The Method of Fictitious Negative Sources to Model Diffusive Channels With Absorbing Boundaries

Abstract

This paper presents an approach to address the diffusion equation in scenarios involving multiple absorbing boundary conditions, commonly found in diffusive molecular communication (MC) channels. Instead of using multiple mirror images of the source, fictitious sources with time-varying release rates are introduced to replace the boundaries. This transformation enables the calculation of the expected cumulative number of absorbed particles (CNAP) by multiple absorbing boundaries with finite volume. To compute the expected CNAP, the concept of barycenter, which represents the spatial mean of particles the receiver absorbs is introduced. Substituting absorbing objects with their barycenters leads to model the CNAP in scenarios with convex geometry of absorbers. In a one-dimensional (1D) space, the proposed approach yields the same expression as the method of images for describing the expected CNAP by an absorber. However, in three-dimensional (3D) space, where using the method of images is challenging or even impossible, the proposed approach enables substituting the objects with fictitious sources and compute the expected CNAP. In 1D, an extension of this approach to the case in which one boundary exhibits an absorption characteristic while the other has zero-flux characteristic is demonstrated. This research direction is valuable for modeling channels where not all objects are particle receptors.