Conditional Mean First Passage Times to Small Traps in a 3-D Domain with a Sticky Boundary: Applications to T Cell Searching Behavior in Lymph Nodes

Calculating the time required for a diffusing object to reach a small target within a larger domain is a feature of a large class of modeling and simulation efforts in biology. Here, we are motivated by the motion of a T cell of the immune system seeking a particular antigen-presenting cell within a large lymph node. The precise nature of the cell motion at the outer boundary of the lymph node is not completely understood in terms of how cells choose to remain within a given lymph node, or exit. In previous work, we and others have studied diffusive motion to a small trap. We extend this previous work to analyze models where the diffusing object may exit the outer boundary of the domain (in this case, the lymph node). This is modeled by a Robin boundary condition on the surface of the lymph node. For the general problem of small traps inside a three-dimensional domain that has a partially sticky or absorbent domain boundary, the method of matched asymptotic expansions is used to calculate the mean and variance of the conditional first passage time for the T cell to reach a specific target trap. Our results are illustrated explicitly for the idealized situation of a spherical lymph node containing small spherically shaped traps, and are verified for a radially symmetric geometry with one trap at the origin where exact solutions are available. Mathematically, our analysis extends previous work on the calculation of the mean first passage time by allowing for a sticky boundary and by calculating conditional statistics of the diffusion process. Finally, our results are interpreted and applied to the context of T cell biology.