Global and local persistent homology for the shape and classification of biological data

Persistent homology (PH) is an algorithmic method that allows one to study shape and higher-order interactions in high-dimensional data. Over the last decade, PH has been used in a wide variety of applications, including biology. PH considers topological invariants, such as connected components, loops, and holes, and their changes across a filtration which one can imagine as observing the data through multiple scales or resolutions. The filtration determines the questions that can be answered about the data and, in many cases, needs to be developed specifically for the problem of interest. There are also many other practical challenges when applying PH such as computational complexity and interpretation of PH output. This thesis has two parts: In the first part, we showcase how PH can be applied to two types of biological data: tumour blood vessel networks and functional neuronal networks. For tumour blood vessel networks, we develop a novel filtration that spatially characterises their structural abnormality. We show that the number of vessel loops and their distribution in the networks change over time when tumours undergo treatment with vascular targeting agents and radiation therapy. In functional neuronal networks, we find that PH can provide insight into dynamical processes in motor-learning data as well as in working-memory data from healthy versus schizophrenic human subjects. We highlight what type of information we can gain by applying persistence landscapes and persistence images to analyse and interpret the output from PH. In the second part of this thesis, we develop novel methods that consider PH locally around data points. To address computational issues when applying PH to large and noisy data sets -- both traits are commonly found in biological data -- we develop a novel landmark selection technique for point clouds. In contrast to existing methods, our subsampling process is robust to outliers and is developed specifically for PH. We further introduce a novel method that can detect geometric anomalies, such as intersections or boundaries, in point cloud data sampled from intersecting surfaces. Our detection is based on the computation of PH in local annular neighbourhoods around points and is less sensitive to the size of the local neighbourhood and surface curvature than an existing method.</p

Multiscale topology characterises dynamic tumour vascular networks

Advances in imaging techniques enable high-resolution three-dimensional (3D) visualization of vascular networks over time and reveal abnormal structural features such as twists and loops, and their quantification is an active area of research. Here, we showcase how topological data analysis, the mathematical field that studies the “shape” of data, can characterize the geometric, spatial, and temporal organization of vascular networks. We propose two topological lenses to study vasculature, which capture inherent multiscale features and vessel connectivity, and surpass the single-scale analysis of existing methods. We analyze images collected using intravital and ultramicroscopy modalities and quantify spatiotemporal variation of twists, loops, and avascular regions (voids) in 3D vascular networks. This topological approach validates and quantifies known qualitative trends such as dynamic changes in tortuosity and loops in response to antibodies that modulate vessel sprouting; furthermore, it quantifies the effect of radiotherapy on vessel architecture