Numerical Analysis of the Immersed Boundary
نویسندگان
چکیده
Mathematical modelling provides a useful framework within which to investigate 4 the organization of biological tissues. With advances in experimental biology leading to increasingly 5 detailed descriptions of cellular behaviour, models that consider cells as individual objects are be6 coming a common tool to study how processes at the single-cell level affect collective dynamics and 7 determine tissue size, shape and function. However, there often remains no comprehensive account 8 of these models, their method of solution, computational implementation or analysis of parameter 9 scaling, hindering our ability to utilise and accurately compare different models. Here we present 10 an efficient, open-source implementation of the immersed boundary method (IBM), tailored to sim11 ulate the dynamics of cell populations. This approach considers the dynamics of elastic membranes, 12 representing cell boundaries, immersed in a viscous Newtonian fluid. The IBM enables complex and 13 emergent cell shape dynamics, spatially heterogeneous cell properties and precise control of growth 14 mechanisms. We solve the model numerically using an established algorithm, based on the fast 15 Fourier transform, providing full details of all technical aspects of our implementation. The imple16 mentation is undertaken within Chaste, an open-source C++ library that allows one to easily change 17 constitutive assumptions. Our implementation scales linearly with time step, and subquadratically 18 with mesh spacing and immersed boundary node spacing. We identify the relationship between the 19 immersed boundary node spacing and fluid mesh spacing required to ensure fluid volume conserva20 tion within immersed boundaries, and the scaling of cell membrane stiffness and cell-cell interaction 21 strength required when refining the immersed boundary discretization. This study provides a recipe 22 allowing consistent parametrisation of IBM models. 23
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