Topological Data Analysis of Task-Based fMRI Data from Experiments on Schizophrenia

We use methods from computational algebraic topology to study functional brain networks, in which nodes represent brain regions and weighted edges encode the similarity of fMRI time series from each region. With these tools, which allow one to characterize topological invariants such as loops in high-dimensional data, we are able to gain understanding into low-dimensional structures in networks in a way that complements traditional approaches that are based on pairwise interactions. In the present paper, we use persistent homology to analyze networks that we construct from task-based fMRI data from schizophrenia patients, healthy controls, and healthy siblings of schizophrenia patients. We thereby explore the persistence of topological structures such as loops at different scales in these networks. We use persistence landscapes and persistence images to create output summaries from our persistent-homology calculations, and we study the persistence landscapes and images using $k$-means clustering and community detection. Based on our analysis of persistence landscapes, we find that the members of the sibling cohort have topological features (specifically, their 1-dimensional loops) that are distinct from the other two cohorts. From the persistence images, we are able to distinguish all three subject groups and to determine the brain regions in the loops (with four or more edges) that allow us to make these distinctions

Geometric anomaly detection in data

This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can optimally stratify a given dataset into clusters, each of which is in turn well-approximable by a suitable submanifold of the ambient space. Since these approximating submanifolds might have different dimensions, we are able to detect non-manifold like singular regions in data even when none of the data points have been sampled from those singularities. We showcase this method by identifying the intersection of two surfaces in the 24-dimensional space of cyclo-octane conformations, and by locating all the self-intersections of a Henneberg minimal surface immersed in 3-dimensional space. Due to the local nature of the required topological computations, the algorithmic burden of performing such data stratification is readily distributable across several processors

Enhanced perfusion following exposure to radiotherapy: a theoretical investigation

Tumour angiogenesis leads to the formation of blood vessels that are structurally and spatially heterogeneous. Poor blood perfusion, in conjunction with increased hypoxia and oxygen heterogeneity, impairs a tumour’s response to radiotherapy. The optimal strategy for enhancing tumour perfusion remains unclear, preventing its regular deployment in combination therapies. In this work, we first identify vascular architectural features that correlate with enhanced perfusion following radiotherapy, using in vivo imaging data from vascular tumours. Then, we present a novel computational model to determine the relationship between these architectural features and blood perfusion in silico. If perfusion is defined to be the proportion of vessels that support blood flow, we find that vascular networks with small mean diameters and large numbers of angiogenic sprouts show the largest increases in perfusion post-irradiation for both biological and synthetic tumours. We also identify cases where perfusion increases due to the pruning of hypoperfused vessels, rather than blood being rerouted. These results indicate the importance of considering network composition when determining the optimal irradiation strategy. In the future, we aim to use our findings to identify tumours that are good candidates for perfusion enhancement and to improve the efficacy of combination therapies

Outlier-robust subsampling techniques for persistent homology

In recent years, persistent homology (PH) has been successfully applied to real-world data in many different settings. Despite significant computational advances, PH algorithms do not yet scale to large datasets preventing interesting applications. One approach to address computational issues posed by PH is to select a set of landmarks by subsampling from the data. Currently, these landmark points are chosen either at random or using the maxmin algorithm. Neither is ideal as random selection tends to favour dense areas of the data while the maxmin algorithm is very sensitive to noise. Here, we propose a novel approach to select landmarks specifically for PH that preserves coarse topological information of the original dataset. Our method is motivated by the Mayer-Vietoris sequence and requires only local PH calculations thus enabling efficient computation. We test our landmarks on artificial data sets which contain different levels of noise and compare them to standard landmark selection techniques. We demonstrate that our landmark selection outperforms standard methods as well as a subsampling technique based on an outlier-robust version of the k-means algorithm for low sampling densities in noisy data with respect to robustness to outliers

Enhanced perfusion following exposure to radiotherapy: a theoretical investigation (revised manuscript data)

<p>Dataset and software supporting the revised manuscript: "Enhanced perfusion following exposure to radiotherapy: a theoretical investigation." See the enclosed README and manuscript for further information.</p&gt

Quantification of vascular networks in photoacoustic mesoscopy.

Mesoscopic photoacoustic imaging (PAI) enables non-invasive visualisation of tumour vasculature. The visual or semi-quantitative 2D measurements typically applied to mesoscopic PAI data fail to capture the 3D vessel network complexity and lack robust ground truths for assessment of accuracy. Here, we developed a pipeline for quantifying 3D vascular networks captured using mesoscopic PAI and tested the preservation of blood volume and network structure with topological data analysis. Ground truth data of in silico synthetic vasculatures and a string phantom indicated that learning-based segmentation best preserves vessel diameter and blood volume at depth, while rule-based segmentation with vesselness image filtering accurately preserved network structure in superficial vessels. Segmentation of vessels in breast cancer patient-derived xenografts (PDXs) compared favourably to ex vivo immunohistochemistry. Furthermore, our findings underscore the importance of validating segmentation methods when applying mesoscopic PAI as a tool to evaluate vascular networks in vivo