An adaptive numerical method for multi–cellular simulations of tissue development

Abstract

AbstractIn 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 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 betweenL∞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 Osborneet. al. [2017,https://doi.org/10.1371/journal.pcbi.1005387] and a further speed up by a factor of 4 can be achieved by using an adaptive time–step.