An adaptive numerical method for multi-cellular simulations of tissue development and maintenance. (English) Zbl 07931367
Summary: In recent years, multi-cellular models, where cells are represented as individual interacting entities, are becoming ever popular. This has led to a proliferation of novel methods and simulation tools. The first aim of this paper is to review the numerical methods utilised by multi-cellular modelling tools and to demonstrate which numerical methods are appropriate for simulations of tissue and organ development, maintenance, and disease. The second aim is to introduce an adaptive time-stepping algorithm and to demonstrate it’s efficiency and accuracy. We focus on off-lattice, mechanics based, models where cell movement is defined by a series of first order ordinary differential equations, derived by assuming over-damped motion and balancing forces. We see that many numerical methods have been used, ranging from simple forward Euler approaches through to higher order single-step methods like Runge-Kutta 4 and multi-step methods like Adams-Bashforth 2. Through a series of exemplar multi-cellular simulations, we see that if: care is taken to have events (births deaths and re-meshing/re-arrangements) occur on common time-steps; and boundaries are imposed on all sub-steps of numerical methods or implemented using forces, then all numerical methods can converge with the correct order. We introduce an adaptive time-stepping method and demonstrate that the best compromise between \(L_\infty\) error and run-time is to use Runge-Kutta 4 with an increased time-step and moderate adaptivity. We see that a judicious choice of numerical method can speed the simulation up by a factor of 10–60 from the forward Euler methods seen in [J. M. Osborne et al. [PLoS Comput. Biol. 13, No. 2, 1–34 (2017; doi:10.1371/journal.pcbi.1005387)], and a further speed up by a factor of 4 can be achieved by using an adaptive time-step.
MSC:
| 92C37 | Cell biology |
| 92C15 | Developmental biology, pattern formation |
| 65L99 | Numerical methods for ordinary differential equations |
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