proxy of the minimal one for sufficiently complex networks.

Robust radiogenomics approach to the identification of EGFR mutations among patients with NSCLC from three different countries using topologically invariant Betti numbers

Objectives

To propose a novel robust radiogenomics approach to the identification of epidermal growth factor receptor (EGFR) mutations among patients with non-small cell lung cancer (NSCLC) using Betti numbers (BNs).

Materials and methods

Contrast enhanced computed tomography (CT) images of 194 multi-racial NSCLC patients (79 EGFR mutants and 115 wildtypes) were collected from three different countries using 5 manufacturers’ scanners with a variety of scanning parameters. Ninety-nine cases obtained from the University of Malaya Medical Centre (UMMC) in Malaysia were used for training and validation procedures. Forty-one cases collected from the Kyushu University Hospital (KUH) in Japan and fifty-four cases obtained from The Cancer Imaging Archive (TCIA) in America were used for a test procedure. Radiomic features were obtained from BN maps, which represent topologically invariant heterogeneous characteristics of lung cancer on CT images, by applying histogram- and texture-based feature computations. A BN-based signature was determined using support vector machine (SVM) models with the best combination of features that maximized a robustness index (RI) which defined a higher total area under receiver operating characteristics curves (AUCs) and lower difference of AUCs between the training and the validation. The SVM model was built using the signature and optimized in a five-fold cross validation. The BN-based model was compared to conventional original image (OI)- and wavelet-decomposition (WD)-based models with respect to the RI between the validation and the test.

Results

The BN-based model showed a higher RI of 1.51 compared with the models based on the OI (RI: 1.33) and the WD (RI: 1.29).

Conclusion

The proposed model showed higher robustness than the conventional models in the identification of EGFR mutations among NSCLC patients. The results suggested the robustness of the BN-based approach against variations in image scanner/scanning parameters.

Topological Feature Extraction and Visualization of Whole Slide Images using Graph Neural Networks

Whole-slide images (WSI) are digitized representations of thin sections of stained tissue from various patient sources (biopsy, resection, exfoliation, fluid) and often exceed 100,000 pixels in any given spatial dimension. Deep learning approaches to digital pathology typically extract information from sub-images (patches) and treat the sub-images as independent entities, ignoring contributing information from vital large-scale architectural relationships. Modeling approaches that can capture higher-order dependencies between neighborhoods of tissue patches have demonstrated the potential to improve predictive accuracy while capturing the most essential slide-level information for prognosis, diagnosis and integration with other omics modalities. Here, we review two promising methods for capturing macro and micro architecture of histology images, Graph Neural Networks, which contextualize patch level information from their neighbors through message passing, and Topological Data Analysis, which distills contextual information into its essential components. We introduce a modeling framework, WSI-GTFE that integrates these two approaches in order to identify and quantify key pathogenic information pathways. To demonstrate a simple use case, we utilize these topological methods to develop a tumor invasion score to stage colon cancer.

The shape of things to come: Topological data analysis and biology, from molecules to organisms

Shape is data and data is shape. Biologists are accustomed to thinking about how the shape of biomolecules, cells, tissues, and organisms arise from the effects of genetics, development, and the environment. Less often do we consider that data itself has shape and structure, or that it is possible to measure the shape of data and analyze it. Here, we review applications of topological data analysis (TDA) to biology in a way accessible to biologists and applied mathematicians alike. TDA uses principles from algebraic topology to comprehensively measure shape in data sets. Using a function that relates the similarity of data points to each other, we can monitor the evolution of topological features-connected components, loops, and voids. This evolution, a topological signature, concisely summarizes large, complex data sets. We first provide a TDA primer for biologists before exploring the use of TDA across biological sub-disciplines, spanning structural biology, molecular biology, evolution, and development. We end by comparing and contrasting different TDA approaches and the potential for their use in biology. The vision of TDA, that data are shape and shape is data, will be relevant as biology transitions into a data-driven era where the meaningful interpretation of large data sets is a limiting factor.

Enhanced resolution 3D digital cytology and pathology with dual-view inverted selective plane illumination microscopy

The current gold-standard histopathology for tissue analysis is destructive, time consuming, and limited to 2D slices. Light sheet microscopy has emerged as a powerful tool for 3D imaging of tissue biospecimens with its fast speed and low photo-damage, but usually with worse axial resolution and complicated configuration for sample imaging. Here, we utilized inverted selective plane illumination microscopy for easy sample mounting and imaging, and dual-view imaging and deconvolution to overcome the anisotropic resolution. We have rendered 3D images of fresh cytology cell blocks and millimeter- to centimeter-sized fixed tissue samples with high resolution in both lateral and axial directions. More accurate cellular quantification, higher image sharpness, and more image details have been achieved with the dual-view method compared with single-view imaging.