Coupling vs decoupling approaches for PDE/ODE systems modeling intercellular signaling

Abstract

We consider PDE/ODE systems for the simulation of intercellular signaling in multicellular environments. The intracellular processes for each cell described here by ODEs determine the long-time dynamics, but the PDE part dominates the solving effort. Thus, it is not clear if commonly used decoupling methods can outperform a coupling approach. Based on a sensitivity analysis, we present a systematic comparison between coupling and decoupling approaches for this class of problems and show numerical results. For biologically relevant configurations of the model, our quantitative study shows that a coupling approach performs much better than a decoupling one.

Introduction

Cellular signaling has been mathematically described by a variety of models mostly relying on large systems of ordinary differential equations (ODE) [1]. These earlier models were extended by partial differential equations (PDE) to accurately consider concentration gradients and their effect [2], [3], [4], [5]. In our intercellular model we consider the diffusion and action of small signaling proteins in the intercellular space described by PDE, i.e. reaction–diffusion equations, coupled on the cell surfaces with ODEs responsible for the intracellular dynamics. This coupling of mixed differential equations allows mainly two strategies for an implicit solver: (1) nonlinear methods among them the nonlinear multigrid method also called “full approximation scheme” (FAS) [6], [7], (2) linearization based approaches (Newton-type). These methods can be used in a combined approach, where for example a Newton-type method can be used as smoother for a FAS and a linear or a nonlinear multigrid can be used as a preconditioner for a Newton-type method. The comparison and discussion of advantages and disadvantages of these strategies that depend on many aspects like, e.g. the accuracy of the Jacobian approximation [8], is not the focus of our work. Moreover, since in the considered coupled PDE/ODE system the linearization is not a critical point we choose a Newton-type method preconditioned by a linear multigrid and study the effect of splitting the linearization. A decoupling solution approach, based on a fixed-point method, is often used when restrictions on accuracy can be relaxed in order to allow an easier numerical treatment of complicated problems. Such an approach makes it possible to reuse existing solvers and is widely used in numerical methods for coupled systems, see [9], [10], [11], [12], [13], [14]. In case of strongly coupled equations (see Section 4 for the definition of strong coupling used here), this strategy leads usually to high computational costs through very small time steps needed to reduce the strength of the coupling. In fact, the convergence of the fixed-point iterations is typically linear with the convergence rate depending on the used block-iterative method and on the strength of the coupling since it depends on the spectral radius of the matrices involved [15]. Therefore a drawback of decoupling solvers is their low convergence and possibly divergence. On the contrary, the drawback of fully implicit solvers is mainly that they demand for the implementation of a special-purpose code.

We consider PDE/ODE systems for the simulation of intercellular signaling in multicellular environments. Since the ODE part does not lead to a large discretization system like the PDE part, it is not clear if a decoupling method can outperform a coupling approach. In fact other works with PDE/ODE models have used decoupling approaches for systems arising in biological applications, e.g. blood flow including chemical interaction [16], [17], signal transduction [4], cardiovascular flow [9], cancer invasion [18]. With the exception of [9] these works do not consider a comparison with a monolithic approach. In addition, for the most of these applications it is not clear whether the strength of the coupling is strong or weak.

In this context, the scope of our work is to present a systematic comparison between coupling and decoupling approaches for this class of problems. The method is based on a sensitivity analysis to compute the strength of the coupling. Additionally, we compare a multigrid method in which the coupling is considered only at the coarsest level to a fully coupling approach. There are few works that deal with the fully coupling solution process of dimensionally heterogeneous systems such as [19]. Furthermore, we focus on the solution of local microenvironments. Therefore, this solution process can be used for example as local solver for nonlinear preconditioner of Newton-type methods [20] or domain decomposition methods [21].

Outline  The paper is organized as follows. In Section 2 we give an abstract description of the model. We present the mathematical formulation and the functional setting. We discretize the coupled PDE/ODE system by the finite element method (FEM) in Section 3. We use a sensitivity approach in Section 4 to analyze the coupling of the PDE/ODE system and in Section 5 we present different solving approaches for the coupled system. We present the numerical results exemplarily for a particular application and discuss numerical aspects in Section 6. In Section 7 we describe a realistic configuration and give a biological interpretation to the results obtained.