On the approximation and efficient evaluation of integral terms in PDE models of cell adhesion

Recently, a nonlocal term has been introduced in time-dependent partial differential equation (PDE) models of cell migration in tissue. This term is used to model adhesive effects between cells and also between cells and the extracellular matrix. We assume periodic boundary conditions for the model and that the PDE system is discretized following the method of lines and using a finite-volume scheme on a uniform grid in space. For high-resolution simulations of the PDE system an efficient evaluation of the approximation of the nonlocal term is crucial. For one and two spatial dimensions we develop suitable approximations of the nonlocal term and evaluate these using fast Fourier transform (FFT) techniques. Comprehensive numerical tests show the accuracy and efficiency of our approach. We also demonstrate the impact of the proposed scheme for the treatment of the nonlocal term on simulation times for a differential cell adhesion model. We discuss extensions and applicability of our work to systems with nonperiodic boundary conditions and for other nonlocal PDE models from mathematical biology.