Statistical and topological summaries aid disease detection for segmented retinal vascular images.

OBJECTIVE: Disease complications can alter vascular network morphology and disrupt tissue functioning. Microvascular diseases of the retina are assessed by visual inspection of retinal images, but this can be challenging when diseases exhibit silent symptoms or patients cannot attend in-person meetings. We examine the performance of machine learning algorithms in detecting microvascular disease when trained on statistical and topological summaries of segmented retinal vascular images. METHODS: We compute 13 separate descriptor vectors (5 statistical, 8 topological) to summarize the morphology of retinal vessel segmentation images and train support vector machines to predict each image's disease classification from the summary vectors. We assess the performance of each descriptor vector, using five-fold cross validation to estimate their accuracy. We apply these methods to four datasets that were assembled from four existing data repositories; three datasets contain segmented retinal vascular images from one of the repositories, whereas the fourth "All" dataset combines images from four repositories. RESULTS: Among the 13 total descriptor vectors considered, either a statistical Box-counting descriptor vector or a topological Flooding descriptor vector achieves the highest accuracy levels. On the combined "All" dataset, the Box-counting vector outperforms all other descriptors, including the topological Flooding vector which is sensitive to differences in the annotation styles between the different datasets. CONCLUSION: Our work represents a first step to establishing which computational methods are most suitable for identifying microvascular disease and assessing their current limitations. These methods could be incorporated into automated disease assessment tools.

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

Angiogenesis is the process by which blood vessels form from pre-existing vessels. It plays a key role in many biological processes, including embryonic development and wound healing, and contributes to many diseases including cancer and rheumatoid arthritis. The structure of the resulting vessel networks determines their ability to deliver nutrients and remove waste products from biological tissues. Here we simulate the Anderson-Chaplain model of angiogenesis at different parameter values and quantify the vessel architectures of the resulting synthetic data. Specifically, we propose a topological data analysis (TDA) pipeline for systematic analysis of the model. TDA is a vibrant and relatively new field of computational mathematics for studying the shape of data. We compute topological and standard descriptors of model simulations generated by different parameter values. We show that TDA of model simulation data stratifies parameter space into regions with similar vessel morphology. The methodologies proposed here are widely applicable to other synthetic and experimental data including wound healing, development, and plant biology.

Biologically-informed neural networks guide mechanistic modeling from sparse experimental data.

Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].

Learning Equations from Biological Data with Limited Time Samples.

Equation learning methods present a promising tool to aid scientists in the modeling process for biological data. Previous equation learning studies have demonstrated that these methods can infer models from rich datasets; however, the performance of these methods in the presence of common challenges from biological data has not been thoroughly explored. We present an equation learning methodology comprised of data denoising, equation learning, model selection and post-processing steps that infers a dynamical systems model from noisy spatiotemporal data. The performance of this methodology is thoroughly investigated in the face of several common challenges presented by biological data, namely, sparse data sampling, large noise levels, and heterogeneity between datasets. We find that this methodology can accurately infer the correct underlying equation and predict unobserved system dynamics from a small number of time samples when the data are sampled over a time interval exhibiting both linear and nonlinear dynamics. Our findings suggest that equation learning methods can be used for model discovery and selection in many areas of biology when an informative dataset is used. We focus on glioblastoma multiforme modeling as a case study in this work to highlight how these results are informative for data-driven modeling-based tumor invasion predictions.

Analyzing collective motion with machine learning and topology.

We use topological data analysis and machine learning to study a seminal model of collective motion in biology [M. R. D'Orsogna et al., Phys. Rev. Lett. 96, 104302 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based on topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters.

The influence of numerical error on parameter estimation and uncertainty quantification for advective PDE models.

Advective partial differential equations can be used to describe many scientific processes. Two significant sources of error that can cause difficulties in inferring parameters from experimental data on these processes include (i) noise from the measurement and collection of experimental data and (ii) numerical error in approximating the forward solution to the advection equation. How this second source of error alters parameter estimation and uncertainty quantification during an inverse problem methodology is not well understood. As a step towards a better understanding of this problem, we present both analytical and computational results concerning how a least squares cost function and parameter estimator behave in the presence of numerical error in approximating solutions to the underlying advection equation. We investigate residual patterns to derive an autocorrelative statistical model that can improve parameter estimation and confidence interval computation for first order methods. Building on our results and their general nature, we provide guidelines for practitioners to determine when numerical or experimental error is the main source of error in their inference, along with suggestions of how to efficiently improve their results.

Modeling keratinocyte wound healing dynamics: Cell-cell adhesion promotes sustained collective migration.

The in vitro migration of keratinocyte cell sheets displays behavioral and biochemical similarities to the in vivo wound healing response of keratinocytes in animal model systems. In both cases, ligand-dependent Epidermal Growth Factor Receptor (EGFR) activation is sufficient to elicit collective cell migration into the wound. Previous mathematical modeling studies of in vitro wound healing assays assume that physical connections between cells have a hindering effect on cell migration, but biological literature suggests a more complicated story. By combining mathematical modeling and experimental observations of collectively migrating sheets of keratinocytes, we investigate the role of cell-cell adhesion during in vitro keratinocyte wound healing assays. We develop and compare two nonlinear diffusion models of the wound healing process in which cell-cell adhesion either hinders or promotes migration. Both models can accurately fit the leading edge propagation of cell sheets during wound healing when using a time-dependent rate of cell-cell adhesion strength. The model that assumes a positive role of cell-cell adhesion on migration, however, is robust to changes in the leading edge definition and yields a qualitatively accurate density profile. Using RNAi for the critical adherens junction protein, α-catenin, we demonstrate that cell sheets with wild type cell-cell adhesion expression maintain migration into the wound longer than cell sheets with decreased cell-cell adhesion expression, which fails to exhibit collective migration. Our modeling and experimental data thus suggest that cell-cell adhesion promotes sustained migration as cells pull neighboring cells into the wound during wound healing.

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.

Learning differential equation models from stochastic agent-based model simulations.

Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth-death-migration model commonly used to explore cell biology experiments and a susceptible-infected-recovered model of infectious disease spread.

Learning partial differential equations for biological transport models from noisy spatio-temporal data.

We investigate methods for learning partial differential equation (PDE) models from spatio-temporal data under biologically realistic levels and forms of noise. Recent progress in learning PDEs from data have used sparse regression to select candidate terms from a denoised set of data, including approximated partial derivatives. We analyse the performance in using previous methods to denoise data for the task of discovering the governing system of PDEs. We also develop a novel methodology that uses artificial neural networks (ANNs) to denoise data and approximate partial derivatives. We test the methodology on three PDE models for biological transport, i.e. the advection-diffusion, classical Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) and nonlinear Fisher-KPP equations. We show that the ANN methodology outperforms previous denoising methods, including finite differences and both local and global polynomial regression splines, in the ability to accurately approximate partial derivatives and learn the correct PDE model.

A tutorial review of mathematical techniques for quantifying tumor heterogeneity.

Intra-tumor and inter-patient heterogeneity are two challenges in developing mathematical models for precision medicine diagnostics. Here we review several techniques that can be used to aid the mathematical modeller in inferring and quantifying both sources of heterogeneity from patient data. These techniques include virtual populations, nonlinear mixed effects modeling, non-parametric estimation, Bayesian techniques, and machine learning. We create simulated virtual populations in this study and then apply the four remaining methods to these datasets to highlight the strengths and weak-nesses of each technique. We provide all code used in this review at https://github.com/jtnardin/Tumor-Heterogeneity/ so that this study may serve as a tutorial for the mathematical modelling community. This review article was a product of a Tumor Heterogeneity Working Group as part of the 2018-2019 Program on Statistical, Mathematical, and Computational Methods for Precision Medicine which took place at the Statistical and Applied Mathematical Sciences Institute.

INVESTIGATION OF A STRUCTURED FISHER'S EQUATION WITH APPLICATIONS IN BIOCHEMISTRY.

Recent biological research has sought to understand how biochemical signaling pathways, such as the mitogen-activated protein kinase (MAPK) family, influence the migration of a population of cells during wound healing. Fisher's Equation has been used extensively to model experimental wound healing assays due to its simple nature and known traveling wave solutions. This partial differential equation with independent variables of time and space cannot account for the effects of biochemical activity on wound healing, however. To this end, we derive a structured Fisher's Equation with independent variables of time, space, and biochemical pathway activity level and prove the existence of a self-similar traveling wave solution to this equation. We exhibit that these methods also apply to a general structured reaction-diffusion equation and a chemotaxis equation. We also consider a more complicated model with different phenotypes based on MAPK activation and numerically investigate how various temporal patterns of biochemical activity can lead to increased and decreased rates of population migration.