Andreas Buttenschön

Andreas Buttenschön

Assistant Professor in Applied Mathematics at the University of Massachusetts Amherst

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publications

Agent-Based Lattice Models of Multicellular Systems: Numerical Methods, Implementation, and Applications

Published in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2017

Agent-based models (ABMs) of multicellular systems are models in which each cell is represented individually. These models allow taking the variability between individual cells and the spatial heterogeneity of tissues on histological scales into account. In this chapter we present an overview and methodology of ABMs that are used to simulate mechanical and physiological phenomena in cells and tissues. Read more

Recommended citation: Drasdo, D., Buttenschön, A. and Van Liedekerke, P., 2018. Agent-based lattice models of multicellular systems: numerical methods, implementation, and applications. In Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes (pp. 223-238). Academic Press.
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Off-Lattice Agent-Based Models for Cell and Tumor Growth: Numerical Methods, Implementation, and Applications

Published in Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes, 2017

Lattice-free agent-based models (ABMs) of multicellular systems are mathematical models in which each cell is represented individually and can move continuously in space. In this chapter we present an overview of the methodology of ABMs that are used to simulate mechanical and physiological phenomena in cells and tissues. Read more

Recommended citation: Van Liedekerke, P., Buttenschön, A. and Drasdo, D., 2018. Off-lattice agent-based models for cell and tumor growth: numerical methods, implementation, and applications. In Numerical Methods and Advanced Simulation in Biomechanics and Biological Processes (pp. 245-267). Academic Press.
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Integro-partial differential equation models for cell-cell adhesion and its application

Published in University of Alberta, 2018

In my dissertation, I explore models of non-local effects in biological systems. To study these nonlocal phenomena, I use iPDEs. The thesis has three chapters: (1) a derivation of non-local models from an underlying stochastic random walk; (2) an analysis of the steady-states of a non-local model of cell-cell adhesion; and (3) the construction of no-flux boundary conditions for a non-local model of cell-cell adhesion. Read more

Recommended citation: Buttenschoen, A. (2018). Integro-partial differential equation models for cell-cell adhesion and its application.
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Cops-on-the-Dots: The Linear Stability of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime

Published in European Journal of Applied Mathematics, 2018

In a singularly perturbed limit, we analyze the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system … Read more

Recommended citation: Buttenschoen, A., Kolokolnikov, T., Ward, M.J. and Wei, J., 2019. Cops-on-the-dots: The linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime. European Journal of Applied Mathematics, pp.1-47.
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Nonlocal Adhesion Models for Microorganisms on Bounded Domains

Published in SIAM Journal on Applied Mathematics, 2019

For the one dimensional Armstrong model of cell-cell adhesion, we derive various types of adhesive, repulsive, and no-flux boundary conditions. We prove local and global existence and uniqueness for the resulting integro-differential equations. Read more

Recommended citation: Hillen T., Buttenschön, A. SIAM Journal on Applied Mathematics (2020)
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Correlated random walks inside a cell: actin branching and microtubule dynamics

Published in Journal of Mathematical Biology, 2019

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. Read more

Recommended citation: Buttenschön, A. & Edelstein-Keshet, L. J. Math. Biol. (2019)
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Adhesion-driven patterns in a calcium-dependent model of cancer cell movement

Published in arXiv, 2020

Cancer cells exhibit increased motility and proliferation, which are instrumental in the formation of tumours and metastases. These pathological changes can be traced back to malfunctions of cellular signalling pathways, and calcium signalling plays a prominent role in these. We formulate a new model for cancer cell movement which for the first time explicitly accounts for the dependence of cell proliferation and cell-cell adhesion on calcium. Read more

Recommended citation: Kaouri, K., Bitsouni, V., Buttenschön, A., & Thul, R. arXiv preprint arXiv:2003.00612. (2020)

Bridging from single to collective cell migration: A review of models and links to experiments

Published in PLOS Computational Biology, 2020

Mathematical and computational models can assist in gaining an understanding of cell behavior at many levels of organization. Here, we review models in the literature that focus on eukaryotic cell motility at 3 size scales: intracellular signaling that regulates cell shape and movement, single cell motility, and collective cell behavior from a few cells to tissues. Read more

Recommended citation: Buttenschön A, Edelstein-Keshet L PLOS Computational Biology 16(12): e1008411 (2020)
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How cells stay together; a mechanism for maintenance of a robust cluster explored by local and nonlocal continuum models

Published in Bulletin of Mathematical Biology, 2024

Formation of organs and specialized tissues in embryonic development requires migration of cells to specific targets. In some instances, such cells migrate as a robust cluster. We here explore a recent local approximation of nonlocal continuum models by Falcó, Baker, and Carrillo (2023). We apply their theoretical results by specifying biologically-based cell-cell interactions, showing how such cell communication results in an effective attraction-repulsion Morse potential. We then explore the clustering instability, the existence and size of the cluster, and its stability. We also extend their work by investigating the accuracy of the local approximation relative to the full nonlocal model. Read more

Recommended citation: Buttenschön A, Sinclair S, Edelstein-Keshet L Bull. Math. Bio. (2024)
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Support Graph Preconditioners for Off-Lattice Cell-Based Models

Published in arxiv, 2024

Off-lattice agent-based models (or cell-based models) of multicellular systems are increasingly used to create in-silico models of in-vitro and in-vivo experimental setups of cells and tissues, such as cancer spheroids, neural crest cell migration, and liver lobules. These applications, which simulate thousands to millions of cells, require robust and efficient numerical methods. At their core, these models necessitate the solution of a large friction-dominated equation of motion, resulting in a sparse, symmetric, and positive definite matrix equation. The conjugate gradient method is employed to solve this problem, but this requires a good preconditioner for optimal performance. In this study, we develop a graph-based preconditioning strategy that can be easily implemented in such agent-based models. Our approach centers on extending support graph preconditioners to block-structured matrices. We prove asymptotic bounds on the condition number of these preconditioned friction matrices. We then benchmark the conjugate gradient method with our support graph preconditioners and compare its performance to other common preconditioning strategies. Read more

Recommended citation: Steinman J, Buttenschön A arxiv (2024)

research

Modelling of intra-cellular cell decision mechanisms

Published:

I consider models of intra-cellular signalling in cell polarization, that control how a cell determines its front and back. The objective is to uncover how cell-cell interactions, intracellular signalling, forces, and adhesion affect cell polarity and lead to the emergence of collective cell behaviour, in small to large cell groups. Read more

Non-local tissue models

Published:

When cells in a tissue exert traction forces, they pull on one another and on the underlying matrix, much like a tug of war. I describe this process using non-local PDEs. Read more

software

Adhesion Random Walk

Published:

This framework implements a stochastic simulation algorithm for cell adhesion models where cells perform biased random walks based on non-local sensing of their environment. It provides a microscopic foundation for continuum adhesion models. Read more

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TDR Solver

Published:

TDR Solver is a specialized finite volume solver designed for taxis-diffusion-reaction systems commonly found in mathematical biology. It efficiently handles non-local terms arising from cell-cell adhesion and long-range sensing, using the ROWMAP integrator for robust time-stepping of stiff equations. Read more

talks

teaching

Applied Scientific Computing (MATH 552)

Undergraduate course, University of Massachusetts Amherst, Department of Mathematics and Statistics, 2023

Introduction to the application of computational methods to models arising in science and engineering, concentrating mainly on the solution of partial differential equations. Read more