Review on Some Boundary Value Problems Defining the Mean First-Passage Time in Cell Migration

• Published in: Axioms Vol. 13; no. 8; p. 537
• Main Authors: Serrano, Hélia, Álvarez-Estrada, Ramón F.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.08.2024
• Subjects: Angiogenesis; Boundary conditions; Boundary value problems; Cell adhesion & migration; Cell migration; Diffusion; diffusion equation; Dirichlet boundary condition; Integral equations; Mathematical models; mean first-passage time; mixed boundary conditions; Motility; Neumann boundary condition; Partial differential equations; Position (location); Random walk theory; Robin boundary condition; Spain
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13080537

The mean first-passage time represents the average time for a migrating cell within its environment, starting from a certain position, to reach a specific location or target for the first time. In this feature article, we provide an overview of the characterization of the mean first-passage time of cells moving inside two- or three-dimensional domains, subject to various boundary conditions (Dirichlet, Neumann, Robin, or mixed), through the so-called adjoint diffusion equation. We concentrate on reducing the latter to inhomogeneous linear integral equations for certain density functions on the boundaries. The integral equations yield the mean first-passage time exactly for a very reduced set of boundaries. For various boundary surfaces, which include small deformations of the exactly solvable boundaries, the integral equations provide approximate solutions. Moreover, the method also allows to deal approximately with mixed boundary conditions, which constitute a genuine long-standing and open problem. New plots, figures, and discussions are presented, aimed at clarifying the analysis.