An integro-differential non-local model for cell migration and its efficient numerical solution

Cell migration is fundamental in a wide variety of physiological and pathological phenomena, being exploited in biomedical engineering as well. In this respect, we here present a hybrid non-local integro-differential model where a representative cell, reproduced by a point particle with an orientation, moves on a planar domain upon signals coming from environmental variables. From a numerical point of view, non-locality implies the need to evaluate integral terms which may present non-regular integrand functions because of heterogeneities in the environmental conditions and/or in cell sensing region. Having in mind multicellular applications, we here propose a robust computational method able to handle such non-regularities. The procedure is based on low order Runge–Kutta methods and on an ad hoc application of the Gauss–Legendre quadrature rule. The accuracy and efficiency of the resulting computational method is then tested by selected benchmark settings. In this context, the ad hoc application of the quadrature rule reveals to be crucial to obtain a high accuracy with a remarkably low number of quadrature nodes with respect to the standard Gauss–Legendre quadrature formula, and which thus results in a reduced overall computational cost. Finally, the proposed method is further coupled with the cubic spline interpolation scheme which allows to deal also with possible poor (i.e., point-wise defined) molecular spatial information. The performed simulations (which accounts also for different scenarios) show how the interpolation of the molecular variables affects the efficiency of the overall method and further justify the proposed procedure.