Qiyao Peng (1*), Sander C. Hille (1), Fred J. Vermolen (2)
1. Mathematical Institute, Leiden University, Neils Borhweg 1, 2333 CA, Leiden, The Netherlands.
2. Computational Mathematics Group, Discipline group Mathematics and Statistics, Faculty of Science, Hasselt University, Belgium.
*Correspondence: q.peng@math.leidenuniv.nl (Qiyao Peng).
In biomedical applications, there are many interactions between single objects (e.g. cells) and their direct environment, for instance, the diffusion of the compounds, such as signalling molecules, and the mechanical forces exerted by the (skin) cells to the extracellular matrix. For example, in wound healing, immune cells release cytokines to attract the skin cells to migrate towards the wound region, and then the skin cells will pull the collagen bundles to close the wound.
For the sake of computational efficiency, in mathematical modelling and for theoretical purposes, the Dirac Delta distributions are often utilized as a replacement for cells or vesicles, since the size of cells or vesicles is much smaller than the size of the surrounding tissues. Typically, one separates the intracellular and extracellular environment and uses homogeneous Neumann boundary conditions for the cell boundary (so-called spatial exclusion model), while the point source/force model neglects the intracellular environment.
In this talk, we will discuss the so-called spatial exclusion model and the point source/force model in a diffusion equation and momentum balance equation. In the diffusion equation, numerical simulation suggests a time-delay discrepancy between the solutions to the two approaches; whereas in the diffusion equation, Dirac delta distribution leads to a singular solution, and there is a significant difference in the order of the solutions between the spatial exclusion model and the point force model. Hence, to maintain the consistency between these two types of models, different extra conditions are required in different equations.