NDLI: Numerical Analysis of the Immersed Boundary Method for Cell-Based Simulation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Baker, Ruth E. Cooper, Fergus R. Fletcher, Alexander G.
Copyright Year	2017
Abstract	Mathematical modelling provides a useful framework within which to investigate the organization of biological tissues. With advances in experimental biology leading to increasingly detailed descriptions of cellular behavior, models that consider cells as individual objects are becoming a common tool to study how processes at the single-cell level affect collective dynamics and determine tissue size, shape, and function. However, there often remains no comprehensive account of these models, their method of solution, computational implementation, or analysis of parameter scaling, hindering our ability to utilize and accurately compare different models. Here we present an efficient, open-source implementation of the immersed boundary (IB) method, tailored to simulate the dynamics of cell populations. This approach considers the dynamics of elastic membranes, representing cell boundaries, immersed in a viscous Newtonian fluid. The IB method enables complex and emergent cell shape dynamics, spatially heterogeneous cell properties, and precise control of growth mechanisms. We solve the model numerically using an established algorithm, based on the fast Fourier transform, providing full details of all technical aspects of our implementation. The implementation is undertaken within Chaste, an open-source C++ library that allows one to easily change constitutive assumptions. Our implementation scales linearly with time step, and subquadratically with mesh spacing and immersed boundary node spacing. We identify the relationship between the immersed boundary node spacing and fluid mesh spacing required to ensure fluid volume conservation within immersed boundaries, and the scaling of cell membrane stiffness and cell-cell interaction strength required when refining the immersed boundary discretization. Finally, we present a simulation study of a growing epithelial tissue to demonstrate the applicability of our implementation to relevant biological questions, highlighting several features of the IB method that make it well suited to address certain questions in epithelial morphogenesis.
Sponsorship	Engineering and Physical Sciences Research Council
Starting Page	B943
Ending Page	B967
Page Count	25
File Format	PDF
ISSN	10648275
DOI	10.1137/16M1092246
e-ISSN	10957197
Issue Number	5
Volume Number	39
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-10-03
Access Restriction	Subscribed
Subject Keyword	Chaste convergence Developmental biology, pattern formation Cell biology immersed boundary method Cell movement Spectral methods cell-based modelling Finite difference methods Navier-Stokes equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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