Numerical integration of mechanical forces in center-based models for biological cell populations

Abstract

Center-based models are used to simulate the mechanical behavior of biological cells during embryonic development or cancer growth. To allow for the simulation of biological populations potentially growing from a few individual cells to many thousands or more, these models have to be numerically efficient, while being reasonably accurate on the level of individual cell trajectories. In this work, we increase the robustness, accuracy, and efficiency of the simulation of center-based models by choosing the time steps adaptively in the numerical method and comparing five different integration methods. We investigate the gain in using single rate time stepping based on local estimates of the numerical errors for the forward and backward Euler methods of first order accuracy and a Runge-Kutta method and the trapezoidal method of second order accuracy. Properties of the analytical solution such as convergence to steady state and conservation of the center of gravity are inherited by the numerical solutions. Furthermore, we propose a multirate time stepping scheme that simulates regions with high local force gradients (e.g. as they happen after cell division) with multiple smaller time steps within a larger single time step for regions with smoother forces. These methods are compared for a model system in numerical experiments. We conclude, for example, that the multirate forward Euler method performs better than the Runge-Kutta method for low accuracy requirements but for higher accuracy the latter method is preferred. Only with frequent cell divisions the method with a fixed time step is the best choice.