Adaptive Finite Element Simulation of Coupled PDE/ODE Systems Modeling Intercellular Signaling

Abstract

Intercellular signaling is a defining property of multicellular organisms, yet the spatio-temporal dynamics remain poorly understood. The subject of this work is the design of an efficient numerical algorithm for simulations of intercellular signaling in multicellular 3D environments modeled by coupled systems of partial differential equations (PDE) and ordinary differential equations (ODE). The PDE part of these systems consists of reaction-diffusion equations and describes the concentration distribution of diffusible messengers, e.g. cytokines. Intracellular dynamics are described by a small number of ODEs per cell. Thus, every single iteration of a commonly used decoupling scheme has similar computational costs than solving the coupled PDE/ODE system at once. We therefore develop an efficient multilevel preconditioner for the coupled system. The computational cost of both coupled and decoupled solution methods are investigated for model problems of different coupling strength. To keep the computational costs of the 3D simulations moderate, we use methods for adaptive mesh refinement. We discretize the system by different time meshes for the PDE and the ODE part to reduce the number of computationally expensive PDE time steps. Reliable a posteriori error estimations for coupled PDE/ODE systems are derived by means of the ’Dual Weighted Residual’ (DWR) method. The discretization error is split into the contributions of the PDE and the ODE part. We compute local error indicators in space and time and set up an efficient adaptive mesh refinement method. The described methods are validated by numerical tests for several biologically motivated model problems. We apply the developed numerical methods and simulate cytokine signaling between T cells in lymph nodes, which regulates the adaptive immune response in the human body. The numerical results show that, despite the high diffusivity of cytokines, highly localized cytokine concentrations with large gradients occur, which enables short-range cell-to-cell communication.