Adaptive time 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. We investigate the gain in using single rate time stepping for the forward and backward Euler methods, based on local estimates of the numerical errors and the stability of the method in the case of the explicit forward Euler method. 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 different model systems in numerical experiments. We conclude that the adaptive single rate forward Euler method results in significant gains in terms of reduced wall clock times for the simulation of a linearly growing tissue, while at the same time eliminating the need for manual determination of a suitable time step size.