Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis

Quantifying collective motion patterns in mesenchymal cell populations using topological data analysis and agent-based modeling

Fibroblasts in a confluent monolayer are known to adopt elongated morphologies in which cells are oriented parallel to their neighbors. We collected and analyzed new microscopy movies to show that confluent fibroblasts are motile and that neighboring cells often move in anti-parallel directions in a collective motion phenomenon we refer to as "fluidization" of the cell population. We used machine learning to perform cell tracking for each movie and then leveraged topological data analysis (TDA) to show that time-varying point-clouds generated by the tracks contain significant topological information content that is driven by fluidization, i.e., the anti-parallel movement of individual neighboring cells and neighboring groups of cells over long distances. We then utilized the TDA summaries extracted from each movie to perform Bayesian parameter estimation for the D'Orsgona model, an agent-based model (ABM) known to produce a wide array of different patterns, including patterns that are qualitatively similar to fluidization. Although the D'Orsgona ABM is a phenomenological model that only describes inter-cellular attraction and repulsion, the estimated region of D'Orsogna model parameter space was consistent across all movies, suggesting that a specific level of inter-cellular repulsion force at close range may be a mechanism that helps drive fluidization patterns in confluent mesenchymal cell populations.

Computational modeling of angiogenesis: The importance of cell rearrangements during vascular growth

Angiogenesis is the process wherein endothelial cells (ECs) form sprouts that elongate from the pre-existing vasculature to create new vascular networks. In addition to its essential role in normal development, angiogenesis plays a vital role in pathologies such as cancer, diabetes and atherosclerosis. Mathematical and computational modeling has contributed to unraveling its complexity. Many existing theoretical models of angiogenic sprouting are based on the "snail-trail" hypothesis. This framework assumes that leading ECs positioned at sprout tips migrate toward low-oxygen regions while other ECs in the sprout passively follow the leaders' trails and proliferate to maintain sprout integrity. However, experimental results indicate that, contrary to the snail-trail assumption, ECs exchange positions within developing vessels, and the elongation of sprouts is primarily driven by directed migration of ECs. The functional role of cell rearrangements remains unclear. This review of the theoretical modeling of angiogenesis is the first to focus on the phenomenon of cell mixing during early sprouting. We start by describing the biological processes that occur during early angiogenesis, such as phenotype specification, cell rearrangements and cell interactions with the microenvironment. Next, we provide an overview of various theoretical approaches that have been employed to model angiogenesis, with particular emphasis on recent in silico models that account for the phenomenon of cell mixing. Finally, we discuss when cell mixing should be incorporated into theoretical models and what essential modeling components such models should include in order to investigate its functional role. This article is categorized under: Cardiovascular Diseases > Computational Models Cancer > Computational Models.

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.

Topological Data Analysis Captures Task-Driven fMRI Profiles in Individual Participants: A Classification Pipeline Based on Persistence

BOLD-based fMRI is the most widely used method for studying brain function. The BOLD signal while valuable, is beset with unique vulnerabilities. The most notable of these is the modest signal to noise ratio, and the relatively low temporal and spatial resolution. However, the high dimensional complexity of the BOLD signal also presents unique opportunities for functional discovery. Topological Data Analyses (TDA), a branch of mathematics optimized to search for specific classes of structure within high dimensional data may provide particularly valuable applications. In this investigation, we acquired fMRI data in the anterior cingulate cortex (ACC) using a basic motor control paradigm. Then, for each participant and each of three task conditions, fMRI signals in the ACC were summarized using two methods: a) TDA based methods of persistent homology and persistence landscapes and b) non-TDA based methods using a standard vectorization scheme. Finally, using machine learning (with support vector classifiers), classification accuracy of TDA and non-TDA vectorized data was tested across participants. In each participant, TDA-based classification out-performed the non-TDA based counterpart, suggesting that our TDA analytic pipeline better characterized task- and condition-induced structure in fMRI data in the ACC. Our results emphasize the value of TDA in characterizing task- and condition-induced structure in regional fMRI signals. In addition to providing our analytical tools for other users to emulate, we also discuss the unique role that TDA-based methods can play in the study of individual differences in the structure of functional brain signals in the healthy and the clinical brain.

Topological data analysis of spatial patterning in heterogeneous cell populations: clustering and sorting with varying cell-cell adhesion

Different cell types aggregate and sort into hierarchical architectures during the formation of animal tissues. The resulting spatial organization depends (in part) on the strength of adhesion of one cell type to itself relative to other cell types. However, automated and unsupervised classification of these multicellular spatial patterns remains challenging, particularly given their structural diversity and biological variability. Recent developments based on topological data analysis are intriguing to reveal similarities in tissue architecture, but these methods remain computationally expensive. In this article, we show that multicellular patterns organized from two interacting cell types can be efficiently represented through persistence images. Our optimized combination of dimensionality reduction via autoencoders, combined with hierarchical clustering, achieved high classification accuracy for simulations with constant cell numbers. We further demonstrate that persistence images can be normalized to improve classification for simulations with varying cell numbers due to proliferation. Finally, we systematically consider the importance of incorporating different topological features as well as information about each cell type to improve classification accuracy. We envision that topological machine learning based on persistence images will enable versatile and robust classification of complex tissue architectures that occur in development and disease.

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.

Zigzag persistence for coral reef resilience using a stochastic spatial model

A complex interplay between species governs the evolution of spatial patterns in ecology. An open problem in the biological sciences is characterizing spatio-temporal data and understanding how changes at the local scale affect global dynamics/behaviour. Here, we extend a well-studied temporal mathematical model of coral reef dynamics to include stochastic and spatial interactions and generate data to study different ecological scenarios. We present descriptors to characterize patterns in heterogeneous spatio-temporal data surpassing spatially averaged measures. We apply these descriptors to simulated coral data and demonstrate the utility of two topological data analysis techniques-persistent homology and zigzag persistence-for characterizing mechanisms of reef resilience. We show that the introduction of local competition between species leads to the appearance of coral clusters in the reef. We use our analyses to distinguish temporal dynamics stemming from different initial configurations of coral, showing that the neighbourhood composition of coral sites determines their long-term survival. Using zigzag persistence, we determine which spatial configurations protect coral from extinction in different environments. Finally, we apply this toolkit of multi-scale methods to empirical coral reef data, which distinguish spatio-temporal reef dynamics in different locations, and demonstrate the applicability to a range of datasets.

Characterizing emerging features in cell dynamics using topological data analysis methods

Filament-motor interactions inside cells play essential roles in many developmental as well as other biological processes. For instance, actin-myosin interactions drive the emergence or closure of ring channel structures during wound healing or dorsal closure. These dynamic protein interactions and the resulting protein organization lead to rich time-series data generated by using fluorescence imaging experiments or by simulating realistic stochastic models. We propose methods based on topological data analysis to track topological features through time in cell biology data consisting of point clouds or binary images. The framework proposed here is based on computing the persistent homology of the data at each time point and on connecting topological features through time using established distance metrics between topological summaries. The methods retain aspects of monomer identity when analyzing significant features in filamentous structure data, and capture the overall closure dynamics when assessing the organization of multiple ring structures through time. Using applications of these techniques to experimental data, we show that the proposed methods can describe features of the emergent dynamics and quantitatively distinguish between control and perturbation experiments.

Algebra, Geometry and Topology of ERK Kinetics

The MEK/ERK signalling pathway is involved in cell division, cell specialisation, survival and cell death (Shaul and Seger in Biochim Biophys Acta (BBA)-Mol Cell Res 1773(8):1213–1226, 2007). Here we study a polynomial dynamical system describing the dynamics of MEK/ERK proposed by Yeung et al. (Curr Biol 2019, 10.1016/j.cub.2019.12.052) with their experimental setup, data and known biological information. The experimental dataset is a time-course of ERK measurements in different phosphorylation states following activation of either wild-type MEK or MEK mutations associated with cancer or developmental defects. We demonstrate how methods from computational algebraic geometry, differential algebra, Bayesian statistics and computational algebraic topology can inform the model reduction, identification and parameter inference of MEK variants, respectively. Throughout, we show how this algebraic viewpoint offers a rigorous and systematic analysis of such models.

Multiscale topology characterizes dynamic tumor 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.

Multiscale topology characterizes dynamic tumor 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.

OUP accepted manuscript

Motivation

Inferring the parameters of models describing biological systems is an important problem in the reverse engineering of the mechanisms underlying these systems. Much work has focused on parameter inference of stochastic and ordinary differential equation models using Approximate Bayesian Computation (ABC). While there is some recent work on inference in spatial models, this remains an open problem. Simultaneously, advances in topological data analysis (TDA), a field of computational mathematics, have enabled spatial patterns in data to be characterized.

Results

Here, we focus on recent work using TDA to study different regimes of parameter space for a well-studied model of angiogenesis. We propose a method for combining TDA with ABC to infer parameters in the Anderson–Chaplain model of angiogenesis. We demonstrate that this topological approach outperforms ABC approaches that use simpler statistics based on spatial features of the data. This is a first step toward a general framework of spatial parameter inference for biological systems, for which there may be a variety of filtrations, vectorizations and summary statistics to be considered.

Availability and implementation

All code used to produce our results is available as a Snakemake workflow from github.com/tt104/tabc_angio.