MS04 - MFBM-18

Geometrical and Topological Methods for Data-Driven Modeling (Part 1)

Tuesday, July 15 at 3:50pm

SMB2025 Follow

Note: this minisymposia has multiple sessions. The other session is: and Part-2.

Dhananjay Bhaskar (Yale University), Bernadette Stolz-Pretzer

Description:

The increasing complexity of modern biomedical datasets necessitates advanced mathematical frameworks that reveal intrinsic structures beyond traditional statistical approaches. Geometrical and topological methods provide powerful tools to extract robust and interpretable patterns from high-dimensional, noisy, and multimodal data, enabling more effective data-driven modeling. This minisymposium will explore recent advances at the intersection of geometry, topology, and mathematical modeling, highlighting techniques such as persistent homology, sheaf theory, optimal transport, manifold learning, and geometric deep learning. Speakers will present applications to diverse fields ranging from oncology, liver disease to neuroscience, demonstrating how these methods enhance our understanding of complex biological systems, disease progression, and cellular organization. By fostering interdisciplinary dialogue, this session aims to showcase novel approaches that push the boundaries of data-driven discovery and advance mathematical techniques for biomedical research.

Katherine Benjamin

University of Oxford

"Topological methods for subcellular spatial transcriptomics"

Spatial transcriptomics technologies produce gene expression measurements at millions of locations across a tissue sample. An open problem in this area is the inference of spatial information about single cells. Here we present a multiscale machine learning method to pinpoint the locations of individual sparsely dispersed cells from subcellular spatial transcriptomics data. We integrate this approach with multiparameter persistence landscapes, a state of the art tool in topological data analysis, to identify a loop structure in infiltrating glomerular immune cells in a mouse model of lupus nephritis.

Veronica Ciocanel

Duke University

"Unraveling aster and ring structures in cell models of dynamic actin filaments using topological data analysis"

Actomyosin is a dynamic network of interacting proteins that reshapes and organizes in a variety of structures that are essential in cell movement, cell division, as well as in wound healing. Agent-based models can simulate realistic dynamic interactions between actin filaments and myosin motor proteins inside cells. These stochastic simulations reproduce bundles, clusters, and contractile rings that resemble biological observations. We have developed techniques based on topological data analysis to extract insights from spatio-temporal data in these protein network interactions. Recently, we have been interested in adapting the framework of vines and vineyards in order to track topological and geometrical features through time-parameterized stacks of persistence diagrams. This approach allows us to quantify characteristics of formation and maintenance of relevant actin structures such as rings and asters in simulated datasets. This is joint work with Niny Arcila-Maya.

Robert McDonald

University of Oxford

"Topological model selection: a case-study in tumour-induced angiogenesis"

Comparing mathematical models offers a means to evaluate competing scientific theories. However, exact methods of model calibration are not applicable to many probabilistic models which simulate high-dimensional spatio-temporal data. Approximate Bayesian Computation is a widely-used method for parameter inference and model selection in such scenarios, and it may be combined with Topological Data Analysis to study models which simulate data with fine spatial structure. We develop a flexible pipeline for parameter inference and model selection in spatio-temporal models. Our pipeline identifies topological summary statistics which quantify spatio-temporal data and uses them to approximate parameter and model posterior distributions. We validate our pipeline on models of tumour-induced angiogenesis, inferring four parameters in three established models and identifying the correct model in synthetic test-cases.

Nan Wu

University of Texas at Dallas

"Adaptive Bayesian regression on manifold"

We investigate how the posterior contraction rate under a Gaussian process prior is influenced by the intrinsic dimension of the domain and the smoothness of the regression function. Specifically, we consider the setting where the domain is a d-dimensional manifold and the regression function is intrinsically s-Hölder smooth on the manifold. We establish the optimal posterior contraction rate of O(n^{-s/(2s + d)}), up to a logarithmic factor. To eliminate the need for prior knowledge of the manifold's dimension, we propose an empirical Bayes prior on the kernel bandwidth, leveraging kernel affinity and k-nearest neighbor statistics. This talk is based on joint work with Tao Tang, Xiuyuan Cheng, and David Dunson.

#SMB2025 Follow

Annual Meeting for the Society for Mathematical Biology, 2025.