Mean first-passage time of cell migration in confined domains

A number of key biological processes involving cell migration in which cells traverse boundaries separating well-defined tissues can be modeled in terms of mean first-passage time (MFPT) problems in confined domains. Motivated by this scenario, we consider suitable three-dimensional domains omega$$ \Omega $$ on which MFPT functions T$$ T $$, fulfilling a Poisson-like equation and different boundary conditions on the surface S$$ S $$ enclosing omega$$ \Omega $$, are studied. By extending methods coming from potential theory, the calculation of T$$ T $$ boils down to dealing with inhomogeneous linear integral equations having singular kernels on S$$ S $$. The latter are solved compactly and yield consistent T$$ T $$'s for several domains and homogeneous boundary conditions. Moreover, the integral equation approach allows us to analyze the MFPT with mixed (Dirichlet-Neumann) boundary conditions on S$$ S $$ for the case of a closed spherical surface with Dirichlet conditions on most of S$$ S $$, except for a small complementary surface domain with Neumann conditions.