Mean first‐passage time of cell migration in confined domains

Abstract

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 Ω on which MFPT functions T, fulfilling a Poisson-like equation and different boundary conditions on the surface S enclosing Ω, are studied. By extending methods coming from potential theory, the calculation of T boils down to dealing with inhomogeneous linear integral equations having singular kernels on S. The latter are solved compactly and yield consistent 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 for the case of a closed spherical surface with Dirichlet conditions on most of S, except for a small complementary surface domain with Neumann conditions.

Characterization of the Mean First-Passage Time Function Subject to Advection in Annular-like Domains

“…in Ω, which is subject to the boundary conditions (B1) and (B2) and the finiteness condition 0 ≤ T(y) < +∞ in Ω. This is a direct generalization of the result, without drift (|ν| = 0), as stated in [15] (Proposition 2.3). See also [16,17].…”

“…Such an integral equation provides, at least, a mathematical basis for all cases where the shape of the surface prevents separability (in practice, this includes almost all geometries). In a previous publication [15], the random motion of a tumor cell in a tissue had given rise to the study of a three-dimensional Poisson equation for the MFPT function of the tumor cell inside a domain limited by one or two surfaces. By nontrivially extending [21] to that Poisson equation, the analysis of the three-dimensional MFPT yielded suitable systems of coupled inhomogeneous Fredholm linear integral equations for the corresponding density functions.…”

“…The density µ 2 has supposed consistency, which depends only on ρ = |ξ|. By using (A1), (A5), and (15), noticing that the normal derivative is the inner one (towards the interior of the domain), and integrating over the angles (so that only g 0 (ρ, ρ ) contributes), it follows that…”

“…∂ρ is ambiguous so that some prescription will be required to handle it. By applying the "averaging" recipe in [15], we have…”

“…The study of the asymptotic behavior of the MFPT in annulus geometries with inner and outer regions has been carried out in [14]. The characterization of the MFPT function in three-dimensional simply and doubly connected domains subject to Dirichlet and Neumann boundary conditions has been achieved previously in [15] by means of a system of inhomogeneous linear integral equations.…”